A Generative (Dynamic Causal) Model

This example of a generative model is deliberately very simple, restricting itself to modeling a limited differential diagnosis that includes schizophrenia, affective disorder, schizoaffective disorder, and a remitted state. These diagnostic outcomes are accompanied by six symptom scores that have been normalized to lie between plus and minus one. Furthermore, we consider only a limited number of exogenous causes and latent variables: Specifically, a single therapeutic cause perturbs the evolution of three physiological states, from which two psychopathological states are derived. At any one time, the location in the psychopathological state-space determines a symptom profile (through a linear mixture of psychopathological states that is passed through a sigmoid function). Diagnostic outcomes are, here, encoded by a probability profile over the differential diagnosis (i.e., the relative confidence that a clinician places in the differential diagnoses afforded by some accepted nosological scheme; e.g., DSM or ICD). In this model, the probability of any diagnosis corresponds to its relative proximity to the current point in a psychopathological state-space. In other words, as the disorder progresses, a trajectory is traced out in a two-dimensional psychopathological state-space, where, at any one time, the prevalent diagnosis is determined by the diagnosis point the current state is closest to. Technically, this has been modeled by a softmax function of diagnostic potential, which is the (negative) Euclidean distance between the current state and the locations associated with the different diagnostic categories (encoded by ${\theta }_i^{\Delta }$—the color dots in Figure 1).

The trajectory of psychopathology is determined by the corresponding trajectory through a pathophysiological state-space that has its own dynamics. These are encoded by equations of motion or flow—based, in this example, on a Lorenz attractor (Lorenz, 1963). See van de Leemput et al. (2014) for an example of modeling mood using a (simplified) Lotka–Volterra system. The Lorenz form for the dynamics or kinetics of pathophysiology is a somewhat arbitrary choice, but it provides a ubiquitous model of chaotic dynamics in the physical (Poland, 1993) and biological (de Boer & Perelson, 1991) sciences. Interestingly, it arose from the modeling of convection dynamics, which speaks to the analogy between the present modeling proposal and weather forecasting. In this setting, the ensuing dynamics can be regarded as a canonical form for nonlinear coupled processes that might underlie the pathophysiology in psychosis. It has a canonical form because, as we see in Figure 1 (on the right), one can regard the parameters as specifying coupling coefficients or connections that mediate the influence of one physiological state on the others. Crucially, some of these connections are state-dependent. This is important because it means that one can model fluctuations in pathophysiology in terms of self-organized (chaotic) dynamics that have an underlying attracting set. In other words, the Lorenz form provides a canonical summary of slow fluctuations in neuronal (or hormonal) states that show homoeostatic or allostatic tendencies (e.g., Leyton & Vezina, 2014; Misiak, Frydecka, Zawadzki, Krefft, & Kiejna, 2014; Oglodek, Szota, Just, Mos, & Araszkiewicz, 2014; Pettorruso et al., 2014).

The equations on the right of Figure 1 all include random fluctuations. These fluctuations render the generative model a probabilistic statement about how a number of variables could influence each other. Here, the random fluctuations can be regarded as observation noise (when assessing symptoms) and as random fluctuations or perturbations to psychological or physiological processing. These fluctuations stand in for transient changes in daily experiences and neurophysiological processes that are not predicted by the slower dynamics of psychological and physiological states (modeled by the equations of motion). In this article, these random fluctuations are smooth processes with a Gaussian correlation function and a correlation length of half an assessment interval (i.e., a few days).

We make no pretense that any of these states map in a simple way onto the physiological variables; rather, they stand in for mixtures¹ of physiological variables that have relatively simple dynamics. The existence of mixtures is assured by technical theorems such as the center manifold theorem and the slaving principle in physics (Carr, 1981; Davis, 2006; Frank, 2004; Haken, 1983). Crucially, we have constructed this generative model such that the therapeutic intervention changes the state-dependent coupling between the first and second pathophysiological states (in fluid dynamics, this control parameter is known as a Rayleigh number and reflects the degree of turbulent flow). This means that we can regard the intervention as a pharmacotherapy that changes the coupling between different neuronal (or hormonal) systems—for instance, the influence of an atypical antipsychotic (Hrdlicka & Dudova, 2015) on the dopaminergic and 5HT receptor function responsible for both monoaminergic tone in the ventral striatum and serotoninergic projections from the amygdala to the paraventricular nucleus (Muzerelle, Scotto-Lomassese, Bernard, Soiza-Reilly, & Gaspar, 2014; Wieland et al., 2015). Furthermore, we have introduced a parameter ${\theta }_3^p$ that determines the sensitivity to the intervention, which may be important in determining a patient’s responsiveness to therapy (Brennan, 2014). Note that although we have portrayed the intervention as therapeutic, we could have treated it as some known change in life circumstances—for example, a life event (Kendler & Karkowski-Shuman, 1997) that has a persistent effect on neuroendocrine function or neurophysiology.

The middle panels of Figure 1 provide an illustration of how a patient might present over time under this particular model. Imagine that we wanted to model six symptom scores and a probabilistic differential diagnosis over four diagnoses (schizophrenia, schizoaffective, affective, and remitted) when assessing an outpatient on a weekly basis for 64 weeks. In this illustrative example, the symptom scores could correspond to (normalized) clinical ratings from the PANSS (Kay, Fiszbein, & Opler, 1987) and the Beck Depression Inventory:

• •   Delusions [PANSS: minimum score = 1, maximum score = 7]
• •   Conceptual disorganization [PANSS: minimum score = 1, maximum score = 7]
• •   Blunted affect [PANSS: minimum score = 1, maximum score = 7]
• •   Emotional withdrawal [PANSS: minimum score = 1, maximum score = 7]
• •   Anxiety [PANSS: minimum score = 1, maximum score = 7]
• •   Beck Depression Inventory [BDI: minimum score = 0, maximum score = 63]

with diagnoses according to the ICD-10 Classification of Mental and Behavioural Disorders: Diagnostic Criteria for Research (World Health Organization, 1993):

• •   F20 Schizophrenia
• •   F25 Schizoaffective disorder
• •   F32 Depressive episode
• •   No symptoms

A therapeutic intervention—say, an atypical antipsychotic—is introduced at 32 weeks, and we want to model the response. This therapeutic input is shown in the lower left panel as a dot-dashed line, and it affects the evolution of physiological states according to the equations of motion on the right. These equations generate chaotic fluctuations in (three) pathological states, shown at lower right in the middle group of panels. Two of these states are then mixed to produce a trajectory in a psychopathological state-space. This trajectory is shown in the center panels as a function of time (left panel) and as a trajectory in state-space (right panel). In other words, the trajectory (red line) is defined by plotting the two psychopathological variables against each other. In turn, the psychopathology generates symptom scores (shown as colored lines at the upper left) and diagnostic probabilities (shown at the upper right). The relationship between the continuous (dimensional) latent space of psychopathology and the (categorical) differential diagnosis is determined by diagnostic parameters ${\theta }_i^{\Delta }$, defining the characteristic location of the ith diagnosis.

These locations are shown as dots in the state-space, where the blue dot corresponds to a diagnosis of remission, and the green, cyan, and red locations correspond to diagnoses of schizophrenia and of schizoaffective and affective disorders, respectively. One can see that initial oscillations between schizophrenia and schizoaffective diagnoses are subverted by the therapeutic intervention. At this point, the latent pathophysiology is drawn to its (point) attractor at zero, the most likely diagnosis becomes remission, and the symptom scores regress to their normal values of zero. In short, this models a successful intervention in a pathophysiological process that shows chaotic oscillations expressed in terms of fluctuating symptoms and differential diagnosis. We will see later that—in the absence of therapy—these chaotic oscillations would otherwise produce a relapsing–remitting progression with an ambiguous diagnosis that fluctuates between schizophrenic and schizoaffective. This intervention is formally similar to what is known anecdotally as chaos control (e.g., Rose, 2014). This example suggests that the goal of therapy is not so much countering the pathological deviations as the more subtle problem of suppressing chaotic or turbulent neurohormonal processes that are equipped with many self-organizing feedback mechanisms. Heuristically, the role of the clinician becomes like the captain of a ship who uses prevailing winds to navigate toward calmer waters.

This particular example is not meant to be definitive or valid in any sense. It is just one of a universe of potential models (or hypotheses) about the way that psychiatric morbidity is generated. We will return to procedures for comparing models in the next section. However, this example allows us to make a few key points about the nature of pathology and its expression. First, in any generative model of psychopathology there is a fundamental distinction between (time-invariant) parameters and (time-sensitive) states. This distinction can be regarded as the formal homologue of the distinction between trait and state abnormalities. For example, the patient illustrated above had a particular set of parameters ${\theta }_i^p$ determining the family of trajectories (and their attracting sets) of pathophysiology. However, simply knowing these parameters does not tell us anything about the patient’s pathological state at a particular time. To determine this, we have to infer the latent pathophysiology in terms of the current state x(t), by using model fitting or inversion. This presents a difficult (but solvable) problem, because we have to estimate both the parameters (traits) and states of a patient in order to determine that patient’s trajectory in the short term.

The second distinction this sort of model brings to the table is between parameters that are patient-specific and parameters that are conserved over the population to which the model applies. In statistical terms, this corresponds to the difference between random and fixed effects, in which patient-specific effects model random variations in traits that may reflect predisposing factors (e.g., genetic predisposition). Conversely, other parameters may be fixed over patients and determine the canonical form of nosology.

In the example above, this distinction is illustrated by the difference between parameters that are specific to each patient or pathophysiology, θ_i^p, and those that are inherent in the nosology, ${\theta }_i^n$. The nosological parameters define a generic mapping from pathophysiology to psychopathology that is conserved over patients. Understanding this distinction is important practically, because nosological parameters can only be estimated from group data. We will see examples of this process in the next section.

Model Inversion and Bayesian Filtering

The problem of estimating unknown parameters and latent states from time-series data is known as deconvolution or filtering in the modeling literature. Because we have to estimate both parameters and states, this presents a dual estimation problem that is usually accom modated by treating parameters as very slowly fluctuating states. We will illustrate Bayesian filtering using an established procedure called dynamic expectation maximization(DEM). DEM was originally devised to infer latent neuronal states and the connectivity parameters generating neurophysiological signals in distributed brain networks, and it has been applied in a number of different contexts (Friston, Trujillo-Barreto, & Daunizeau, 2008). Special cases of DEM include Kalman filtering (when the states are known and the state-space model is linear).

Figure 2 shows the results of Bayesian filtering when applied to the symptom and diagnostic time series shown in the previous figure. The format is similar to the middle panels of Figure 1; however, here the colored lines correspond not to the true values generating the data, but to the estimated trajectories based on Bayesian filtering (as implemented with the MATLAB routine spm_DEM.m). In this example, we simulated a clinical progression in the absence of any therapeutic intervention (as shown by the flat line in the lower left panel). In the absence of any check on the pathophysiology, chaotic oscillations of slowly increasing amplitude emerge over a period of 64 weeks (middle right panel). The resulting fluctuations in psychopathology are shown as a function of time and as a trajectory in state-space (middle panels). In the lower right panel, the posterior expectations (the most likely trajectories) of the pathophysiological states are contained within 90% Bayesian confidence intervals (gray areas). Notice that the first pathophysiological state has greater confidence intervals and is therefore estimated with slightly less precision than the remaining two states. Interestingly, the uncertainty about the first pathophysiological state itself fluctuates, peaking at certain points in the cycle (due to the nonlinearities in the generative model). In this example, the true (not shown) and estimated values were almost identical. This is because we used very low levels of random fluctuations, modeling random effects at the levels of outcomes, psychopathology, and pathophysiology (see Figure 1).

The upper panels show the resulting fluctuations in symptom and diagnostic scores as a function of time in graphical format (upper left panel) and in image format (upper right panel). One can see clearly that the differential diagnoses of schizophrenia and schizoaffective disorder vary every few months, reflecting an unstable and ambiguous diagnostic picture. We then repeated the filtering, but with a therapeutic intervention at 32 weeks. The simulated response and inferred latent states are shown in Figure 3. These reproduce the results of Figure 1 and show the success of the intervention—as indicated by the emergence of a remitted diagnosis as time progresses (solid blue line on the upper left and cyan circle on the upper right).

In these illustrations, we estimated both the unknown states generating (simulated) clinical data and the patient-specific (trait) parameters governing pathophysiological dynamics. The estimated and true parameters are shown in the upper left panel of Figure 4: The estimated values are shown as gray bars (with pink 90% confidence intervals), and the true values are shown in black. The accuracy of these estimates is self-evident, with a slight overconfidence that is characteristic of the approximate Bayesian inference implicit in DEM (MacKay, 1995). Although these estimates show that, in principle, one can recover the traits and states of a particular subject at a particular time, we used the true values of the nosological parameters coupling pathophysiology to psychopathology (and generating clinical outcomes) during model inversion. One might ask, can these parameters also be estimated?

To illustrate this estimation, the upper right panel of Figure 4 reports the estimates (and 90% confidence intervals) based on an empirical Bayesiananalysis (Kass & Steffey, 1989) of eight simulated patients (using spm_dcm_peb.m). Empirical Bayesian analysis refers to the hierarchical modeling of within- and between-subjects effects that may or may not be treated as random effects. In other words, empirical Bayes allows one to extend dynamic causal modeling of the subject-specific time series into group studies (Friston et al., 2016). Again, the (group mean) estimates are remarkably accurate, suggesting that, in principle, it is possible to recover parameters that are conserved over subjects. Note that the third nosological parameter has an estimated value of zero. This is important, because it leads us into the realm of Bayesian model comparison and evidence-based hypothesis testing.

Bayesian Model Comparison and Hypothesis Testing

Above, we have seen that this sort of model can, in principle, be inverted such that underlying (latent) psychopathological and pathophysiological states can be inferred, in the context of (unknown) subject-specific parameters or traits. However, this does not mean that the model itself has any validity or will generalize to real clinical data. In other words, how do we know we have a good model?

This is a question of model comparison. In short, the best model provides an accurate explanation for the data with minimum complexity. The model evidence reflects this, because a model’s evidence is equal to accuracy minus complexity. The complexity term is important, because it ensures that models do not overfit the data—and will therefore generalize to new data. The model evidence is evaluated by marginalizing (i.e., averaging over) unknown parameters and states to provide the probability of some data under a particular model. The model here is defined in terms of the number of states (and parameters) and how they depend upon each other. A simple example of model comparison is provided in Figure 4 (lower panels).

In our model, the physiological states could influence psychopathology in a number of ways. Our model presupposes two physiological states that can influence two pathophysiological states, creating four possible dependencies that may or may not exist. This leads to 2⁴ = 16 models that cover all combinations of the nosological parameters (see the lower left panel of Figure 4). We can evaluate the evidence for each of these 16 models by inverting all 16 and evaluating the evidence, or, as illustrated here, inverting the model with all four parameters in place and computing the evidence of all the reduced models with one or more parameters missing. This is known as Bayesian model reduction, which is an efficient way of performing Bayesian model comparison (Friston & Penny, 2011). The results of this model comparison are shown in the lower right panel of Figure 4 and suggest that the posterior probability is much greater for Model 14 than for any of the others. In this model, the influence of the second pathophysiological state on the first psychopathological state has been removed. Removing this parameter reduces the model complexity without any loss in accuracy, and therefore increases the model evidence. We might have guessed that this was the case by inspecting the posterior density of the third nosological parameter, which mediates this model component (see the upper right panel).

This is a rather trivial example of model comparison, but it illustrates an important aspect of dynamic causal modeling: namely, the ability to test and compare different models or hypotheses. Although it is not illustrated here, one can imagine comparing models with different numbers of pathophysiological states and different forms of dynamics. One could even imagine comparing models with different graphical structures. One interesting example here would be the modeling of psychotherapeutic interventions that might influence pathophysiology through experience-dependent plasticity. This would necessitate comparing models in which a therapeutic intervention influenced psychopathology, which couples back to pathophysiology through the parameters of its dynamics. This is illustrated by the dotted arrows at the left of Figure 1.

We could speculate about many other examples: Crucial examples here involve an increasingly mechanistic interpretation of pathophysiology, in which pathophysiological states could be mapped onto neurotransmitter systems through careful (generative) modeling of electrophysiological and psychophysical measurements (Stephan & Mathys, 2014). One could also contemplate comparing models with different sorts of inputs or causes, ranging from social or environmental perturbations (e.g., traumatic events) to genetic factors (or their proxies, such as family history). Questions about whether and where genetic polymorphisms affect pathophysiology are formalized by simply comparing different generative models that accommodate effects on different states or parameters. For example, do models that include genetic biases on physiological parameters have greater evidence than models that do not?

The potential importance of model comparison should not be underestimated. We have tried to give a flavor of its potential. It is also worth noting that this field is an area of active research, with fast and improved schemes for scoring large model spaces being developed year by year (see also Viceconti, Hunter, & Hose, 2015). One can construe an exploration of model space as a greedy search through a field of competing hypotheses and a formal statement of the scientific process. This may be especially relevant for psychiatry, which deals with the special problem of integrating both physiological and psychological therapies—calling for generative models that map between these two levels of description. In the final section, we turn to the more pragmatic issue of predicting a response to treatment for an individual patient.

Relationship to Existing Frameworks

One might ask why dynamical systems predominate in our treatment. The answer rests upon the fact that neuronal dynamics, physiology, neuroendocrinology, and their behavioral (and autonomic) correlates are dynamic processes that can only be modeled with a dynamic or state-space model. In other words, clinical outcomes are generated by dynamic processes that necessarily call for a dynamic causal model. Any generative model that does not entail dynamics will, at best, only describe clusters (and associations) of symptoms (and signs). An emphasis on dynamic models is reflected in the modeling of infectious diseases (Paynter, 2016; White et al., 2007) and of immunological responses. Interestingly, these initiatives also have to contend with the balance between model complexity and accuracy (Thakar, Poss, Albert, Long, & Zhang, 2010)—supported by Bayesian model selection—and indeed with the very nature of causality (Vineis & Kriebel, 2006)—supported by dynamic causal models (Valdes-Sosa, Roebroeck, Daunizeau, & Friston, 2011).

It is interesting to contrast the computational nosology described here with the network analysis proposed in Borsboom and Cramer (2013). There are clear points of overlap, as well as some fundamental differences. Borsboom and Cramer use network analysis to dissolve some of the problems with traditional research strategies. They consider the utility of symptom networks, in which symptoms cause symptoms. The ensuing arguments about the ability to mimic symptom trajectories and model therapeutic interventions are compelling. Both the dynamic-causal-modeling and network analysis approaches can be regarded as responding to the limitations of current nosology, and both appeal to nonlinear, state-dependent interactions among the causes of symptoms. On the other hand, the notion that symptoms cause symptoms is antithetical to an underlying generative model—in which latent pathophysiology and psychopathology cause symptoms. In other words, if one subscribes to the approach described in this article, one commits to a formal distinction between measurable signs and symptoms (including clinical diagnoses) and the causes of those symptoms. For example, there is a fundamental distinction between a measurement (e.g., a temperature of 38.2 °C) and the causes of that measurement (e.g., bacterial infection). It is almost self-evident that to generate the (profile of) measurements available to a clinician, it is necessary to model their latent causes, whether or not they are ontologically well-defined.

Neuroimaging provides a cautionary tale here, which may usefully inform computational psychiatry. In neuroimaging, network analyses have been applied uncritically to measurements of hemodynamic or metabolic brain responses in order to furnish measures of functional connectivity. Although functional connectivity can be useful for phenotyping various individuals or cohorts, it does not provide insight into how the statistical dependencies among measurements are generated (Valdes-Sosa et al., 2011). To characterize the underlying network, one has to acknowledge that hemodynamic signals in one part of the brain do not cause hemodynamics in another, but rather are caused by—or are symptoms of—the underlying (latent) neuronal activity. Network analysis of the effective connectivity among neuronal systems (i.e., the causes) generating measurements (i.e., the consequences) almost invariably calls on some form of dynamic causal modeling (Moran, Symmonds, Stephan, Friston, & Dolan, 2011). This cautionary tale suggests that we should not look for correlations among symptoms. Rather, we should try to identify the causal (network) architecture among the symptoms’ latent causes: namely, the best generative model.

Both symptom network analysis (Borsboom & Cramer, 2013) and generative modeling eschew the common-cause framework—namely, the assumption that symptoms and signs can be uniquely attributed to a common cause. Indeed, the ill-posed nature of differential diagnosis can be particularly severe in psychiatry (e.g., Brendel & Stern, 2005). Any many-to-one mapping between latent causes and symptoms (and diagnostic constructs) can, in principle, be resolved through the Bayesian inversion of generative models that call on prior assumptions to resolve the implicit indeterminacy. The role of prior assumptions or constraints could, therefore, be regarded as finessing ill-posed problems that present beyond the common-cause framework.

The Challenges of Generative Modeling

The choice of generative model is clearly crucial in terms of providing valid and useful predictions of treatment responses. In introducing dynamic causal models of pathophysiology and psychopathology, we have emphasized that the example used in this article is purely illustrative. So, how should one choose a generative model? Several issues make this choice simpler than it might appear at first glance. First, one does not have to choose a single model: The key benefit of Bayesian model comparison is that one can score the quality of different models in terms of their evidence—and thereby use strictly evidence-based criteria to select among plausible models. Second, at a phenomenological level, dynamic systems can be described in terms of a normal form (Murdock, 2003). In other words, some minimum description exists that preserves the essential dynamic behavior using a relatively simple set of differential equations. The notion of a normal form has proved extremely useful in the modeling of neu ronal dynamics—for example, in epilepsy (Roberts, Iyer, Finnigan, Vanhatalo, & Breakspear, 2014). This means that one does not need differential equations that describe every detailed fluctuation in the embodied brain; it is sufficient to summarize key fluctuations in terms of mixtures of physiological states with relatively simple differential equations (Freyer, Roberts, Ritter, & Breakspear, 2012). Finally, having established a normal form or minimal model, it is possible to successively refine the equations to move from a phenomenological to a more informed and mechanistic description. A nice example is the use of dynamic causal models in neuroimaging. These models started with a single, lumped neuronal state for each brain region—and a bilinear approximation to the underlying dynamics (Friston et al., 2003). These models were then progressively refined (through Bayesian model comparison and more informative data) to provide realistic descriptions of the synaptic activity and depolarization of specific neuronal populations (Moran et al., 2011). The hope here is that computational psychiatry could witness the same refinement of simple (e.g., normal-form) models over the same 5- to 10-year timescale seen in neuroimaging. The end result could be canonical models for particular illnesses, whose increasing complexity and neuroendocrinological realism is matched by an increase in the accuracy with which patient data are modeled in a continuous, medical setting. This long-term refinement is exemplified nicely by dynamic causal models of the mismatch negativity (Garrido, Kilner, Kiebel, & Friston, 2009) that now form the basis of routine applications—and that have found their way into psychiatric research (e.g., Ranlund et al., 2016).

Although we have emphasized the provisional nature of this approach, it should be acknowledged that one could analyze existing clinical data using the model described in this article with existing algorithms. Indeed, hundreds of publications in the neuroimaging literature have used dynamic causal modeling to infer the functional coupling among hidden neuronal states. In other words, it would be relatively simple to apply the techniques described above to existing data at the present time. However, the real challenge will lie in searching the vast model space to find models that are sufficiently comprehensive to account for the diverse range of clinical measures—in a way that generalizes from patient to patient—yet still parsimonious. This challenge is not necessarily insurmountable: One might argue that if we invested the same informatics resources in psychiatry that have been invested in weather forecasting and geophysical modeling, considerable progress could be made. Ultimately, one could imagine such model-based psychiatric prognoses being received with the same confidence that we currently accept daily weather forecasts. Of course, there are differences between psychiatric and meteorological forecasting. The latter has to deal with the “big data problem” in a relatively small model space. Conversely, psychiatry may have to contend with a “big theory problem” in a relatively large model space, but with more manageable datasets.

Having said this, perhaps the more important contribution of a formal nosology will not be its pragmatic application to precision medicine (i.e., through the introduction of prognostic apps for clinicians), but the use of Bayesian model selection to test increasingly mechanistic hypotheses and pursue a deeper understanding of pathogenesis in psychiatry. This is the way in which dynamic causal modeling has been applied in computational neuroscience and, as such, is just a formal operationalization of the scientific process.