Model-Based Cognitive Neuroscience: A Conceptual Introduction · 2 Birte U. Forstmann and Eric-Jan...

Model-Based Cognitive Neuroscience: AConceptual Introduction

Birte U. Forstmann and Eric-Jan Wagenmakers

Abstract This tutorial chapter shows how the separate fields of mathematical psy-chology and cognitive neuroscience can interact to their mutual benefit. Historically,the field of mathematical psychology is mostly concerned with formal theories ofbehavior, whereas cognitive neuroscience is mostly concerned with empirical mea-surements of brain activity. Despite these superficial differences in method, the ul-timate goal of both disciplines is the same: to understand the workings of humancognition. In recognition of this common purpose, mathematical psychologists haverecently started to apply their models in cognitive neuroscience, and cognitive neu-roscientists have borrowed and extended key ideas that originated from mathemati-cal psychology. This chapter consists of three main sections: the first describes thefield of mathematical psychology, the second describes the field of cognitive neu-roscience, and the third describes their recent combination: model-based cognitiveneuroscience.

1 Introduction

The griffin is a creature with the body of a lion and the head and wings of an eagle.This mythical hybrid is thought to symbolize the rule over two empires, one onthe earth (the lion part) and the other in the skies (the eagle part). The precedingsix tutorial chapters may have given the impression that the field of model-basedcognitive neuroscience is similar to a griffin in that it represents the union of twofundamentally incompatible disciplines. After all, the methods and concepts from

Birte U. ForstmannUniversity of Amsterdam, Cognitive Science Center Amsterdam, Nieuwe Achtergracht 129, 1018WS Amsterdam, The Netherlands e-mail: [email protected]

Eric-Jan WagenmakersUniversity of Amsterdam, Department of Psychological Methods, Weesperplein 4, 1018 XA Am-sterdam, The Netherlands e-mail: [email protected]

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2 Birte U. Forstmann and Eric-Jan Wagenmakers

the field of formal modeling, explained in Chapters 1, 2, and 3, appear to have littlein common with the methods and concepts from the field of cognitive neuroscienceas discussed in Chapters 4, 5, and 6. The goal of this tutorial chapter is to explainthat this impression is mistaken – the griffin analogy is apt because it highlights theadded possibilities and novel insights that can be obtained when formal models forbehavior are combined with methods from cognitive neuroscience [24].

Fig. 1 The griffin—part lion,part eagle—as depicted inJonston (1660); copper en-graving by Matthius Merian.

In this chapter we explain why it is natural to combine behavioral modeling withcognitive neuroscience; furthermore, we illustrate the benefits of the symbiotic re-lationship between the two disciplines by means of concrete examples. However,before we discuss our model-neuroscience griffin in detail, it is informative to firstdiscuss its component disciplines separately.

2 Mathematical Psychology

Mathematical psychologists are concerned with the formal analysis of human be-havior. Objects of study include perception, decision-making, learning, memory, at-tention, categorization, preference judgments, and emotion. Whenever researcherspropose, extend, or test formal models of human behavior they are practising math-ematical psychology. Thus, the field of mathematical psychology is relatively broad,and defined more by method than by topic or subject matter. To give you an impres-sion of the work done by mathematical psychologists, Table 1 provides an overviewof the articles published in the June 2012 issue of the Journal of Mathematical Psy-chology.

The inner core of card-carrying mathematical psychologists is comprised of onlyabout a few hundred researchers, and consequently progress in the field can be ago-nizingly slow. In his 2008 editorial in the Journal of Mathematical Psychology, thesociety’s president Jim Townsend wrote:

It can prove a frustrating experience to compare psychology’s pace of advance with progressin the ‘hard’ sciences. [...] steps in filling in data about a phenomenon not to mention testing

Model-Based Cognitive Neuroscience: A Conceptual Introduction 3

Table 1 Articles published in the June 2012 issue of the Journal of Mathematical Psychology.

Title Reference

A tutorial on the Bayesian approach for analyzing structural equation models. [62]Symmetry axiom of Haken-Kelso-Bunz coordination dynamics revisited in thecontext of cognitive activity. [26]Quantum-like generalization of the Bayesian updating scheme for objective andsubjective mental uncertainties. [2]Torgerson’s conjecture and Luce’s magnitude production representation implyan empirically false property. [36]A predictive approach to nonparametric inference for adaptive sequentialsampling of psychophysical experiments. [49]On a signal detection approach to m-alternative forced choice with bias, withmaximum likelihood and Bayesian approaches to estimation. [15]How to measure post-error slowing: A confound and a simple solution. [21]

of major theoretical issues and models, seem to occur with all the urgency of a glacier. Onemay wait years, before a modeler picks up the scent of an intriguing theoretical problem andcarries it ahead. It is disheartening to contrast our situation with, say, that of microbiology.[66, p. 270]

One solution to this glacier-like progress is for mathematical psychologists tocollaborate with researchers from other disciplines; when more researchers are in-terested in a particular phenomenon this greatly increases the speed with which newdiscoveries are made. This is in fact exactly what happened when cognitive neu-roscientists became interested in quantitative models for speeded decision making(e.g., [28, 29, 30]); prior to this development, such models were proposed, adjusted,and tested only by a handful of mathematical psychologists—for example, from1978 to 2001 Roger Ratcliff stood alone in his persistent efforts to promote thedrift diffusion model as a comprehensive account of human performance in speededtwo-choice tasks.

2.1 The Drift Diffusion Model

In the drift diffusion model (DDM), shown in Figure 2, noisy information is accu-mulated over time until a decision threshold is reached and a response is initiated.The DDM provides a formal account of how people make speeded decisions be-tween two choice alternatives. In other words, the model yields parameter estimates(e.g., for drift rate and boundary separation) that represent specific psychologicalprocesses (e.g., ability and response caution) in order to account for error rates aswell as response time distributions for both correct choices and errors. Put differ-ently, the DDM takes observed behavior –which may be difficult to interpret– anddecomposes it into psychological processes that are easier to interpret. For instance,the boundary separation parameter in the DDM reflects the amount of information


that a participant seeks to accumulate before being confident enough to respond.Higher levels of boundary separation reflect a more cautious response regime, onein which responding is slow but errors are few.

Drift Rate

Starting Point

Bou

ndar

y S

epar

atio

n

’Word!’

’Non−word!’

StimulusEncoding

DecisionTime

ResponseExecution

Fig. 2 A drift diffusion model for the lexical decision task. In this task, the participant is showna letter string and has to decide quickly whether it is an existing word (e.g., tiger) or not (e.g.,drapa). Noisy evidence is accumulated over time until a boundary is reached and the correspond-ing response is initiated. Drift rate quantifies decision difficulty and boundary separation quan-tifies response caution. Predicted response time equals the decision time plus the time requiredfor non-decision processes such as stimulus encoding and response execution. Figure as originallypublished in [70].

Throughout the years, Ratcliff repeatedly demonstrated how the DDM allows fordeeper insight in the processes that underlie observed behavior (e.g., [71]). Consider,for instance, the finding that older adults respond more slowly than younger adults,a general empirical regularity that holds even in relatively simple tasks such as lex-ical decision. The once-dominant explanation of this age-related slowing holds thatolder adults have a reduced rate of information processing, perhaps as a result ofneural degradation; hence, the age-related slowing was assumed to hold generally,across a wide range of different tasks and processes [7, 9, 60]. However, when theDDM was applied to the data from older adults, Ratcliff and colleagues discoveredsomething surprising [54, 56]: in most speeded two-choice tasks, drift rates did notdiffer between the young and the old. That is, older adults were accumulating di-agnostic information as efficiently as the young. Instead, the age-related slowdownwas usually due to a combination of two factors: (1) an increase in non-decisiontime, that is, the time needed for encoding and response execution, and (2) an in-


crease in response caution. These results suggest that the age-related slowing can beundone, at least in part, by encouraging the elderly to adopt a more risky responsestrategy (for a confirmation of this prediction see for instance [55]).

Currently, the DDM can be considered one of the most successful quantitativemodels in mathematical psychology: not only does it provide fits to empirical datathat are consistently good, it has also driven theoretical progress in fields tradition-ally dominated by verbal or quasi-formal accounts. These intuitive accounts wereoften unable to withstand quantitative scrutiny (e.g., [72]).

The main weakness of the DDM is that it provides a decomposition of perfor-mance that is relatively abstract, that is, the DDM does not commit to any represen-tational assumptions. This makes the model less interesting from a psychologicalpoint of view. The main weakness of the DDM, however, is also its main strength:because its account is relatively abstract it can be applied to a wide range of differenttasks and paradigms.

For the first 25 years, the development and application of the DDM was guidedby statistical and pragmatic considerations; Of particular relevance here is that thedynamics of decision-making in neural circuits is remarkably similar to that postu-lated by the DDM (e.g., [30]) in that neurons appear to accumulate noisy evidenceuntil threshold. Thus, the DDM does not only capture behavioral data but holds thepromise to capture underlying neural dynamics as well. This may not be accidental:the DDM describes performance of a decision-maker who is statistically optimal inthe sense of minimizing mean response time for a fixed level of accuracy (e.g., [5])and it is plausible that for simple perceptual tasks, evolution and individual learn-ing has curtailed those neural dynamics that lead to suboptimal outcomes. Cogni-tive neuroscientists have not only applied the DDM to neural data, they have alsoproposed theoretical extensions to the model. For instance, high-profile extensionsconcern the generalization to more than two choice-alternatives [18, 68], collaps-ing bounds [20], urgency-gating [12], and drift rates that change during stimulusprocessing [35].

2.2 Ambivalence Towards Neuroscience

Although mathematical psychologists are increasingly interested in the neural un-derpinnings of cognition, the overall attitude towards the neurosciences is one ofambivalence or even open distrust.1 Some of this ambivalence stems from the con-cern that brain measurements alone may not be theoretically meaningful. For in-stance, Coltheart claimed that “no functional neuroimaging research to date hasyielded data that can be used to distinguish between competing psychological theo-ries” [13, p. 323] (see the exercise at the end of this chapter).

1 At the 2009 annual meeting of the Society for Mathematical Psychology, one of the plenaryspeakers discussed some of his beginning exploits in cognitive neuroscience. Following his talk,the first question from the audience was whether he had now “joined the dark force”.


To demonstrate the limitations of neuroscientific methods, Ulrich presented thefollowing thought experiment [67]. Suppose you are intrigued by the ability of acomputer program to provide analytic solutions to integrals. In the Maple program,for instance, you can enter the integral

∫x sinx dx as int(x*sin(x),x); and

Maple will immediately return the solution: sin(x)-x*cos(x). How can welearn more about how Maple accomplishes these and other computations?

Ulrich argues that neuroscientists may tackle this problem in different ways, asillustrated in Figure 3: analogous to functional brain imaging, one might performa heat scan on the laptop as it computes integrals, and compare this with a controlcondition where it is just waiting for input (Figure 3, top left panel). Analogousto EEG measurements, one could attach surface electrodes to the laptop, have thelaptop repeatedly perform integrals, and compute a stimulus-locked or a response-locked event-related potential (Figure 3, top right panel). Analogous to single-cellrecordings in monkeys, one might implant electrodes and register the activity ofsmall components within the laptop (Figure 3, lower left panel). Finally, analogousto neurosurgical methods, one might lesion the laptop, for instance by hitting itwith a hammer. With luck, one might even discover a double dissociation, that is,lesioning one part of the laptop harms the computation of integrals but does notharm word processing, whereas lesioning another part of the laptop harms wordprocessing but not the computation of integrals (Figure 3, lower right panel).

Ulrich ([67, p. 29]) concludes that “(...) none of these fancy neuroscience tech-niques can directly unravel the hidden mechanisms of this symbolic math program”and hence, brain measurement techniques alone cannot replace formal theories ofcognition. We suspect that most mathematical psychologists subscribe to the Ulrichlaptop metaphor of neuroscience. The laptop metaphor is insightful and thought-provoking, but it should not be misinterpreted to mean that neuroscientific methodsare by definition uninformative. For example, consider a race of aliens who discovera refrigerator and wish to learn how it works. They may first conduct behavioral ex-periments and conclude that parts of the refrigerator are cooler then others. Theymay study the speed of cooling in the different compartments, and its relation to ahost of relevant factors (e.g., the extent to which the refrigerator door is left open,the temperature of various products just before they are put inside, and the vol-ume occupied by the products). The aliens may develop theoretical concepts suchas homeostasis, they may propose sets of axioms, and they may develop compet-ing formal models about how the refrigerator does what it does. Unfortunately, thebehavioral data are rather sparse and therefore they will fail to falsify many of themore complicated theories. It is evident that “neuroscientific” measures of study-ing the refrigerator (e.g., examining its underlying circuitry) will yield additionalinsights that can be used either to adjudicate between the competing theories or todevelop new theories that are more appropriate.

We will leave it up to philosophers to decide whether the study of cognitionis similar to a laptop or to a refrigerator. This decision may well depend on thecognitive phenomenon under study. For instance, perceptual illusions are perhapsbest understood by taking into account the neural processes that subserve percep-


Fig. 3 Illustration of Ulrich’s thought experiment. The operations of a computer program are stud-ied with methods from neuroscience. Top left panel: heat radiation scan; top right panel: event-related potentials; lower left panel: single-unit recordings; lower right panel: experimental lesions.Figure reprinted with permission from [67].

tion, whereas a different approach is warranted when one wants to understand loss-aversion in gambling.

Pragmatically, any approach is worthwhile as long as it yields theoretical progress,that is, a deeper understanding of human cognition. It is undeniably true that brainmeasurements, just as response times or error rates, constitute data that are poten-tially informative about the underlying cognitive process. The main difficulty, there-fore, is to develop formal models that allow the brain measurements to make contactwith putative cognitive processes.

3 Cognitive Neuroscience

The annual meetings of the Society for Neuroscience attract up to 40,000 partici-pants, and plenary lectures are given by celebrities such as the Dalai Lama. Based onthe attendance to their respective annual meeting, neuroscientists outnumber math-ematical psychologist by a factor of 200 to 1. Cognitive neuroscientists use brainmeasurement techniques to study cognitive processes such as perception, attention,learning, emotion, decision-making, etc. Most of this work involves an empirical


comparison between groups, treatments, or experimental conditions. For instance,Rouw and Scholte [57] compared a group of control participants with a group ofgrapheme-color synesthetes, people who experience a specific color whenever theysee a particular letter or number (e.g., “T is bright red”). Diffusion tensor imag-ing confirmed the hypothesis that the added sensations in synesthesia are associatedwith more coherent white matter tracts in various brain areas in frontal, parietal,and temporal cortex. In another example, Jepma and Nieuwenhuis [34] used a rein-forcement learning task in which participants have to maximize rewards by makinga series of choices with an uncertain outcome. The main result was that baselinepupil diameter was larger preceding exploratory choices (i.e., choices associatedwith a large uncertainty in outcome) than it was preceding exploitative choices (i.e.,choices associated with a small uncertainty in outcome). Pupil diameter is an in-direct marker for the activity of the locus coeruleus, a nucleus that modulates thenorepinephrine system. Hence, the results are consistent with adaptive gain theory,according to which activity in the locus coeruleus regulates the balance betweenexploration and exploitation. A final example concerns the work by Ding and Gold[17], who showed that electrical microstimulation of the monkey caudate nucleusbiases performance in a random-dot motion task.2 This result suggests that the cau-date has a causal role in perceptual decision making.

To give you a further impression of the work done by cognitive neuroscientists,Table 2 provides an overview of the articles published in the June 2012 issue of theJournal of Cognitive Neuroscience. Compared to the mathematical psychology ap-proach, the cognitive neuroscience approach is geared towards understanding cog-nition on a relatively concrete level of implementation: what brain areas, neuralprocesses, and circuits are involved in a particular cognitive process?

Table 2 First seven articles published in the June 2012 issue of the Journal of Cognitive Neuro-science.

Title Reference

Focal brain lesions to critical locations cause widespread disruption of the modularorganization of the brain.

[31]

Playing a first-person shooter video game induces neuroplastic change. [73]Closing the gates to consciousness: Distractors activate a central inhibition process. [44]TMS of the FEF interferes with spatial conflict. [4]Local field potential activity associated with temporal expectations in the macaquelateral intraparietal area.

[50]

Spatio-temporal brain dynamics mediating post-error behavioral adjustments. [37]Hippocampal involvement in processing of indistinct visual motion stimuli. [25]

It is tempting to believe that the level of implementation is the level that is some-how appropriate for the study of cognition. This is suggested, for example, by theadage “the mind is what the brain does”. However, Ulrich’s laptop metaphor shows

2 In this popular perceptual task, the participant has to judge the apparent direction of a cloud ofmoving dots.


that such a conclusion is premature; clearly, the analytical integration that Mapleaccomplishes is “what the laptop does”, but it does not follow that we need or wantto study the properties of the laptop in order to understand how Maple handles in-tegrals analytically. Thus, even though “the mind is what the brain does”, it is notautomatically the case that when we measure the brain we learn a great deal aboutthe mind. Readers who doubt this statement are advised to read the contributionsthat follow the article by Coltheart [13]; here, the discussants have to put in hardwork to come up with just a single example of how functional neuroimaging hasprovided data to discriminate between competing psychological theories.

In order for cognitive neuroscience to have impact on psychological theory, it isimportant that the two are linked [8, 61, 65]. One way to accomplish such linking isby elaborating the psychological theory such that it becomes explicit about the brainprocesses involved [3]; another way is by using formal models to connect findingsfrom neuroscience to the cognitive processes at hand. For instance, a mathematicalpsychologist may use the DDM to state that, when prompted to respond quickly,participants become less cautious, that is, they require less evidence before theyare willing to make a decision. This description of cognition is relatively abstractand does not speak to how the brain implements the process. A neuroscientist maymake this more concrete and suggest that the instruction to respond quickly leadsto an increase of the baseline level of activation in the striatum, such that less inputfrom cortex is needed to suppress the output nuclei of the basal ganglia, therebyreleasing the brain from tonic inhibition and allowing an action to be executed [23,42]. Thus, the DDM may provide an estimate of a latent cognitive process (e.g.,response caution) which may then be compared against activation patterns in thebrain. By using formal models that estimate psychological processes, this particularneuroscience approach furthers real theoretical progress and potentially bridges thedivide between the implementational level and the algorithmic level [38].

4 Model-Based Cognitive Neuroscience: Symbiosis of Disciplines

The goal of model-based cognitive neuroscience is to bridge the gap betweenbrain measurements and cognitive process with the help of formal models (e.g.,[19, 27, 28, 32, 39, 48]). This interdisciplinary approach is illustrated in Figure 4.The figure shows that experimental psychology, mathematical psychology, and cog-nitive neuroscience all pursue a common goal: a better understanding of humancognition. It is often difficult, however, to learn about the relevant cognitive pro-cesses from the data directly – often, one first needs a mathematical model to pro-vide quantitative estimates for the cognitive processes involved. Next, the estimatesof the cognitive processes can be related to the brain measurements.

The “model-in-the-middle” [14] symbiosis of disciplines is useful in severalways. Rather than discuss the advantages abstractly, the next two sections provideconcrete illustrations of the mutually beneficial relationship between mathematicalmodels and brain measurements (see also [24]).


Behavioral data Cognitive process

Brain data

Formal model

Formal model

Experimental

Psychology

thematical

logy

ional

sTraditional

Neurosciences

Mathematical

Psychology

Fig. 4 The model-in-the-middle approach unites experimental psychology, mathematical psychol-ogy, and cognitive neuroscience, as the main goal of all three disciplines is to understand moredeeply the processes and mechanisms that constitute human cognition. The red arrow indicates thereciprocal relation between measuring the brain and modeling behavioral data. Figure reprintedwith permission from [24].

4.1 Use of Mathematical Models for Cognitive Neuroscience

Mathematical models are useful in many ways. First, they decompose observed be-havioral data into latent cognitive processes. Brain measurements can then be asso-ciated with particular cognitive processes instead of behavioral data. For example,Chapter 2 explained how the LBA model –just as the DDM model discussed inSection 2 and in Chapter 3– decomposes response time distributions and error ratesinto underlying concepts such as response caution and the speed of informationprocessing. This decomposition can be used to demonstrate that a particular experi-mental manipulation had the desired effect. For instance, participants in a study byForstmann and colleagues [23] performed a random-dot motion task under variouscue-induced levels of speed-stress. That is, before each stimulus a cue indicatedwhether the stimulus needed to be classified accurately or quickly. Because the au-thors were interested in the neural basis of the speed-accuracy tradeoff, they hopedthat the cue would selectively affect the LBA response caution parameter. And in-deed, the model decomposition confirmed that this was the case. Note that withoutconcrete model fitting, this conclusion had been premature, unwarranted, and poten-tially misleading – it is certainly possible that instructions to respond more quicklycan, for some tasks, also induce a change in speed of processing, or a change in thetime required for peripheral processes.

Another advantage that the model-based decomposition brings is that, even whena particular manipulation is not process-pure, one may associate brain measure-


ments specifically with the parameter of interest. For instance, suppose that in thespeed-accuracy experiment by Forstmann and colleagues [23], a cue to respondmore quickly had also lowered the speed of information processing. This meansthat the critical fMRI contrasts are contaminated because they reflect a combinationof two effects: the change in response caution associated with the speed-accuracytradeoff, and the change in drift rate associated with the lower speed of process-ing. One method to address this complication is to correlate the contaminated brainmeasures (e.g., the average change in the BOLD response for each participant) withthe process of interest (e.g., the individual estimates of the change in response cau-tion), perhaps after partialling out the effects of the nuisance process. This methodidentifies those voxels that relate to the cognitive construct of response caution.

Finally, the model-based decomposition allows one to take into account indi-vidual differences. For instance, Forstmann and colleagues [23] found that speedcues activated the right anterior striatum and the right pre-supplementary motorarea (pre-SMA). This result was corroborated by an analysis of individual differ-ences: participants with a relatively large cue-induced decrease in response cautionalso showed a relatively large increase in activation in the right anterior striatumand right pre-SMA. Of course, such an analysis is only meaningful if there are sub-stantial individual differences to begin with; if all participants respond to the cue inapproximately the same way then the group-average result will be highly significantbut the individual difference analysis may not be significant at all.

In another example of the importance of individual differences, Forstmann andcolleagues [22] studied the neural basis of prior knowledge in perceptual decision-making. As before, participant performed a random-dot motion task; this time, thecue gave prior information about the likely direction of movement of the upcomingstimulus. The cue “L9”, for example, indicated that the probability was 90% that theupcoming stimulus would move to the left (see also [43]). The cue-induced bias wasclearly visible in the behavioral data: responses were much faster and more oftencorrect when the cue was reliable and informative. Surprisingly, however, the fMRIcontrast did not reveal any significant results. After including an LBA response biasparameter as a covariate in the fMRI analysis, however, the results showed signifi-cant cue-related activation in regions that generally matched the theoretical predic-tions (e.g., putamen and orbitofrontal cortex). The reason for the discrepancy is thatby adding the response bias parameter we can account for individual differences inpeople’s reactions to the cue. Some participants exhibited a lot of bias, and othersonly a little. These individual differences in the latent cognitive process are usu-ally not incorporated the fMRI analysis and hence add to the error term instead.By explicitly accounting for individual differences the error term is reduced andexperimental power is increased.

Mathematical models are also useful because they can drive the search for brainareas involved in a particular cognitive function. In fMRI research, for instance, thismeans that a model’s predictions are convolved with the hemodynamic responsefunction. Next, the predicted blood oxygenation level dependent signal (BOLD)response profiles are used to search for areas in the brain with similar activationprofiles. The search can be exploratory or more confirmatory. A prominent example


of the latter approach is the recent work that attempts to link the components of theACT-R model to different brain regions (see the final chapter of this book as well as[1, 6]).

Another way in which mathematical models are useful is that they may helpdemonstrate that performance in superficially different tasks may be driven by thesame cognitive process. For example, the cognitive neuroscience literature suggeststhat perceptual categorization and old-new recognition are subserved by separateneural systems. In contrast, mathematical models of these tasks suggests that similarprocesses are involved. Specifically, exemplar models posit that people categorizea test object by comparing it to exemplars stored in memory. Both categorizationand recognition decisions are thought to be based on the summed similarity of thetest object to the exemplars [45, 47]. These conflicting perspectives were recentlyreconciled by Nosofsky and colleagues [46], who argued that categorization andrecognition differ not in terms of the underlying process, but in terms of the under-lying criterion settings: in recognition, the criterion needs to be strict, since the testobject needs to match one of the study items exactly; in categorization, the criterioncan be more lax, as exact matches are not needed. In an fMRI experiment, Nosof-sky and colleagues [46] induced participants to use different criterion settings; theresulting data were then fit by an exemplar model. Results confirmed that (1) theexemplar model provides a good account of both categorization and recognition,with only criterion settings free to vary; (2) the average task-related differences inbrain activation can be explained by differences in evidence accumulation causedby systematically varying criterion settings; and (3) participants with high crite-rion settings show large BOLD differences between old and random stimuli in thefrontal eye fields and the anterior insular cortex. Hence, Nosofsky and colleagues[46] concluded that there is little evidence that categorization and recognition aresubserved by separate memory systems. The most important lesson to be learnedfrom this work is that differences in brain activation do not necessarily indicatedifferent underlying mechanisms or processes. Differences in brain activation canalso come about through differences in stimulus surface features (which Nosofskyand colleagues controlled for) and differences in criterion settings. A mathematicalmodel can estimate these criterion settings and allow a statistical assessment of theirimportance.

In addition to the above, mathematical models have general worth because theyprovide (1) a concrete implementation of a theoretical framework; (2) a coherentinterpretive framework; and (3) a guide to experimental manipulations that are par-ticularly informative. In sum, it is evident that for cognitive neuroscience, the use ofmathematical models comes with considerable advantages. The reverse –the advan-tages of cognitive neuroscience for mathematical models– is the topic of the nextsection.


4.2 Use of Cognitive Neuroscience for Mathematical Models

Until recently, findings from cognitive neuroscience had little impact on model de-velopment in mathematical psychology. Exceptions that confirm the rule are paralleldistributed processing models [58, 59] and neurocomputational models developedto take into account the details of neural processing [3, 52, 64, 63]. This state ofaffairs is changing, and for good reason: response times, error rates, and confidencejudgements ultimately provide little information to tell apart mathematical modelswith incompatible assumptions and architectures [61]. For instance, Ditterich [18]showed that behavioral data are insufficient to discriminate between multiple-choiceresponse time models that have evidence integrators with and without leakage, withand without feedforward and feedback inhibition, and with and without linear andnon-linear mechanisms for combining information across choice alternatives. Neu-ral data, however, can able to adjudicate between the hypothesized mechanisms, atleast in potentia [18].

As a specific example, consider a generic response time model with N evidenceaccumulators, one for each choice alternative, that race to a threshold. The modelcan account for the speed-accuracy tradeoff by changing the distance from baselineto threshold. However, the model is mute on whether instructions to respond accu-rately increase threshold or decrease baseline; in fact, these mechanisms are mathe-matically equivalent. Nevertheless, the mechanisms are not conceptually equivalent,and neural data could discriminate between the two accounts. A similar exampleconsiders the change in processing that occurs as the number of incorrect choicealternatives increases. Such an increase in task difficulty requires a longer period ofevidence accumulation in order to reach an acceptable level of accuracy in identify-ing the target alternative. The evidence accumulation process can be extended eitherby increasing thresholds or by decreasing the baseline. Using behavioral data, thereis no way to tell these two mechanisms apart. However, Churchland and colleagues[11] used single-cell recordings to show that, confronted with the prospect of hav-ing to chose between four instead of two random-dot choice alternatives, monkeyshad decreased firing rates in the lateral intraparietal area. These single cell record-ings are consistent with a changing baseline account rather than a shifting thresholdaccount.

Hence, the general promise is that data from cognitive neuroscience may pro-vide additional constraints. An interesting illustration of this principle is providedby Purcell and colleagues [51], who studied how monkeys perform a visual searchtask in which they have to make an eye movement toward a single target presentedamong seven distractors. Several models were fit to the data, and initial constraintwas gained by using the measured spike trains as input to the evidence accumulators.This creates an immediate challenge: the models must determine when the accumu-lators start to be driven by the stimulus, because the neural activity that precedesstimulus onset is uninformative and its accumulation can only harm performance.Hence, models with perfect integration failed, as they were overly impacted by earlyspiking activity that was unrelated to the stimulus. Models with leaky integration didnot suffer from early spiking activity, but their predictions were inconsistent with an-


other neural constraint: the spiking activity from movement neurons. In the end, theonly class of models that survived the test were gated integration models, modelsthat block the influence of noise inputs until a certain threshold level of activation isreached.

In general, it is clear that neuroscience data hold tremendous potential for an-swering questions that mathematical psychologists can never address with behav-ioral data alone.

5 Open Challenges

The above examples have only scratched the surface of the work conducted withinmodel-based cognitive neuroscience. Nevertheless, the field can greatly expand byconsidering a broader range of mathematical models, and a broader range of brainmeasures (e.g., structural MRI, event-related potentials in EEG, genetics, pharma-cology, etc.). Also, model dynamics can be linked more closely to brain dynamics,either by constructing a single overarching statistical model, or by developing sin-gle trial estimates of cognitive processes. For example, van Maanen and colleagues[69] extended the LBA model to estimate drift rate and thresholds on a trial-by-trialbasis. This allows a more direct comparison with neurophysiological data, whichalso vary on a trial-by-trial basis.

Another challenge is to balance the desire for parsimonious models (i.e., mod-els with few parameters and clear mechanisms) against the reality of the brain’soverwhelming complexity. The appropriate level of model complexity depends verymuch on the goals of the researchers. If the goal is to obtain estimates of latent cog-nitive processes, then the model needs to be relatively simple – the behavioral datasimply do not provide sufficient support for models that are relatively complex. Onthe other hand, if the goal is to create a model that accounts for the detailed interac-tions between neurons or brain systems, the model needs to be more intricate.

A final challenge is that, despite the intuitive attractiveness of results from ratsand monkeys, we should remain aware of the possibility that some of the resultsobtained with these species may not carry over to humans. This may be due to dif-ferences in anatomy, but other factors can contribute as well. For instance, recentwork suggests that monkeys who perform a speeded choice task may experiencean increased urgency to respond [12] that can expresses itself in response thresh-olds that decreases over time [16, 20]. Before concluding that response urgency orcollapsing bounds are a universal signature of human decision making, however,we need to make sure that the pattern in monkeys is obtained in humans as well.This requires a careful modeling exercise in which benchmark data sets are fit withtwo versions of the same sequential sampling model: one that has constant thresh-olds and one that has collapsing bounds [41]. It is entirely possible, for instance,that collapsing bounds are used by monkeys because they want to maximize rewardrate [16]; first-year psychology undergraduates, however, are usually not reinforcedwith squirts of orange juice and may approach the task with a different goal. The


collapsing-bound hypothesis shows promise and is worth exploring, but its general-ity is pending investigation.

6 Concluding Comments

The examples in this chapter have shown how mathematical models can advancecognitive neuroscience, and how cognitive neuroscience can provide constraint formathematical models. The increasing collaboration between these historically sep-arate fields of study is an exciting new development that we believe will continue inthe future.

7 Exercises

1. Section 2: in what sense is the DDM similar to signal-detection theory?2. Section 2: Can you find a concrete example to refute Coltheard’s claim that

“no functional neuroimaging research to date has yielded data that can beused to distinguish between competing psychological theories”?

3. Section 3: Read the articles by Miller [40] and by Insel [33] on the impactof neuroscience on psychiatry and clinical psychology. Who do you agreewith, and why?

4. Section 2: Read the Gold and Shadlen [30] article and prepare a 30-minutepresentation on it, critically summarizing and explaining its content.

5. Describe a mathematical model (not discussed in this chapter) that couldfind application in cognitive neuroscience.

6. Mention one pro and one con for each of the following brain measures:single-cell recordings, ERP, fMRI, and DWI.

7. Can you think of concrete research questions in cognitive neuroscience thatcould profit from a model-based approach?

8 Further Reading

1. Ratcliff and McKoon [53] offer an overview of the drift diffusion model and itsrelation to cognitive neuroscience.

2. http://neuroskeptic.blogspot.com/2012/02/mystery-joker-parodies-neuroscience.html tells a tale aboutneuroscience and Sigmund Freud.


3. Churchland and Ditterich [10] discuss recent developments in models for achoice between multiple alternatives.

4. We consider our work on bias [22] as one of our better efforts. Unfortunately, thereviewers did not agree, and one even commented “Flawed design, faulty logic,and limited scholarship engender no confidence or enthusiasm whatsoever”.

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