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• How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

Table of contents

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

• An independent variable is something the researcher changes or controls.
• A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias  will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

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Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

• The relevant variables
• The specific group being studied
• The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in  if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

• H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
• H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

• Sampling methods
• Simple random sampling
• Stratified sampling
• Cluster sampling
• Likert scales
• Reproducibility

 Statistics

• Null hypothesis
• Statistical power
• Probability distribution
• Effect size
• Poisson distribution

Research bias

• Optimism bias
• Cognitive bias
• Implicit bias
• Hawthorne effect
• Anchoring bias
• Explicit bias

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A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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What is a Research Hypothesis: How to Write it, Types, and Examples

Any research begins with a research question and a research hypothesis . A research question alone may not suffice to design the experiment(s) needed to answer it. A hypothesis is central to the scientific method. But what is a hypothesis ? A hypothesis is a testable statement that proposes a possible explanation to a phenomenon, and it may include a prediction. Next, you may ask what is a research hypothesis ? Simply put, a research hypothesis is a prediction or educated guess about the relationship between the variables that you want to investigate.  

It is important to be thorough when developing your research hypothesis. Shortcomings in the framing of a hypothesis can affect the study design and the results. A better understanding of the research hypothesis definition and characteristics of a good hypothesis will make it easier for you to develop your own hypothesis for your research. Let’s dive in to know more about the types of research hypothesis , how to write a research hypothesis , and some research hypothesis examples .  

Table of Contents

What is a hypothesis ?  

A hypothesis is based on the existing body of knowledge in a study area. Framed before the data are collected, a hypothesis states the tentative relationship between independent and dependent variables, along with a prediction of the outcome.  

What is a research hypothesis ?  

Young researchers starting out their journey are usually brimming with questions like “ What is a hypothesis ?” “ What is a research hypothesis ?” “How can I write a good research hypothesis ?”   

A research hypothesis is a statement that proposes a possible explanation for an observable phenomenon or pattern. It guides the direction of a study and predicts the outcome of the investigation. A research hypothesis is testable, i.e., it can be supported or disproven through experimentation or observation.     

Characteristics of a good hypothesis  

Here are the characteristics of a good hypothesis :  

• Clearly formulated and free of language errors and ambiguity  
• Concise and not unnecessarily verbose  
• Has clearly defined variables  
• Testable and stated in a way that allows for it to be disproven  
• Can be tested using a research design that is feasible, ethical, and practical   
• Specific and relevant to the research problem  
• Rooted in a thorough literature search  
• Can generate new knowledge or understanding.  

How to create an effective research hypothesis  

A study begins with the formulation of a research question. A researcher then performs background research. This background information forms the basis for building a good research hypothesis . The researcher then performs experiments, collects, and analyzes the data, interprets the findings, and ultimately, determines if the findings support or negate the original hypothesis.  

Let’s look at each step for creating an effective, testable, and good research hypothesis :  

• Identify a research problem or question: Start by identifying a specific research problem.   
• Review the literature: Conduct an in-depth review of the existing literature related to the research problem to grasp the current knowledge and gaps in the field.   
• Formulate a clear and testable hypothesis : Based on the research question, use existing knowledge to form a clear and testable hypothesis . The hypothesis should state a predicted relationship between two or more variables that can be measured and manipulated. Improve the original draft till it is clear and meaningful.  
• State the null hypothesis: The null hypothesis is a statement that there is no relationship between the variables you are studying.   
• Define the population and sample: Clearly define the population you are studying and the sample you will be using for your research.  
• Select appropriate methods for testing the hypothesis: Select appropriate research methods, such as experiments, surveys, or observational studies, which will allow you to test your research hypothesis .  

Remember that creating a research hypothesis is an iterative process, i.e., you might have to revise it based on the data you collect. You may need to test and reject several hypotheses before answering the research problem.  

How to write a research hypothesis  

When you start writing a research hypothesis , you use an “if–then” statement format, which states the predicted relationship between two or more variables. Clearly identify the independent variables (the variables being changed) and the dependent variables (the variables being measured), as well as the population you are studying. Review and revise your hypothesis as needed.  

An example of a research hypothesis in this format is as follows:  

“ If [athletes] follow [cold water showers daily], then their [endurance] increases.”  

Population: athletes  

Independent variable: daily cold water showers  

Dependent variable: endurance  

You may have understood the characteristics of a good hypothesis . But note that a research hypothesis is not always confirmed; a researcher should be prepared to accept or reject the hypothesis based on the study findings.  

Research hypothesis checklist  

Following from above, here is a 10-point checklist for a good research hypothesis :  

• Testable: A research hypothesis should be able to be tested via experimentation or observation.  
• Specific: A research hypothesis should clearly state the relationship between the variables being studied.  
• Based on prior research: A research hypothesis should be based on existing knowledge and previous research in the field.  
• Falsifiable: A research hypothesis should be able to be disproven through testing.  
• Clear and concise: A research hypothesis should be stated in a clear and concise manner.  
• Logical: A research hypothesis should be logical and consistent with current understanding of the subject.  
• Relevant: A research hypothesis should be relevant to the research question and objectives.  
• Feasible: A research hypothesis should be feasible to test within the scope of the study.  
• Reflects the population: A research hypothesis should consider the population or sample being studied.  
• Uncomplicated: A good research hypothesis is written in a way that is easy for the target audience to understand.  

By following this research hypothesis checklist , you will be able to create a research hypothesis that is strong, well-constructed, and more likely to yield meaningful results.  

Types of research hypothesis  

Different types of research hypothesis are used in scientific research:  

1. Null hypothesis:

A null hypothesis states that there is no change in the dependent variable due to changes to the independent variable. This means that the results are due to chance and are not significant. A null hypothesis is denoted as H0 and is stated as the opposite of what the alternative hypothesis states.   

Example: “ The newly identified virus is not zoonotic .”  

2. Alternative hypothesis:

This states that there is a significant difference or relationship between the variables being studied. It is denoted as H1 or Ha and is usually accepted or rejected in favor of the null hypothesis.  

Example: “ The newly identified virus is zoonotic .”  

3. Directional hypothesis :

This specifies the direction of the relationship or difference between variables; therefore, it tends to use terms like increase, decrease, positive, negative, more, or less.   

Example: “ The inclusion of intervention X decreases infant mortality compared to the original treatment .”   

4. Non-directional hypothesis:

While it does not predict the exact direction or nature of the relationship between the two variables, a non-directional hypothesis states the existence of a relationship or difference between variables but not the direction, nature, or magnitude of the relationship. A non-directional hypothesis may be used when there is no underlying theory or when findings contradict previous research.  

Example, “ Cats and dogs differ in the amount of affection they express .”  

5. Simple hypothesis :

A simple hypothesis only predicts the relationship between one independent and another independent variable.  

Example: “ Applying sunscreen every day slows skin aging .”  

6 . Complex hypothesis :

A complex hypothesis states the relationship or difference between two or more independent and dependent variables.   

Example: “ Applying sunscreen every day slows skin aging, reduces sun burn, and reduces the chances of skin cancer .” (Here, the three dependent variables are slowing skin aging, reducing sun burn, and reducing the chances of skin cancer.)  

7. Associative hypothesis:  

An associative hypothesis states that a change in one variable results in the change of the other variable. The associative hypothesis defines interdependency between variables.  

Example: “ There is a positive association between physical activity levels and overall health .”  

8 . Causal hypothesis:

A causal hypothesis proposes a cause-and-effect interaction between variables.  

Example: “ Long-term alcohol use causes liver damage .”  

Note that some of the types of research hypothesis mentioned above might overlap. The types of hypothesis chosen will depend on the research question and the objective of the study.  

Research hypothesis examples  

Here are some good research hypothesis examples :  

“The use of a specific type of therapy will lead to a reduction in symptoms of depression in individuals with a history of major depressive disorder.”  

“Providing educational interventions on healthy eating habits will result in weight loss in overweight individuals.”  

“Plants that are exposed to certain types of music will grow taller than those that are not exposed to music.”  

“The use of the plant growth regulator X will lead to an increase in the number of flowers produced by plants.”  

Characteristics that make a research hypothesis weak are unclear variables, unoriginality, being too general or too vague, and being untestable. A weak hypothesis leads to weak research and improper methods.   

Some bad research hypothesis examples (and the reasons why they are “bad”) are as follows:  

“This study will show that treatment X is better than any other treatment . ” (This statement is not testable, too broad, and does not consider other treatments that may be effective.)  

“This study will prove that this type of therapy is effective for all mental disorders . ” (This statement is too broad and not testable as mental disorders are complex and different disorders may respond differently to different types of therapy.)  

“Plants can communicate with each other through telepathy . ” (This statement is not testable and lacks a scientific basis.)  

Importance of testable hypothesis  

If a research hypothesis is not testable, the results will not prove or disprove anything meaningful. The conclusions will be vague at best. A testable hypothesis helps a researcher focus on the study outcome and understand the implication of the question and the different variables involved. A testable hypothesis helps a researcher make precise predictions based on prior research.  

To be considered testable, there must be a way to prove that the hypothesis is true or false; further, the results of the hypothesis must be reproducible.  

Frequently Asked Questions (FAQs) on research hypothesis  

1. What is the difference between research question and research hypothesis ?  

A research question defines the problem and helps outline the study objective(s). It is an open-ended statement that is exploratory or probing in nature. Therefore, it does not make predictions or assumptions. It helps a researcher identify what information to collect. A research hypothesis , however, is a specific, testable prediction about the relationship between variables. Accordingly, it guides the study design and data analysis approach.

2. When to reject null hypothesis ?

A null hypothesis should be rejected when the evidence from a statistical test shows that it is unlikely to be true. This happens when the test statistic (e.g., p -value) is less than the defined significance level (e.g., 0.05). Rejecting the null hypothesis does not necessarily mean that the alternative hypothesis is true; it simply means that the evidence found is not compatible with the null hypothesis.  

3. How can I be sure my hypothesis is testable?  

A testable hypothesis should be specific and measurable, and it should state a clear relationship between variables that can be tested with data. To ensure that your hypothesis is testable, consider the following:  

• Clearly define the key variables in your hypothesis. You should be able to measure and manipulate these variables in a way that allows you to test the hypothesis.  
• The hypothesis should predict a specific outcome or relationship between variables that can be measured or quantified.   
• You should be able to collect the necessary data within the constraints of your study.  
• It should be possible for other researchers to replicate your study, using the same methods and variables.   
• Your hypothesis should be testable by using appropriate statistical analysis techniques, so you can draw conclusions, and make inferences about the population from the sample data.  
• The hypothesis should be able to be disproven or rejected through the collection of data.  

4. How do I revise my research hypothesis if my data does not support it?  

If your data does not support your research hypothesis , you will need to revise it or develop a new one. You should examine your data carefully and identify any patterns or anomalies, re-examine your research question, and/or revisit your theory to look for any alternative explanations for your results. Based on your review of the data, literature, and theories, modify your research hypothesis to better align it with the results you obtained. Use your revised hypothesis to guide your research design and data collection. It is important to remain objective throughout the process.  

5. I am performing exploratory research. Do I need to formulate a research hypothesis?  

As opposed to “confirmatory” research, where a researcher has some idea about the relationship between the variables under investigation, exploratory research (or hypothesis-generating research) looks into a completely new topic about which limited information is available. Therefore, the researcher will not have any prior hypotheses. In such cases, a researcher will need to develop a post-hoc hypothesis. A post-hoc research hypothesis is generated after these results are known.  

6. How is a research hypothesis different from a research question?

A research question is an inquiry about a specific topic or phenomenon, typically expressed as a question. It seeks to explore and understand a particular aspect of the research subject. In contrast, a research hypothesis is a specific statement or prediction that suggests an expected relationship between variables. It is formulated based on existing knowledge or theories and guides the research design and data analysis.

7. Can a research hypothesis change during the research process?

Yes, research hypotheses can change during the research process. As researchers collect and analyze data, new insights and information may emerge that require modification or refinement of the initial hypotheses. This can be due to unexpected findings, limitations in the original hypotheses, or the need to explore additional dimensions of the research topic. Flexibility is crucial in research, allowing for adaptation and adjustment of hypotheses to align with the evolving understanding of the subject matter.

8. How many hypotheses should be included in a research study?

The number of research hypotheses in a research study varies depending on the nature and scope of the research. It is not necessary to have multiple hypotheses in every study. Some studies may have only one primary hypothesis, while others may have several related hypotheses. The number of hypotheses should be determined based on the research objectives, research questions, and the complexity of the research topic. It is important to ensure that the hypotheses are focused, testable, and directly related to the research aims.

9. Can research hypotheses be used in qualitative research?

Yes, research hypotheses can be used in qualitative research, although they are more commonly associated with quantitative research. In qualitative research, hypotheses may be formulated as tentative or exploratory statements that guide the investigation. Instead of testing hypotheses through statistical analysis, qualitative researchers may use the hypotheses to guide data collection and analysis, seeking to uncover patterns, themes, or relationships within the qualitative data. The emphasis in qualitative research is often on generating insights and understanding rather than confirming or rejecting specific research hypotheses through statistical testing.

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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Muhammad Hassan

Researcher, Academic Writer, Web developer

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The Craft of Writing a Strong Hypothesis

Table of Contents

Writing a hypothesis is one of the essential elements of a scientific research paper. It needs to be to the point, clearly communicating what your research is trying to accomplish. A blurry, drawn-out, or complexly-structured hypothesis can confuse your readers. Or worse, the editor and peer reviewers.

A captivating hypothesis is not too intricate. This blog will take you through the process so that, by the end of it, you have a better idea of how to convey your research paper's intent in just one sentence.

What is a Hypothesis?

The first step in your scientific endeavor, a hypothesis, is a strong, concise statement that forms the basis of your research. It is not the same as a thesis statement , which is a brief summary of your research paper .

The sole purpose of a hypothesis is to predict your paper's findings, data, and conclusion. It comes from a place of curiosity and intuition . When you write a hypothesis, you're essentially making an educated guess based on scientific prejudices and evidence, which is further proven or disproven through the scientific method.

The reason for undertaking research is to observe a specific phenomenon. A hypothesis, therefore, lays out what the said phenomenon is. And it does so through two variables, an independent and dependent variable.

The independent variable is the cause behind the observation, while the dependent variable is the effect of the cause. A good example of this is “mixing red and blue forms purple.” In this hypothesis, mixing red and blue is the independent variable as you're combining the two colors at your own will. The formation of purple is the dependent variable as, in this case, it is conditional to the independent variable.

Different Types of Hypotheses‌

Types of hypotheses

Some would stand by the notion that there are only two types of hypotheses: a Null hypothesis and an Alternative hypothesis. While that may have some truth to it, it would be better to fully distinguish the most common forms as these terms come up so often, which might leave you out of context.

Apart from Null and Alternative, there are Complex, Simple, Directional, Non-Directional, Statistical, and Associative and casual hypotheses. They don't necessarily have to be exclusive, as one hypothesis can tick many boxes, but knowing the distinctions between them will make it easier for you to construct your own.

1. Null hypothesis

A null hypothesis proposes no relationship between two variables. Denoted by H 0 , it is a negative statement like “Attending physiotherapy sessions does not affect athletes' on-field performance.” Here, the author claims physiotherapy sessions have no effect on on-field performances. Even if there is, it's only a coincidence.

2. Alternative hypothesis

Considered to be the opposite of a null hypothesis, an alternative hypothesis is donated as H1 or Ha. It explicitly states that the dependent variable affects the independent variable. A good  alternative hypothesis example is “Attending physiotherapy sessions improves athletes' on-field performance.” or “Water evaporates at 100 °C. ” The alternative hypothesis further branches into directional and non-directional.

• Directional hypothesis: A hypothesis that states the result would be either positive or negative is called directional hypothesis. It accompanies H1 with either the ‘<' or ‘>' sign.
• Non-directional hypothesis: A non-directional hypothesis only claims an effect on the dependent variable. It does not clarify whether the result would be positive or negative. The sign for a non-directional hypothesis is ‘≠.'

3. Simple hypothesis

A simple hypothesis is a statement made to reflect the relation between exactly two variables. One independent and one dependent. Consider the example, “Smoking is a prominent cause of lung cancer." The dependent variable, lung cancer, is dependent on the independent variable, smoking.

4. Complex hypothesis

In contrast to a simple hypothesis, a complex hypothesis implies the relationship between multiple independent and dependent variables. For instance, “Individuals who eat more fruits tend to have higher immunity, lesser cholesterol, and high metabolism.” The independent variable is eating more fruits, while the dependent variables are higher immunity, lesser cholesterol, and high metabolism.

5. Associative and casual hypothesis

Associative and casual hypotheses don't exhibit how many variables there will be. They define the relationship between the variables. In an associative hypothesis, changing any one variable, dependent or independent, affects others. In a casual hypothesis, the independent variable directly affects the dependent.

6. Empirical hypothesis

Also referred to as the working hypothesis, an empirical hypothesis claims a theory's validation via experiments and observation. This way, the statement appears justifiable and different from a wild guess.

Say, the hypothesis is “Women who take iron tablets face a lesser risk of anemia than those who take vitamin B12.” This is an example of an empirical hypothesis where the researcher  the statement after assessing a group of women who take iron tablets and charting the findings.

7. Statistical hypothesis

The point of a statistical hypothesis is to test an already existing hypothesis by studying a population sample. Hypothesis like “44% of the Indian population belong in the age group of 22-27.” leverage evidence to prove or disprove a particular statement.

Characteristics of a Good Hypothesis

Writing a hypothesis is essential as it can make or break your research for you. That includes your chances of getting published in a journal. So when you're designing one, keep an eye out for these pointers:

• A research hypothesis has to be simple yet clear to look justifiable enough.
• It has to be testable — your research would be rendered pointless if too far-fetched into reality or limited by technology.
• It has to be precise about the results —what you are trying to do and achieve through it should come out in your hypothesis.
• A research hypothesis should be self-explanatory, leaving no doubt in the reader's mind.
• If you are developing a relational hypothesis, you need to include the variables and establish an appropriate relationship among them.
• A hypothesis must keep and reflect the scope for further investigations and experiments.

Separating a Hypothesis from a Prediction

Outside of academia, hypothesis and prediction are often used interchangeably. In research writing, this is not only confusing but also incorrect. And although a hypothesis and prediction are guesses at their core, there are many differences between them.

A hypothesis is an educated guess or even a testable prediction validated through research. It aims to analyze the gathered evidence and facts to define a relationship between variables and put forth a logical explanation behind the nature of events.

Predictions are assumptions or expected outcomes made without any backing evidence. They are more fictionally inclined regardless of where they originate from.

For this reason, a hypothesis holds much more weight than a prediction. It sticks to the scientific method rather than pure guesswork. "Planets revolve around the Sun." is an example of a hypothesis as it is previous knowledge and observed trends. Additionally, we can test it through the scientific method.

Whereas "COVID-19 will be eradicated by 2030." is a prediction. Even though it results from past trends, we can't prove or disprove it. So, the only way this gets validated is to wait and watch if COVID-19 cases end by 2030.

Finally, How to Write a Hypothesis

Quick tips on writing a hypothesis

1.  Be clear about your research question

A hypothesis should instantly address the research question or the problem statement. To do so, you need to ask a question. Understand the constraints of your undertaken research topic and then formulate a simple and topic-centric problem. Only after that can you develop a hypothesis and further test for evidence.

2. Carry out a recce

Once you have your research's foundation laid out, it would be best to conduct preliminary research. Go through previous theories, academic papers, data, and experiments before you start curating your research hypothesis. It will give you an idea of your hypothesis's viability or originality.

Making use of references from relevant research papers helps draft a good research hypothesis. SciSpace Discover offers a repository of over 270 million research papers to browse through and gain a deeper understanding of related studies on a particular topic. Additionally, you can use SciSpace Copilot , your AI research assistant, for reading any lengthy research paper and getting a more summarized context of it. A hypothesis can be formed after evaluating many such summarized research papers. Copilot also offers explanations for theories and equations, explains paper in simplified version, allows you to highlight any text in the paper or clip math equations and tables and provides a deeper, clear understanding of what is being said. This can improve the hypothesis by helping you identify potential research gaps.

3. Create a 3-dimensional hypothesis

Variables are an essential part of any reasonable hypothesis. So, identify your independent and dependent variable(s) and form a correlation between them. The ideal way to do this is to write the hypothetical assumption in the ‘if-then' form. If you use this form, make sure that you state the predefined relationship between the variables.

In another way, you can choose to present your hypothesis as a comparison between two variables. Here, you must specify the difference you expect to observe in the results.

4. Write the first draft

Now that everything is in place, it's time to write your hypothesis. For starters, create the first draft. In this version, write what you expect to find from your research.

Clearly separate your independent and dependent variables and the link between them. Don't fixate on syntax at this stage. The goal is to ensure your hypothesis addresses the issue.

5. Proof your hypothesis

After preparing the first draft of your hypothesis, you need to inspect it thoroughly. It should tick all the boxes, like being concise, straightforward, relevant, and accurate. Your final hypothesis has to be well-structured as well.

Research projects are an exciting and crucial part of being a scholar. And once you have your research question, you need a great hypothesis to begin conducting research. Thus, knowing how to write a hypothesis is very important.

Now that you have a firmer grasp on what a good hypothesis constitutes, the different kinds there are, and what process to follow, you will find it much easier to write your hypothesis, which ultimately helps your research.

Now it's easier than ever to streamline your research workflow with SciSpace Discover . Its integrated, comprehensive end-to-end platform for research allows scholars to easily discover, write and publish their research and fosters collaboration.

It includes everything you need, including a repository of over 270 million research papers across disciplines, SEO-optimized summaries and public profiles to show your expertise and experience.

If you found these tips on writing a research hypothesis useful, head over to our blog on Statistical Hypothesis Testing to learn about the top researchers, papers, and institutions in this domain.

Frequently Asked Questions (FAQs)

1. what is the definition of hypothesis.

According to the Oxford dictionary, a hypothesis is defined as “An idea or explanation of something that is based on a few known facts, but that has not yet been proved to be true or correct”.

2. What is an example of hypothesis?

The hypothesis is a statement that proposes a relationship between two or more variables. An example: "If we increase the number of new users who join our platform by 25%, then we will see an increase in revenue."

3. What is an example of null hypothesis?

A null hypothesis is a statement that there is no relationship between two variables. The null hypothesis is written as H0. The null hypothesis states that there is no effect. For example, if you're studying whether or not a particular type of exercise increases strength, your null hypothesis will be "there is no difference in strength between people who exercise and people who don't."

4. What are the types of research?

• Fundamental research

• Applied research

• Qualitative research

• Quantitative research

• Mixed research

• Exploratory research

• Longitudinal research

• Cross-sectional research

• Field research

• Laboratory research

• Fixed research

• Flexible research

• Action research

• Policy research

• Classification research

• Comparative research

• Causal research

• Inductive research

• Deductive research

5. How to write a hypothesis?

• Your hypothesis should be able to predict the relationship and outcome.

• Avoid wordiness by keeping it simple and brief.

• Your hypothesis should contain observable and testable outcomes.

• Your hypothesis should be relevant to the research question.

6. What are the 2 types of hypothesis?

• Null hypotheses are used to test the claim that "there is no difference between two groups of data".

• Alternative hypotheses test the claim that "there is a difference between two data groups".

7. Difference between research question and research hypothesis?

A research question is a broad, open-ended question you will try to answer through your research. A hypothesis is a statement based on prior research or theory that you expect to be true due to your study. Example - Research question: What are the factors that influence the adoption of the new technology? Research hypothesis: There is a positive relationship between age, education and income level with the adoption of the new technology.

8. What is plural for hypothesis?

The plural of hypothesis is hypotheses. Here's an example of how it would be used in a statement, "Numerous well-considered hypotheses are presented in this part, and they are supported by tables and figures that are well-illustrated."

9. What is the red queen hypothesis?

The red queen hypothesis in evolutionary biology states that species must constantly evolve to avoid extinction because if they don't, they will be outcompeted by other species that are evolving. Leigh Van Valen first proposed it in 1973; since then, it has been tested and substantiated many times.

10. Who is known as the father of null hypothesis?

The father of the null hypothesis is Sir Ronald Fisher. He published a paper in 1925 that introduced the concept of null hypothesis testing, and he was also the first to use the term itself.

11. When to reject null hypothesis?

You need to find a significant difference between your two populations to reject the null hypothesis. You can determine that by running statistical tests such as an independent sample t-test or a dependent sample t-test. You should reject the null hypothesis if the p-value is less than 0.05.

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

What is a scientific hypothesis?

It's the initial building block in the scientific method.

Hypothesis basics

What makes a hypothesis testable.

• Types of hypotheses
• Hypothesis versus theory

Additional resources

Bibliography.

A scientific hypothesis is a tentative, testable explanation for a phenomenon in the natural world. It's the initial building block in the scientific method . Many describe it as an "educated guess" based on prior knowledge and observation. While this is true, a hypothesis is more informed than a guess. While an "educated guess" suggests a random prediction based on a person's expertise, developing a hypothesis requires active observation and background research. 

The basic idea of a hypothesis is that there is no predetermined outcome. For a solution to be termed a scientific hypothesis, it has to be an idea that can be supported or refuted through carefully crafted experimentation or observation. This concept, called falsifiability and testability, was advanced in the mid-20th century by Austrian-British philosopher Karl Popper in his famous book "The Logic of Scientific Discovery" (Routledge, 1959).

A key function of a hypothesis is to derive predictions about the results of future experiments and then perform those experiments to see whether they support the predictions.

A hypothesis is usually written in the form of an if-then statement, which gives a possibility (if) and explains what may happen because of the possibility (then). The statement could also include "may," according to California State University, Bakersfield .

Here are some examples of hypothesis statements:

• If garlic repels fleas, then a dog that is given garlic every day will not get fleas.
• If sugar causes cavities, then people who eat a lot of candy may be more prone to cavities.
• If ultraviolet light can damage the eyes, then maybe this light can cause blindness.

A useful hypothesis should be testable and falsifiable. That means that it should be possible to prove it wrong. A theory that can't be proved wrong is nonscientific, according to Karl Popper's 1963 book " Conjectures and Refutations ."

An example of an untestable statement is, "Dogs are better than cats." That's because the definition of "better" is vague and subjective. However, an untestable statement can be reworded to make it testable. For example, the previous statement could be changed to this: "Owning a dog is associated with higher levels of physical fitness than owning a cat." With this statement, the researcher can take measures of physical fitness from dog and cat owners and compare the two.

Types of scientific hypotheses

In an experiment, researchers generally state their hypotheses in two ways. The null hypothesis predicts that there will be no relationship between the variables tested, or no difference between the experimental groups. The alternative hypothesis predicts the opposite: that there will be a difference between the experimental groups. This is usually the hypothesis scientists are most interested in, according to the University of Miami .

For example, a null hypothesis might state, "There will be no difference in the rate of muscle growth between people who take a protein supplement and people who don't." The alternative hypothesis would state, "There will be a difference in the rate of muscle growth between people who take a protein supplement and people who don't."

If the results of the experiment show a relationship between the variables, then the null hypothesis has been rejected in favor of the alternative hypothesis, according to the book " Research Methods in Psychology " (​​BCcampus, 2015). 

There are other ways to describe an alternative hypothesis. The alternative hypothesis above does not specify a direction of the effect, only that there will be a difference between the two groups. That type of prediction is called a two-tailed hypothesis. If a hypothesis specifies a certain direction — for example, that people who take a protein supplement will gain more muscle than people who don't — it is called a one-tailed hypothesis, according to William M. K. Trochim , a professor of Policy Analysis and Management at Cornell University.

Sometimes, errors take place during an experiment. These errors can happen in one of two ways. A type I error is when the null hypothesis is rejected when it is true. This is also known as a false positive. A type II error occurs when the null hypothesis is not rejected when it is false. This is also known as a false negative, according to the University of California, Berkeley . 

A hypothesis can be rejected or modified, but it can never be proved correct 100% of the time. For example, a scientist can form a hypothesis stating that if a certain type of tomato has a gene for red pigment, that type of tomato will be red. During research, the scientist then finds that each tomato of this type is red. Though the findings confirm the hypothesis, there may be a tomato of that type somewhere in the world that isn't red. Thus, the hypothesis is true, but it may not be true 100% of the time.

Scientific theory vs. scientific hypothesis

The best hypotheses are simple. They deal with a relatively narrow set of phenomena. But theories are broader; they generally combine multiple hypotheses into a general explanation for a wide range of phenomena, according to the University of California, Berkeley . For example, a hypothesis might state, "If animals adapt to suit their environments, then birds that live on islands with lots of seeds to eat will have differently shaped beaks than birds that live on islands with lots of insects to eat." After testing many hypotheses like these, Charles Darwin formulated an overarching theory: the theory of evolution by natural selection.

"Theories are the ways that we make sense of what we observe in the natural world," Tanner said. "Theories are structures of ideas that explain and interpret facts." 

• Read more about writing a hypothesis, from the American Medical Writers Association.
• Find out why a hypothesis isn't always necessary in science, from The American Biology Teacher.
• Learn about null and alternative hypotheses, from Prof. Essa on YouTube .

Encyclopedia Britannica. Scientific Hypothesis. Jan. 13, 2022. https://www.britannica.com/science/scientific-hypothesis

Karl Popper, "The Logic of Scientific Discovery," Routledge, 1959.

California State University, Bakersfield, "Formatting a testable hypothesis." https://www.csub.edu/~ddodenhoff/Bio100/Bio100sp04/formattingahypothesis.htm  

Karl Popper, "Conjectures and Refutations," Routledge, 1963.

Price, P., Jhangiani, R., & Chiang, I., "Research Methods of Psychology — 2nd Canadian Edition," BCcampus, 2015.‌

University of Miami, "The Scientific Method" http://www.bio.miami.edu/dana/161/evolution/161app1_scimethod.pdf  

William M.K. Trochim, "Research Methods Knowledge Base," https://conjointly.com/kb/hypotheses-explained/  

University of California, Berkeley, "Multiple Hypothesis Testing and False Discovery Rate" https://www.stat.berkeley.edu/~hhuang/STAT141/Lecture-FDR.pdf  

University of California, Berkeley, "Science at multiple levels" https://undsci.berkeley.edu/article/0_0_0/howscienceworks_19

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• Education Resources Information Center - Understanding Hypotheses, Predictions, Laws, and Theories
• Simply Psychology - Research Hypothesis: Definition, Types, & Examples
• Cornell University - The Learning Strategies Center - Hypothesis
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• BCCampus Publishing - Research Methods for the Social Sciences: An Introduction - Hypotheses

hypothesis , something supposed or taken for granted, with the object of following out its consequences (Greek hypothesis , “a putting under,” the Latin equivalent being suppositio ).

In planning a course of action, one may consider various alternatives , working out each in detail. Although the word hypothesis is not typically used in this case, the procedure is virtually the same as that of an investigator of crime considering various suspects. Different methods may be used for deciding what the various alternatives may be, but what is fundamental is the consideration of a supposal as if it were true, without actually accepting it as true. One of the earliest uses of the word in this sense was in geometry . It is described by Plato in the Meno .

The most important modern use of a hypothesis is in relation to scientific investigation . A scientist is not merely concerned to accumulate such facts as can be discovered by observation: linkages must be discovered to connect those facts. An initial puzzle or problem provides the impetus , but clues must be used to ascertain which facts will help yield a solution. The best guide is a tentative hypothesis, which fits within the existing body of doctrine. It is so framed that, with its help, deductions can be made that under certain factual conditions (“initial conditions”) certain other facts would be found if the hypothesis were correct.

The concepts involved in the hypothesis need not themselves refer to observable objects. However, the initial conditions should be able to be observed or to be produced experimentally, and the deduced facts should be able to be observed. William Harvey ’s research on circulation in animals demonstrates how greatly experimental observation can be helped by a fruitful hypothesis. While a hypothesis can be partially confirmed by showing that what is deduced from it with certain initial conditions is actually found under those conditions, it cannot be completely proved in this way. What would have to be shown is that no other hypothesis would serve. Hence, in assessing the soundness of a hypothesis, stress is laid on the range and variety of facts that can be brought under its scope. Again, it is important that it should be capable of being linked systematically with hypotheses which have been found fertile in other fields.

If the predictions derived from the hypothesis are not found to be true, the hypothesis may have to be given up or modified. The fault may lie, however, in some other principle forming part of the body of accepted doctrine which has been utilized in deducing consequences from the hypothesis. It may also lie in the fact that other conditions, hitherto unobserved, are present beside the initial conditions, affecting the result. Thus the hypothesis may be kept, pending further examination of facts or some remodeling of principles. A good illustration of this is to be found in the history of the corpuscular and the undulatory hypotheses about light .

Research Hypothesis In Psychology: Types, & Examples

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

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Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

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A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

How to Develop a Good Research Hypothesis

The story of a research study begins by asking a question. Researchers all around the globe are asking curious questions and formulating research hypothesis. However, whether the research study provides an effective conclusion depends on how well one develops a good research hypothesis. Research hypothesis examples could help researchers get an idea as to how to write a good research hypothesis.

This blog will help you understand what is a research hypothesis, its characteristics and, how to formulate a research hypothesis

Table of Contents

What is Hypothesis?

Hypothesis is an assumption or an idea proposed for the sake of argument so that it can be tested. It is a precise, testable statement of what the researchers predict will be outcome of the study.  Hypothesis usually involves proposing a relationship between two variables: the independent variable (what the researchers change) and the dependent variable (what the research measures).

What is a Research Hypothesis?

Research hypothesis is a statement that introduces a research question and proposes an expected result. It is an integral part of the scientific method that forms the basis of scientific experiments. Therefore, you need to be careful and thorough when building your research hypothesis. A minor flaw in the construction of your hypothesis could have an adverse effect on your experiment. In research, there is a convention that the hypothesis is written in two forms, the null hypothesis, and the alternative hypothesis (called the experimental hypothesis when the method of investigation is an experiment).

Characteristics of a Good Research Hypothesis

As the hypothesis is specific, there is a testable prediction about what you expect to happen in a study. You may consider drawing hypothesis from previously published research based on the theory.

A good research hypothesis involves more effort than just a guess. In particular, your hypothesis may begin with a question that could be further explored through background research.

To help you formulate a promising research hypothesis, you should ask yourself the following questions:

• Is the language clear and focused?
• What is the relationship between your hypothesis and your research topic?
• Is your hypothesis testable? If yes, then how?
• What are the possible explanations that you might want to explore?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate your variables without hampering the ethical standards?
• Does your research predict the relationship and outcome?
• Is your research simple and concise (avoids wordiness)?
• Is it clear with no ambiguity or assumptions about the readers’ knowledge
• Is your research observable and testable results?
• Is it relevant and specific to the research question or problem?

The questions listed above can be used as a checklist to make sure your hypothesis is based on a solid foundation. Furthermore, it can help you identify weaknesses in your hypothesis and revise it if necessary.

Source: Educational Hub

How to formulate a research hypothesis.

A testable hypothesis is not a simple statement. It is rather an intricate statement that needs to offer a clear introduction to a scientific experiment, its intentions, and the possible outcomes. However, there are some important things to consider when building a compelling hypothesis.

1. State the problem that you are trying to solve.

Make sure that the hypothesis clearly defines the topic and the focus of the experiment.

2. Try to write the hypothesis as an if-then statement.

Follow this template: If a specific action is taken, then a certain outcome is expected.

3. Define the variables

Independent variables are the ones that are manipulated, controlled, or changed. Independent variables are isolated from other factors of the study.

Dependent variables , as the name suggests are dependent on other factors of the study. They are influenced by the change in independent variable.

4. Scrutinize the hypothesis

Evaluate assumptions, predictions, and evidence rigorously to refine your understanding.

Types of Research Hypothesis

The types of research hypothesis are stated below:

1. Simple Hypothesis

It predicts the relationship between a single dependent variable and a single independent variable.

2. Complex Hypothesis

It predicts the relationship between two or more independent and dependent variables.

3. Directional Hypothesis

It specifies the expected direction to be followed to determine the relationship between variables and is derived from theory. Furthermore, it implies the researcher’s intellectual commitment to a particular outcome.

4. Non-directional Hypothesis

It does not predict the exact direction or nature of the relationship between the two variables. The non-directional hypothesis is used when there is no theory involved or when findings contradict previous research.

5. Associative and Causal Hypothesis

The associative hypothesis defines interdependency between variables. A change in one variable results in the change of the other variable. On the other hand, the causal hypothesis proposes an effect on the dependent due to manipulation of the independent variable.

6. Null Hypothesis

Null hypothesis states a negative statement to support the researcher’s findings that there is no relationship between two variables. There will be no changes in the dependent variable due the manipulation of the independent variable. Furthermore, it states results are due to chance and are not significant in terms of supporting the idea being investigated.

7. Alternative Hypothesis

It states that there is a relationship between the two variables of the study and that the results are significant to the research topic. An experimental hypothesis predicts what changes will take place in the dependent variable when the independent variable is manipulated. Also, it states that the results are not due to chance and that they are significant in terms of supporting the theory being investigated.

Research Hypothesis Examples of Independent and Dependent Variables

Research Hypothesis Example 1 The greater number of coal plants in a region (independent variable) increases water pollution (dependent variable). If you change the independent variable (building more coal factories), it will change the dependent variable (amount of water pollution).

Research Hypothesis Example 2 What is the effect of diet or regular soda (independent variable) on blood sugar levels (dependent variable)? If you change the independent variable (the type of soda you consume), it will change the dependent variable (blood sugar levels)

You should not ignore the importance of the above steps. The validity of your experiment and its results rely on a robust testable hypothesis. Developing a strong testable hypothesis has few advantages, it compels us to think intensely and specifically about the outcomes of a study. Consequently, it enables us to understand the implication of the question and the different variables involved in the study. Furthermore, it helps us to make precise predictions based on prior research. Hence, forming a hypothesis would be of great value to the research. Here are some good examples of testable hypotheses.

More importantly, you need to build a robust testable research hypothesis for your scientific experiments. A testable hypothesis is a hypothesis that can be proved or disproved as a result of experimentation.

Importance of a Testable Hypothesis

To devise and perform an experiment using scientific method, you need to make sure that your hypothesis is testable. To be considered testable, some essential criteria must be met:

• There must be a possibility to prove that the hypothesis is true.
• There must be a possibility to prove that the hypothesis is false.
• The results of the hypothesis must be reproducible.

Without these criteria, the hypothesis and the results will be vague. As a result, the experiment will not prove or disprove anything significant.

What are your experiences with building hypotheses for scientific experiments? What challenges did you face? How did you overcome these challenges? Please share your thoughts with us in the comments section.

Frequently Asked Questions

The steps to write a research hypothesis are: 1. Stating the problem: Ensure that the hypothesis defines the research problem 2. Writing a hypothesis as an 'if-then' statement: Include the action and the expected outcome of your study by following a ‘if-then’ structure. 3. Defining the variables: Define the variables as Dependent or Independent based on their dependency to other factors. 4. Scrutinizing the hypothesis: Identify the type of your hypothesis

Hypothesis testing is a statistical tool which is used to make inferences about a population data to draw conclusions for a particular hypothesis.

Hypothesis in statistics is a formal statement about the nature of a population within a structured framework of a statistical model. It is used to test an existing hypothesis by studying a population.

Research hypothesis is a statement that introduces a research question and proposes an expected result. It forms the basis of scientific experiments.

The different types of hypothesis in research are: • Null hypothesis: Null hypothesis is a negative statement to support the researcher’s findings that there is no relationship between two variables. • Alternate hypothesis: Alternate hypothesis predicts the relationship between the two variables of the study. • Directional hypothesis: Directional hypothesis specifies the expected direction to be followed to determine the relationship between variables. • Non-directional hypothesis: Non-directional hypothesis does not predict the exact direction or nature of the relationship between the two variables. • Simple hypothesis: Simple hypothesis predicts the relationship between a single dependent variable and a single independent variable. • Complex hypothesis: Complex hypothesis predicts the relationship between two or more independent and dependent variables. • Associative and casual hypothesis: Associative and casual hypothesis predicts the relationship between two or more independent and dependent variables. • Empirical hypothesis: Empirical hypothesis can be tested via experiments and observation. • Statistical hypothesis: A statistical hypothesis utilizes statistical models to draw conclusions about broader populations.

Wow! You really simplified your explanation that even dummies would find it easy to comprehend. Thank you so much.

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I enjoy reading the post. Hypotheses are actually an intrinsic part in a study. It bridges the research question and the methodology of the study.

Useful piece!

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It very interesting to read the topic, can you guide me any specific example of hypothesis process establish throw the Demand and supply of the specific product in market

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It is really a useful for me Kindly give some examples of hypothesis

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Six Steps of the Scientific Method

Learn What Makes Each Stage Important

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The scientific method is a systematic way of learning about the world around us. The key difference between the scientific method and other ways of acquiring knowledge is that, when using the scientific method, we make hypotheses and then test them with an experiment.

Anyone can use the scientific method to acquire knowledge by asking questions and then working to find the answers to those questions. Below are the six steps involved in the scientific method and variables you may encounter when working with this method.

The Six Steps

The number of steps in the scientific method can vary from one description to another (which mainly happens when data and analysis are separated into separate steps), however, below is a fairly standard list of the six steps you'll likely be expected to know for any science class:

• Purpose/Question Ask a question.
• Research Conduct background research. Write down your sources so you can cite your references. In the modern era, you might conduct much of your research online. As you read articles and papers online, ensure you scroll to the bottom of the text to check the author's references. Even if you can't access the full text of a published article, you can usually view the abstract to see the summary of other experiments . Interview experts on a topic. The more you know about a subject, the easier it'll be to conduct your investigation.
• Hypothesis Propose a hypothesis . This is a sort of educated guess about what you expect your research to reveal. A hypothesis is a statement used to predict the outcome of an experiment. Usually, a hypothesis is written in terms of cause and effect. Alternatively, it may describe the relationship between two phenomena. The null hypothesis or the no-difference hypothesis is one type of hypothesis that's easy to test because it assumes changing a variable will not affect the outcome. In reality, you probably expect a change, but rejecting a hypothesis may be more useful than accepting one.
• Experiment Design and experiment to test your hypothesis. An experiment has an independent and dependent variable. You change or control the independent variable and record the effect it has on the dependent variable . It's important to change only one variable for an experiment rather than try to combine the effects of variables in an experiment. For example, if you want to test the effects of light intensity and fertilizer concentration on the growth rate of a plant, you're looking at two separate experiments.
• Data/Analysis Record observations and analyze the meaning of the data. Often, you'll prepare a table or graph of the data. Don't throw out data points you think are bad or that don't support your predictions. Some of the most incredible discoveries in science were made because the data looked wrong! Once you have the data, you may need to perform a mathematical analysis to support or refute your hypothesis.
• Conclusion Conclude whether to accept or reject your hypothesis. There's no right or wrong outcome to an experiment, so either result is fine. Accepting a hypothesis doesn't necessarily mean it's correct! Sometimes repeating an experiment may give a different result. In other cases, a hypothesis may predict an outcome, yet you might draw an incorrect conclusion. Communicate your results. You can compile your results into a lab report or formally submit them as a paper . Whether you accept or reject the hypothesis, you likely learned something about the subject and may wish to revise the original hypothesis or form a new one for a future experiment.

Understanding Science

How science REALLY works...

• Understanding Science 101
• Misconceptions
• Testing ideas with evidence is at the heart of the process of science.
• Scientific testing involves figuring out what we would  expect  to observe if an idea were correct and comparing that expectation to what we  actually  observe.

Misconception:  Science proves ideas.

Misconception:  Science can only disprove ideas.

Correction:  Science neither proves nor disproves. It accepts or rejects ideas based on supporting and refuting evidence, but may revise those conclusions if warranted by new evidence or perspectives.  Read more about it.

Testing scientific ideas

Testing ideas about childbed fever.

As a simple example of how scientific testing works, consider the case of Ignaz Semmelweis, who worked as a doctor on a maternity ward in the 1800s. In his ward, an unusually high percentage of new mothers died of what was then called childbed fever. Semmelweis considered many possible explanations for this high death rate. Two of the many ideas that he considered were (1) that the fever was caused by mothers giving birth lying on their backs (as opposed to on their sides) and (2) that the fever was caused by doctors’ unclean hands (the doctors often performed autopsies immediately before examining women in labor). He tested these ideas by considering what expectations each idea generated. If it were true that childbed fever were caused by giving birth on one’s back, then changing procedures so that women labored on their sides should lead to lower rates of childbed fever. Semmelweis tried changing the position of labor, but the incidence of fever did not decrease; the actual observations did not match the expected results. If, however, childbed fever were caused by doctors’ unclean hands, having doctors wash their hands thoroughly with a strong disinfecting agent before attending to women in labor should lead to lower rates of childbed fever. When Semmelweis tried this, rates of fever plummeted; the actual observations matched the expected results, supporting the second explanation.

Testing in the tropics

Let’s take a look at another, very different, example of scientific testing: investigating the origins of coral atolls in the tropics. Consider the atoll Eniwetok (Anewetak) in the Marshall Islands — an oceanic ring of exposed coral surrounding a central lagoon. From the 1800s up until today, scientists have been trying to learn what supports atoll structures beneath the water’s surface and exactly how atolls form. Coral only grows near the surface of the ocean where light penetrates, so Eniwetok could have formed in several ways:

Hypothesis 2: The coral that makes up Eniwetok might have grown in a ring atop an underwater mountain already near the surface. The key to this hypothesis is the idea that underwater mountains don’t sink; instead the remains of dead sea animals (shells, etc.) accumulate on underwater mountains, potentially assisted by tectonic uplifting. Eventually, the top of the mountain/debris pile would reach the depth at which coral grow, and the atoll would form.

Which is a better explanation for Eniwetok? Did the atoll grow atop a sinking volcano, forming an underwater coral tower, or was the mountain instead built up until it neared the surface where coral were eventually able to grow? Which of these explanations is best supported by the evidence? We can’t perform an experiment to find out. Instead, we must figure out what expectations each hypothesis generates, and then collect data from the world to see whether our observations are a better match with one of the two ideas.

If Eniwetok grew atop an underwater mountain, then we would expect the atoll to be made up of a relatively thin layer of coral on top of limestone or basalt. But if it grew upwards around a subsiding island, then we would expect the atoll to be made up of many hundreds of feet of coral on top of volcanic rock. When geologists drilled into Eniwetok in 1951 as part of a survey preparing for nuclear weapons tests, the drill bored through more than 4000 feet (1219 meters) of coral before hitting volcanic basalt! The actual observation contradicted the underwater mountain explanation and matched the subsiding island explanation, supporting that idea. Of course, many other lines of evidence also shed light on the origins of coral atolls, but the surprising depth of coral on Eniwetok was particularly convincing to many geologists.

• Take a sidetrip

Visit the NOAA website to see an animation of coral atoll formation according to Hypothesis 1.

• Teaching resources

Scientists test hypotheses and theories. They are both scientific explanations for what we observe in the natural world, but theories deal with a much wider range of phenomena than do hypotheses. To learn more about the differences between hypotheses and theories, jump ahead to  Science at multiple levels .

• Use our  web interactive  to help students document and reflect on the process of science.
• Learn strategies for building lessons and activities around the Science Flowchart: Grades 3-5 Grades 6-8 Grades 9-12 Grades 13-16
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Observation beyond our eyes

The logic of scientific arguments

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ACH , Analysis of Competing Hypotheses , Hypothesis mapping , Hypothesis Testing

Hypothesis Investigation – overview

Hypothesis investigation (short for “hypothesis-based investigation”) is simply attempting to determine “what is going on” in some situation by assessing various hypotheses or “guesses”.  The goal is to determine which hypothesis is most likely to be true. 

Hypothesis investigation can concern

• Factual situations – e.g. what are current Saudi oil reserves?
• Causes – e.g. what killed the dinosaurs?
• Functions or roles – e.g. what was the Antikythera mechanism for?
• Future events – e.g. how will the economy be affected by Peak Oil?
• States of mind – e.g. what are the enemy planning to do?
• Perpetrators – e.g. Who murdered Professor Plum?

Most investigation is to some extent hypothesis-based.  The exception is situations where the outcome is pre-determined in some way (e.g., a political show trial) and the point of the investigation is simply to amass evidence supporting that determination. 

A related, though subtly different notion is that of “hypothesis driven investigation” (Rasiel, 1999), in which a single hypothesis is selected relatively early in the process, and most effort is then devoted to substantiating this hypothesis.   It is hypothesis-based investigation with all attention focused on one guess, at least while not forced to reject it and consider another. 

Hypothesis investigation is comprised of three main activities

• Hypothesis generation  – coming up with hypotheses;
• Hypothesis evaluation – assessing relative plausibility of hypotheses given the available evidence; and
• Hypothesis testing – seeking further evidence.

Traps in Hypothesis Investigation

Hypothesis investigation fails, at its simplest, when we get (take as true) the wrong hypothesis.  This can have dismal consequences if costly actions are then taken.  Hypothesis investigation also fails when

• there is misplaced or excessive confidence in a hypothesis (even if it happens to be correct);
•  no conclusion is reached, when more careful investigation might have revealed that one hypothesis was most plausible. 

There are three main traps leading to these failures.

Tunnel vision

Not considering the full range of reasonable hypotheses.   Lots of effort is put into investigating one or a few hypotheses, usually obvious ones, while other possibilities are not considered at all.  All too often one of those others is in fact the right one. 

Abusing the evidence

Here the evidence already at hand is not evaluated properly, leading to erroneous assessments of the plausibility of hypotheses.

A particular item of evidence might be regarded as stronger or more significant than it really is, especially if it appears to support your preferred hypothesis.  Conversely, a “negative” piece of evidence – one that directly undercuts your preferred hypothesis, or appears to strongly support another – is regarded as weak or worthless.    

Further, the whole body of evidence bearing upon a hypothesis might be mis-rated.  A few scraps of dismal evidence might be taken as collectively amounting to a strong case. 

Looking in the wrong places

When seeking additional evidence, you instinctively look for information that in fact is useless or at least not very helpful in terms of helping you determine the truth.

In particular we are prone to “confirmation bias,” which is seeking information that would lend weight to our favoured hypothesis.  We tend to think that by accumulating lots of such supporting evidence, we’re rigorously testing the hypothesis.  But this is a classic mistake. We need to know not only that there’s lots of evidence consistent with our favoured hypothesis, but also that there is evidence inconsistent with alternatives.   You need to seek the right kind of evidence in relation to your whole hypothesis set, rather than just lots of evidence consistent with one hypothesis.  

This can have two unfortunate consequences.  The search may be

• Ineffective – you never find evidnce which could have very strongly ruled one or more hypotheses “in” or “out”. 
• Inefficient – the hypothesis testing process may take much more time and resources than it really should have. 

We fall for these traps because of basic facts of human psychology, hard-wired “features” of our thinking tracing back to our evolutionary origins as hunter-gatherers in small tribal units: 

• We dislike disorder, confusion and uncertainty.  Our brains strive to find the simple pattern that makes sense of a complex or noisy reality. 
• We don’t like changing our minds.  We find it easier to stick with our current opinion than to upend things and take  Further, we have undue preference for hypotheses that are consistent with our general background beliefs, and so don’t force us to question or modify those beliefs.  
• We become emotionally engaged in the issues, and build affection for one hypothesis and loathing for others.   Hypothesis investigation becomes a matter of protecting one’s young rather than culling the pack (Chamberlin, 1965).
• Social pressure.  We become publicly committed to a position, and feel that changing our minds would mean losing face. 

And of course we are frequently under time pressure, exacerbating the above tendencies.    

General Guidelines for Good Hypothesis Investigation

Canvass a wide range of hypotheses.

Our natural tendency is to grab hold of the first plausible hypothesis that comes to mind and start shaking it hard.  This should be resisted.  From the outset you should canvass as wide a range of hypotheses as you reasonably can.  It is impossible to canvass all hypotheses and absurd to even try ( Maybe 9/11 was the work of the Jasper County Beekeepers! ).   But you can and should keep in mind a broad selection of hypotheses, including at least some “long shots.”   In generating this hypothesis set, diversity is at least as important as quantity.

You should continue seeking additional hypotheses throughout the investigation.   Incoming information can suggest interesting new possibilities, but only if you’re in a suitably “suggestible” state of mind.   

Actively investigate multiple hypotheses

At any given time you should keep a number of hypotheses “in play”.   In hypothesis testing, i.e. seeking new information, you should seek information which discriminates which will be “telling” in relation to multiple hypotheses at once. 

Seek disconfirming evidence       Instead of trying to prove that some hypothesis is correct, you should be trying to prove that it is false.   As philosopher Karl Popper famously observed, the best hypotheses are those that survive numerous attempts at refutation.   Ideally, you should seek to disconfirm multiple hypotheses at the same.   This can be easier if your hypothesis set is hierarchically organised, allowing you to seek evidence knocking out whole groups of hypotheses at a time.  

Instead of trying to prove that some hypothesis is correct, you should be trying to prove that it is false.   As philosopher Karl Popper famously observed, the best hypotheses are those that survive numerous attempts at refutation.  

Ideally, you should seek to disconfirm multiple hypotheses at the same.   This can be easier if your hypothesis set is hierarchically organised, allowing you to seek evidence knocking out whole groups of hypotheses at a time.  

Structured methodologies.

Some methodologies have been developed to help with hypothesis investigation.  The methodologies have some important advantages over proceeding in an “intuitive” or spontaneous fashion. 

• They are designed to help us avoid the traps, and do so by building in, to some extent, the general guidelines above.
• They provide distinctive external representations which help us organize and comprehend the hypothesis sets and the evidence.   These external representations reduce the cognitive load involved in keeping lots of information related in complex ways in our heads.

Some structured methodologies are:

• Analysis of Competing Hypotheses (Heuer, 1999), designed especially for intelligence analysis
• Hypothesis Mapping
• Root Cause Analysis

4 thoughts on “ Hypothesis Investigation – overview ”

Add Comment

The point there is misplaced or excessive confidence in a hypothesis (even if it happens to be correct) could do with a little extra explication–why is excessive confidence in a correct hypothesis a problem?

‘evidence’ is misspelled.

There are some words missing from We find it easier to stick with our current opinion than to upend things and take Further,

But it seems nice and clear otherwise.

It’s only peripherally related, but you might be interested in this latest twist on second-guessing yourself.

I like the title too: “You know more than you think” http://www.scientificamerican.com/article.cfm?id=you-know-more-than-you-think

cheers, RdR

I would argue for a activity between 2 and 3 above – “hypothesis framing”, in which the hypothesis is expressed in such a way that it can be tested.

Hi, nice job! Like the lack of jargon :)

Hope my comment does not lead you the other direction, but you may want to take a look at the literature on “abduction,” a term coined by philosopher Charles Pierce, if you have not already.

Abduction is defined by most as the process of generating hypotheses or generating and evaluating hypotheses.

There is a diverse set of academic literature that touches on abduction, including philosophy, the history of science (e.g., scientific discovery), management (e.g., product development), and healthcare (e.g., medical diagnosis).

I dug into this pretty deep so if you have any follow up questions, pls feel free to ping me.

Regards, Michael

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5.5 Introduction to Hypothesis Tests

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter.

Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealership advertises that its new small truck gets 35 miles per gallon on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that female managers in their company earn an average of $60,000 per year. A statistician may want to make a decision about or evaluate these claims. A hypothesis test can be used to do this.

A hypothesis test involves collecting data from a sample and evaluating the data. Then the statistician makes a decision as to whether or not there is sufficient evidence to reject the null hypothesis based upon analyses of the data.

In this section, you will conduct hypothesis tests on single means when the population standard deviation is known.

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will perform some variation of these steps:

• Define hypotheses.
• Collect and/or use the sample data to determine the correct distribution to use.
• Calculate test statistic.
• Make a decision.
• Write a conclusion.

Defining your hypotheses

The actual test begins by considering two hypotheses: the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints.

The null hypothesis ( H 0 ) is often a statement of the accepted historical value or norm. This is your starting point that you must assume from the beginning in order to show an effect exists.

The alternative hypothesis ( H a ) is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision . There are two options for a decision. They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “decline to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

The following table shows mathematical symbols used in H 0 and H a :

NOTE: H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol in the alternative hypothesis depends on the wording of the hypothesis test. Despite this, many researchers may use =, ≤, or ≥ in the null hypothesis. This practice is acceptable because our only decision is to reject or not reject the null hypothesis.

We want to test whether the mean GPA of students in American colleges is 2.0 (out of 4.0). The null hypothesis is: H 0 : μ = 2.0. What is the alternative hypothesis?

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Using the Sample to Test the Null Hypothesis

Once you have defined your hypotheses, the next step in the process is to collect sample data. In a classroom context, the data or summary statistics will usually be given to you.

Then you will have to determine the correct distribution to perform the hypothesis test, given the assumptions you are able to make about the situation. Right now, we are demonstrating these ideas in a test for a mean when the population standard deviation is known using the z distribution. We will see other scenarios in the future.

Calculating a Test Statistic

Next you will start evaluating the data. This begins with calculating your test statistic , which is a measure of the distance between what you observed and what you are assuming to be true. In this context, your test statistic, z ο , quantifies the number of standard deviations between the sample mean, x, and the population mean, µ . Calculating the test statistic is analogous to the previously discussed process of standardizing observations with z -scores:

where µ o   is the value assumed to be true in the null hypothesis.

Making a Decision

Once you have your test statistic, there are two methods to use it to make your decision:

• Critical value method (discussed further in later chapters)
• p -value method (our current focus)

p -Value Method

To find a p -value , we use the test statistic to calculate the actual probability of getting the test result. Formally, the p -value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.

A large p -value calculated from the data indicates that we should not reject the null hypothesis. The smaller the p -value, the more unlikely the outcome and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.

Draw a graph that shows the p -value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.

Suppose a baker claims that his bread height is more than 15 cm on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes ten loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm and the distribution of heights is normal.

The null hypothesis could be H 0 : μ ≤ 15.

The alternate hypothesis is H a : μ > 15.

The words “is more than” calls for the use of the > symbol, so “ μ > 15″ goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.

Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then, is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p -value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.

This means that the p -value is the probability that a sample mean is the same or greater than 17 cm when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means.

A p -value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm on average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm purely by CHANCE had the population mean height really been 15 cm. Because the outcome of 17 cm is so unlikely (meaning it is happening NOT by chance alone), we conclude that the evidence is strongly against the null hypothesis that the mean height would be at most 15 cm. There is sufficient evidence that the true mean height for the population of the baker’s loaves of bread is greater than 15 cm.

A normal distribution has a standard deviation of one. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

Find the p -value.

Decision and Conclusion

A systematic way to decide whether to reject or not reject the null hypothesis is to compare the p -value and a preset or preconceived α (also called a significance level ). A preset α is the probability of a type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem. If there is no given preconceived α , then use α = 0.05.

When you make a decision to reject or not reject H 0 , do as follows:

• If α > p -value, reject H 0 . The results of the sample data are statistically significant . You can say there is sufficient evidence to conclude that H 0 is an incorrect belief and that the alternative hypothesis, H a , may be correct.
• If α ≤ p -value, fail to reject H 0 . The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis, H a , may be correct.

After you make your decision, write a thoughtful conclusion in the context of the scenario incorporating the hypotheses.

NOTE: When you “do not reject H 0 ,” it does not mean that you should believe that H 0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H o .

When using the p -value to evaluate a hypothesis test, the following rhymes can come in handy:

If the p -value is low, the null must go.

If the p -value is high, the null must fly.

This memory aid relates a p -value less than the established alpha (“the p -value is low”) as rejecting the null hypothesis and, likewise, relates a p -value higher than the established alpha (“the p -value is high”) as not rejecting the null hypothesis.

Fill in the blanks:

• Reject the null hypothesis when              .
• The results of the sample data             .
• Do not reject the null when hypothesis when             .

It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

• H 0 : p = 0.50, H a : p > 0.50
• p -value = 0.025

Interpret the results and state a conclusion in simple, non-technical terms.

Click here for more multimedia resources, including podcasts, videos, lecture notes, and worked examples.

Figure References

Figure 5.11: Alora Griffiths (2019). dalmatian puppy near man in blue shorts kneeling. Unsplash license. https://unsplash.com/photos/7aRQZtLsvqw

Figure 5.13: Kindred Grey (2020). Bread height probability. CC BY-SA 4.0.

A decision-making procedure for determining whether sample evidence supports a hypothesis

The claim that is assumed to be true and is tested in a hypothesis test

A working hypothesis that is contradictory to the null hypothesis

A measure of the difference between observations and the hypothesized (or claimed) value

The probability that an event will occur, assuming the null hypothesis is true

Probability that a true null hypothesis will be rejected, also known as type I error and denoted by α

Finding sufficient evidence that the observed effect is not just due to variability, often from rejecting the null hypothesis

Significant Statistics Copyright © 2024 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

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• Published: 06 September 2024

Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks

• Omid G. Sani   ORCID: orcid.org/0000-0003-3032-5669 1 ,
• Bijan Pesaran   ORCID: orcid.org/0000-0003-4116-0038 2 &
• Maryam M. Shanechi   ORCID: orcid.org/0000-0002-0544-7720 1 , 3 , 4 , 5  

Nature Neuroscience ( 2024 ) Cite this article

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• Brain–machine interface
• Dynamical systems
• Machine learning
• Neural decoding
• Neural encoding

Understanding the dynamical transformation of neural activity to behavior requires new capabilities to nonlinearly model, dissociate and prioritize behaviorally relevant neural dynamics and test hypotheses about the origin of nonlinearity. We present dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling approach that enables these capabilities with a multisection neural network architecture and training approach. Analyzing cortical spiking and local field potential activity across four movement tasks, we demonstrate five use-cases. DPAD enabled more accurate neural–behavioral prediction. It identified nonlinear dynamical transformations of local field potentials that were more behavior predictive than traditional power features. Further, DPAD achieved behavior-predictive nonlinear neural dimensionality reduction. It enabled hypothesis testing regarding nonlinearities in neural–behavioral transformation, revealing that, in our datasets, nonlinearities could largely be isolated to the mapping from latent cortical dynamics to behavior. Finally, DPAD extended across continuous, intermittently sampled and categorical behaviors. DPAD provides a powerful tool for nonlinear dynamical modeling and investigation of neural–behavioral data.

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Understanding how neural population dynamics give rise to behavior is a major goal in neuroscience. Many methods that relate neural activity to behavior use static mappings or embeddings, which do not describe the temporal structure in how neural population activity evolves over time 1 . In comparison, dynamical models can describe these temporal structures in terms of low-dimensional latent states embedded in the high-dimensional space of neural recordings. Prior dynamical models have often been linear or generalized linear 1 , 2 , 3 , 4 , 5 , 6 , 7 , thus motivating recent work to develop support for piece-wise linear 8 , locally linear 9 , switching linear 10 , 11 , 12 , 13 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 models of neural dynamics, especially in applications such as single-trial smoothing of neural population activity 9 , 14 , 15 , 16 , 17 , 18 , 19 and decoding behavior 20 , 21 , 22 , 23 , 24 , 26 . Once trained, the latent states of these models can subsequently be mapped to behavior 1 , 25 to learn an overall dynamical transformation from neural activity to behavior. However, multiple challenges hinder the dynamical modeling and interpretation of neural–behavioral transformations.

First, the neural–behavioral transformation can exhibit nonlinearities, which the dynamical model should capture. Moreover, these nonlinearities can be in one or more different elements within the dynamical model, for example, in the dynamics of the latent state or in its embedding. Enabling hypothesis testing regarding the origin of nonlinearity (that is, where the nonlinearity can be isolated to within the model) is important for interpreting neural computations and developing neurotechnology but remains largely unaddressed in current nonlinear models. Second, neural dynamics related to a given behavior often constitute a minority of the total neural variance 28 , 29 , 30 , 31 , 32 , 33 . To avoid missing or confounding these dynamics, nonlinear dynamical models need to dissociate behaviorally relevant neural dynamics from other neural dynamics and prioritize the learning of the former, which is currently not possible. Indeed, existing nonlinear methods for modeling neural activity either do not explicitly model temporal dynamics 34 , 35 , 36 or do not prioritize behaviorally relevant dynamics 16 , 37 , 38 , or have a mixed objective 18 that may mix behaviorally relevant and other neural dynamics in the same latent states ( Discussion and Extended Data Table 1 ). Our prior method, termed PSID 6 , has enabled prioritized dissociation of behaviorally relevant neural dynamics but for linear dynamical models. Third, for broad applicability, in addition to continuous behaviors, dynamical models should admit categorical (for example, choices) or intermittently sampled behaviors (for example, mood reports), which are not supported by existing dynamical methods with a mixed objective 18 or by PSID. To date, learning nonlinear dynamical models of neural population activity that can address the above challenges has not been achieved.

Here, we develop dissociative prioritized analysis of dynamics (DPAD), a nonlinear dynamical modeling framework using recurrent neural networks (RNNs) that addresses all the above challenges. DPAD models both behaviorally relevant and other neural dynamics but dissociates them into separate latent states and prioritizes the learning of the former. To do so, we formulate a two-section RNN as the DPAD nonlinear dynamical model and develop a four-step optimization algorithm to train it. The first RNN section learns the behaviorally relevant latent states with priority, and the second section learns any remaining neural dynamics (Fig. 1a and Supplementary Fig. 1 ). Moreover, DPAD adjusts these optimization steps as needed to admit continuous-valued, categorical or intermittently sampled data ( Methods ). Furthermore, to capture nonlinearity in the neural–behavioral transformation and enable hypothesis testing regarding its origins, DPAD decomposes this transformation into the following four interpretable elements and allows each element to become linear or nonlinear (Fig. 1a,b ): the mapping from neural activity to the latent space (neural input), the latent state dynamics within this space (recursion) and the mappings of the state to neural activity and behavior (neural and behavior readouts). Finally, we formulate the DPAD model in predictor form such that the learned model can be directly used for inference, enabling causal and computationally efficient decoding for data, whether with or without a fixed-length trial structure ( Methods ).

a , DPAD decomposes the neural–behavioral transformation into four interpretable mapping elements. It learns the mapping of neural activity ( y k ) to latent states ( x k ), termed neural input in the model; learns the dynamics or temporal structure of the latent states, termed recursion in the model; dissociates the behaviorally relevant latent states ( \({x}_{k}^{\left(1\right)}\) ) that are relevant to a measured behavior ( z k ) from other states ( \({x}_{k}^{\left(2\right)}\) ); learns the mapping of the latent states to behavior and to neural activity, termed behavior and neural readouts in the model; and allows flexible linear or nonlinear mappings in any of its elements. DPAD additionally prioritizes the learning of behaviorally relevant neural dynamics to learn them accurately. b , Computation graph of the DPAD model consists of a two-section RNN whose input is neural activity at the current time step and whose outputs are the predicted behavior and neural activity in the next time step ( Methods ). This graph assumes that computations are Markovian, that is, with a high enough dimension, latent states can summarize the information from past neural data that is useful for predicting future neural–behavioral data. Each of the four mapping elements from a has a corresponding parameter in each section of the RNN model, indicated by the same colors and termed as introduced in a . c , We developed a four-step optimization method to learn all the model parameters from training neural–behavioral data (Supplementary Fig. 1a ). Further, each model parameter can be specified via the ‘nonlinearity setting’ to be linear or nonlinear with various options to implement the nonlinearity (Supplementary Fig. 1b,c ). After a model is learned, only past neural activity is used to decode behavior and predict neural activity using the computation graph in b . d , DPAD also has the option of automatically selecting the ‘nonlinearity setting’ for the data by fitting candidate models and comparing them in terms of both behavior decoding and neural self-prediction accuracy ( Methods ). In this work, we chose among 90 candidate models with various nonlinearity settings ( Methods ). We refer to this automatic selection of nonlinearity as ‘DPAD with flexible nonlinearity’.

To show its broad utility, we demonstrate five distinct use-cases for DPAD across four diverse nonhuman primate (NHP) datasets consisting of both population spiking activity and local field potentials (LFPs). First, DPAD more accurately models the overall neural–behavioral data than alternative nonlinear and linear methods. This is due both to DPAD’s prioritized and dynamical modeling of behaviorally relevant neural dynamics and to its nonlinearity. Second, DPAD can automatically uncover nonlinear dynamical transformations of raw LFP that are more predictive of behavior than traditional LFP power band features and in some datasets can even outperform population spiking activity in terms of behavior prediction. Further, DPAD reveals that among the neural modalities, the degree of nonlinearity is greatest for the raw LFP. Third, DPAD enables nonlinear and dynamical neural dimensionality reduction while preserving behavior information, thus extracting lower-dimensional yet more behavior-predictive latent states from past neural activity. Fourth, DPAD enables hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation. Consistently across our movement-related datasets, doing so revealed that summarizing the nonlinearities just in the behavior readout from the latent state is largely sufficient for predicting the neural–behavioral data (see Discussion ). Fifth, DPAD extends to categorical and intermittently observed behaviors, which is important for cognitive neuroscience 11 , 39 and neuropsychiatry 40 , 41 , 42 . Together, these results highlight DPAD’s broad utility as a dynamical modeling tool to investigate the nonlinear and dynamical transformation of neural activity to specific behaviors across various domains of neuroscience.

Overview of DPAD

Formulation.

We model neural activity and behavior jointly and nonlinearly ( Methods ) as

where k is the time index, \({y}_{k}\in {{\mathbb{R}}}^{{n}_{y}}\) and \({z}_{k}\in {{\mathbb{R}}}^{{n}_{z}}\) denote the neural activity and behavior time series, respectively, \({x}_{k}\in {{\mathbb{R}}}^{{n}_{x}}\) is the latent state, and e k and \({{\epsilon }}_{k}\) denote neural and behavior dynamics that are unpredictable from past neural activity. Multi-input–multi-output functions A ′ (recursion), K (neural input), C y (neural readout) and C z (behavior readout) are parameters that fully specify the model and have interpretable descriptions ( Methods , Supplementary Note 1 and Fig. 1a,b ). The adjusted formulation for intermittently sampled and noncontinuous-valued (for example, categorical) data is provided in Methods . DPAD supports both linear and nonlinear modeling, which will be termed linear DPAD and nonlinear DPAD (or just DPAD), respectively.

Dissociative and prioritized learning

We further expand the model in Eq. ( 1 ) in two sections, as depicted in Fig. 1b (Eq. ( 2 ) in Methods and Supplementary Note 2 ). The first and second sections describe the behaviorally relevant neural dynamics and the other neural dynamics with latent states \({x}_{k}^{(1)}\in {{\mathbb{R}}}^{{n}_{1}}\) and \({x}_{k}^{(2)}\in {{\mathbb{R}}}^{{n}_{x}-{n}_{1}}\) , respectively. We specify the parameters of the two RNN sections with superscripts (for example, K (1) and K (2) ) and learn them all sequentially via a four-step optimization ( Methods , Supplementary Fig. 1a and Fig. 1b ). The first two steps exclusively learn neural dynamics that are behaviorally relevant with the objective of behavior prediction, whereas the optional last two steps learn any remaining neural dynamics with the objective of residual neural prediction ( Methods and Supplementary Fig. 1 ). We implement DPAD in Tensorflow and use an ADAM 43 optimizer ( Methods ).

Comparison baselines

As a baseline, we compare DPAD with standard nonlinear RNNs fitted to maximize neural prediction, unsupervised with respect to behavior. We refer to this baseline as nonlinear neural dynamical modeling (NDM) 6 or as linear NDM if all RNN parameters are linear. NDM is nondissociative and nonprioritized, so comparisons with NDM show the benefit of DPAD’s prioritized dissociation of behaviorally relevant neural dynamics. We also compare DPAD with latent factor analysis via dynamical systems (LFADS) 16 and with two concurrently 44 developed methods with DPAD named targeted neural dynamical modeling (TNDM) 18 and consistent embeddings of high-dimensional recordings using auxiliary variables (CEBRA) 36 in terms of neural–behavioral prediction; however, as summarized in Extended Data Table 1 , these and other existing methods differ from DPAD in key goals and capabilities and do not enable some of DPAD’s use-cases (see Discussion ).

Decoding using past neural data

Given DPAD’s learned parameters, the latent states can be causally extracted from neural activity by iterating through the RNN in Eq. ( 1 ) ( Methods and Supplementary Note 1 ). Note that this decoding always only uses neural activity without seeing the behavior data.

Flexible control of nonlinearities

We allow each model parameter (for example, C z ) to be an arbitrary multilayer neural network (Supplementary Fig. 1c ), which can universally approximate any smooth nonlinear function or implement linear matrix multiplications ( Methods and Supplementary Fig. 1b ). Users can manually specify which parameters will be learned as nonlinear and with what architecture (Fig. 1c ; see application in use-case 4). Alternatively, DPAD can automatically determine the best nonlinearity setting for the data by conducting a search over nonlinearity options (Fig. 1d and Methods ), a process that we refer to as flexible nonlinearity. For a fair comparison, we also implement this flexible nonlinearity for NDM. To show the benefits of nonlinearity, we also compare with linear DPAD, where all parameters are set to be linear, in which case Eq. ( 1 ) formulates a standard linear state-space model in predictor form ( Methods ).

Evaluation metrics

We evaluate how well the models can use the past neural activity to predict the next sample of behavior (termed ‘decoding’) or the next sample of neural activity itself (termed ‘neural self-prediction’ or simply ‘self-prediction’). Thus, decoding and self-prediction assess the one-step-ahead prediction accuracies and reflect the learning of behaviorally relevant and overall neural dynamics, respectively. Both performance measures are always computed with cross-validation ( Methods ).

Our primary interest is to find models that simultaneously reach both accurate behavior decoding and accurate neural self-prediction. But in some applications, only one of these metrics may be of interest. Thus, we use the term ‘performance frontier’ to refer to the range of performances achievable by those models that compared to every other model are better in neural self-prediction and/or behavior decoding or are similar in terms of both metrics ( Methods ).

Diverse neural–behavioral datasets

We used DPAD to study the behaviorally relevant neural dynamics in four NHPs performing four different tasks (Fig. 2 and Methods ). In the first task, the animal made naturalistic three-dimensional (3D) reach, grasp and return movements to diverse locations while the joint angles in the arm, elbow, wrist and fingers were tracked as the behavior (Fig. 2a ) 6 , 45 . In the second task, the animal made saccadic eye movements to one of eight possible targets on a screen, with the two-dimensional (2D) eye position tracked as the behavior (Fig. 2d ) 6 , 46 . In the third task, the animal made sequential 2D reaches on a screen using a cursor controlled with a manipulandum while the 2D cursor position and velocity were tracked as the behavior (Fig. 2g ) 47 , 48 . In the fourth task, the animal made 2D reaches to random targets in a virtual-reality-presented grid via a cursor that mirrored the animal’s fingertip movements, for which the 2D position and velocity were tracked as the behavior (Fig. 2i ) 49 . In tasks 1 and 4, primary motor cortical activity was modeled. For tasks 2 and 3, prefrontal cortex and dorsal premotor cortical activities were modeled, respectively.

a , The 3D reach task, along with example true and decoded behavior dimensions, decoded from spiking activity using DPAD, with more example trajectories for all modalities shown in Supplementary Fig. 3 . b , Cross-validated decoding accuracy correlation coefficient (CC) achieved by linear and nonlinear DPAD. Results are shown for spiking activity, raw LFP activity and LFP band power activity ( Methods ). For nonlinear DPAD, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (that is, flexible nonlinearity). The latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in the training data among all state dimensions ( Methods ). Bars show the mean, whiskers show the s.e.m., and dots show all data points ( N  = 35 session-folds). Asterisks (*) show significance level for a one-sided Wilcoxon signed-rank test (* P  < 0.05, ** P  < 0.005 and *** P  < 0.0005); NS, not significant. c , The difference between the nonlinear and linear results from b shown with the same notations. d – f , Same as a – c for the second dataset with saccadic eye movements ( N  = 35 session-folds). g , h , Same as a and b for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). Behavior consists of the 2D position and velocity of the cursor, denoted as ‘hand kinematics’ in the figure. i – k , Same as a – c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip movement ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics.

Source data

In all datasets, we modeled the Gaussian smoothed spike counts as the main neural modality ( Methods ). In three datasets that had LFP, we also modeled the following two additional modalities: (1) raw LFP, downsampled to the sampling rate of behavior (that is, 50-ms time steps), which in the motor cortex is known as the local motor potential 50 , 51 , 52 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 ; and (2) LFP power in standard frequency bands from delta (0.1–4 Hz) to high gamma (130–170 Hz (refs. 5 , 6 , 40 ); Methods ). Similar results held for all three modalities.

Numerical simulations validate DPAD

We first validate DPAD with linear simulations here (Extended Data Fig. 1 ) and then present nonlinear simulations under use-case 4 below (Extended Data Fig. 2 and Supplementary Fig. 2 ). We simulated general random linear models (not emulating any real data) in which only a subset of state dimensions contributed to generating behavior and thus were behaviorally relevant ( Methods ). We found that with a state dimension equal to that of the true model, DPAD achieved ideal cross-validated prediction (that is, similar to the true model) for both behavior and neural signals (Extended Data Fig. 1b,d ). Moreover, even given a minimal state dimension equal to the true behaviorally relevant state dimension, DPAD still achieved ideal prediction for behavior (Extended Data Fig. 1c ). Finally, across various regimens of training samples, linear DPAD performed similarly to the linear-algebraic-based PSID 6 from our prior work (Extended Data Fig. 1 ). Thus, hereafter, we use linear DPAD as our linear modeling benchmark.

Use-case 1: DPAD enables nonlinear neural–behavioral modeling across modalities

Dpad captures nonlinearity in behaviorally relevant dynamics.

We modeled each neural modality (spiking, raw LFP or LFP power) along with behavior using linear and nonlinear DPAD and compared their cross-validated behavior decoding (Fig. 2b,e,h,j and Supplementary Fig. 3 ). Across all neural modalities in all datasets, nonlinear DPAD achieved significantly higher decoding accuracy than linear DPAD. This result suggests that there is nonlinearity in the dynamical neural–behavioral transformation, which DPAD successfully captures (Fig. 2b,e,h,j ).

DPAD better predicts the neural–behavioral data

Across all datasets and modalities, compared to nonlinear NDM or linear DPAD, nonlinear DPAD reached higher behavior decoding accuracy while also being as accurate or better in terms of neural self-prediction (Fig. 3 , Extended Data Fig. 3 and Supplementary Fig. 4 ). Indeed, compared to these, DPAD was always on the best performance frontier for predicting the neural–behavioral data (Fig. 3 and Extended Data Fig. 3 ). Additionally, DPAD was always on the best performance frontier for predicting the neural–behavioral data compared to long short-term memory (LSTM) networks as well as a concurrently 44 developed method with DPAD termed CEBRA 36 on our four datasets (Fig. 4a–h ) in addition to a fifth movement dataset 54 analyzed in the CEBRA paper (Fig. 4i,j ). These results suggest that DPAD provides a more accurate description for neural–behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by each method shown on the horizontal axis versus the corresponding behavior decoding accuracy on the vertical axis for modeling spiking activity. Latent state dimension for each method in each session, and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). The plus on the plot shows the mean self-prediction and decoding accuracy across sessions and folds ( N  = 35 session-folds), and the horizontal and vertical whiskers show the s.e.m. for these two measures, respectively. Capital letter annotations denote the methods according to the legend to make the plots more accessible. Models whose self-prediction and decoding accuracy measures lead to values toward the top-rightmost corner of the plot lie on the best performance frontier (indicated by red arrows) as they have better performance in both measures and thus better explain the neural–behavioral data ( Methods ). c , d , Same as a and b for the second dataset with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h , Same as a and b for the fourth dataset with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). For nonlinear DPAD/NDM, we fit models with different combinations of nonlinearities and then select a final model among these fitted models based on either decoding or self-prediction accuracy in the training data and report both sets of results (Supplementary Fig. 1 and Methods ). DPAD with nonlinearity selected based on neural self-prediction was better than all other methods overall ( b , d , f and h ).

a – h , Figure content is parallel to Fig. 3 (with pluses and whiskers defined in the same way) but instead of NDM shows CEBRA and LSTM networks as baselines ( Methods ). i , j , Here, we also add a fifth dataset 54 ( Methods ), where in each trial an NHP moves a cursor from a center point to one of eight peripheral targets ( i ). In this fifth dataset ( N  = 5 folds), we use the exact CEBRA hyperparameters that were used for this dataset from the paper introducing CEBRA 36 . In the other four datasets ( N  = 35 session-folds in b , d and h and N  = 15 session-folds in f ), we also show CEBRA results for when hyperparameters are picked based on an extensive search ( Methods ). Two types of LSTM networks are shown, one fitted to decode behavior from neural activity and another fitted to predict the next time step of neural activity (self-prediction). We also show the results for DPAD when only using the first two optimization steps. Note that CEBRA-Behavior (denoted by D and F), LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior-related objectives (for example, prediction or contrastive loss), whereas other methods dedicate some or all latent states to neural self-prediction. As in Fig. 3 , the final latent dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Across all datasets, DPAD outperforms baseline methods in terms of cross-validated neural–behavioral prediction and lies on the best performance frontier. For a summary of the fundamental differences in goals and capabilities of these methods, see Extended Data Table 1 .

Beyond one-step-ahead predictions, we next evaluated DPAD in terms of multistep-ahead prediction of neural–behavioral data, also known as forecasting. To do this, starting with one-step-ahead predictions (that is, m  = 1), we pass m -step-ahead predictions of neural data using the learned models as the neural observation in the next time step to obtain ( m  + 1)-step-ahead predictions ( Methods ). Nonlinear DPAD was consistently better than nonlinear NDM and linear dynamical systems (LDS) modeling in multistep-ahead forecasting of behavior (Extended Data Fig. 4 ). For neural self-prediction, we used a naive predictor as a conservative forecasting baseline, which reflects how easy it is to predict the future in a model-free way purely based on the smoothness of neural data. DPAD significantly outperformed this baseline in terms of one-step-ahead and multistep-ahead neural self-predictions (Supplementary Fig. 5 ).

Use-case 2: DPAD extracts behavior-predictive nonlinear transformations from raw LFP

We next used DPAD to compare the amount of nonlinearity in the neural–behavioral transformation across different neural modalities (Fig. 2 and Supplementary Fig. 3 ). To do so, we compared the gain in behavior decoding accuracy when going from linear to nonlinear DPAD modeling in each modality. In all datasets, raw LFP activity had the highest gain from nonlinearity in behavior decoding accuracy (Fig. 2c,f,k ). Notably, using nonlinear DPAD, raw LFP reached more accurate behavior decoding than traditional LFP band powers in all tasks (Fig. 2b,e,j ). In one dataset, raw LFP even significantly surpassed spiking activity in terms of behavior decoding accuracy (Fig. 2e ). Note that computing LFP powers involves a prespecified nonreversible nonlinear transformation of raw LFP, which may be discarding important behaviorally relevant information that DPAD can uncover directly from raw LFP. Interestingly, linear dynamical modeling did worse for raw LFP than LFP powers in most tasks (compare linear DPAD for raw LFP versus LFP powers), suggesting that nonlinearity, captured by DPAD, was required for uncovering the extra behaviorally relevant information in raw LFP.

We next examined the spatial pattern of behaviorally relevant information across recording channels. For different channels, we compared the neural self-prediction of DPAD’s low-dimensional behaviorally relevant latent states (Extended Data Fig. 5 ). We computed the coefficient of variation (defined as standard deviation divided by mean) of the self-prediction over recording channels and found that the spatial distribution of behaviorally relevant information was less variable in raw LFP than spiking activity ( P  ≤ 0.00071, one-sided signed-rank test, N  = 35 for all three datasets with LFP). This could suggest that raw LFPs reflect large-scale network-level behaviorally relevant computations, which are thus less variable within the same spatial brain area than spiking, which represents local, smaller-scale computations 55 .

Use-case 3: DPAD enables behavior-predictive nonlinear dynamical dimensionality reduction

We next found that DPAD extracted latent states that were lower dimensional yet more behavior predictive than both nonlinear NDM and linear DPAD (Fig. 5 ). Specifically, we inspected the dimension required for nonlinear DPAD to reach almost (within 5% of) peak behavior decoding accuracy in each dataset (Fig. 5b,g,l,o ). At this low latent state dimension, linear DPAD and nonlinear and linear NDM all achieved much lower behavior decoding accuracy than nonlinear DPAD across all neural modalities (Fig. 5c–e,h–j,m,p–r ). The lower decoding accuracy of nonlinear NDM suggests that the dominant dynamics in spiking and LFP modalities can be unrelated to the modeled behavior. Thus, behaviorally relevant dynamics can be missed or confounded unless they are prioritized during nonlinear learning, as is done by DPAD. Moreover, we visualized the 2D latent state trajectories learned by each method (Extended Data Fig. 6 ). Consistent with the above results, DPAD extracted latent states from neural activity that were clearly different for different behavior/movement conditions (Extended Data Fig. 6b,e,h,k ). In comparison, NDM extracted latent states that did not as clearly dissociate different conditions (Extended Data Fig. 6c,f,i,l ). These results highlight the capability of DPAD for nonlinear dynamical dimensionality reduction in neural data while preserving behaviorally relevant neural dynamics.

a , The 3D reach task. b , Cross-validated decoding accuracy (CC) achieved by variations of linear/nonlinear DPAD/NDM for different latent state dimensions. For nonlinear DPAD/NDM, the nonlinearities are selected automatically based on the training data to maximize behavior decoding accuracy (flexible nonlinearity). Solid lines show the average across sessions and folds ( N  = 35 session-folds), and the shaded areas show the s.e.m.; Low-dim., low-dimensional. c , Decoding accuracy of nonlinear DPAD versus linear DPAD and nonlinear/linear NDM at the latent state dimension for which DPAD reaches within 5% of its peak decoding accuracy in the training data across all latent state dimensions. Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , Same as c for modeling of raw LFP ( N  = 35 session-folds). e , Same as c for modeling of LFP band power activity ( N  = 35 session-folds). f – j , Same as a – e for the second dataset with saccadic eye movements ( N  = 35 session-folds). k – m , Same as a – c for the third dataset, which did not include LFP data, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). n – r , Same as a – e for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, only the first two optimization steps were used in this figure (that is, n 1  =  n x ) to only focus on learning behaviorally relevant neural dynamics in the dimensionality reduction regimen.

Next, we found that at low dimensions, nonlinearity could improve the accuracy of both behavior decoding (Fig. 5b,g,l,o ) and neural self-prediction (Extended Data Fig. 7 ). However, as the state dimension was increased, linear methods reached similar neural self-prediction performance as nonlinear methods across modalities (Fig. 3 and Extended Data Fig. 3 ). This was in contrast to behavior decoding, which benefited from nonlinearity regardless of how high the dimension was (Figs. 2 and 3 ).

Use-case 4: DPAD localizes the nonlinearity in the neural–behavioral transformation

Numerical simulations validate dpad’s localization.

To demonstrate that DPAD can correctly find the origin of nonlinearity in the neural–behavioral transformation (Extended Data Fig. 2 and Supplementary Fig. 2 ), we simulated random models where only one of the parameters was set to a random nonlinear function ( Methods ). DPAD identifies a parameter as the origin if models with nonlinearity only in that parameter are on the best performance frontier when compared to alternative models, that is, models with nonlinearity in other parameters, models with flexible/full nonlinearity and fully linear models (Fig. 6a ). DPAD enables this assessment due to (1) its flexible control over nonlinearities to train alternative models and (2) its simultaneous neural–behavioral modeling and evaluation ( Methods ). In all simulations, DPAD identified that the model with the correct nonlinearity origin was on the best performance frontier compared to alternative nonlinear models (Extended Data Fig. 2 and Supplementary Fig. 2 ), thus correctly revealing the origin of nonlinearity.

a , The process of determining the origin of nonlinearity via hypothesis testing shown with an example simulation. Simulation results are taken from Extended Data Fig. 2b , and the origin is correctly identified as K . Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). b , The 3D reach task. c , DPAD’s hypothesis testing. Cross-validated neural self-prediction accuracy (CC) for each nonlinearity and the corresponding decoding accuracy. DPAD variations that have only one nonlinear parameter (for example, C z ) use a nonlinear neural network for that parameter and keep all other parameters linear. Linear and flexible nonlinear results are as in Fig. 3 . Latent state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Annotated arrows indicate any individual nonlinearities that are on the best performance frontier compared to all other models. Results are shown for spiking activity here and for raw LFP and LFP power activity in Supplementary Fig. 6 . d , e , Same as b and c for the second dataset with saccadic eye movements ( N  = 35 session-folds). f , g , Same as b and c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as b and c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps, and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

DPAD consistently localized nonlinearities in the behavior readout

Having validated the localization of nonlinearity in simulations, we used DPAD to find where in the model nonlinearities could be isolated to in our real datasets. We found that having the nonlinearity only in the behavior readout parameter C z was largely sufficient for achieving high behavior decoding and neural self-prediction accuracies across all our datasets and modalities (Fig. 6b–i and Supplementary Fig. 6 ). First, for spiking activity, models with nonlinearity only in the behavior readout parameter C z reached the best behavior decoding accuracy compared to models with other individual nonlinearities (Fig. 6c,e,i ) while reaching almost the same decoding accuracy as fully nonlinear models (Fig. 6c,e,g,i ). Second, these models with nonlinearity only in the behavior readout also reached a self-prediction accuracy that was unmatched by other types of individual nonlinearity (Fig. 6c,e,g,i ). Overall, this meant that models with nonlinearity only in the behavior readout parameter C z were always on the best performance frontier when compared to all other linear or nonlinear models (Fig. 6c,e,g,i ). This result interestingly also held for both LFP modalities (Supplementary Fig. 6 ).

Consistent with the above localization results, DPAD with flexible nonlinearity also, very frequently, automatically selected models with nonlinearity in the behavior readout parameter (Supplementary Fig. 7 ). However, critically, this observation on its own cannot conclude that nonlinearities can be isolated in the behavior readout parameter. This is because in the flexible nonlinearity approach, parameters may be selected as nonlinear as long as this nonlinearity does not hurt the prediction accuracies, which does not imply that such nonlinearities are necessary ( Methods ); this is why we need the hypothesis testing procedure above (Fig. 6a ). Of note, using an LSTM for the recursion parameter A ′ is one of the nonlinearity options that is automatically considered in DPAD (Extended Data Fig. 3 ), but we found that LSTM was rarely selected in our datasets as the recursion dynamics in the flexible search over nonlinearities (Supplementary Fig. 7 ). Finally, note that fitting models with a nonlinear behavior readout via a post hoc nonlinear refitting of linear DPAD models (1) cannot identify the origin of nonlinearity in general (for example, other brain regions or tasks) and (2) even in our datasets resulted in significantly worse decoding than the same models being fitted end-to-end as done by nonlinear DPAD ( P  ≤ 0.0027, one-sided signed-rank test, N  ≥ 15).

Together, these results highlight the application of DPAD in enabling investigations of nonlinear processing in neural computations underlying specific behaviors. DPAD’s machinery can not only fit fully nonlinear models but also provide evidence for the location in the model where the nonlinearity can be isolated ( Discussion ).

Use-case 5: DPAD extends to noncontinuous and intermittent data

Dpad extends to intermittently sampled behavior observations.

DPAD also supports intermittently sampled behaviors ( Methods ) 56 , that is, when behavior is measured only during a subset of time steps. We first confirmed in numerical simulations with random models that DPAD correctly learns the model with intermittently sampled behavioral data (Supplementary Fig. 8 ). Next, in each of our neural datasets, we emulated intermittent sampling by randomly discarding up to 90% of behavior samples during learning. DPAD learned accurate nonlinear models even in this case (Extended Data Fig. 8 ). This capability is important, for example, in affective neuroscience or neuropsychiatry applications where the behavior consists of sparsely sampled momentary ecological assessments of mental states such as mood 40 . We next simulated a mood decoding application and found that with as low as one behavioral (for example, mood survey) sample per day, DPAD still outperformed NDM even when NDM had access to continuous behavior samples (Extended Data Fig. 9 ). These results suggest the potential utility of DPAD in such applications, although substantial future validation in data is needed 7 , 40 , 41 , 42 .

DPAD extends to noncontinuous-valued observations

DPAD also extends to modeling of noncontinuous-valued (for example, categorical) behaviors ( Methods ). To demonstrate this, we modeled the transformation from neural activity to the momentary phase of the task in the 3D reach task: reach, hold, return or rest (Fig. 7 ). Compared to nonlinear NDM (which is dynamic) or nonlinear nondynamic methods such as support vector machines, DPAD more accurately predicted the task phase at each point in time (Fig. 7 ). This capability can extend the utility of DPAD to categorical behaviors such as decision choices in cognitive neuroscience 39 .

a , In the 3D reach dataset, we model spiking activity along with the epoch of the task as discrete behavioral data ( Methods and Fig. 2a ). The epochs/classes are (1) reaching toward the target, (2) holding the target, (3) returning to resting position and (4) resting until the next reach. b , DPAD’s predicted probability for each class is shown in a continuous segment of the test data. Most of the time, DPAD predicts the highest probability for the correct class. c , The cross-validated behavior classification performance, quantified as the area under curve (AUC) for the four-class classification, is shown for different methods at different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). AUC of 1 and 0.5 indicate perfect and chance-level classification, respectively. We include three nondynamic/static classification methods that map neural activity for a given time step to class label at the same time step (Extended Data Table 1 ): (1) multilayer neural network, (2) nonlinear support vector machine (SVM) and (3) linear discriminant analysis (LDA). d , Cross-validated behavior classification performance (AUC) achieved by each method when choosing the state dimension in each session and fold as the smallest that reaches peak classification performance in the training data among all state dimensions with that method ( Methods ). Bars, whiskers, dots and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). e , Same as d when all methods use the same latent state dimension as DPAD (best nonlinearity for decoding) does in d ( N  = 35 session-folds). c and e show DPAD’s benefit for dimensionality reduction. f , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior classification performance. Here, the latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds).

Finally, we applied DPAD to nonsmoothed spike counts, where we compared the results with two noncausal sequential autoencoder methods, termed LFADS 16 and TNDM 18 (Supplementary Fig. 9 ), both of which have Poisson observations that model nonsmoothed spike counts 16 , 18 . TNDM 18 , which was developed after LFADS 16 and concurrently with our work 44 , 56 , adds behavioral terms to the objective function for a subset of latents but unlike DPAD does so with a mixed objective and thus does not completely dissociate or prioritize behaviorally relevant dynamics (Extended Data Table 1 and Supplementary Note 3 ). Compared to both LFADS and TNDM, DPAD remained on the best performance frontier for predicting the neural–behavioral data (Supplementary Fig. 9a ) and more accurately predicted behavior using low-dimensional latent states (Supplementary Fig. 9b ). Beyond this, TNDM and LFADS also have fundamental differences with DPAD and do not address some of DPAD’s use-cases ( Discussion and Extended Data Table 1 ).

We developed DPAD for nonlinear dynamical modeling and investigation of neural dynamics underlying behavior. DPAD can dissociate the behaviorally relevant neural dynamics and prioritize their learning over other neural dynamics, enable hypothesis testing regarding the origin of nonlinearity in the neural–behavioral transformation and achieve causal decoding. DPAD enables prioritized dynamical dimensionality reduction by extracting lower-dimensional yet more behavior-predictive latent states from neural population activity and supports modeling noncontinuous-valued (for example, categorical) and intermittently sampled behavioral data. These attributes make DPAD suitable for diverse use-cases across neuroscience and neurotechnology, some of which we demonstrated here.

We found similar results for three neural modalities: spiking activity, LFP band powers and raw LFP. For all modalities, nonlinear DPAD more accurately learned the behaviorally relevant neural dynamics than linear DPAD and linear/nonlinear NDM as reflected in its better decoding while also reaching the best performance frontier when considering both behavior decoding and neural self-prediction. Notably, the raw LFP activity benefited the most from nonlinear modeling using DPAD and outperformed LFP powers in all tasks in terms of decoding. This suggests that automatic learning of nonlinear models from raw LFP using DPAD reveals behaviorally relevant information that may be discarded when extracting traditionally used features such as LFP band powers. Also, nonlinearity was necessary to recover the extra information in raw LFP, as, unlike DPAD modeling, linear dynamical modeling of raw LFP did not outperform that of LFP powers in most datasets. These results highlight another use-case of DPAD for automatic dynamic feature extraction from LFP data.

As another use-case, DPAD enabled an investigation of which element in the neural–behavioral transformation was nonlinear. Interestingly, consistently across our four movement-related datasets, DPAD models with nonlinearity only in the behavior readout performed similarly to fully nonlinear models, reaching the best performance frontier for predicting future behavior and neural data using past neural data. The consistency of this result across our datasets is interesting because, as demonstrated in simulations (Extended Data Fig. 2 , Supplementary Fig. 2 and Fig. 6a ), the detected origin of nonlinearity could have technically been in any one (or more) of the following four elements (Fig. 1a,b ): neural input, recurrent dynamics and neural or behavior readouts, all of which were correctly localized in simulations (Extended Data Fig. 2 and Supplementary Fig. 2 ). Thus, the consistent localization results on our neural datasets provide evidence that across these four tasks, neural dynamics in these recorded cortical areas may be largely describable with linear dynamics of sufficiently high dimension, with additional nonlinearities introduced somewhere between the neural state and behavior. This finding may be consistent with (1) introduction of nonlinear processing along the downstream neuromuscular pathway that goes from the recorded cortical area to the measured behavior or any of the convergent inputs along this pathway 57 , 58 , 59 or (2) cognition intervening nonlinearly between these latent neural states and behavior, for example, by implementing context-dependent computations 60 . This result illustrates how DPAD can provide new hypotheses and the machinery to test them in future experiments that would record from multiple additional brain regions (for example, both motor and cognitive regions) and use DPAD to model them together. Such analyses may narrow down or revise the origin of nonlinearity for the wider neural–behavioral measurement set; for example, the state dynamics may be found to be nonlinear once additional brain regions are added. Localization of nonlinearity could also guide the design of competitive deep learning architectures that are more flexible or easier to implement in neurotechnologies such as brain–computer interfaces 61 .

Interestingly, the behavior decoding aspect of the localization finding here is consistent with a prior study 22 that explored the mapping of the motor cortex to an electromyogram (EMG) during a one-dimensional movement task with varying forces and found that a fully linear model was worse than a nonlinear EMG readout in decoding the EMG 22 . However, as our simulations show (Extended Data Fig. 2b and Fig. 6a ), comparing a linear model to a model that has nonlinear behavior readout is not sufficient to conclude the origin of nonlinearity, and a stronger test is needed (see Fig. 6a for a counter example and details in Methods ). Further, this previous study 22 used a specific condition-dependent nonlinearity for behavior readout rather than a universal nonlinear function approximator that DPAD enables. Finally, to conclude localization, the model with that specific nonlinearity should perform similarly to fully nonlinear models; however, unlike our results, a fully nonlinear LSTM model in some cases appears to outperform models with nonlinear readout in this prior study (see Fig. 7a,b in ref. 22 versus Fig. 9c in ref. 22 ); it is unclear if this result is due to this prior study’s specific readout nonlinearity being suboptimal or to the nonlinear origin being different in its dataset 22 . DPAD can address such questions by (1) allowing for training and comparison of alternative models with different nonlinear origins and (2) enabling a general (versus specific) nonlinearity in model parameters.

When hypothesis testing about where in the model nonlinearity can be isolated to, it may be possible to equivalently explain the same data with multiple types of nonlinearities (for example, with either a nonlinear neural input or a nonlinear readout). Such nonidentifiability is a common limitation for latent models. However, when such equivalence exists, we expect all equivalent nonlinear models to have similar performance and thus lie on the best performance frontier. But this was not the case in our datasets. Instead, we found that the nonlinear behavior readout was in most cases the only individual nonlinear parameter on the best performance frontier, providing evidence that no other individual nonlinear parameter was as suitable in our datasets. Alternatively, the best model describing the data may require two or more of the four parameters to be nonlinear. But in our datasets, models with nonlinearity only in the behavior readout were always on the best performance frontier and could not be considerably outperformed by models with more than one nonlinearity (Fig. 6 ). Nevertheless, we note that ultimately our analysis simply provides evidence for one location of nonlinearity resulting in a better fit to data with a parsimonious model, but it does not rule out other possibilities for explaining the data. For example, one could reformulate a nonlinear readout model by adding latent states and representing the readout nonlinearity as a recursion nonlinearity for the additional states, although such an equivalent but less parsimonious model may need more data to be learned as accurately. Finally, we also note that our conclusions were based on the datasets and family of nonlinear models (recursive RNNs) considered here, and thus we cannot rule out different conclusions in other scenarios and/or brain regions. Nevertheless, by providing evidence for a nonlinearity configuration, DPAD can provide testable hypotheses for future experiments that record from more brain regions.

Sequential autoencoders, spearheaded by LFADS 16 , have been used to smooth single-trial neural activity 16 without considering relevance to behavior, which is a distinct goal as we showed in comparison to PSID in our prior work 6 . Notably, another sequential autoencoder, termed TNDM, has been developed concurrently with our work 44 , 56 that adds a behavior term to the optimization objective 18 . However, these approaches do not enable several of the use-cases of DPAD here. First, unlike DPAD’s four-step learning approach, TNDM and LFADS use a single learning step with a neural-only objective (LFADS) 16 or a mixed neural–behavioral objective (TNDM) 18 that does not fully prioritize the behaviorally relevant neural dynamics (Extended Data Table 1 and Supplementary Note 3 ). DPAD’s prioritization is important for accurate learning of behaviorally relevant neural dynamics and for preserving them in dimensionality reduction, as our results comparing DPAD to TNDM/LFADS suggest (Supplementary Fig. 9 ). Second, TNDM and LFADS 16 , 18 , like other prior works 16 , 18 , 20 , 23 , 24 , 26 , 61 , do not provide flexible nonlinearity or explore hypotheses regarding the origin of nonlinearities because they use fixed nonlinear network structures (use-case 4). Third, TNDM considers spiking activity and continuous behaviors 18 , whereas DPAD extends across diverse neural and behavioral modalities: spiking, raw LFP and LFP powers and continuous, categorical or intermittent behavioral modalities. Fourth, in contrast to these noncausal sequential autoencoders 16 , 18 and some other nonlinear methods 8 , 14 , DPAD can process the test data causally and without expensive computations such as iterative expectation maximization 8 , 14 or sampling and averaging 16 , 18 . This causal efficient processing is also important for real-time closed-loop brain–computer interfaces 62 , 63 . Of note, noncausal processing is also implemented in the DPAD code library as an option ( Methods ), although it is not shown in this work. Finally, unlike these prior methods 14 , 16 , 18 , DPAD does not require fixed-length trials or trial structure, making it suitable for modeling naturalistic behaviors 5 and neural dynamics with trial-to-trial variability in the alignment to task events 64 .

Several methods can in some ways prioritize behaviorally relevant information while extracting latent embeddings from neural data but are distinct from DPAD in terms of goals and capabilities. One group includes nondynamic/static methods that do not explicitly model temporal dynamics 1 . These methods build linear maps (for example, as in demixed principal component analysis (dPCA) 34 ) or nonlinear maps, such as convolutional maps in a concurrently 44 developed method with DPAD named CEBRA 36 , to extract latent embeddings that can be guided by behavior either as a trial condition 34 or indirectly as a contrastive loss 36 . These nondynamic mappings only use a single sample or a small fixed window around each sample of neural data to extract latent embeddings (Extended Data Table 1 ). By contrast, DPAD can recursively aggregate information from all past neural data by explicitly learning a model of temporal dynamics (recursion), which also enables forecasting unlike in static/nondynamic methods. These differences may be one reason why DPAD outperformed CEBRA in terms of neural–behavioral prediction (Fig. 4 ). Another approach is used by task aligned manifold estimation (TAME-GP) 9 , which uses a Gaussian process prior (as in Gaussian process factor analysis (GPFA) 14 ) to expand the window of neural activity used for extracting the embedding into a complete trial. Unlike DPAD, methods with a Gaussian process prior have limited support for nonlinearity, often do not have closed-forms for inference and thus necessitate numerical optimization even for inference 9 and often operate noncausally 9 . Finally, the above methods do not provide flexible nonlinearity or hypothesis testing to localize the nonlinearity.

Other prior works have used RNNs either causally 20 , 22 , 23 , 24 , 26 or noncausally 16 , 18 , for example, for causal decoding of behavior from neural activity 20 , 22 , 23 , 24 , 26 . These works 20 , 22 , 23 , 24 , 26 have similarities to the first step of DPAD’s four-step optimization (Supplementary Fig. 1a ) in that the RNNs in these works learn dynamical models by solely optimizing behavior prediction. However, these works do not learn the mapping from the RNN latent states to neural activity, which is done in DPAD’s second optimization step to enable neural self-prediction (Supplementary Fig. 1a ). In addition, unlike what the last two optimization steps in DPAD enable, these prior works do not model additional neural dynamics beyond those that decode behavior and thus do not dissociate the two types of neural dynamics (Extended Data Table 1 ). Finally, as noted earlier, these prior works 9 , 20 , 23 , 24 , 26 , 36 , 61 , similar to prior sequential autoencoders 16 , 18 , have fixed nonlinear network structures and thus cannot explore hypotheses regarding the origin of nonlinearities or flexibly learn the best nonlinear structure for the training data (Fig. 1c,d and Extended Data Table 1 ).

DPAD’s optimization objective functions are not convex, similar to most nonlinear deep learning methods. Thus, as usual with nonconvex optimizations, convergence to a global optimum is not guaranteed. Moreover, as with any method, quality and neural–behavioral prediction of the learned models depend on dataset properties such as signal-to-noise ratio. Thus, we compare alternative methods within each dataset, suggesting that (for example, Fig. 4 ) across the multiple datasets here, DPAD learns more accurate models of neural–behavioral data. However, models in other datasets/scenarios may not be as accurate.

Here, we focused on using DPAD to model the transformation of neural activity to behavior. DPAD can also be used to study the transformation between other signals. For example, when modeling data from multiple brain regions, one region can be taken as the primary signal ( y k ) and another as the secondary signal ( z k ) to dissociate their shared versus distinct dynamics. Alternatively, when modeling the brain response to electrical 7 , 41 , 42 or sensory 41 , 65 , 66 stimulation, one could take the primary signal ( y k ) to be the stimulation and the secondary signal ( z k ) to be neural activity to dissociate and predict neural dynamics that are driven by stimulation. Finally, one may apply DPAD to simultaneously recorded brain activity from two subjects as primary and secondary signals to find shared intersubject dynamics during social interactions.

Model formulation

Equation ( 1 ) simplifies the DPAD model by showing both of its RNN sections as one, but the general two-section form of the model is as follows:

This equation separates the latent states of Eq. ( 1 ) into the following two parts: \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) denotes the latent states of the first RNN section that summarize the behaviorally relevant dynamics, and \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) , with \({n}_{2}={n}_{x}-{n}_{1}\) , denotes those of the second RNN section that represent the other neural dynamics (Supplementary Fig. 1a ). Here, A ′(1) , A ′(2) , K (1) , K (2) , \({C}_{y}^{\,\left(1\right)}\) , \({C}_{y}^{\,\left(2\right)}\) , \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) are multi-input–multi-output functions that parameterize the model, which we learn using a four-step numerical optimization formulation expanded on in the next section (Supplementary Fig. 1a ). DPAD also supports learning the initial value of the latent states at time 0 (that is, \({x}_{0}^{\left(1\right)}\) and \({x}_{0}^{\left(2\right)}\) ) as a parameter, but in all analyses in this paper, the initial states are simply set to 0 given their minimal impact when modeling long data sequences. Each pair of superscripted parameters (for example, A ′(1) and A ′(2) ) in Eq. ( 2 ) is a dissociated version of the corresponding nonsuperscripted parameter in Eq. ( 1 ) (for example, A ′). The computation graph for Eq. ( 2 ) is provided in Fig. 1b (and Supplementary Fig. 1a ). In Eq. ( 2 ), the recursions for computing \({x}_{k}^{\left(1\right)}\) are not dependent on \({x}_{k}^{\left(2\right)}\) , thus allowing the former to be computed without the latter. By contrast, \({x}_{k}^{\left(2\right)}\) can depend on \({x}_{k}^{\left(1\right)}\) , and this dependence is modeled via K (2) (see Supplementary Note 2 ). Note that such dependence of \({x}_{k}^{\left(2\right)}\) on \({x}_{k}^{\left(1\right)}\) via K (2) does not introduce new dynamics to \({x}_{k}^{\left(2\right)}\) because it does not involve the recursion parameter A ′(2) , which describes the dynamics of \({x}_{k}^{\left(2\right)}\) . This two-section RNN formulation is mathematically motivated by equivalent representations of a dynamical system model in different bases and by the relation between the predictor and stochastic forms of dynamical systems (Supplementary Notes 1 and 2 ).

For the RNN formulated in Eq. ( 1 ) or ( 2 ), neural activity y k constitutes the input, and predictions of neural and behavioral signals are the outputs (Fig. 1b ) given by

Note that each x k is estimated purely using all past y k (that is, y 1 , …, y k   –  1 ), so the predictions in Eq. ( 3 ) are one-step-ahead predictions of y k and z k using past neural observations (Supplementary Note 1 ). Once the model parameters are learned, the extraction of latent states x k involves iteratively applying the first line from Eq. ( 2 ), and predicting behavior or neural activity involves applying Eq. ( 3 ) to the extracted x k . As such, by writing the nonlinear model in predictor form 67 , 68 (Supplementary Note 1 ), we enable causal and computationally efficient prediction.

Learning: four-step numerical optimization approach

Unlike nondynamic models 1 , 34 , 35 , 36 , 69 , dynamical models explicitly model temporal evolution in time series data. Recent dynamical models have gone beyond linear or generalized linear dynamical models 2 , 3 , 4 , 5 , 6 , 7 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 , 79 , 80 , 81 to incorporate switching linear 10 , 11 , 12 , 13 , locally linear 37 or nonlinear 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 23 , 24 , 26 , 27 , 38 , 61 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 dynamics, often using deep learning methods 25 , 91 , 92 , 93 , 94 . But these recent nonlinear/switching works do not aim to localize nonlinearity or allow for flexible nonlinearity and do not enable fully prioritized dissociation of behaviorally relevant neural dynamics because they either do not consider behavior in their learning objective at all 14 , 16 , 37 , 38 , 61 , 95 , 96 or incorporate it with a mixed neural–behavioral objective 9 , 18 , 35 , 61 (Extended Data Table 1 ).

In DPAD, we develop a four-step learning method for training our two-section RNN in Eq. ( 1 ) and extracting the latent states that (1) enables dissociation and prioritized learning of the behaviorally relevant neural dynamics in the nonlinear model, (2) allows for flexible modeling and localization of nonlinearities, (3) extends to data with diverse distributions and (4) does all this while also achieving causal decoding and being applicable to data both with and without a trial structure. DPAD is for nonlinear modeling, and its multistep learning approach, in each step, uses numerical optimization tools that are rooted in deep learning. Thus, DPAD is mathematically distinct from our prior PSID work for linear models, which is an analytical and linear technique. PSID is based on analytical linear algebraic projections rooted in control theory 6 , which are thus not extendable to nonlinear modeling or to non-Gaussian, noncontinuous or intermittently sampled data. Thus, even when we restrict DPAD to linear modeling as a special case, it is still mathematically different from PSID 6 .

To dissociate and prioritize the behaviorally relevant neural dynamics, we devise a four-step optimization approach for learning the two-section RNN model parameters (Supplementary Fig. 1a ). This approach prioritizes the extraction and learning of the behaviorally relevant dynamics in the first two steps with states \({x}_{k}^{\left(1\right)}\in {{\mathbb{R}}}^{{n}_{1}}\) while also learning the rest of the neural dynamics in the last two steps with states \({x}_{k}^{\left(2\right)}\in {{\mathbb{R}}}^{{n}_{2}}\) and dissociating the two subtypes of dynamics. This prioritization is important for accurate learning of behaviorally relevant neural dynamics and is achieved because of the multistep learning approach; the earlier steps learn the behaviorally relevant dynamics first, that is, with priority, and then the subsequent steps learn the other neural dynamics later so that they do not mask or confound the behaviorally relevant dynamics. Importantly, each optimization step is independent of subsequent steps so all steps can be performed in order, with no need to iteratively repeat any step. We define the neural and behavioral prediction losses that are used in the optimization steps based on the negative log-likelihoods (NLLs) associated with the neural and behavior distributions, respectively. This approach benefits from the statistical foundation of maximum likelihood estimation and facilitates generalizability across behavioral distributions. We now expand on each of the four optimization steps for RNN training.

Optimization step 1

In the first two optimization steps (Supplementary Fig. 1a ), the objective is to learn the behaviorally relevant latent states \({x}_{k}^{\left(1\right)}\) and their associated parameters. In the first optimization step, we learn the parameters A ′(1) , \({C}_{z}^{\,\left(1\right)}\) and K (1) of the RNN

and estimate its latent state \({x}_{k}^{\left(1\right)}\) while minimizing the NLL of the behavior z k given by \({x}_{k}^{\left(1\right)}\) . For continuous-valued (Gaussian) behavioral data, we minimize the following sum of squared prediction error 69 , 97 given by

where the sum is over all available samples of behavior z k , and \({\Vert .\Vert }_{2}\) indicates the two-norm operator. This objective, which is typically used when fitting models to continuous-valued data 69 , 97 , is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals (that is, ∑ 𝜖  = σ 𝜖 I ) 69 , 97 . If desired, a general nonisotropic residual covariance ∑ 𝜖 can be empirically computed from model residuals after the above optimization is solved (see Learning noise statistics ), although having ∑ 𝜖 is mainly useful for simulating new data and is not needed when using the learned model for inference. Similarly, in the subsequent optimization steps detailed later, the same points hold regarding how the appropriate mean squared error used for continuous-valued data is proportional to the Gaussian NLL if we assume isotropic Gaussian residuals and how the residual covariance can be computed empirically after the optimization if desired.

Optimization step 2

The second optimization step uses the extracted latent state \({x}_{k}^{\left(1\right)}\) from the RNN and fits the parameter \({C}_{y}^{\left(1\right)}\) in

while minimizing the NLL of the neural activity y k given by \({x}_{k}^{(1)}\) . For continuous-valued (Gaussian) neural activity y k , we minimize the following sum of squared prediction error 69 :

where the sum is over all available samples of y k . Optimization steps 1 and 2 conclude the prioritized extraction and modeling of behaviorally relevant latent states \({x}_{k}^{(1)}\) (Fig. 1b ) and the learning of the first section of the RNN model (Supplementary Fig. 1a ).

Optimization step 3

In optimization steps 3 and 4 (Supplementary Fig. 1a ), the objective is to learn any additional dynamics in neural activity that are not learned in the first two optimization steps, that is, \({x}_{k}^{\left(2\right)}\) and the associated parameters. To do so, in the third optimization step, we learn the parameters A ′(2) , \({C}_{y}^{\,\left(2\right)}\) and K (2) of the RNN

and estimate its latent state \({x}_{k}^{\left(2\right)}\) while minimizing the aggregate NLL of y k given both latent states, that is, by also taking into account the NLL obtained from step 2 via the \({C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) term in Eq. ( 6 ). The notations \({y}_{k}^{{\prime} }\) and \({e}_{k}^{{\prime} }\) in the second line of Eq. ( 8 ) signify the fact that it is not y k that is predicted by the RNN of Eq. ( 8 ), rather it is the yet unpredicted parts of y k (that is, unpredicted after extracting \({x}_{k}^{(1)}\) ) that are being predicted. In the case of continuous-valued (Gaussian) neural activity y k , we minimize the following loss:

where the sum is over all available samples of y k . Note that in the continuous-valued (Gaussian) case, this loss is equivalent to minimizing the error in predicting the residual neural activity given by \({y}_{k}-{C}_{y}^{\,\left(1\right)}\left({x}_{k}^{\left(1\right)}\right)\) and is computed using the previously learned parameter \({C}_{y}^{\,\left(1\right)}\) and the previously extracted states \({x}_{k}^{\left(1\right)}\) in steps 1 and 2. Also, the input to the RNN in Eq. ( 8 ) includes both y k and the extracted \({x}_{k+1}^{\left(1\right)}\) from optimization step 1. The above shows how the optimization steps are appropriately linked together to compute the aggregate likelihoods.

Optimization step 4

If we assume that the second set of states \({x}_{k}^{\left(2\right)}\) do not contain any information about behavior, we could stop the modeling. However, this may not be the case if the dimension of the states extracted in the first optimization step (that is, n 1 ) is selected to be very small such that some behaviorally relevant neural dynamics are not learned in the first step. To be robust to such selections of n 1 , we can use another final numerical optimization to determine based on the data whether and how \({x}_{k}^{\left(2\right)}\) should affect behavior prediction. Thus, a fourth optimization step uses the extracted latent state in optimization steps 1 and 3 and fits C z in

while minimizing the negative log-likelihood of behavior given both latent states. In the case of continuous-valued (Gaussian) behavior z k , we minimize the following loss:

The parameter C z that is learned in this optimization step will replace both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) in Eq. ( 2 ). Optionally, in a final optimization step, a similar nonlinear mapping from \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) can also be learned, this time to predict y k , which allows DPAD to support nonlinear interactions of \({x}_{k}^{\left(1\right)}\) and \({x}_{k}^{\left(2\right)}\) in predicting neural activity. In this case, the resulting learned C y parameter will replace both \({C}_{y}^{\,\left(1\right)}\) and \({C}_{y}^{\,\left(2\right)}\) in Eq. ( 2 ). This concludes the learning of both model sections (Supplementary Fig. 1a ) and all model parameters in Eq. ( 2 ).

In this work, when optimization steps 1 and 3 are both used to extract the latent states (that is, when 0 <  n 1  <  n x ), we do not perform the additional fourth optimization step in Eq. ( 10 ), and the prediction of behavior is done solely using the \({x}_{k}^{\left(1\right)}\) states extracted in the first optimization step. Note that DPAD can also cover NDM as a special case if we only use the third optimization step to extract the states (that is, n 1  = 0, in which case the first two steps are not needed). In this case, we use the fourth optimization step to learn C z , which is the mapping from the latent states to behavior. Also, in this case, we simply have a unified state x k as there is no dissociation in NDM, and the only goal is to extract states that predict neural activity accurately.

Additional generalizations of state dynamics

Finally, the first lines of Eqs. ( 4 ) and ( 8 ) can also be written more generally as

where instead of an additive relation between the two terms of the righthand side, both terms are combined in nonlinear functions \({{A}^{{\prime} {\prime} }}^{\left(1\right)}\) and \({{A}^{{\prime} {\prime} }}^{\left(2\right)}\) , which as a special case can still learn the additive relation in Eqs. ( 4 ) and ( 8 ). Whenever both the state recursion A and neural input K parameters (with the appropriate superscripts) are specified to be nonlinear, we use the more general architecture in Eqs. ( 12 ) and ( 13 ), and if any one of A or K or both are linear, we use Eqs. ( 4 ) and ( 8 ).

As another option, both RNN sections can be made bidirectional, which enables noncausal prediction for DPAD by using future data in addition to past data, with the goal of improving prediction, especially in datasets with stereotypical trials. Although this option is not reported in this work, it is implemented and available for use in DPAD’s public code library.

Learning noise statistics

Once the learning is complete, we also compute the covariances of the neural and behavior residual time series e k and 𝜖 k as ∑ e and ∑ 𝜖 , respectively. This allows the learned model in Eq. ( 1 ) to be usable for generating new simulated data. This application is not the focus of this work, but an explanation of it is provided in Numerical simulations .

Regularization

Adding norm 1 or norm 2 regularization for any set of parameters and the option to automatically select the regularization weight with inner cross-validation is implemented in the DPAD code. However, we did not use regularization in any of the analyses presented here.

Forecasting

DPAD also enables the capability to predict neural–behavioral data more than one time step into the future. To obtain two-step-ahead prediction, we pass the one-step-ahead neural predictions of the model as neural observations into it. This allows us to perform one state update iteration, that is, line 1 of Eq. ( 2 ), with y k being replaced with \({\hat{y}}_{k}\) from Eq. ( 3 ). Repeating this procedure m times gives the ( m  + 1)-step-ahead prediction of the latent state and neural–behavioral data.

Extending to intermittently measured behaviors

We also extend DPAD to modeling intermittently measured behavior time series (Extended Data Figs. 8 and 9 and Supplementary Fig. 8 ). To do so, when forming the behavior loss (Eqs. ( 5 ) and ( 11 )), we only compute the loss on samples where the behavior is measured and solve the optimization with this loss.

Extending to noncontinuous-valued data observations

We can also extend DPAD to noncontinuous-valued (non-Gaussian) observations by devising modified loss functions and observation models. Here, we demonstrate this extension for categorical behavioral observations, for example, discrete choices or epochs/phases during a task (Fig. 7 ). A similar approach could be used in the future to model other non-Gaussian behaviors and non-Gaussian (for example, Poisson) neural modalities, as shown in a thesis 56 .

To model categorical behaviors, we devise a new behavior observation model for DPAD by making three changes. First, we change the behavior loss (Eqs. ( 5 ) and ( 11 )) to the NLL of a categorical distribution, which we implement using the dedicated class in the TensorFlow library (that is, tf.keras.losses.CategoricalCrossentropy). Second, we change the behavior readout parameter C z to have an output dimension of n z  ×  n c instead of n z , where n c denotes the number of behavior categories or classes. Third, we apply Softmax normalization (Eq. ( 14 )) to the output of the behavior readout parameter C z to ensure that for each of the n z behavior dimensions, the predicted probabilities for all the n c classes add up to 1 so that they represent valid probability mass functions. Softmax normalization can be written as

where \({l}_{k}\in {{\mathbb{R}}}^{{n}_{z}\times {n}_{c}}\) is the output of C z at time k , and the superscript ( m , n ) denotes the element of l k associated with the behavior dimension m and the class/category number n . With these changes, we obtain a new RNN architecture with categorical behavioral outputs. We then learn this new RNN architecture with DPAD’s four-step prioritized optimization approach as before but now incorporating the modified NLL losses for categorical data. Together, with these changes, DPAD extends to modeling categorical behavioral measurements.

Behavior decoding and neural self-prediction metrics and performance frontier

Cross-validation.

To evaluate the learning, we perform a cross-validation with five folds (unless otherwise noted). We cut the data from the recording session into five equal continuous segments, leave these segments out one by one as the test data and train the model only using the data in the remaining segments. Once the model is trained using the neural and behavior training data, we pass the neural test data to the model to get the latent states in the test data using the first line of Eq. ( 1 ) (or Eq. ( 2 ), equivalently). We then pass the extracted latent states to Eq. ( 3 ) to get the one-step-ahead prediction of the behavior and neural test data, which we refer to as behavior decoding and neural self-prediction, respectively. Note that only past neural data are used to get the behavior and neural predictions. Also, the behavior test data are never used in predictions. Given the predicted behavior and neural time series, we compute the CC between each dimension of these time series and the actual behavior and neural test time series. We then take the mean of CC across dimensions of behavior and neural data to get one final cross-validated CC value for behavior decoding and one final CC value for neural self-prediction in each cross-validation fold.

Selection of the latent state dimension

We often need to select a latent state dimension to report an overall behavior decoding and/or neural self-prediction accuracy for each model/method (for example, Figs. 2 – 7 ). By latent state dimension, we always refer to the total latent state dimension of the model, that is, n x . For DPAD, unless otherwise noted, we always used n 1  = 16 to extract the first 16 latent state dimensions (or all latent state dimensions when n x  ≤ 16) using steps 1 and 2 and any remaining dimensions using steps 3 and 4. We chose n 1  = 16 because dedicating more, even all, latent state dimensions to behavior prediction only minimally improved it across datasets and neural modalities. For all methods, to select a state dimension n x , in each cross-validation fold, we fit models with latent state dimensions 1, 2, 4, 16,…and 128 (powers of 2 from 1 to 128) and select one of these models based on their decoding and neural self-prediction accuracies within the training data of that fold. We then report the decoding/self-prediction of this selected model computed in the test data of that fold. Our goal is often to select a model that simultaneously explains behavior and neural data well. For this goal, we pick the state dimension that reaches the peak neural self-prediction in the training data or the state dimension that reaches the peak behavior decoding in the training data, whichever is larger; we then report both the neural self-prediction and the corresponding behavior decoding accuracy of the same model with the selected state dimension in the test data (Figs. 3 – 4 , 6 and 7f , Extended Data Figs. 3 and 4 and Supplementary Figs. 4 – 7 and 9 ). Alternatively, for all methods, when our goal is to find models that solely aim to optimize behavior prediction, we report the cross-validated prediction performances for the smallest state dimension that reaches peak behavior decoding in training data (Figs. 2 , 5 and 7d , Extended Data Fig. 8 and Supplementary Fig. 3 ). We emphasize that in all cases, the reported performances are always computed in the test data of the cross-validation fold, which is not used for any other purpose such as model fitting or selection of the state dimension.

Performance frontier

When comparing a group of alternative models, we use the term ‘performance frontier’ to describe the best performances reached by models that in every comparison with any alternative model are in some sense better than or at least comparable to the alternative model. More precisely, when comparing a group \({\mathcal{M}}\) of models, model \({\mathcal{A}}\in {\mathcal{M}}\) will be described as reaching the best performance frontier when compared to every other model \({\mathcal{B}}{\mathscr{\in }}{\mathcal{M}}\) , \({\mathcal{A}}\) is significantly better than \({\mathcal{B}}\) in behavior decoding or in neural self-prediction or is comparable to \({\mathcal{B}}\) in both. Note that \({\mathcal{A}}\) may be better than some model \({{\mathcal{B}}}_{1}\in {\mathcal{M}}\) in decoding while being better than another model \({{\mathcal{B}}}_{2}\in {\mathcal{M}}\) in self-prediction; nevertheless \({\mathcal{A}}\) will be on the frontier as long as in every comparison one of the following conditions hold: (1) there is at least one measure for which \({\mathcal{A}}\) is more performant and (2) \({\mathcal{A}}\) is at least equally performant in both measures. To avoid exclusion of models from the best performance frontier due to very minimal performance differences, in this analysis, we only declare a difference in performance significant if in addition to resulting in P  ≤ 0.05 in a one-sided signed-rank test there is also at least 1% relative difference in the mean performance measures.

DPAD with flexible nonlinearity: automatic determination of appropriate nonlinearity

Fine-grained control over nonlinearities.

Each parameter in the DPAD model represents an operation in the computation graph of DPAD (Fig. 1b and Supplementary Fig. 1a ). We solve the numerical optimizations involved in model learning in each step of our multistep learning via standard stochastic gradient descent 43 , which remains applicable for any modification of the computation graph that remains acyclic. Thus, the operation associated with each model parameter (for example, A ′, K , C y and C z ) can be replaced with any multilayer neural network with an arbitrary number of hidden units and layers (Supplementary Fig. 1c ), and the model remains trainable with the same approach. Having no hidden layers implements the special case of a linear mapping (Supplementary Fig. 1b ). Of course, given that the training data are finite, the typical trade-off between model capacity and generalization error remains 69 . Given that neural networks can approximate any continuous function (with a compact domain) 98 , replacing model parameters with neural networks should have the capacity to learn any nonlinear function in their place 99 , 100 , 101 . The resulting RNN in Eq. ( 1 ) can in turn approximate any state-space dynamics (under mild conditions) 102 . In this work, for nonlinear parameters, we use multilayer feed-forward networks with one or two hidden layers, each with 64 or 128 units. For all hidden layers, we always use a rectified linear unit (ReLU) nonlinear activation (Supplementary Fig. 1c ). Finally, when making a parameter (for example, C z ) nonlinear, we always do so for that parameter in both sections of the RNN (for example, both \({C}_{z}^{\,\left(1\right)}\) and \({C}_{z}^{\,\left(2\right)}\) ; see Supplementary Fig. 1a ) and using the same feed-forward network structure. Given that no existing RNN implementation allowed individual RNN elements to be independently set to arbitrary multilayer neural networks, we developed a custom TensorFlow RNN cell to implement the RNNs in DPAD (Eqs. ( 4 ) and ( 8 )). We used the Adam optimizer to implement gradient descent for all optimization steps 43 . We continued each optimization for up to 2,500 epochs but stopped earlier if the objective function did not improve in three consecutive epochs (convergence criteria).

Automatic selection of nonlinearity settings

We devise a procedure for automatically determining the most suitable combination of nonlinearities for the data, which we refer to as DPAD with flexible nonlinearity. In this procedure, for each cross-validation fold in each recording session of each dataset, we try a series of nonlinearities within the training data and select one based on an inner cross-validation within the training data (Fig. 1d ). Specifically, we consider the following options for the nonlinearity. First, each of the four main parameters (that is, A ′, K , C y and C z ) can be linear or nonlinear, resulting in 16 cases (that is, 2 4 ). In cases with nonlinearity, we consider four network structures for the parameters, that is, having one or two hidden layers and having 64 or 128 units in each hidden layer (Supplementary Fig. 1c ), resulting in 61 cases (that is, 15 × 4 + 1, where 1 is for the fully linear model) overall. Finally, specifically for the recursion parameter A ′, we also consider modeling it as an LSTM, with the other parameters still having the same nonlinearity options as before, resulting in another 29 cases for when this LSTM recursion is used (that is, 7 × 4 + 1, where 1 is for the case where the other three model parameters are all linear), bringing the total number of considered cases to 90. For each of these 90 considered linear or nonlinear architectures, we perform a twofold inner cross-validation within the training data to compute an estimate of the behavior decoding and neural self-prediction of each architecture using the training data. Note that although this process for automatic selection of nonlinearities is computationally expensive, it is parallelizable because each candidate model can be fitted independently on a different processor. Once all candidate architectures are fitted and evaluated within the training data, we select one final architecture purely based on training data to be used for that cross-validation fold based on one of the following two criteria: (1) decoding focused: pick the architecture with the best neural self-prediction in training data among all those that reach within 1 s.e.m. of the best behavior decoding; or (2) self-prediction focused: pick the architecture with the best behavior decoding in training data among all those that reach within 1 s.e.m. of the best neural self-prediction. The first criterion prioritizes good behavior decoding in the selection, and the second criterion prioritizes good neural self-prediction. Note that these two criteria are used when selecting among different already-learned models with different nonlinearities and thus are independent of the four internal objective functions used in learning the parameters for a given model with the four-step optimization approach (Supplementary Fig. 1a ). For example, in the first optimization step of DPAD, model parameters are always learned to optimize behavior decoding (Eq. ( 5 )). But once the four-step optimization is concluded and different models (with different combinations of nonlinearities) are learned, we can then select among these already-learned models based on either neural self-prediction or behavior decoding. Thus, whenever neural self-prediction is also of interest, we report the results for flexible nonlinearity based on both model selection criteria (for example, Figs. 3 , 4 and 6 ).

Localization of nonlinearities

DPAD enables an inspection of where nonlinearities can be localized to by providing two capabilities, without either of which the origin of nonlinearities may be incorrectly found. As the first capability, DPAD can train alternative models with different individual nonlinearities and then compare these alternative nonlinear models not only with a fully linear model but also with each other and with fully nonlinear models (that is, flexible nonlinearity). Indeed, our simulations showed that simply comparing a linear model to a model with nonlinearity in a given parameter may incorrectly identify the origin of nonlinearity (Extended Data Fig. 2b and Fig. 6a ). For example, in Fig. 6a , although the nonlinearity is just in the neural input parameter, a linear model does worse than a model with a nonlinear behavior readout parameter. Thus, just a comparison of the latter model to a linear model would incorrectly find the origin of nonlinearity to be the behavior readout. This issue is avoided in DPAD because it can also train a model with the neural input being nonlinear, thus finding it to be more predictive than models with any other individual nonlinearity and as predictive as a fully nonlinear model (Fig. 6a ). As the second capability, DPAD can compare alternative nonlinear models in terms of overall neural–behavioral prediction rather than either behavior decoding or neural prediction alone. Indeed, our simulations showed that comparing the models in terms of just behavior decoding (Extended Data Fig. 2d,f ) or just neural self-prediction (Extended Data Fig. 2d,h ) may lead to incorrect conclusions about the origin of nonlinearities; this is because a model with the incorrect origin may be equivalent in one of these metrics to the one with the correct origin. DPAD avoids this problem by jointly evaluating both neural–behavioral metrics. Here, when comparing models with nonlinearity in different individual parameters for localization purposes (for example, Fig. 6 ), we only consider one network architecture for the nonlinearity, that is, having one hidden layer with 64 units.

Numerical simulations

To validate DPAD in numerical simulations, we perform two sets of simulations. One set validates linear modeling to show the correctness of the four-step numerical optimization for learning. The other set validates nonlinear modeling. In the linear simulation, we randomly generate 100 linear models with various dimensionality and noise statistics, as described in our prior work 6 . Briefly, the neural and behavior dimensions are selected from 5 ≤  n y , n z  ≤ 10 randomly with uniform probability. The state dimension is selected as n x  = 16, of which n 1  = 4 latent state dimensions are selected to drive behavior. Eigenvalues of the state transition matrix are selected randomly as complex conjugate pairs with uniform probability within the unit disk. Each element in the behavior and neural readout matrices is generated as a random Gaussian variable. State and neural observation noise covariances are generated as random positive definite matrices and scaled randomly with a number between 0.003 and 0.3 or between 0.01 and 100, respectively, to obtain a wide range of relative noises across random models. A separate random linear state-space model with four latent state dimensions is generated to produce the behavior readout noise 𝜖 k , representing the behavior dynamics that are not encoded in the recorded neural activity. Finally, the behavior readout matrix is scaled to set the ratio of the signal standard deviation to noise standard deviation in each behavior dimension to a random number from 0.5 to 50. We perform model learning and evaluation with twofold cross-validation (Extended Data Fig. 1 ).

In the nonlinear simulations that are used to validate both DPAD and the hypothesis testing procedure it enables to find the origin of nonlinearity, we start by generating 20 random linear models ( n y  =  n z  = 1) either with n x  =  n z  =  n y (Extended Data Fig. 2 ) or n x  = 2 latent states, only one of which drives behavior (Supplementary Fig. 2 ). We then introduce nonlinearity in one of the four model parameters (that is, A ′, K , C y or C z ) by replacing that parameter with a nonlinear trigonometric function, such that roughly one period of the trigonometric function is visited by the model (while keeping the rest of the parameters linear). To do this, we first scale each latent state in the initial random linear model to find a similarity transform for it where the latent state has a 95% confidence interval range of 2 π . We then add a sine function to the original parameter that is to be changed to nonlinear and scale the amplitude of the sine such that its output reaches roughly 0.25 of the range of the outputs from the original linear parameter. This was done to reduce the chance of generating unrealistic unstable nonlinear models that produce outputs with infinite energy, which is likely when A ′ is nonlinear. Changing one parameter to nonlinear can change the range of the statistics of the latent states in the model; thus, we generate some simulated data from the model and redo the scaling of the nonlinearity until ratio conditions are met.

To generate data from any nonlinear model in Eq. ( 1 ), we first generate a neural noise time series e k based on its covariance ∑ e in the model and initialize the state as x 0  = 0. We then iteratively apply the second and first lines of Eq. ( 1 ) to get the simulated neural activity y k from line 2 and then the next state \({x}_{k+1}\) from line 1, respectively. Finally, once the state time series is produced, we generate a behavior noise time series 𝜖 k based on its covariance ∑ 𝜖 in the model and apply the third line of Eq. ( 1 ) to get the simulated behavior z k . Similar to linear simulations, we perform the modeling and evaluation of nonlinear simulations with twofold cross-validation (Extended Data Fig. 2 and Supplementary Fig. 2 ).

Neural datasets and behavioral tasks

We evaluate DPAD in five datasets with different behavioral tasks, brain regions and neural recording modalities to show the generality of our conclusions. For each dataset, all animal procedures were performed in compliance with the National Research Council Guide for Care and Use of Laboratory Animals and were approved by the Institutional Animal Care and Use Committee at the respective institution, namely New York University (datasets 1 and 2) 6 , 45 , 46 , Northwestern University (datasets 3 and 5) 47 , 48 , 54 and University of California San Francisco (dataset 4) 21 , 49 .

Across all four main datasets (datasets 1 to 4), the spiking activity was binned with 10-ms nonoverlapping bins, smoothed with a Gaussian kernel with standard deviation of 50 ms (refs. 6 , 14 , 34 , 103 , 104 ) and downsampled to 50 ms to be used as the neural signal to be modeled. The behavior time series was also downsampled to a matching 50 ms before modeling. In the three datasets where LFP activity was also available, we also studied two types of features extracted from LFP. As the first LFP feature, we considered raw LFP activity itself, which was high-pass filtered above 0.5 Hz to remove the baseline, low-pass filtered below 10 Hz (that is, antialiasing) and downsampled to the behavior sampling rate of a 50-ms time step (that is, 20 Hz). Note that in the context of the motor cortex, low-pass-filtered raw LFP is also referred to as the local motor potential 50 , 51 , 52 , 105 , 106 and has been used to decode behavior 6 , 50 , 51 , 52 , 53 , 105 , 106 , 107 . As the second feature, we computed the LFP log-powers 5 , 6 , 7 , 40 , 77 , 79 , 106 , 108 , 109 in eight standard frequency bands (delta: 0.1–4 Hz, theta: 4–8 Hz, alpha: 8–12 Hz, low beta: 12–24 Hz, mid-beta: 24–34 Hz, high beta: 34–55 Hz, low gamma: 65–95 Hz and high gamma: 130–170 Hz) in sliding 300-ms windows at a time step of 50 ms using Welch’s method (using eight subwindows with 50% overlap) 6 . The median analyzed data length for each session across the datasets ranged from 4.6 to 9.9 min.

First dataset: 3D reaches to random targets

In the first dataset, the animal (named J) performed reaches to a target randomly positioned in 3D space within the reach of the animal, grasped the target and returned its hand to resting position 6 , 45 . Kinematic data were acquired using the Cortex software package (version 5.3) to track retroreflective markers in 3D (Motion Analysis) 6 , 45 . Joint angles were solved from the 3D marker data using a Rhesus macaque musculoskeletal model via the SIMM toolkit (version 4.0, MusculoGraphics) 6 , 45 . Angles of 27 joints in the shoulder, elbow, wrist and fingers in the active hand (right hand) were taken as the behavior signal 6 , 45 . Neural activity was recorded with a 137-electrode microdrive (Gray Matter Research), of which 28 electrodes were in the contralateral primary motor cortex M1. The multiunit spiking activity in these M1 electrodes was used as the neural signal. For LFP analyses, LFP features were also extracted from the same M1 electrodes. We analyzed the data from seven recording sessions.

To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we determined the periods of reach and return movements in the data (Fig. 7a ), resampled them to have similar number of time samples and averaged the latent states across those resampled trials. Given the redundancy in latent descriptions (that is, any scaling, rotation and so on on the latent states still gives an equivalent model), before averaging trials across cross-validation folds and sessions, we devised the following procedure to standardize the latent states for each fold in the case of 2D latent states (Extended Data Fig. 6 ). (1) We z score all state dimensions to have zero mean and unit variance. (2) We rotate the 2D latent states such that the average 2D state trajectory for the first condition (here, the reach epochs) starts from an angle of 0. (3) We estimate the direction of the rotation for the average 2D state trajectory of the first condition, and if it is not counterclockwise, we multiply the second state dimension by –1 to make it so. Note that in each step, the same mapping is applied to the latent states during the whole test data, regardless of condition, so this procedure does not alter the relative differences in the state trajectory across different conditions. The procedure also does not change the learned model and simply corresponds to a similarity transform that changes the basis of the model. This procedure only removes the redundancies for describing a 2D latent state-space model and standardizes the extracted latent states so that trials across different test sets can be averaged together.

Second dataset: saccadic eye movements

In the second dataset, the animal (named A) performed saccadic eye movements to one of eight targets on a display 6 , 46 . The visual stimuli in the task with saccadic eye movements were controlled via custom LabVIEW (version 9.0, National Instruments) software executed on a real-time embedded system (NI PXI-8184, National Instruments) 46 . The 2D position of the eye was tracked and taken as the behavior signal. Neural activity was recorded with a 32-electrode microdrive (Gray Matter Research) covering the prefrontal cortex 6 , 46 . Single-unit activity from these electrodes, ranging from 34 to 43 units across different recording sessions, was used as the neural signal. For LFP analyses, LFP features were also extracted from the same 32 electrodes. We analyzed the data from the first 7 days of recordings. We only included data from successful trials where the animal performed the task correctly by making a saccadic eye movement to the specified target. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials based on their target position. Standardization across folds before averaging was done as in the first dataset.

Third dataset: sequential reaches with a 2D cursor controlled with a manipulandum

In the third dataset, which was collected and made publicly available by the laboratory of L. E. Miller 47 , 48 , the animal (named T) controlled a cursor on a 2D screen using a manipulandum and performed a sequential reach task 47 , 48 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded using a 100-electrode microelectrode array (Blackrock Microsystems) in the dorsal premotor cortex 47 , 48 . Single-unit activity, recorded from 37 to 49 units across recording sessions, was used as the neural signal. This dataset did not include any LFP recordings, so LFP features could not be considered. We analyzed the data from all three recording sessions. To visualize the low-dimensional latent state trajectories for each behavioral condition (Extended Data Fig. 6 ), we grouped the trials into eight different conditions based on the angle of the direction of movement (that is, end position minus starting position) during the trial, with each condition covering movement directions within a 45° (that is, 360/8) range. Standardization across folds before averaging was performed as in the first dataset.

Fourth dataset: virtual reality random reaches with a 2D cursor controlled with the fingertip

In the fourth dataset, which was collected and made publicly available by the laboratory of P. N. Sabes 49 , the animal (named I) controlled a cursor based on the fingertip position on a 2D surface within a 3D virtual reality environment 21 , 49 . The 2D cursor position and velocity were taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) 21 , 49 covering M1. We selected a random subset of 32 of these electrodes, which had 77 to 99 single units across the recording sessions, as the neural signal. LFP features were also extracted from the same 32 electrodes. We analyzed the data for the first seven sessions for which the wideband activity was also available (sessions 20160622/01 to 20160921/01). Grouping into conditions for visualization of low-dimensional latent state trajectories (Extended Data Fig. 6 ) was done as in the third dataset. Standardization across folds before averaging was done as in the first dataset.

Fifth dataset: center-out cursor control reaching task

In the fifth dataset, which was collected and made publicly available by the laboratory of L. E. Miller 54 , the animal (named H) controlled a cursor on a 2D screen using a manipulandum and performed reaches from a center point to one of eight peripheral targets (Fig. 4i ). The 2D cursor position was taken as the behavior signal. Neural activity was recorded with a 96-electrode microelectrode array (Blackrock Microsystems) covering area 2 of the somatosensory cortex 54 . Preprocessing for this dataset was done as in ref. 36 . Specifically, the spiking activity was binned with 1-ms nonoverlapping bins and smoothed with a Gaussian kernel with a standard deviation of 40 ms (ref. 110 ), with the behavior also being sampled with the same 1-ms sampling rate. Trials were also aligned as in the same prior work 110 with data from –100 to 500 ms around movement onset of each trial being used for modeling 36 .

Additional details for baseline methods

For the fifth dataset, which has been analyzed in ref. 36 and introduces CEBRA, we used the exact same CEBRA hyperparameters as those reported in ref. 36 (Fig. 4i,j ). For each of the other four datasets (Fig. 4a–h ), when learning a CEBRA-Behavior or CEBRA-Time model for each session, fold and latent dimension, we also performed an extensive search over CEBRA hyperparameters and picked the best value with the same inner cross-validation approach as we use for the automatic selection of nonlinearities in DPAD. We considered 30 different sets of hyperparameters: 3 options for the ‘time-offset’ hyperparameter (1, 2 or 10) and 10 options for the ‘temperature’ hyperparameter (from 0.0001 to 0.01), which were designed to include all sets of hyperparameters reported for primate data in ref. 36 . We swept the CEBRA latent dimension over the same values as DPAD, that is, powers of 2 up to 128. In all cases, we used a k -nearest neighbors regression to map the CEBRA-extracted latent embeddings to behavior and neural data as done in ref. 36 because CEBRA itself does not learn a reconstruction model 36 (Extended Data Table 1 ).

It is important to note that CEBRA and DPAD have fundamentally different architectures and goals (Extended Data Table 1 ). CEBRA uses a small ten-sample window (when ‘model_architecture’ is ‘offset10-model’) around each datapoint to extract a latent embedding via a series of convolutions. By contrast, DPAD learns a dynamical model that recursively aggregates all past neural data to extract an embedding. Also, in contrast to CEBRA-Behavior, DPAD’s embedding includes and dissociates both behaviorally relevant neural dimensions and other neural dimensions to predict not only the behavior but also the neural data well. Finally, CEBRA does not automatically map its latent embeddings back to neural data or to behavior during learning but does so post hoc, whereas DPAD learns these mappings for all its latent states. Given these differences, several use-cases of DPAD are not targeted by CEBRA, including explicit dynamical modeling of neural–behavioral data (use-case 1), flexible nonlinearity, hypothesis testing regarding the origin of nonlinearity (use-case 4) and forecasting.

We used the Wilcoxon signed-rank test for all paired statistical tests.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Data availability

Three of the datasets used in this work are publicly available 47 , 48 , 49 , 54 . The other two datasets used to support the results are available upon reasonable request from the corresponding author. Source data are provided with this paper.

Code availability

The code for DPAD is available at https://github.com/ShanechiLab/DPAD .

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Acknowledgements

This work was supported, in part, by the following organizations and grants: the Office of Naval Research (ONR) Young Investigator Program under contract N00014-19-1-2128, National Institutes of Health (NIH) Director’s New Innovator Award DP2-MH126378, NIH R01MH123770, NIH BRAIN Initiative R61MH135407 and the Army Research Office (ARO) under contract W911NF-16-1-0368 as part of the collaboration between the US DOD, the UK MOD and the UK Engineering and Physical Research Council (EPSRC) under the Multidisciplinary University Research Initiative (MURI) (to M.M.S.) and a University of Southern California Annenberg Fellowship (to O.G.S.).

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Ming Hsieh Department of Electrical and Computer Engineering, Viterbi School of Engineering, University of Southern California, Los Angeles, CA, USA

Omid G. Sani & Maryam M. Shanechi

Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, USA

Bijan Pesaran

Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, USA

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O.G.S. and M.M.S. conceived the study, developed the DPAD algorithm and wrote the manuscript, and O.G.S. performed all the analyses. B.P. designed and performed the experiments for two of the NHP datasets and provided feedback on the manuscript. M.M.S. supervised the work.

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Correspondence to Maryam M. Shanechi .

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University of Southern California has a patent related to modeling and decoding of shared dynamics between signals in which M.M.S. and O.G.S. are inventors. The other author declares no competing interests.

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Extended data

Extended data fig. 1 dpad dissociates and prioritizes the behaviorally relevant neural dynamics while also learning the other neural dynamics in numerical simulations of linear models..

a , Example data generated from one of 100 random models ( Methods ). These random models do not emulate real data but for terminological consistency, we still refer to the primary signal (that is, y k in Eq. ( 1 )) as the ‘neural activity’ and to the secondary signal (that is, z k in Eq. ( 1 )) as the ‘behavior’. b , Cross-validated behavior decoding accuracy (correlation coefficient, CC) for each method as a function of the number of training samples when we use a state dimension equal to the total state dimension of the true model. The performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Performance for the true model is shown in black. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 100 random models). c , Same as b but when learned models have low-dimensional latent states with enough dimensions just for the behaviorally relevant latent states (that is, n x  =  n 1 ). d - e , Same as b - c showing the cross-validated normalized neural self-prediction accuracy. Linear NDM, which learns the parameters using a numerical optimization, performs similarly to a linear algebraic subspace-based implementation of linear NDM 67 , thus validating NDM’s numerical optimization implementation. Linear DPAD, just like PSID 6 , achieves almost ideal behavior decoding even with low-dimensional latent states ( c ); this shows that DPAD correctly dissociates and prioritizes behaviorally relevant dynamics, as opposed to aiming to simply explain the most neural variance as non-prioritized methods such as NDM do. For this reason, with a low-dimensional state, non-prioritized NDM methods can explain neural activity well ( e ) but prioritized methods can explain behavior much better ( c ). Nevertheless, using the second stage of PSID and the last two optimization steps in DPAD, these two prioritized techniques are still able to learn the overall neural dynamics accurately if state dimension is high enough ( d ). Overall, the performance of linear DPAD and PSID 6 are similar for the special case of linear modeling.

Extended Data Fig. 2 DPAD successfully identifies the origin of nonlinearity and learns it in numerical simulations.

DPAD can perform hypothesis testing regarding the origin of nonlinearity by considering both behavior decoding (vertical axis) and neural self-prediction (horizontal axis). a , True value for nonlinear neural input parameter K in an example random model with nonlinearity only in K and the nonlinear value that DPAD learned for this parameter when only K in the learned model was set to be nonlinear. The true and learned mappings match and almost exactly overlap. b , Behavior decoding and neural self-prediction accuracy achieved by DPAD models with different locations of nonlinearities. These accuracies are for data generated from 20 random models that only had nonlinearity in the neural input parameter K . Performance measures for each random model are normalized by their ideal values that were achieved by the true model itself. Pluses and whiskers are defined as in Fig. 3 ( N  = 20 random models). c , d , Same as a , b for data simulated from models that only have nonlinearity in the recursion parameter A ′. e - f , Same as a , b for data simulated from models that only have nonlinearity in the neural readout parameter C y . g , h , Same as a , b for data simulated from models that only have nonlinearity in the behavior readout parameter C z . In each case ( b , d , f , h ), the nonlinearity option that reaches closest to the upper-rightmost corner of the plot, that is, has both the best behavior decoding and the best neural self-prediction, is chosen as the model that specifies the origin of nonlinearity. Regardless of the true location of nonlinearity ( b , d , f , h ), always the correct location (for example, K in b ) achieves the best performance overall compared with all other locations of nonlinearities. These results provide evidence that by fitting and comparing DPAD models with different nonlinearities, we can correctly find the origin of nonlinearity in simulated data.

Extended Data Fig. 3 Across spiking and LFP neural modalities, DPAD is on the best performance frontier for neural-behavioral prediction unlike LSTMs, which are fitted to explain neural data or behavioral data.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy achieved by each method versus the corresponding behavior decoding accuracy on the vertical axis. Latent state dimension for each method in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak neural self-prediction in training data or reaches peak decoding in training data, whichever is larger ( Methods ). Pluses and whiskers are defined as in Fig. 3 ( N  = 35 session-folds). Note that DPAD considers an LSTM as a special case ( Methods ). Nevertheless, results are also shown for LSTM networks fitted to decode behavior from neural activity (that is, RNN decoders in Extended Data Table 1 ) or to predict the next time step of neural activity (self-prediction). Also, note that LSTM for behavior decoding (denoted by H) and DPAD when only using the first two optimization steps (denoted by G) dedicate all their latent states to behavior prediction, whereas other methods dedicate some or all latent states to neural self-prediction. Compared with all methods including these LSTM networks, DPAD always reaches the best performance frontier for predicting the neural-behavioral data whereas LSTM does not; this is partly due to the four-step optimization algorithm in DPAD that allows for overall neural-behavioral description rather than one or the other, and that prioritizes the learning of the behaviorally relevant neural dynamics. c , Same as b for raw LFP activity ( N  = 35 session-folds). d , Same as b for LFP band power activity ( N  = 35 session-folds). e - h , Same as a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). i , j , Same as a and b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). Results and conclusions are consistent across all datasets.

Extended Data Fig. 4 DPAD can also be used for multi-step-ahead forecasting of behavior.

a , The 3D reach task. b , Cross-validated behavior decoding accuracy for various numbers of steps into the future. For m -step-ahead prediction, behavior at time step k is predicted using neural activity up to time step k − m . All models are taken from Fig. 3 , without any retraining or finetuning, with m -step-ahead forecasting done by repeatedly ( m −1 times) passing the neural predictions of the model as its neural observation in the next time step ( Methods ). Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across the number of steps ahead, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD vs nonlinear NDM is shown with the orange top horizontal line with p-value indicated by asterisks next to the line as defined in Fig. 2b (N = 35 session-folds). Similar pairwise comparison between nonlinear DPAD vs linear dynamical system (LDS) modeling is shown with the purple top horizontal line. c - d , Same as a - b for the second dataset, with saccadic eye movements ( N  = session-folds). e - f , Same as a - b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g - h , Same as a - b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds).

Extended Data Fig. 5 Neural self-prediction accuracy of nonlinear DPAD across recording electrodes for low-dimensional behaviorally relevant latent states.

a , The 3D reach task. b , Average neural self-prediction correlation coefficient (CC) achieved by nonlinear DPAD for analyzed smoothed spiking activity is shown for each recording electrode ( N  = 35 session-folds; best nonlinearity for decoding). c , Same as b for modeling of raw LFP activity. d , Same as b for modeling of LFP band power activity. Here, prediction accuracy averaged across all 8 band powers ( Methods ) of a given recording electrode is shown for that electrode. e-h , Same a - d for the second dataset, with saccadic eye movements ( N  = 35 session-folds). For datasets with single-unit activity ( Methods ), spiking self-prediction of each electrode is averaged across the units associated with that electrode. i - j , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). White areas are due to electrodes that did not have a neuron associated with them in the data. k - n , Same as a - d for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all results, the latent state dimension was 16, and all these dimensions were learned using the first optimization step (that is, n 1  = 16).

Extended Data Fig. 6 Nonlinear DPAD extracted distinct low dimensional latent states from neural activity for all datasets, which were more behaviorally relevant than those extracted using nonlinear NDM.

a , The 3D reach task. b , The latent state trajectory for 2D states extracted from spiking activity using nonlinear DPAD, averaged across all reach and return epochs across sessions and folds. Here only optimization steps 1-2 of DPAD are used to just extract 2D behaviorally relevant states. c , Same as b for 2D states extracted using nonlinear NDM (special case of using just DPAD optimization steps 3-4). d , Saccadic eye movement task. Trials are averaged depending on the eye movement direction. e , The latent state trajectory for 2D states extracted using DPAD (extracted using optimizations steps 1-2), averaged across all trials of the same movement direction condition across sessions and folds. f , Same as d for 2D states extracted using nonlinear NDM. g-i , Same as d - f for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum. j - l , Same as d - f for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position. Overall, in each dataset, latent states extracted by DPAD were clearly different for different behavior conditions in that dataset ( b , e , h , k ), whereas NDM’s extracted latent states did not as clearly dissociate different conditions ( c , f , i , l ). Of note, in the first dataset, DPAD revealed latent states with rotational dynamics that reversed direction during reach versus return epochs, which is consistent with the behavior roughly reversing direction. In contrast, NDM’s latent states showed rotational dynamics that did not reverse direction, thus were less congruent with behavior. In this first dataset, in our earlier work 6 , we had compared PSID and a subspace-based linear NDM method and, similar to b and c here, had found that only PSID uncovers reverse-directional rotational patterns across reach and return movement conditions. These results thus also complement our prior work 6 by showing that even nonlinear NDM models may not uncover the distinct reverse-directional dynamics in this dataset, thus highlighting the need for dissociative and prioritized learning even in nonlinear modeling, as enabled by DPAD.

Extended Data Fig. 7 Neural self-prediction across latent state dimensions.

a , The 3D reach task. b , Cross-validated neural self-prediction accuracy (CC) achieved by variations of nonlinear and linear DPAD/NDM, for different latent state dimensions. Solid lines and shaded areas are defined as in Fig. 5b ( N  = 35 session-folds). Across latent state dimensions, the statistical significance of a one-sided pairwise comparison between nonlinear DPAD/NDM (with best nonlinearity for self-prediction) vs linear DPAD/NDM is shown with a horizontal green/orange line with p-value indicated by asterisks next to the line as defined in Fig. 2b ( N  = 35 session-folds). c , d , Same as a , b for the second dataset, with saccadic eye movements ( N  = 35 session-folds). e , f , Same as a , b for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). g , h Same as a , b for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16). As expected, at low state dimensions, DPAD’s latent states achieve higher behavior decoding (Fig. 5 ) but lower neural self-prediction than NDM because DPAD prioritizes the behaviorally relevant neural dynamics in these dimensions. However, by increasing the state dimension and utilizing optimization steps 3-4, DPAD can reach similar neural self-prediction to NDM while doing better in terms of behavior decoding (Fig. 3 ). Also, for low dimensional latent states, nonlinear DPAD/NDM consistently result in significantly more accurate neural self-prediction than linear DPAD/NDM. For high enough state dimensions, linear DPAD/NDM eventually reach similar neural self-prediction accuracy to nonlinear DPAD/NDM. Given that NDM solely aims to optimize neural self-prediction (irrespective of the relevance of neural dynamics to behavior), the latter result suggests that the overall neural dynamics can be approximated with linear dynamical models but only with high-dimensional latent states. Note that in contrast to neural self-prediction, behavior decoding of nonlinear DPAD is higher than linear DPAD even at high state dimensions (Fig. 3 ).

Extended Data Fig. 8 DPAD accurately learns the mapping from neural activity to behavior dynamics in all datasets even if behavioral samples are intermittently available in the training data.

Nonlinear DPAD can perform accurately and better than linear DPAD even when as little as 20% of training behavior samples are kept. a , The 3D reach task. b , Examples are shown from one of the joints in the original behavior time series (light gray) and intermittently subsampled versions of it (cyan) where a subset of the time samples of the behavior time series are randomly chosen to be kept for use in training. In each subsampling, all dimensions of the behavior data are sampled together at the same time steps; this means that at any given time step, either all behavior dimensions are kept or all are dropped to emulate the realistic case with intermittent measurements. c , Cross-validated behavior decoding accuracy (CC) achieved by linear DPAD and by nonlinear DPAD with nonlinearity in the behavior readout parameter C z . For this nonlinear DPAD, we show the CC when trained with different percentage of behavior samples kept (that is, we emulate different rates of intermittent sampling). The state dimension in each session and fold is chosen (among powers of 2 up to 128) as the smallest that reaches peak decoding in training data. Bars, whiskers, dots, and asterisks are defined as in Fig. 2b ( N  = 35 session-folds). d , e , Same as a , c for the second dataset, with saccadic eye movements ( N  = 35 session-folds). f , g , Same as a , c for the third dataset, with sequential cursor reaches controlled via a 2D manipulandum ( N  = 15 session-folds). h , i , Same as a , c for the fourth dataset, with random grid virtual reality cursor reaches controlled via fingertip position ( N  = 35 session-folds). For all DPAD variations, the first 16 latent state dimensions are learned using the first two optimization steps and the remaining dimensions are learned using the last two optimization steps (that is, n 1  = 16).

Extended Data Fig. 9 Simulations suggest that DPAD may be applicable with sparse sampling of behavior, for example with behavior being a self-reported mood survey value collected once per day.

a , We simulated the application of decoding self-reported mood variations from neural signals 40 , 41 . Neural data is simulated based on linear models fitted to intracranial neural data recorded from epilepsy subjects. Each recorded region in each subject is simulated as a linear state-space model with a 3-dimensional latent state, with the same parameters as those fitted to neural recordings from that region. Simulated latent states from a subset of regions were linearly combined to generate a simulated mood signal (that is, biomarker). As the simulated models were linear, we used the linear versions of DPAD and NDM (NDM used the subspace identification method that we found does similarly to numerical optimization for linear models in Extended Data Fig. 1 ). We generated the equivalent of 3 weeks of intracranial recordings, which is on the order the time-duration of the real intracranial recordings. We then subsampled the simulated mood signal (behavior) to emulate intermittent behavioral measures such as mood surveys. b , Behavior decoding results in unseen simulated test data, across N  = 87 simulated models, for different sampling rates of behavior in the training data. Box edges show the 25 th and 75 th percentiles, solid horizontal lines show the median, whiskers show the range of data, and dots show all data points ( N  = 87 simulated models). Asterisks are defined as in Fig. 2b . DPAD consistently outperformed NDM regardless of how sparse behavior measures were, even when these measures were available just once per day ( P  < 0.0005, one-sided signed-rank, N  = 87).

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Sani, O.G., Pesaran, B. & Shanechi, M.M. Dissociative and prioritized modeling of behaviorally relevant neural dynamics using recurrent neural networks. Nat Neurosci (2024). https://doi.org/10.1038/s41593-024-01731-2

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Green development of the country: Role of macroeconomic stability

• Lyulyov, Oleksii
• Pimonenko, Tetyana
• Kwilinski, Aleksy

The intensification of ecological issues provokes to search for the appropriate mechanism and resources to solve them without declining the economic growth. This requires moving from resources oriented to green economic development. It could be realised through two goals: achieving macroeconomic stability – core driver of economic growth; declining environmental degradation and increasing efficiency of resources using – core requirements for green development. The paper aims to check the hypothesis on macroeconomic stability's impact on the green development of the countries. The object of investigation is European Union countries from 2000 to 2020. The study applied the following methods: the Global Malmquist-Luenberger productivity index – to estimate the green development of the countries; Macroeconomic Stabilisation Pentagon model – to estimate macroeconomic stability; Kernel density estimation and Tobit model – to check the macroeconomic stability impact on the green development of the countries. The empirical findings show that Malta from the 'Green Group' and Estonia from the 'Yellow group' have the highest value of green development, and Sweden and Greece have the highest value of macroeconomic stability. Besides, the findings allow confirming the research hypothesis. Thus, the growth of external dimensions of macroeconomic stability by 1 point led to the growth of green economic development by 0.085 (among 'Green group') and 0.195 (among 'Yellow group'). It confirms that harmonising macroeconomic stability among all EU members allows for achieving the synergy effect.