Presentation for the Cognitive Control course (DGCN25) of the Research Master Cognitive Neuroscience at Radboud University on the topic of Cognitive Modeling
2. Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
3. Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
4. 1.1 Why ask the question in the first place?
Source: Google
Bram Zandbelt
8. 1.3 Definition of a model
Fum et al. (2007) Cog Sys Res 8:135
“A model is a simpler and more abstract version of a
system that keeps its essential features
while omitting unnecessary details”
–Howard Skipper
“A model is a lie that helps you see the truth”
Bram Zandbelt
9. Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
10. 2.1 Why use models?
Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions
Models are tools serving various purposes, including
description, prediction, and explanation
… lead to new experiments
[…] leading to new hypotheses, guiding experiments,
and findings
… and are often complex
Abstractions help to see the big picture
… promote a scientific habit
Formulating models forces you to think logically and
clearly about what you know and don’t know
Bram Zandbelt
12. Sources: fieltriptoolbox.org
Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
… and are often complex
Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
13. Sources: Van Belle et al. (2014) Neuroimage; fieltriptoolbox.org
Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
… and are often complex
Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
14. Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions
Models are tools serving various purposes, including
description, prediction, and explanation
… and are often complex
Abstractions help to see the big picture
2.1 Why use models?
Bram Zandbelt
15. Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions
Models are tools serving various purposes, including
description, prediction, and explanation
… and are often complex
Abstractions help to see the big picture
… promote a scientific habit
Formulating models forces you to think logically and
clearly about what you know and don’t know
2.1 Why use models?
Bram Zandbelt
16. Data never speak for themselves
A framework, theory, causal model, logical construct,
perception of the world, etc. is necessary to make
sense of data
Models address scientific questions
Models are tools serving various purposes, including
description, prediction, and explanation
… lead to new experiments
They suggest novel hypotheses, predicting future
findings, and guide experimental design
… and are often complex
Abstractions help to see the big picture
… promote a scientific habit
Formulating models forces you to think logically and
clearly about what you know and don’t know
2.1 Why use models?
Bram Zandbelt
17. 2.2 Why use (formal) models in cognitive neuroscience?
The brain is a complex system
Complex systems need to be understood at
multiple levels
Bram Zandbelt
19. Marr’s level Question
1
Computational/
Functional
What is the function?
Why is it performed?
2 Algorithm
What algorithm achieves this function?
How are inputs and outputs represented?
f(i)input i output o
Sources: Marr (1982)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
20. Marr’s level Question
1
Computational/
Functional
What is the function?
Why is it performed?
2 Algorithm
What algorithm achieves this function?
How are inputs and outputs represented?
3 Implementation
How are algorithm and representation
realized physically?
Sources: Marr (1982)
input i output o
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
21. 2.2 Why use (formal) models in cognitive neuroscience?
The brain is a complex system
Complex systems need to be understood at
multiple levels
Models can help to bridge the gap
between brain and behavior
Understanding computation guides research in
the underlying circuits and provides a language
for theories of behavior
Bram Zandbelt
23. The brain is a complex system
Complex systems need to be understood at
multiple levels
Models can take different forms
Verbal: words or box-and-arrow diagrams
Formal: axioms, equations, computer code
2.2 Why use (formal) models in cognitive neuroscience?
Models can help to bridge the gap
between brain and behavior
Understanding computation guides research in
the underlying circuits and provides a language
for theories of behavior
Bram Zandbelt
24. Cognitive process
input i output o = f(i)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
25. Cognitive processBrain… …
Model inputs
Predicted
behavior
Unobserved
neural/mental
process
N
Brain
signal
detection … …
response
preparation
response
execution
qualitative
fit
RT
Data/Reality
2.2 Why use (formal) models in cognitive neuroscience?
Verbal model
Observed
neural
process
1
Unobserved
neural/mental
process
i
qualitative
constraints
Bram Zandbelt
27. Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
28. Does the conclusion logically follow from the premises?
Premise 1
No police dogs are
vicious
No nutritional things
are inexpensive
No addictive things are
inexpensive
No millionaires are
hard workers
Premise 2
Some highly trained
dogs are vicious
Some vitamin tablets
are inexpensive
Some cigarettes are
inexpensive
Some rich people are
hard works
Conclusion
Therefore, some highly
trained dogs are not
police dogs
Therefore, some
vitamin tablets are not
nutritional
Therefore, some
addictive things are not
cigarettes
Therefore, some
millionaires are not rich
people
Valid
Believable
Valid
Unbelievable
Invalid
Believable
Invalid
Unbelievable
Sources: Evans et al. (1983) Mem Cogn
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
29. Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Limits of human thinking
(imagination, working memory)
e.g. reasoning about complex systems
Computational power
(in-depth exploration, no memory issues)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
30. Sources: Marder (2014) Ann Rev Neurosci
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
31. Sources: Farrell, S., & Lewandowsky, S. (2010). Curr Dir Psych Sci; Fum et al. (2007) Cog Sys Res; Hintzman (1991)
Potential problems of
verbal models
Solutions from
formal models
Flawed reasoning
(inconsistencies, contradictions, gaps)
e.g. belief bias
Formal system
(clarity, coherence, completeness)
Limits of human thinking
(imagination, working memory)
e.g. reasoning about complex systems
Computational power
(in-depth exploration, no memory issues)
Misunderstanding
(hidden assumptions, vague definitions)
e.g. concept of inhibition
Computer code and equations
(explicit assumptions, precise definitions)
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
32. Sources: Aron (2007) Neuroscientist; see also MacLeod et al. (2003) in Psychology of learning and motivation, B. Ross, Ed., vol. 43, pp. 163–214.
2.2 Why use (formal) models in cognitive neuroscience?
Bram Zandbelt
33. Sources: Lewandowsky, S. (1993) Psych Sci; Jacobs, A. M., & Grainger, J. (1994). J Exp Psychol um Percept Perform; Ulrich (2009) in: Rösler, Ranganath, Röder,
Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford University Press
… overspecification of irrelevant details
Obscures the discovery of general principles
… less suitable for new research fields
Sometimes we only have vague ideas
… overparameterization
Good fits can be bad; simpler models may exist
… realism comes at a cost
Bonini’s paradox: as a model becomes more realistic, it
becomes increasingly difficult to understand
2.2 Why use (formal) models in cognitive neuroscience?
Formal models have limitations, too:
Bram Zandbelt
35. Expected Utility
Axiomatic models
Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models
Relate physical stimuli to sensation/perception
2.3 Formal models come in different flavors
Source: von Neumann & Morgenstern (1944)
Bram Zandbelt
36. Axiomatic models
Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models
Relate physical stimuli to sensation/perception
Algebraic models
Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
2.3 Formal models come in different flavors
Source: Logan (1988) Psych Rev; Logan (2002) Psych Rev
Bram Zandbelt
37. Algorithmic models
Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models
Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models
Relate physical stimuli to sensation/perception
Algebraic models
Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
2.3 Formal models come in different flavors
Bram Zandbelt
38. Algorithmic models
Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models
Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models
Relate physical stimuli to sensation/perception
Algebraic models
Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
Connectionist models
Describe behavior with multilayer networks of
interconnected units
2.3 Formal models come in different flavors
Bram Zandbelt
39. Schizophrenia is not a rare disorder. It has a lifetime
risk of ~0.7%1
(similar to that of rheumatoid arthritis).
It has a genetic basis, but the importance of social fac-
tors in its emergence is also recognized. Schizophrenia
is devastating for both sufferers and their carers.
Patients are likely to be unemployed or fail to fulfil
their original potential. Contact with the police result-
ing from socially unacceptable behaviour is common,
and the risk of suicide is high. The first episode typi-
cally occurs when patients are in their mid 20s, and
most sufferers never fully recover. Although drug treat-
ment and, more recently, cognitive behavioural therapy
can reduce suffering, there is as yet no cure for this
disorder. Furthermore, although schizophrenia clearly
has a strong biological component (BOX 1), no diagnos-
tic physiological markers have been found. Diagnosis,
therefore, is made on the basis of symptoms described
by the patient, signs observed by the clinician and the
history of the disorder (BOX 2).
The most striking and characteristic features
of the disorder are hallucinations and delusions.
Hallucinations are false perceptions, such as patients
hearing people talking about them or hearing their
thoughts spoken aloud (TABLE 1). Delusions are per-
sistent bizarre or irrational beliefs that are not easily
understood in terms of an individual’s social or cul-
tural background. For example, patients may believe
that other people can hear their thoughts or that
the government is monitoring their every action.
Hallucinations and delusions are examples of positive
symptoms, which are so called because the abnormal-
ity lies in their presence. Positive symptoms contrast
with negative symptoms (also known as signs), which
are defined by the absence of normal functions, as is
the case with reduced speech output (alogia) or loss
of motivation (avolition). There is evidence that posi-
tive and negative symptoms reflect different underlying
physiological disorders2,3
. Although an important chal-
lenge for future work will be to find an explanation
for both positive and negative symptoms, we believe
that the current state of the field and the fact that these
symptoms seem to dissociate across groups of patients
make it sensible to confine our ideas in this Review
to the positive symptoms. Our aim is to consider how
abnormal physiological responses in the brains of peo-
ple with schizophrenia might be linked to the positive
symptoms that they experience. We show that a com-
mon mechanism, involving minimization of predic-
tion error, may underlie perception and inference, and
that a disruption in this mechanism may cause both
abnormal perceptions (hallucinations) and abnormal
beliefs (delusions). We are not concerned with the ulti-
mate causes of the disorder, in which both genetic and
environmental factors play a part.
*University of Cambridge,
Department of Psychiatry,
Addenbrooke’s Hospital,
Hills Road, Cambridge,
CB2 2QQ, UK.
‡
Centre for Functionally
Integrative Neuroscience,
Aarhus University Hospital,
8000 Aarhus C, Denmark.
§
Wellcome Trust Centre for
Neuroimaging, Functional
Imaging Laboratory,
University College London,
London, WC1N 3BG, UK.
Correspondence to C.D.F.
e-mail: c.frith@ucl.ac.uk
doi:10.1038/nrn2536
Published online
3 December 2008
Cognitive behavioural
therapy
A form of psychotherapy in
which the patient is
encouraged to examine the
cognitive processes by which
they arrive at a particular state
of mind, and to change these
processes together with the
accompanying behaviours that
may reinforce them.
Perceiving is believing: a Bayesian
approach to explaining the positive
symptoms of schizophrenia
Paul C. Fletcher* and Chris D. Frith‡§
Abstract | Advances in cognitive neuroscience offer us new ways to understand the
symptoms of mental illness by uniting basic neurochemical and neurophysiological
observations with the conscious experiences that characterize these symptoms. Cognitive
theories about the positive symptoms of schizophrenia — hallucinations and delusions —
have tended to treat perception and belief formation as distinct processes. However, recent
advances in computational neuroscience have led us to consider the unusual perceptual
experiences of patients and their sometimes bizarre beliefs as part of the same core
abnormality — a disturbance in error-dependent updating of inferences and beliefs about
the world. We suggest that it is possible to understand these symptoms in terms of a
disturbed hierarchical Bayesian framework, without recourse to separate considerations of
experience and belief.
REVIEWS
48 | JANUARY 2009 | VOLUME 10 www.nature.com/reviews/neuro
Source: Fletcher & Frith (2009) Nat Rev Neurosci
Algorithmic models
Defined in terms of a computer simulation that
describes how processes interact to produce behavior
Axiomatic models
Replace the phenomenon to be modeled with logical
propositions from which behavior can be derived
Psychophysical models
Relate physical stimuli to sensation/perception
Algebraic models
Simple equations that describe how input stimuli and
model parameters are combined to produce behavior
Connectionist models
Describe behavior with multilayer networks of
interconnected units
Bayesian models
Assume that we make inferences using Bayesian
statistics
2.3 Formal models come in different flavors
Bram Zandbelt
40. Preview
Cognitive
modeling
1. What is a model?
1.1. Why ask this question in the first place?
1.2. Examples of models
1.3. Definition of a model
2. Why use models?
2.1. Why use models in general?
2.2. Why use models in cognitive neuroscience?
2.3. Formal models come in di erent flavors
3. How to use models?
3.1. How to formulate a model?
3.2. How to estimate a model?
3.3. How to evaluate a model?
Bram Zandbelt
41. Steps in cognitive modeling
Formulation
Estimation
Evaluation
Bram Zandbelt
42. 3.1 How to formulate a model?
Core assumptions (A)
Based on conceptual theory of
underlying mechanism
Auxilliary assumptions
Conceptual theories often lack
important details
Definitions
Of dependent variables, such as RT
Theorems (T)
Combine assumptions & definitions to
derive abstract predictions
Predictions (P)
Add parameters for concrete predictions
that can be compared with data
Parameters
Tuning knobs of the model
Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford
University Press
Bram Zandbelt
43. Sources: Ulrich (2009) in: Rösler, Ranganath, Röder, Kluwe (Eds.), Neuroimaging of human memory:linking cognitive processes to neural systems. New York: Oxford
University Press
Model of cross modal temporal discrimination
3.1 How to formulate a model?
Bram Zandbelt
45. 3.2 How to estimate a model?
Main estimation methods: LSE & MLE
LSE: finds parameters that most accurately
describe the data
MLE: finds parameters that most likely have
generated the data
Least-squares estimation (LSE)
Maximum likelihood estimation (LSE)
Bram Zandbelt
46. 3.2 How to estimate a model?
Source: Lewandowsky, S., & Farrell, S. (2010). Computational modeling in cognition: Principles and practice. Sage.
Main estimation methods: LSE & MLE
LSE: finds parameters that most accurately
describe the data
MLE: finds parameters that most likely have
generated the data
Various approaches to find best fit
Grid search - easy but laborious
Simplex - efficient but risk ending in local
minimum
Simulated annealing, genetic algorithm - likely
to end in global minimum but time-consumingparam X
param Y
cost
fun
Bram Zandbelt
47. 3.3 How to evaluate a model?
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little; see also Jacobs & Grainger (1994) J Exp
Psychol Hum Percept Perform
Bram Zandbelt
48. Goodness of fit can be quantified with
likelihood or root mean squared error
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
3.3 How to evaluate a model?
Bram Zandbelt
49. Complexity can be quantified with
Akaike and Bayesian Information Criterion (AIC,BIC)
Goodness of fit
Penalty for
free parameters
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
3.3 How to evaluate a model?
Bram Zandbelt
50. Generalizability can be quantified with
cross validation
Source: Cavagnaro, Myung, Pitt (2010) in: Oxford Handbook of Quantitative Methods, Volume 1: Foundations, Ed. T. Little
Same model,
new data
3.3 How to evaluate a model?
Bram Zandbelt
51. Further reading
Lewandowsky, S., & Farrell, S. (2010). Computational
modeling in cognition: Principles and practice. Sage.
Cavagnaro, D. R., Myung, J. I., & Pitt, M. A. (2010).
Mathematical modeling. In T. D. Little (Ed.), The Oxford
Handbook of Quantitative Methods (Vol. 1, pp. 438–
453). New York, NY: Oxford University Press.
C H A P T E R
21
Mathematical Modeling
Daniel R. Cavagnaro, Jay I. Myung, and Mark A. Pitt
Abstract
Explanations of human behavior are most often presented in a verbal form as theories. Psychologists
can also harness the power and precision of mathematics by explaining behavior quantitatively. This
chapter introduces the reader to how this is done and the advantages of doing so. It begins by
contrasting mathematical modeling with hypothesis testing to highlight how the two methods of
knowledge acquisition differ. The many styles of modeling are then surveyed, along with their
advantages and disadvantages. This is followed by an in-depth example of how to create a
mathematical model and fit it to experimental data. Issues in evaluating models are discussed, including
a survey of quantitative methods of model selection. Particular attention is paid to the concept of
generalizability and the trade-off of model fit with model complexity. The chapter closes by describing
some of the challenges for the discipline in the years ahead.
Key Words: Cognitive modeling, model testing, model evaluation, model comparison
Introduction
Psychologists study behavior. Data, acquired
through experimentation, are used to build theo-
ries that explain behavior, which in turn provide
meaning and understanding. Because behavior is
complex, a complete theory of any behavior (e.g.,
depression, reasoning, motivation) is likely to be
complex as well, having many variables and condi-
tions that influence it.
Mathematical models are tools that assist in the-
ory development and testing. Models are theories, or
parts of theories, formalized mathematically. They
complement theorizing in many ways, as discussed
in the following pages, but their ultimate goal
is to promote understanding of the theory, and
thus behavior, by taking advantage of the precision
offered by mathematics. Although they have been
part of psychology since its inception, their popu-
larity began to rise in the 1950s and has increased
substantially since the 1980s, in part because of the
introduction of personal computers. This interest is
not an accident or fad. Every style of model that
has been introduced has had a significant impact
in its discipline, and sometimes far beyond that.
After reading this chapter, the reader should begin
to understand why.
This chapter is written as a first introduction to
mathematical modeling in psychology for those with
little or no prior experience with the topic. Our aim
is to provide a good conceptual understanding of
the topic and make the reader aware of some of
the fundamental issues in mathematical modeling
but not necessarily to provide an in-depth step-by-
step tutorial on how to actually build and evaluate a
mathematical model from scratch. In doing so, we
assume no more of the reader than a year-long course
in graduate-level statistics. For related publications
on the topic, the reader is directed to Busemeyer and
Diederich (2010), Fum, Del Missier, and Stocco
(2007), and Myung and Pitt (2002). In particular,
437
Bram Zandbelt
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Please attribute Bram Zandbelt with a link to
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Bram Zandbelt