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Formal Innovations in Clinical Cognitive Science and Assessment

Formal Innovations in Clinical Cognitive Science and Assessment Mathematical modeling is increasingly driving progress in clinical cognitive science and assessment. Mathematical modeling is essential for detecting certain effects of psychopathology through comprehensive understanding of telltale cognitive variables, such as workload capacity and efficiency in using capacity, as well as for quantitatively stipulating the subtle but important differences among these variables. The research paradigm guiding this formal clinical science is outlined. A distinctive cognitive abnormality in schizophrenia—taking longer to cognitively represent encountered stimulation—is used as a specific example to illustrate a general quantitative framework for studying intricate phenomena that impair mental health. Developments in mathematical modeling will also benefit symptom description and prediction; provide grounding in cognitive and statistical science for new methods of clinical assessment over time, both for individuals and for treatment regimens; and contribute to refining the cognitive-function side of clinical functional neurophysiology. Keywords clinical mathematical modeling, cognitive assessment, cognition in schizophrenia, formal cognitive neuroimaging, cognitive mixture models Contemporary clinical applications of basic cognitive them through a case study involving the symptom-related science have taken on a new direction involving math- cognitive neuroscience of schizophrenia. ematical modeling of symptom-related cognitive abnor- As illustrated in Figure 1, clinical mathematical cog- malities. The basic research paradigm for this movement nitive neuroscience stands to uniquely contribute to is illustrated in Figure 1. Mathematical models of cogni- mainstream mathematical cognitive neuroscience. It can tive performance among healthy individuals are adjusted do so through model generalization testing (Busemeyer to accommodate deviations from normal performance & Wang, 2000), in which the robustness of a model’s among participants with selected forms of psychopa- performance is evaluated with new experimental para- thology. Such deviations typically center around speed digms or populations. Clinical mathematical cognitive and/or accuracy in performing cognitive tasks. Parts of neuroscience provides key opportunities for generaliza- the model remaining intact after this adjustment are tion testing with respect to extreme individual differ- considered to indicate cognitive functions that are ences, which are associated with psychopathology. spared, and performance deviations that compel modi- Models that readily accommodate performance devia- fication of the model are flagged as signifying disorder- tions are preferred to those that fail to accommodate affected functions. Minimal adjustment of the model is them or that are strained in doing so. That is, a model desired, in the interest of parsimony. In this way, models is supported when observed abnormalities relating to provide a formal framework to determine which cogni- psychopathology can be well predicted without major tive processes do or do not differ between clinical adjustment of the model’s workings. groups and healthy control groups. Such formal theoretical developments can offer mul- Corresponding Author: tiple advantages in explaining and measuring psychopa- Richard W. J. Neufeld, Department of Psychology, Western University thology. This article describes these advantages, illustrating Email: rneufeld@uwo.ca 216 Neufeld, Shanahan Clinical Mathematical Cognitive Mathematical Cognitive Neuroscience Invoking Clinical Neuroscience Invoking Mathematical Modeling and Model Mathematical Models of Adjustment to Accommodate Deviations Experimental Cognitive-Task in Experimental Cognitive-Task Performance Performance Model Generalization Testing Fig. 1. Relations between mathematical cognitive neuroscience and clinical mathematical cognitive neuroscience. The upper arrow indicates the application of mathematical models of normal cognitive-task performance to the understanding of performance changes occurring with psychopathology. The lower arrow indicates that the suc- cess of such application, in turn, bears on the validity of the applied models. Note that clinical mathematical modeling, the subject and then, after a delay, to identify which stimuli in a of this article, involves analytic theorems and proofs, new set were part of the original set. To do well in this as well as algebraic derivations expressing cognitive task, participants must encode stimuli presented in the transactions relevant to clinical disorders. It should be second set into a cognitive format facilitating compari- distinguished from a somewhat related type of model- son with the previously memorized set of stimuli. For ing known as computational psychiatry. Computational example, in the case of basic visual-template matching, psychiatry addresses how cognitive transactions might it may be necessary to cognitively extract the physical be realized at the level of neural organization and oper- features of a presented stimulus, such as its curves, ations. By and large, computational psychiatry dele- lines, and intersections; or in the case of stimulus-name gates disorder-related deviations of neural functioning matching, it may be necessary to tag a presented digit to computer simulation of neural networks and neuro- or letter stimulus with its name, for comparison with dynamics (communication between neurons). (Accounts names of the stimuli in the previously memorized set. and examples of computational psychiatry are available Mathematical modeling enables quantitative dissec- in Montague et  al., 2012; Huys et  al., 2016; Wang & tion of cognitive processes, to help pinpoint specific Krystal, 2014; and Grossberg, 1999. For a rigorous, com- sources of deviation in cognitive performance. With prehensive treatment of artificial neural networks, see respect to encoding, for example, the processes can be Golden, 1996.) Computational psychiatry and clinical broken down as follows. First, the overall process is mathematical modeling potentially are complementary made up of constituent encoding operations—encoding when it comes to quantitative accounts of clinical dis- subprocesses, such as registration of curves, lines, and orders (e.g., Carter & Neufeld, 1999, 2007; extensive intersections of a presented stimulus. Second, the elaboration on distinctions among alternate approaches encoding subprocesses take place at a certain rate, to modeling clinical phenomena can be found in known as subprocess-level cognitive-workload capacity Neufeld, 2007a). (e.g., Neufeld et al., 2007; Wenger & Townsend, 2000). Application of the model-adjustment operation of Figure 1 has repeatedly shown that subprocess-level Case Study: Cognitive Neuroscience of cognitive-workload capacity remains intact in schizo- Stimulus Encoding in Schizophrenia phrenia, but the number of encoding subprocesses Schizophrenia affects approximately 0.5% of the North undertaken is elevated. In other words, cognitive-work- American population. Symptoms can take the form of load capacity escapes impairment, whereas efficiency delusions and hallucinations (thought-content disorder), of its implementation does not. This combination of incoherent speech, reduced cognitive efficiency, and spared and affected components of encoding perfor- impoverished motivation. Research studies applying the mance is analogous to a racehorse striding at a normal research strategy depicted in Figure 1 have shown that pace but closer to the outside rail, which increases the delayed completion of stimulus encoding is a deviation requisite number of paces, and therefore the time in cognitive performance recurrently found in schizo- needed, to complete the course. This combination illus- phrenia, across multiple experimental tasks and levels trates the nature of the model adjustment referred to of patient status (e.g., first episode, never treated, out- in Figure 1: The altered model conforms to the specific patient, inpatient). In this case, stimulus encoding refers pattern of empirical deviations among clinical partici- to cognitively preparing and transforming cognitive- pants (in this case, people with schizophrenia). task stimuli into a format facilitating collateral pro- The studies just mentioned have tracked the abnor- cesses. For instance, participants might be asked to mality in encoding to a specific formalized property—the memorize a set of novel stimuli (e.g., “TZAM,” “CEYP”) subprocess-number parameter of the stimulus-encoding Current Directions in Psychological Science 31(3) 217 process. Identification of such an abnormality stands to response on cognitive-task trials). The methods used open the way to potentially important advances in this include maximum likelihood, distribution-moment domain of psychological clinical science. The abnormal- matching (e.g., Evans et al., 2000), and Bayesian param- ity arguably represents a critical deficit that compromises eter estimation (e.g., Alexandrowicz & Gula, 2020, who activities relying on timely stimulus encoding (e.g., per- applied this method, along with a mathematical model forming daily self-maintenance and meeting environ- of decision and choice, to clinical disorders). Clinically mental stresses and demands). The quantitative apparatus relevant cognitive processing concealed in raw data can in which this property is embedded provides for certain be revealed via mathematical modeling. methodological benefits, including theory-guided mea- Often it may not be reasonable to assume that all surement and clinical assessment, and stipulation of the individuals within a clinical group have (roughly) the cognitive neurophysiological processes taking place. The same level of cognitive processing, as indexed, for abnormality, as quantitatively defined, also can be shown example, by a fixed value for a model parameter. For- to be potentially symptom related, notably with respect tunately, mathematical models can be expanded to to thought-content disorder (delusions and thematic hal- account for this, notably through expansion as mixture lucinations). Such symptomatology is considered to ema- models. Mixture models treat the overall performance nate from failure to encode specifically context-related of a group as a mixture of different levels of perfor- features of a stimulus complex during episodes of infor- mance among individual group members (e.g., Carter mation intake. When the influence of reality-grounding, et al., 1998; Cutler & Neufeld, 2017). objectifying cues is weakened, other information that Mixing distributions can be important not only successfully is taken in during an episode is open to because they make it possible to systematically accom- false interpretation (this mechanism of symptom produc- modate individual differences, but also because the tion is expanded upon quantitatively in Neufeld, 2007b, parameters that mathematically govern the random dis- and Neufeld et al., 2010). This formal theoretical account tributions of model properties (mixing-distribution is in the spirit of the current clinical-science trend hyperparameters) can be clinically meaningful in their toward determining underlying mechanisms of complex own right. For example, mixture-model hyperparame- behaviors, in this case mathematically. It accords, more- ters can convey a particular group’s general level of over, with the currently prominent Research Domain facility with undertaking the elements of the cognitive Criteria initiative (e.g., Kozak & Cuthbert, 2016), inas- process at hand (e.g., encoding subprocesses); they can much as the identified mechanism evidently extends to also be used to indicate susceptibility of this facility to other forms of clinical disturbance (e.g., major depres- impairment during psychological stress (for concrete sive disorder; Taylor et al., 2016), and nonclinical popu- examples, see Neufeld, 2016). lations (Nicholson & Neufeld, 1993). In short, mixture-model expansions, illustrated in Note that model adjustment capturing changes in Figure 2, can increase the span of what a model explains cognition associated with clinical disorder typically by incorporating individual differences. They addition- takes the form of altering the values of model param- ally can tap clinically meaningful constructs, such as eters, such as the number of encoding subprocesses or cognitive-task facility and vulnerability of performance the rate at which subprocesses are completed. Model to psychological stress. architecture (in terms of the number of model param- eters involved or their arrangement in relation to each Measuring better with Bayes other) ordinarily is common to clinical and nonclinical groups alike (see, e.g., Neufeld & Broga, 1981; Wallsten Mixture models allow for the likelihood that individuals et al., 2005). In other words, the basic mental apparatus systematically differ in properties of mathematically meeting a cognitive challenge is common across groups, expressed cognitive performance. They go an important but modification of one or more of its parts (parame- step further, in providing for efficient estimation of ters) accompanies clinical disorder. model properties for the individual. They do so by cus- tomizing the properties to the person, through Bayesian statistical methodology, as follows. Bayes’ theorem, Measurement and Clinical Assessment appropriated to the present context, states that Guided by Formal Theory Pr () APrA ({∗}| ) Samples of individuals’ cognitive performance, for PA r( |{∗}) = , (1) instance, on encoding-intensive tasks, permit estimation Pr () ∗ {} of cognitive-process parameter values. Such estimation is accomplished by applying established methods to where Pr(A|{*}) is the Bayesian probability that a pre- empirical performance data (e.g., observed latencies of dicted entity, such as a cognitive-process parameter 218 Neufeld, Shanahan Mixing Distribution Expressing Mixing Distribution Expressing Mixing Distribution Expressing Individual Differences in the Individual Differences in Encoding- Individual Differences in the Number of Subprocesses: Subprocess Cognitive-Workload Number of Subprocesses: Lower Values Among Healthy Capacity (Rate of Encoding- Higher Values Among Control Participants Subprocess Completion): Common Schizophrenia Participants to Control and Schizophrenia Participants Model of Latencies in Stimulus Encoding, a Function of the Encoding-Subprocess Rate and the Number of Encoding Subprocesses Fig. 2. Design of a stimulus-encoding mixture model accommodating individual differences in parameters of encoding. The rate at which encoding subprocesses unfold is shared by control and schizophrenia participants, whereas the number of encoding subprocesses is greater among schizophrenia than control participants. (e.g., number of encoding subprocesses), has the value Estimates are solidified by feeding into their calculation A given relevant observations {*} (e.g., a sample of specifically that information supplied by a preestab- cognitive performance); Pr ({*}| A) is the likelihood of lished referent, the Bayesian prior represented by the Pr () A the performance sample, given the value A; is mixing distribution. the probability of the value under consideration accord- The operation of mixing-distribution Bayesian priors ing to the relevant mixing distribution, temporarily dis- can help alleviate the problem of small-sample math- regarding the empirical observations {*}; and Pr ({*}) is ematical modeling, which is ubiquitous in applied set- the probability of the observations, all candidate values tings. The approach allows researchers to work with of the predicted entity considered (for accounts of small cognitive-performance sample sizes, which is Bayesian modeling generally, see classic works such as particularly helpful when undertaking person-specific Berger, 1985, and O’Hagan & Forster, 2004). modeling for assessment or research purposes. Valid With Bayes’ theorem and a person’s cognitive- mixing distributions sharpen the estimation of model performance sample in hand, a versatile estimation of properties for the individual. They help compensate individual attributes of clinical interest, based on cogni- for small performance samples by bringing into play tive and statistical science, is possible. Predicted entities performance-relevant information about the group to A actually can be diverse, including, for example, the which the individual at hand belongs. Again, this infor- parameter expressing the number of encoding subpro- mation is conveyed by the mixing distribution that cesses or the symptomatology to which the mathemati- quantifies the relative frequency of the target of predic- cal model and estimated model parameters relate (e.g., tion (e.g., a model-parameter value) in a membership severity of thought-content disorder; for further discus- group. Such a scenario resembles what takes place in sion, see the following section on dynamic assessment a hematology laboratory, where a substantial extant of treatment efficacy). Bayesian estimation stabilizes bank of hematological information, applicable collec- estimated values through the anchoring effects of mix- tively, is brought to bear individually on a modest blood ing distributions, which act as Bayesian priors (e.g., sample from the person at hand. Pr () A , in Equation 1). Variance in estimates (statistical Moreover, dynamic assessment of changes in clinical inefficiency) thus is reduced through the quantitative condition is possible through undertaking Bayesian esti- mechanism formally known as Bayesian shrinkage. mation at designated times of clinical interest (e.g., after Current Directions in Psychological Science 31(3) 219 a selected bout of treatment). Specifically, Equation 1 Implications for Clinical Functional can be used to track changes in the status of symptom- Neurophysiology related (e.g., thought-content symptomatology) cognitive- Mathematical modeling of the cognitive side of vascular model parameters (e.g., number of stimulus-encoding and electrophysiological cognitive neurophysiology subprocesses, which the model-adjustment operation of conveys several methodological assets. Formally anchor- Fig. 1 has identified as inflated in schizophrenia). ing cognitive functions in a viable mathematical model Changes can be monitored as they occur over the natu- is an antidote to a thorny problem in cognitive neuro- ral passage of time, over the course of treatment, or physiology known as reverse inference (Poldrack, 2011). subsequent to an experimental manipulation. In these This problem consists of circularly relying on measured ways, the described formal methodology can be an neurophysiological signals (obtained with fMRI, func- important constituent in the arsenal of clinical assess- tional magnetic resonance spectroscopy, magnetoen- ment. Mixing-distribution Bayesian priors have replaced cephalography, and electroencephalography) to infer the usual population-based norms of multi-item psycho- the cognitive functions whose very neuronal substrates metric inventories. purportedly are being charted. This inferential dilemma Bayesian individualization of model properties also in principle can be overcome as follows. The cognitive allows for evaluation of model performance at the person- functions at work while neurophysiological measure- specific level. Doing so ascertains a model’s validity for ments are taken are quantitatively stipulated in advance, an individual participant; it also affords strong tests of anchored in a formal representation (e.g., Ahn et  al., overall model performance. Fit of model predictions to 2011; White et  al., 2012). That is, cognitive functions empirical observations at both the group and the indi- whose neurophysiological substrates are being exam- vidual levels is an added means of model evaluation. ined are staked out in terms of a quantitative model— This unique form of model evaluation potentially bears one that is a priori freestanding, independent of the on the currently prominent issue of robustness of find- examined neurophysiological activity itself. ings in cognitive modeling (Neufeld & Cutler, 2019). Note, further, that dynamic models of cognitive opera- tions treat the development of cognitive processes as sto- Dynamic assessment of treatment- chastic functions of time (Townsend & Ashby, 1983). The regimen efficacy unfolding of target processes, such as stimulus encoding, With a modest expansion of Equation 1, the present can be overlaid on monitored neurophysiological signals, assessment methodology naturally extends beyond the to produce times of neurophysiological measurement individual; it can be applied to estimating the repre- interest within trials of cognitive-task performance. Such sentation of varying levels of symptom severity in a times of measurement interest can complement brain clinically treated cohort. With A of Equation 1 standing regions of measurement interest (e.g., a region known as for cognition-related symptom severity, and given per- the encoding-intensive dorsal anterior cingulate cortex). formance samples from a random subsample of indi- In this way, mathematical cognitive models can contribute viduals in a treated cohort, changes in proportions of to the calibration of space-time coordinates of neurophysi- relative severity levels can be estimated and monitored ological measurement (illustrated in Neufeld et al., 2010). repeatedly over time. The procedure loosely resembles Isolating critical times to measure a target process (e.g., one from mathematical ecology, in which the stocks of encoding a presented stimulus) has the advantage of various fish species are estimated using netted samples allowing it to function as it would alongside related pro- taken over the course of a fishing season. In the present cesses involved in executing a cognitive task (e.g., com- case, the moving profile of symptom-severity propor- paring a presented stimulus with other stimuli held in tions addresses the efficacy of the treatment regimen memory). The approach, in other words, allows the target in moving the treated cohort toward more healthy cog- process to be examined as it operates in situ—inside its nitive functioning. Note that inferences at the individual cognitive ecological niche. and cohort levels are both centered on a cognitive, Estimating individual differences in model parame- symptom-related mechanism (e.g., parameterized devi- ters, as described above, also can facilitate the forma- ation in cognitive encoding). Such estimation is of spe- tion of parametrically homogeneous groups. Reducing cial interest, for example, when the administered participant-group heterogeneity potentially achieves treatment is a drug targeting the central nervous system greater statistical power for detecting subtle but key (for elaboration on the mathematical and computational neurophysiological anomalies. specifics, assumptions, and methodological caveats of At a broader level, formal cognitive modeling can pro- the assessment procedure described here, see, e.g., vide a cognitive-functional nexus for integrating observa- Neufeld, 2007a; Neufeld et al., 2002, 2010). tions from functional neurophysiology, investigative 220 Neufeld, Shanahan practice. Current Opinion in Behavioral Sciences, 11, 1–7. settings, and experimental sessions. Ascertaining math- https://doi.org/10.1016/j.cobeha.2016.02.001. A discus- ematically that the cognition at play remains stable across sion of the logistics of implementing mathematical model- different sources of data lends assurance that neuro- ing and computational neuroscience in clinical practice. physiological results converge on a shared set of cogni- Busemeyer, J. R., & Diederich, A. (2010). Cognitive model- tive operations. For example, a common mathematical ing. Sage. A generally accessible presentation of tools model of stimulus encoding, as activated by a widely and applications of mathematical cognitive modeling and used cognitive task (the Stroop task), has been shown model testing, with concrete examples. to apply across different levels of cognitive neurophysi- Neufeld, R. W. J. (2007a). (See References). A discussion ological measurement. Investigations first focused on of the distinctions among mathematical, computer- functional magnetic resonance spectroscopy, used to simulation, and statistical modeling in quantitative clini- examine neurochemical mechanisms accompanying cog- cal science, with emphasis on the singular attributes of mathematical modeling. nitive performance, and then on vascular-signal func- Neufeld, R. W. J., & Cutler, C. D. (2019). (See References). A tional MRI, used to examine the specific neuronal circuits delineation of the nature of clinical mathematical model- involved in performing the cognitive task (Taylor et al., ing and its potential contribution to addressing the issue 2015, 2016, 2017). of replicability of findings in the field of cognitive science. By adopting the strategy portrayed in Figure 1, Shanahan, M. J., Townsend, J. T., & Neufeld, R. W. J. (2015). researchers can identify and target cognitive-processing Mathematical models in clinical psychology. In R. L. deviations, as well as estimate the time course of the Cautin & S. O. Lilienfeld (Eds.), The encyclopedia of deviant processing during trials of an experimental task. clinical psychology (Vol. 3, pp. 594–603). John Wiley. A This time course then can be combined with measured description of the fundamentals of clinical mathematical activation of the brain region or regions apt to be modeling for a general audience. involved in the suspected disorder-related cognitive Treat, T. A., McFall, R. M., Viken, R. J., Kruschke, J. K., Nosofsky, R. M., & Wang, S. S. (2007). Clinical cognitive process. The goal is to uncover abnormality in neuronal science: Applying quantitative models of cognitive pro- operations paralleling abnormality in the targeted cog- cessing to examine cognitive aspects of psychopathology. nition. The combination of cognitive-functional and In R. W. J. Neufeld (Ed.), Advances in clinical cognitive neurophysiological information on a disorder, in turn, science: Formal modeling and assessment of processes can profitably feed into clinical assessment and treat- and symptoms (pp. 179–205). American Psychological ment activities. Association. A demonstration that formally assessed per- ceptual organization related to selected clinical issues permeates formally modeled, clinically significant clas- Concluding Comments sification and memory behaviors. Clinical mathematical psychology stands at the ready to contribute to progress in clinical science and assess- Transparency ment (see also Treat & Viken, 2010). Some readers may Action Editor: Teresa A. Treat be put off by the requisite engagement in analytical Editor: Robert L. Goldstone developments (elaborated on in Neufeld, 2007a). How- Declaration of Conflicting Interests ever, behavioral scientists who took advanced statistics The author(s) declared that there were no conflicts of and design courses as undergraduate and graduate stu- interest with respect to the authorship or the publication dents are often in a strong position to grasp the neces- of this article. sary quantitative tools, possibly with the aid of available tutorials (see Recommended Reading). It is motivating ORCID iD to note that the history of science by and large is replete Richard W. J. Neufeld https://orcid.org/0000-0002-3214- with exemplary advances hinging on decidedly formal theoretical developments (necessary propositions; e.g., Braithwaite, 1968; Harper, 2011). The transparency of Acknowledgments mathematically stated accounts of deviations in cogni- The authors thank Colleen Cutler for her comments on this tive processes, moreover, is intrinsically rewarding. It manuscript and two anonymous reviewers for their suggested also can attest to rigor of developments, if justified, but rephrasing of certain sections. as well can throw any flaws into relief—thereby pro- moting scientific self-correction. Note 1. Bayes’ theorem is a landmark contribution to statistical sci- Recommended Reading ence by the Reverend Thomas Bayes, of Tunbridge Wells, Ahn, W.-Y., & Busemeyer, J. R. (2016). Challenges and England, whose theorem was published in the Proceedings of promises for translating computational tools into clinical the Royal Society in 1763. Current Directions in Psychological Science 31(3) 221 Montague, P. R., Dolan, R. J., Friston, K. J., & Dayan, P. (2012). References Computational psychiatry. Trends in Cognitive Sciences, Ahn, W.-Y., Krawitz, A., Kim, W., Busemeyer, J. R., & Brown, 16(1), 72–80. https://doi.org/10.1016/j.tics.2011.11.018 J. W. (2011). A model-based fMRI analysis with hier- Neufeld, R. W. J. (2007a). 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Formal Innovations in Clinical Cognitive Science and Assessment

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Abstract

Mathematical modeling is increasingly driving progress in clinical cognitive science and assessment. Mathematical modeling is essential for detecting certain effects of psychopathology through comprehensive understanding of telltale cognitive variables, such as workload capacity and efficiency in using capacity, as well as for quantitatively stipulating the subtle but important differences among these variables. The research paradigm guiding this formal clinical science is outlined. A distinctive cognitive abnormality in schizophrenia—taking longer to cognitively represent encountered stimulation—is used as a specific example to illustrate a general quantitative framework for studying intricate phenomena that impair mental health. Developments in mathematical modeling will also benefit symptom description and prediction; provide grounding in cognitive and statistical science for new methods of clinical assessment over time, both for individuals and for treatment regimens; and contribute to refining the cognitive-function side of clinical functional neurophysiology. Keywords clinical mathematical modeling, cognitive assessment, cognition in schizophrenia, formal cognitive neuroimaging, cognitive mixture models Contemporary clinical applications of basic cognitive them through a case study involving the symptom-related science have taken on a new direction involving math- cognitive neuroscience of schizophrenia. ematical modeling of symptom-related cognitive abnor- As illustrated in Figure 1, clinical mathematical cog- malities. The basic research paradigm for this movement nitive neuroscience stands to uniquely contribute to is illustrated in Figure 1. Mathematical models of cogni- mainstream mathematical cognitive neuroscience. It can tive performance among healthy individuals are adjusted do so through model generalization testing (Busemeyer to accommodate deviations from normal performance & Wang, 2000), in which the robustness of a model’s among participants with selected forms of psychopa- performance is evaluated with new experimental para- thology. Such deviations typically center around speed digms or populations. Clinical mathematical cognitive and/or accuracy in performing cognitive tasks. Parts of neuroscience provides key opportunities for generaliza- the model remaining intact after this adjustment are tion testing with respect to extreme individual differ- considered to indicate cognitive functions that are ences, which are associated with psychopathology. spared, and performance deviations that compel modi- Models that readily accommodate performance devia- fication of the model are flagged as signifying disorder- tions are preferred to those that fail to accommodate affected functions. Minimal adjustment of the model is them or that are strained in doing so. That is, a model desired, in the interest of parsimony. In this way, models is supported when observed abnormalities relating to provide a formal framework to determine which cogni- psychopathology can be well predicted without major tive processes do or do not differ between clinical adjustment of the model’s workings. groups and healthy control groups. Such formal theoretical developments can offer mul- Corresponding Author: tiple advantages in explaining and measuring psychopa- Richard W. J. Neufeld, Department of Psychology, Western University thology. This article describes these advantages, illustrating Email: rneufeld@uwo.ca 216 Neufeld, Shanahan Clinical Mathematical Cognitive Mathematical Cognitive Neuroscience Invoking Clinical Neuroscience Invoking Mathematical Modeling and Model Mathematical Models of Adjustment to Accommodate Deviations Experimental Cognitive-Task in Experimental Cognitive-Task Performance Performance Model Generalization Testing Fig. 1. Relations between mathematical cognitive neuroscience and clinical mathematical cognitive neuroscience. The upper arrow indicates the application of mathematical models of normal cognitive-task performance to the understanding of performance changes occurring with psychopathology. The lower arrow indicates that the suc- cess of such application, in turn, bears on the validity of the applied models. Note that clinical mathematical modeling, the subject and then, after a delay, to identify which stimuli in a of this article, involves analytic theorems and proofs, new set were part of the original set. To do well in this as well as algebraic derivations expressing cognitive task, participants must encode stimuli presented in the transactions relevant to clinical disorders. It should be second set into a cognitive format facilitating compari- distinguished from a somewhat related type of model- son with the previously memorized set of stimuli. For ing known as computational psychiatry. Computational example, in the case of basic visual-template matching, psychiatry addresses how cognitive transactions might it may be necessary to cognitively extract the physical be realized at the level of neural organization and oper- features of a presented stimulus, such as its curves, ations. By and large, computational psychiatry dele- lines, and intersections; or in the case of stimulus-name gates disorder-related deviations of neural functioning matching, it may be necessary to tag a presented digit to computer simulation of neural networks and neuro- or letter stimulus with its name, for comparison with dynamics (communication between neurons). (Accounts names of the stimuli in the previously memorized set. and examples of computational psychiatry are available Mathematical modeling enables quantitative dissec- in Montague et  al., 2012; Huys et  al., 2016; Wang & tion of cognitive processes, to help pinpoint specific Krystal, 2014; and Grossberg, 1999. For a rigorous, com- sources of deviation in cognitive performance. With prehensive treatment of artificial neural networks, see respect to encoding, for example, the processes can be Golden, 1996.) Computational psychiatry and clinical broken down as follows. First, the overall process is mathematical modeling potentially are complementary made up of constituent encoding operations—encoding when it comes to quantitative accounts of clinical dis- subprocesses, such as registration of curves, lines, and orders (e.g., Carter & Neufeld, 1999, 2007; extensive intersections of a presented stimulus. Second, the elaboration on distinctions among alternate approaches encoding subprocesses take place at a certain rate, to modeling clinical phenomena can be found in known as subprocess-level cognitive-workload capacity Neufeld, 2007a). (e.g., Neufeld et al., 2007; Wenger & Townsend, 2000). Application of the model-adjustment operation of Figure 1 has repeatedly shown that subprocess-level Case Study: Cognitive Neuroscience of cognitive-workload capacity remains intact in schizo- Stimulus Encoding in Schizophrenia phrenia, but the number of encoding subprocesses Schizophrenia affects approximately 0.5% of the North undertaken is elevated. In other words, cognitive-work- American population. Symptoms can take the form of load capacity escapes impairment, whereas efficiency delusions and hallucinations (thought-content disorder), of its implementation does not. This combination of incoherent speech, reduced cognitive efficiency, and spared and affected components of encoding perfor- impoverished motivation. Research studies applying the mance is analogous to a racehorse striding at a normal research strategy depicted in Figure 1 have shown that pace but closer to the outside rail, which increases the delayed completion of stimulus encoding is a deviation requisite number of paces, and therefore the time in cognitive performance recurrently found in schizo- needed, to complete the course. This combination illus- phrenia, across multiple experimental tasks and levels trates the nature of the model adjustment referred to of patient status (e.g., first episode, never treated, out- in Figure 1: The altered model conforms to the specific patient, inpatient). In this case, stimulus encoding refers pattern of empirical deviations among clinical partici- to cognitively preparing and transforming cognitive- pants (in this case, people with schizophrenia). task stimuli into a format facilitating collateral pro- The studies just mentioned have tracked the abnor- cesses. For instance, participants might be asked to mality in encoding to a specific formalized property—the memorize a set of novel stimuli (e.g., “TZAM,” “CEYP”) subprocess-number parameter of the stimulus-encoding Current Directions in Psychological Science 31(3) 217 process. Identification of such an abnormality stands to response on cognitive-task trials). The methods used open the way to potentially important advances in this include maximum likelihood, distribution-moment domain of psychological clinical science. The abnormal- matching (e.g., Evans et al., 2000), and Bayesian param- ity arguably represents a critical deficit that compromises eter estimation (e.g., Alexandrowicz & Gula, 2020, who activities relying on timely stimulus encoding (e.g., per- applied this method, along with a mathematical model forming daily self-maintenance and meeting environ- of decision and choice, to clinical disorders). Clinically mental stresses and demands). The quantitative apparatus relevant cognitive processing concealed in raw data can in which this property is embedded provides for certain be revealed via mathematical modeling. methodological benefits, including theory-guided mea- Often it may not be reasonable to assume that all surement and clinical assessment, and stipulation of the individuals within a clinical group have (roughly) the cognitive neurophysiological processes taking place. The same level of cognitive processing, as indexed, for abnormality, as quantitatively defined, also can be shown example, by a fixed value for a model parameter. For- to be potentially symptom related, notably with respect tunately, mathematical models can be expanded to to thought-content disorder (delusions and thematic hal- account for this, notably through expansion as mixture lucinations). Such symptomatology is considered to ema- models. Mixture models treat the overall performance nate from failure to encode specifically context-related of a group as a mixture of different levels of perfor- features of a stimulus complex during episodes of infor- mance among individual group members (e.g., Carter mation intake. When the influence of reality-grounding, et al., 1998; Cutler & Neufeld, 2017). objectifying cues is weakened, other information that Mixing distributions can be important not only successfully is taken in during an episode is open to because they make it possible to systematically accom- false interpretation (this mechanism of symptom produc- modate individual differences, but also because the tion is expanded upon quantitatively in Neufeld, 2007b, parameters that mathematically govern the random dis- and Neufeld et al., 2010). This formal theoretical account tributions of model properties (mixing-distribution is in the spirit of the current clinical-science trend hyperparameters) can be clinically meaningful in their toward determining underlying mechanisms of complex own right. For example, mixture-model hyperparame- behaviors, in this case mathematically. It accords, more- ters can convey a particular group’s general level of over, with the currently prominent Research Domain facility with undertaking the elements of the cognitive Criteria initiative (e.g., Kozak & Cuthbert, 2016), inas- process at hand (e.g., encoding subprocesses); they can much as the identified mechanism evidently extends to also be used to indicate susceptibility of this facility to other forms of clinical disturbance (e.g., major depres- impairment during psychological stress (for concrete sive disorder; Taylor et al., 2016), and nonclinical popu- examples, see Neufeld, 2016). lations (Nicholson & Neufeld, 1993). In short, mixture-model expansions, illustrated in Note that model adjustment capturing changes in Figure 2, can increase the span of what a model explains cognition associated with clinical disorder typically by incorporating individual differences. They addition- takes the form of altering the values of model param- ally can tap clinically meaningful constructs, such as eters, such as the number of encoding subprocesses or cognitive-task facility and vulnerability of performance the rate at which subprocesses are completed. Model to psychological stress. architecture (in terms of the number of model param- eters involved or their arrangement in relation to each Measuring better with Bayes other) ordinarily is common to clinical and nonclinical groups alike (see, e.g., Neufeld & Broga, 1981; Wallsten Mixture models allow for the likelihood that individuals et al., 2005). In other words, the basic mental apparatus systematically differ in properties of mathematically meeting a cognitive challenge is common across groups, expressed cognitive performance. They go an important but modification of one or more of its parts (parame- step further, in providing for efficient estimation of ters) accompanies clinical disorder. model properties for the individual. They do so by cus- tomizing the properties to the person, through Bayesian statistical methodology, as follows. Bayes’ theorem, Measurement and Clinical Assessment appropriated to the present context, states that Guided by Formal Theory Pr () APrA ({∗}| ) Samples of individuals’ cognitive performance, for PA r( |{∗}) = , (1) instance, on encoding-intensive tasks, permit estimation Pr () ∗ {} of cognitive-process parameter values. Such estimation is accomplished by applying established methods to where Pr(A|{*}) is the Bayesian probability that a pre- empirical performance data (e.g., observed latencies of dicted entity, such as a cognitive-process parameter 218 Neufeld, Shanahan Mixing Distribution Expressing Mixing Distribution Expressing Mixing Distribution Expressing Individual Differences in the Individual Differences in Encoding- Individual Differences in the Number of Subprocesses: Subprocess Cognitive-Workload Number of Subprocesses: Lower Values Among Healthy Capacity (Rate of Encoding- Higher Values Among Control Participants Subprocess Completion): Common Schizophrenia Participants to Control and Schizophrenia Participants Model of Latencies in Stimulus Encoding, a Function of the Encoding-Subprocess Rate and the Number of Encoding Subprocesses Fig. 2. Design of a stimulus-encoding mixture model accommodating individual differences in parameters of encoding. The rate at which encoding subprocesses unfold is shared by control and schizophrenia participants, whereas the number of encoding subprocesses is greater among schizophrenia than control participants. (e.g., number of encoding subprocesses), has the value Estimates are solidified by feeding into their calculation A given relevant observations {*} (e.g., a sample of specifically that information supplied by a preestab- cognitive performance); Pr ({*}| A) is the likelihood of lished referent, the Bayesian prior represented by the Pr () A the performance sample, given the value A; is mixing distribution. the probability of the value under consideration accord- The operation of mixing-distribution Bayesian priors ing to the relevant mixing distribution, temporarily dis- can help alleviate the problem of small-sample math- regarding the empirical observations {*}; and Pr ({*}) is ematical modeling, which is ubiquitous in applied set- the probability of the observations, all candidate values tings. The approach allows researchers to work with of the predicted entity considered (for accounts of small cognitive-performance sample sizes, which is Bayesian modeling generally, see classic works such as particularly helpful when undertaking person-specific Berger, 1985, and O’Hagan & Forster, 2004). modeling for assessment or research purposes. Valid With Bayes’ theorem and a person’s cognitive- mixing distributions sharpen the estimation of model performance sample in hand, a versatile estimation of properties for the individual. They help compensate individual attributes of clinical interest, based on cogni- for small performance samples by bringing into play tive and statistical science, is possible. Predicted entities performance-relevant information about the group to A actually can be diverse, including, for example, the which the individual at hand belongs. Again, this infor- parameter expressing the number of encoding subpro- mation is conveyed by the mixing distribution that cesses or the symptomatology to which the mathemati- quantifies the relative frequency of the target of predic- cal model and estimated model parameters relate (e.g., tion (e.g., a model-parameter value) in a membership severity of thought-content disorder; for further discus- group. Such a scenario resembles what takes place in sion, see the following section on dynamic assessment a hematology laboratory, where a substantial extant of treatment efficacy). Bayesian estimation stabilizes bank of hematological information, applicable collec- estimated values through the anchoring effects of mix- tively, is brought to bear individually on a modest blood ing distributions, which act as Bayesian priors (e.g., sample from the person at hand. Pr () A , in Equation 1). Variance in estimates (statistical Moreover, dynamic assessment of changes in clinical inefficiency) thus is reduced through the quantitative condition is possible through undertaking Bayesian esti- mechanism formally known as Bayesian shrinkage. mation at designated times of clinical interest (e.g., after Current Directions in Psychological Science 31(3) 219 a selected bout of treatment). Specifically, Equation 1 Implications for Clinical Functional can be used to track changes in the status of symptom- Neurophysiology related (e.g., thought-content symptomatology) cognitive- Mathematical modeling of the cognitive side of vascular model parameters (e.g., number of stimulus-encoding and electrophysiological cognitive neurophysiology subprocesses, which the model-adjustment operation of conveys several methodological assets. Formally anchor- Fig. 1 has identified as inflated in schizophrenia). ing cognitive functions in a viable mathematical model Changes can be monitored as they occur over the natu- is an antidote to a thorny problem in cognitive neuro- ral passage of time, over the course of treatment, or physiology known as reverse inference (Poldrack, 2011). subsequent to an experimental manipulation. In these This problem consists of circularly relying on measured ways, the described formal methodology can be an neurophysiological signals (obtained with fMRI, func- important constituent in the arsenal of clinical assess- tional magnetic resonance spectroscopy, magnetoen- ment. Mixing-distribution Bayesian priors have replaced cephalography, and electroencephalography) to infer the usual population-based norms of multi-item psycho- the cognitive functions whose very neuronal substrates metric inventories. purportedly are being charted. This inferential dilemma Bayesian individualization of model properties also in principle can be overcome as follows. The cognitive allows for evaluation of model performance at the person- functions at work while neurophysiological measure- specific level. Doing so ascertains a model’s validity for ments are taken are quantitatively stipulated in advance, an individual participant; it also affords strong tests of anchored in a formal representation (e.g., Ahn et  al., overall model performance. Fit of model predictions to 2011; White et  al., 2012). That is, cognitive functions empirical observations at both the group and the indi- whose neurophysiological substrates are being exam- vidual levels is an added means of model evaluation. ined are staked out in terms of a quantitative model— This unique form of model evaluation potentially bears one that is a priori freestanding, independent of the on the currently prominent issue of robustness of find- examined neurophysiological activity itself. ings in cognitive modeling (Neufeld & Cutler, 2019). Note, further, that dynamic models of cognitive opera- tions treat the development of cognitive processes as sto- Dynamic assessment of treatment- chastic functions of time (Townsend & Ashby, 1983). The regimen efficacy unfolding of target processes, such as stimulus encoding, With a modest expansion of Equation 1, the present can be overlaid on monitored neurophysiological signals, assessment methodology naturally extends beyond the to produce times of neurophysiological measurement individual; it can be applied to estimating the repre- interest within trials of cognitive-task performance. Such sentation of varying levels of symptom severity in a times of measurement interest can complement brain clinically treated cohort. With A of Equation 1 standing regions of measurement interest (e.g., a region known as for cognition-related symptom severity, and given per- the encoding-intensive dorsal anterior cingulate cortex). formance samples from a random subsample of indi- In this way, mathematical cognitive models can contribute viduals in a treated cohort, changes in proportions of to the calibration of space-time coordinates of neurophysi- relative severity levels can be estimated and monitored ological measurement (illustrated in Neufeld et al., 2010). repeatedly over time. The procedure loosely resembles Isolating critical times to measure a target process (e.g., one from mathematical ecology, in which the stocks of encoding a presented stimulus) has the advantage of various fish species are estimated using netted samples allowing it to function as it would alongside related pro- taken over the course of a fishing season. In the present cesses involved in executing a cognitive task (e.g., com- case, the moving profile of symptom-severity propor- paring a presented stimulus with other stimuli held in tions addresses the efficacy of the treatment regimen memory). The approach, in other words, allows the target in moving the treated cohort toward more healthy cog- process to be examined as it operates in situ—inside its nitive functioning. Note that inferences at the individual cognitive ecological niche. and cohort levels are both centered on a cognitive, Estimating individual differences in model parame- symptom-related mechanism (e.g., parameterized devi- ters, as described above, also can facilitate the forma- ation in cognitive encoding). Such estimation is of spe- tion of parametrically homogeneous groups. Reducing cial interest, for example, when the administered participant-group heterogeneity potentially achieves treatment is a drug targeting the central nervous system greater statistical power for detecting subtle but key (for elaboration on the mathematical and computational neurophysiological anomalies. specifics, assumptions, and methodological caveats of At a broader level, formal cognitive modeling can pro- the assessment procedure described here, see, e.g., vide a cognitive-functional nexus for integrating observa- Neufeld, 2007a; Neufeld et al., 2002, 2010). tions from functional neurophysiology, investigative 220 Neufeld, Shanahan practice. Current Opinion in Behavioral Sciences, 11, 1–7. settings, and experimental sessions. Ascertaining math- https://doi.org/10.1016/j.cobeha.2016.02.001. A discus- ematically that the cognition at play remains stable across sion of the logistics of implementing mathematical model- different sources of data lends assurance that neuro- ing and computational neuroscience in clinical practice. physiological results converge on a shared set of cogni- Busemeyer, J. R., & Diederich, A. (2010). Cognitive model- tive operations. For example, a common mathematical ing. Sage. A generally accessible presentation of tools model of stimulus encoding, as activated by a widely and applications of mathematical cognitive modeling and used cognitive task (the Stroop task), has been shown model testing, with concrete examples. to apply across different levels of cognitive neurophysi- Neufeld, R. W. J. (2007a). (See References). A discussion ological measurement. Investigations first focused on of the distinctions among mathematical, computer- functional magnetic resonance spectroscopy, used to simulation, and statistical modeling in quantitative clini- examine neurochemical mechanisms accompanying cog- cal science, with emphasis on the singular attributes of mathematical modeling. nitive performance, and then on vascular-signal func- Neufeld, R. W. J., & Cutler, C. D. (2019). (See References). A tional MRI, used to examine the specific neuronal circuits delineation of the nature of clinical mathematical model- involved in performing the cognitive task (Taylor et al., ing and its potential contribution to addressing the issue 2015, 2016, 2017). of replicability of findings in the field of cognitive science. By adopting the strategy portrayed in Figure 1, Shanahan, M. J., Townsend, J. T., & Neufeld, R. W. J. (2015). researchers can identify and target cognitive-processing Mathematical models in clinical psychology. In R. L. deviations, as well as estimate the time course of the Cautin & S. O. Lilienfeld (Eds.), The encyclopedia of deviant processing during trials of an experimental task. clinical psychology (Vol. 3, pp. 594–603). John Wiley. A This time course then can be combined with measured description of the fundamentals of clinical mathematical activation of the brain region or regions apt to be modeling for a general audience. involved in the suspected disorder-related cognitive Treat, T. A., McFall, R. M., Viken, R. J., Kruschke, J. K., Nosofsky, R. M., & Wang, S. S. (2007). Clinical cognitive process. The goal is to uncover abnormality in neuronal science: Applying quantitative models of cognitive pro- operations paralleling abnormality in the targeted cog- cessing to examine cognitive aspects of psychopathology. nition. The combination of cognitive-functional and In R. W. J. Neufeld (Ed.), Advances in clinical cognitive neurophysiological information on a disorder, in turn, science: Formal modeling and assessment of processes can profitably feed into clinical assessment and treat- and symptoms (pp. 179–205). American Psychological ment activities. Association. A demonstration that formally assessed per- ceptual organization related to selected clinical issues permeates formally modeled, clinically significant clas- Concluding Comments sification and memory behaviors. Clinical mathematical psychology stands at the ready to contribute to progress in clinical science and assess- Transparency ment (see also Treat & Viken, 2010). Some readers may Action Editor: Teresa A. Treat be put off by the requisite engagement in analytical Editor: Robert L. Goldstone developments (elaborated on in Neufeld, 2007a). How- Declaration of Conflicting Interests ever, behavioral scientists who took advanced statistics The author(s) declared that there were no conflicts of and design courses as undergraduate and graduate stu- interest with respect to the authorship or the publication dents are often in a strong position to grasp the neces- of this article. sary quantitative tools, possibly with the aid of available tutorials (see Recommended Reading). It is motivating ORCID iD to note that the history of science by and large is replete Richard W. J. Neufeld https://orcid.org/0000-0002-3214- with exemplary advances hinging on decidedly formal theoretical developments (necessary propositions; e.g., Braithwaite, 1968; Harper, 2011). The transparency of Acknowledgments mathematically stated accounts of deviations in cogni- The authors thank Colleen Cutler for her comments on this tive processes, moreover, is intrinsically rewarding. It manuscript and two anonymous reviewers for their suggested also can attest to rigor of developments, if justified, but rephrasing of certain sections. as well can throw any flaws into relief—thereby pro- moting scientific self-correction. Note 1. Bayes’ theorem is a landmark contribution to statistical sci- Recommended Reading ence by the Reverend Thomas Bayes, of Tunbridge Wells, Ahn, W.-Y., & Busemeyer, J. R. (2016). Challenges and England, whose theorem was published in the Proceedings of promises for translating computational tools into clinical the Royal Society in 1763. Current Directions in Psychological Science 31(3) 221 Montague, P. R., Dolan, R. J., Friston, K. J., & Dayan, P. (2012). References Computational psychiatry. Trends in Cognitive Sciences, Ahn, W.-Y., Krawitz, A., Kim, W., Busemeyer, J. R., & Brown, 16(1), 72–80. https://doi.org/10.1016/j.tics.2011.11.018 J. W. (2011). A model-based fMRI analysis with hier- Neufeld, R. W. J. (2007a). 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Journal

Current Directions in Psychological Science – SAGE

Published: Jun 1, 2022

Keywords: clinical mathematical modeling; cognitive assessment; cognition in schizophrenia; formal cognitive neuroimaging; cognitive mixture models

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