Edited by: Yuka Kotozaki, Iwate Medical University, Japan
Reviewed by: William Sulis, McMaster University, Canada; Chong Chen, Yamaguchi University Graduate School of Medicine, Japan
This article was submitted to Psychopathology, a section of the journal Frontiers in Psychology
†These authors have contributed equally to this work and share first authorship
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These last years, scientific research focuses on the dynamical aspects of psychiatric disorders and their clinical significance. In this article, we proposed a theoretical framework formalized as a generic mathematical model capturing the heterogeneous individual evolutions of psychiatric symptoms. The first goal of this computational model based on differential equations is to illustrate the nonlinear dynamics of psychiatric symptoms. It offers an original approach to nonlinear dynamics to clinical psychiatrists.
In this study, we propose a 3+1 dimensions model (
Constrained by clinical observation of case formulations, the dynamics of psychiatric symptoms is studied through four main psychiatric conditions were modeled: i) a healthy situation, ii) a kind of psychiatric disorder evolving following an outbreak (i.e., schizophrenia spectrum), iii) a kind of psychiatric disorder evolving by kindling and bursts (e.g., bipolar and related disorders); iv) and a kind of psychiatric disorder evolving due to its high susceptibility to the environment (e.g., spersistent complex bereavement disorder). Moreover, we simulate the action of treatments on different psychiatric conditions.
We show that the challenges of dynamical systems allow to understand the interactions of psychiatric symptoms with environmental, descriptive, subjective or biological variables. Although this non-linear dynamical model has limitations (e.g., explanatory scope or discriminant validity), simulations provide at least five main interests for clinical psychiatry, such as a visualization of the potential different evolution of psychiatric disorders, formulation of clinical cases, information about attracting states and bifurcations, or the possibility of a nosological refinement of psychiatric models (e.g., staging and symptom network models).
Contemporary psychiatric nosology is based on categorical and static taxonomic distinctions. Thanks to the International Classification of Diseases, Eleventh Edition (ICD-11) (CIM,
Psychiatric disorders are thus dynamical conditions (Demic and Cheng,
We have identified the existence of at least four main types of non-modeled clinical theoretical proposals in the psychiatric literature. The first one corresponds to the 3P model (Wright et al.,
The second kind of clinical theoretical proposal corresponds to the kindling model. This clinical formulation, from the field of epilepsy, explains the manifestations of a relatively short stage of a psychiatric disorder. At a during fleeting moments of susceptibility (e.g., from a few hours to a few weeks), a triggering factor would lead to expressing the manifestations of this disorder. This psychiatric disorder bursts by successive acute manifestations on a relatively short time scale (Adamec,
The third kind of clinical theoretical proposal corresponds to staging models. Staging models are defined as psychiatric models aimed at distinguishing subgroups evolving by (successive) stages (McGorry et al.,
Finally, the fourth kind of clinical theoretical proposal corresponds to the conception of psychiatric disorders through the prism of dynamical systems, as we are going to explain, develop and use in this work.
In this article, we propose to use dynamical systems in order to build a computational model to apprehend the dynamics of psychiatric symptoms. Three methodological anchors allow this development: case formulations, wealth of dynamic systems and desire for a manipulable model.
First, the main goal of the use of dynamical systems is to computationally validate empirical observations made by clinicians and researchers of the psychiatric field. This computational model is based on case formulations of different psychiatric disorders and conditions and aggregated symptoms. We define (stereotyped) case formulations as the psychiatric disorders and conditions of a given patient. These are typical cases of clinical observation. Case formulations aim to model the characteristics of specific individuals, e.g., “an individual with an autism spectrum disorder.” This computational model is thus designed in a contingent way to exemplify the phenotypes of psychiatric disorders. These stereotyped case formulations are those described in the textbook of clinical psychiatry and transmitted to any clinician in his/her elementary formation, and as he/she can then observe it in his/her daily practice (corresponding to irrefutable and prototypical cases of dynamical evolutions of psychiatric disorders). Such case formulations serve as tools that help organize complex and contradictory information about a person (Eells et al.,
Such dynamical model will allow to learn about non-linear phenomena and instability, major variations related to fluctuations in initial conditions, phenomena of resilience and fragility or the attainment of tipping points (transitions) and steady states, attractors and oscillations between multiple stability in response to internal conditions or external stressors. It also incorporates the elements mentioned in the three previous kinds of clinical modeling, e.g., predisposing, precipitating and perpetuating factors, consideration of different time frames, sensitization (kindling) and stages (staging models) of psychiatric disorders.
First, current computational models found in the literature integrate biological factors, can be predictive, are interested in different time scales, but rare are the models that allow the common integration of all these variables. Studies are restricted to parameters such as noise from the environment (e.g., psychosocial stress) (Huber et al.,
Based on these theoretical considerations, no computational model considering both symptomatology, internal factors, environment and temporality (i.e., the four variables of the model presented below) seems to exist in the scientific literature. We seek to go beyond theoretical contributions by proposing a variable modeling environmental noise (
Here, we propose such a dynamical model as a structure able to receive simulated or empirical data, reproducing the phenomenological dynamics of psychiatric disorders. The whole interest of such a model is precisely to be able to get away from the traditional diagnostic categories to apprehend a multitude of empirical or simulated symptoms in a transdiagnostic way. The “toolbox” constituted by this model can, for instance, integrate both anhedonia and low mood (major depressive disorder category) and acoustico-verbal hallucinations (schizophrenia category). We believe that this granularity at the scale of the symptom is particularly important for clinical psychiatry.
Moreover, because of its desire to be generalizable to different types of disorders and adaptable to different datasets, this toy-model differs from a certain number of other dynamic models of psychiatric disorders found in the scientific literature (Durstewitz et al.,
We propose here a toy-model to study the dynamics of psychiatric symptoms, which reduces complexity by considering aggregates of non-linear relationships between a limited number of variables depending on time: environment, internal elements (e.g., subjective phenomenological experiences or biology) and symptoms. Such a model is based on equations intended to reproduce case formulations. It produces an abstract representation of patients. It is not built on empirical data collected in research. Such a computational model is thus called a toy-model. In a toy-model, abstract values correspond to qualitative behaviors empirically perceived in clinical practice. As we will discuss, such a toy-model could only serve to apprehend, understand, or support debates on the possible dynamics of psychiatric disorders. As its name indicates, a toy-model is deliberately used to explain and make practical a behavioral function, like a box containing balls, which is considered as a toy-model allowing to understand, in a simplified way, both the solar system and the interactions between atoms. As has already been proposed in the context of epilepsy (Depannemaecker et al.,
We propose a model based on dynamical system for psychiatric disorders. This model is based on the theory of dynamical systems, which offers a framework for modeling the evolution of variables as a function of time. We used a structure, able to receive simulated data, reproducing the phenomenological dynamics of psychiatric disorders. In order to avoid starting from psychiatric categories, we isolated variables from case formulations of different psychiatric disorders and conditions and aggregates of symptoms.
The model captures the temporal evolution of a phenomenon using mathematical differential equations, which intrinsically integrate temporality. Differential equations allow to calculate next states given a current state, depending on time. In a first-order differential system, the state of a variable at a time t is calculated based on its variation with respect to time t-Δt. The rate of amplitude changes over time will determine the time scale. Each equation composing the system can vary according to its own time scale. This notion of time scales is an important consideration in psychiatric disorders because psychiatric acute events occur on a much shorter scale (minutes, hour, day, weeks) than the longer time scale of the development and consequences of a psychiatric disorder (months, years, decades). Thus, to model psychiatric disorders, a fast sub-system helps to switch from a basic (healthy or already pathologically latent) state to a state with high level of symptoms. A slower system is needed to drive the transition between these states. Therefore, a slow-fast system is proposed, including these different time scales, in which an external input drives the transition between states.
We seek to infer the dynamical relationships that exist between the variables producing a psychiatric phenomenon. Thus, we describe the expected temporal trajectory of the variables, by including different relevant psychiatric aspects into the equations to obtain the desired phenomenological characteristics. The parameters are identify in order that the simulation dynamics correspond to clinical observations. The modeling of psychiatric disorders could have been carried out in other ways. Our primary goal is to computationally match clinical observations.
As schematized in
Cartoon representation of the computational dynamical toy-model of psychiatric disorders. The three equations are intrinsically linked by their interacting variables. The equation on the left (variable
The computational toy-model is thus described by the following system of equations:
The literal descriptions of the parameters of the toy-model are given in
Description of the parameters of the toy-model.
Maximal level of symptoms. | |
Sensitivity: the difficulty in triggering the system. Rs refers to sensitivity in terms of symptoms. Rb refers to sensitivity in terms of internal elements (or potentiation). | |
λ_{s} and λ_{b} | Slope of the symptom and internal elements (or potentiation) curves. λ_{s} corresponds to the increase in the intensity of symptoms x as a function of the subjective state of the patient y. λ_{b} corresponds to the increase in the internal elements (or potentiation) y as a function of the environment z. |
Maximal rate of internal elements of the systems: the aggregation of biological, psychological or subjective/phenomenological elements. It refers to the fabrication of the semantic configuration of a phenotype from a set of biological signals. It is an element of potentiation of symptoms. | |
Level of predisposing factors: it contributes to a permanent shift in the internal elements (potentiation). | |
λ_{f} | Scaling factor of the slow evolution of the fluctuations affecting |
Overall sensitivity level to the environment. | |
α and β | Weight of the effect of the variable x and y on the perception of the environment |
The first Equation (1) can be understood as: "The intensity of symptoms increases due to subjective state
If nothing participates to maintain high intensity of symptoms, the intensity of symptoms decreases over time (modeled with the exponential decay −
The second Equation (2) refers to the internal elements of a patient. The variable
The third equation refers to the environment (or external world) perceived by a patient, modeled by the variable
A fourth equation can be added to model slower processes of psychiatric disorders. This equation is equivalent to a change of a parameter over time to capture elements on a much longer timescale, especially at the scale of a lifetime (Equation 4):
This equation could be adapted according to the fluctuations of the values of λ_{f}). These fluctuations can create oscillations, or slow evolutions of other variables over the long term. This is a variable of slowness, which interacts at a longer time with the other three variables evolving more rapidly. The variable depends on the internal potentiation
Due to the structure of these equations, we have to consider a set of constraints.
First, we are looking for a system representing several states, accounting for phase transitions: 1) of psychiatric states below a first threshold delimiting a state of health and a pathological state; 2) of psychiatric states above the threshold of psychiatric disorders; 3) of psychiatric states corresponding to the maximum intensity of symptoms, i.e., the most intense state of crisis describable for a disorder.
Secondly, configurations containing negative
Thirdly, the rate of the noise ζ(
Fourthly, the
We use an Euler integration method with
In the following section, we will perform four simulations based on four case formulations to verify that the model captures the following stereotypical dynamics of psychiatric conditions: a healthy condition, a schizophrenia spectrum disorder, a rapidly cycling bipolar disorder, and a persistent complex bereavement disorder.
From the observed dynamics of simulations of these four case formulations of psychiatric disorders, conditions and aggregates of symptoms, and based on this set of equations, we will propose to identify contingent relative threshold values (maximum and minimum) for each of the 13 parameters of the
Finally, in addition to these four simulations, we will propose a fifth simulation in which we visualize the effect of a therapeutic action according to knowledge and stereotyped case formulations.
We used a toy-model built on differential equations to simulate the dynamics of psychiatric conditions and disorders through case formulations. Depending on the variability of the parameters handled in this toy-model, various dynamics of different psychiatric disorder could be modeled. If empirical data from research has been incorporated into the model, although each of these conditions tends to be as stereotyped as possible relative to empirical observations of clinical practice, each condition could be dynamically different according to the interindividual variations of the patients. Thus, in each of the following case formulations, based on observation of stereotypical cases, we can identify that each dynamic could be observed when the value of a parameter increases or decreases.
Constrained by clinical observations, simulations of the four following psychiatric conditions (named here
Time series of the four simulations for different sets of parameters. Negative events from the environment (variable
Values of the 13 parameters for the 4 simulations (or case formulations).
10 | 1 | 0.1 | 14 | 10 | 1.04 | 0.05 | 0.2 | 14 | 4 | 0.5 | 0.5 | 1 | |
10 | 1 | 0.1 | 14 | 10 | 0.904 | 0.05 | 0.2 | 14 | 4 | 0.5 | 0.5 | 1 | |
10 | 1 | 0.1 | 14 | 10 | 1.04 | 0.05 | 1.01 | 14 | 10 | 0.5 | 0.5 | 1 | |
10 | 1 | 0.1 | 14 | 10 | 1 | 0.05 | 0.6 | 14 | 4.5 | 0.5 | 0.5 | 1 |
For all simulations λ_{f} = 1 and τ_{f} = 720.
In this case formulation, described in the panel (a) of the
At a random time of 5,500 days (about 15 years), a potentially destabilizing life event occurs. It could be, for instance, the death of a loved one. However, given the healthy characteristics of the modeled patient (depending on its different values of
In this case formulation, described in the panel (b) of the
We identify that these dynamics could be observed when the
At the random time of 5,500 days, in the absence of any intervention, a potentially destabilizing life event occurs. After this event, the symptoms persist due to different patient parameters simulated. We observe the complete absence of responsiveness to environmental stimuli.
In this case formulation, described in the panel (c) of the
Here, the
Each complete cycle lasts approximately 100 days, with a symptom plateau lasting approximately 15 days, similar to what can be found in clinical practice. Finally, we retrieve that despite the presence of an intense life event, there is no change in the patient's sensitivity to his/her environment or in the intensity of the symptoms.
In this case formulation, described in the panel (d) of the
In the PCBD, it is precisely the onset of an intense life event that causes this disorder. However, unlike the situation (a) in which the intensity of symptoms and the sensitivity to the environment returns to the threshold of normality over time, the patient modeled in this case formulation continues to have mood fluctuations.
This case formulation is thus close to the healthy situation, except for the
In the
Time series of four simulations for different sets of parameters with a therapeutic action on different psychiatric disorders. The black arrow corresponds to the beginning of the treatment.
In the
In the
In the
In this simulation study, we proposed a toy-model (the “psychiator”) that can phenomenologically reproduce the time evolution of the intensity of psychiatric symptoms, interacting with the internal elements and his/her perceived external environmental inputs, while considering different time scales. This computational model enables to understand the effects of non-linear relations between different psychiatric disorders' determinants. It has a set of strengths and limitations that we will detail. The
Example of the dynamics of a simulation of a rapid cycling bipolar disorder.
We retrieve at least five main interests of such a computational toy-model.
First, by varying the values of the parameters, such a model allows visualization of simulations of different case formulations of psychiatric conditions and disorders and aggregates of symptoms. Such visualizations allow to find potentially new endpoints for clinical and research purposes, which in themselves enables computational models to be refined. This model allows to show that the interactions between three relatively simplified variables lead to behaviors that are very difficult to intuitively interpret. This complexity thus demonstrates the need to consider non-linear relationships rather than single variable-phenotype relationships at the clinical level. For instance, for psychotherapy, such non-linear formalization of the patient behaviors can help guide indications for specific drug therapy. It can also constitute a didactic and pedagogical tool to help the patients to understand the (non-linear) factors at the origin of their distress (Burger et al.,
Secondly, this model provides a high flexibility, allowing a large number of concepts to be discussed and made practical. Indeed, its interest lies in the possibility of using a large number of different empirical data from research in psychiatry, clinical psychology or neuroscience, with different actions on the parameters of the toy-model, to observe in particular the inter- and intra-individual differences of psychiatric disorders. For instance, this model is sufficiently generic to be interpreted for different type of symptoms. Moreover, the versatility of this model (i.e., the model can be adapted to many different psychiatric disorders and psychological conditions) allows to compare the differential evolutions of these disorders. This comparison could help to specify their phenotypes and refine precision medicine. Psychiatry is struggling with the issues of differential diagnoses (i.e., distinguishing two disorders whose symptoms overlap) and with the issues of comorbidity (i.e., assessing the need to distinguish two conditions or to combine two of them into one). In recent decades, no diagnostic biomarker, neither predictive nor endotype have been identified to clearly define the boundaries of psychiatric disorders: in this way, hopes lies in the differential evolution of psychiatric disorders themselves, potentially evaluable with such a computational model. The very large number of possible combinations refers to the infinite number of phenotypic variations in psychiatry. Such a model provides access to the variability of psychiatric phenotypes for a same disorder.
Thirdly, on the therapeutic level, such a toy-model provides information on the attracting states (i.e., the states to which the system gravitates). This result allows to understand what stabilizes the patient in a given (healthy) state. The warning signals leading to this attracting state can thus be detected upstream (Hayes and Andrews,
Fourthly, this model proposes a dynamically theoretical framework allowing to constitute longitudinal studies and the use of assessment tools in daily life. This model provides a flexible framework allowing to integrate a large number of heterogeneous data, distinguishing patient-dependent factors, her/his subjective experience and the environment. Certainly, the absence of large cohorts of longitudinal data in psychiatry is due to numerous economic or organizational factors. However, they also relate to a lack of methodological tools. In other words, such a framework constitutes a prerequisite for the collection of longitudinal data in psychiatry. Such methods may be integrated into moment-to-moment ecological macro- or micro-level assessment (depending on the period), and especially a widely used methods such as ecological momentary assessment or joint modeling of time-to-event outcome with time-dependent predictors (i.e., given the temporal nature necessary to predict the onset of a disorder) (McGorry et al.,
Fifthly, such a model could refine at least two kinds of nosological psychiatric models: staging models (McGorry et al.,
Finally, such a ubiquitous and abstract toy-model, in future works, would allow to propose new classifications of psychiatric disorders according to their dynamics. We would find some disorders particularly sensitive to the environment (e.g., OCD), others presenting a rapid rhythmic activity (e.g., rapid cycles in bipolarity), or others with abrupt bifurcations in their trajectory.
This toy-model also has several limitations.
First, the explanatory scope of this model remains limited. There could have been an infinity of models, impeding this model from being considered as predictive. Comparable dynamics could be found with completely different set of parameters, or even with different ordinary equations. Nowadays, the absolute values of parameters are not representative of any physically measurable elements. Indeed, it is important to note that the terms of the equation are contingent, and could have been defined differently. In other words, there could have been several ways of defining the dynamics of a psychiatric disorder, and this model is only one of the answers allowing to visualize their evolution over time. For example, mathematical solutions could be found to reduce the number of model parameters. However, we choose to keep it under this form to maintain intuitive clinical interpretability. To be predictive for a given patient, the model should incorporate her/his specific collected longitudinal values. Unlike digital twins (i.e., data-driven mathematical models of patients that allow for more precise and effective medical interventions), this toy-model is not built, at first, to be personalized. However, the objective of this study is not to select the best model (in terms of the equation structuring), but to propose a systematic formulation of an observed phenotypic behavior, based on the clinically relevant variables and parameters.
Secondly, this model seems reductionist regarding clinical practice. However, it integrates in an original way non-linear relations between qualitatively and clinically interpretable equations. Indeed, we have proposed a model in which it is not the biological mechanistic structures that are modeled, but behaviors (Marr,
Thirdly, it turns out that this model should be tested with experimental data to ensure its discriminative, construct and/or predictive validity. We hypothesize that research in psychiatry waited for such a robust model to collect empirical data, and conversely that no robust model could be built due to a lack of empirical data. The absence of measurements of such values is largely due to the absence of a model as we propose it. We are thus seeking to break this vicious circle with such a toy-model. After empirical validation, the structure of this computational model could serve as an optimized framework for simulating behavior and predicting the course of disorders. In order to choose whether certain other methods could allow to model psychiatric disorders in the same way, a set of models similar to this one should be constructed, with a sorting of these models by an analysis of the choice of the best model (in terms of choice of the free parameters). Future studies will aim to identify the maximum and minimum ranks of the (13) parameter values described in the
Fourthly, the representation of
Fifthly, a last limit concerns the potential difficulty to interpret the dynamics of the models. Indeed, the incorporated variables account for non-linear phenomena, which could be not intuitively explainable to a clinician. More precisely, it could be difficult to know why some stressors and triggers have or not an action on the system (e.g., inducing a dissociation), why certain nonlinear effects occur at particular times (e.g., fluctuations of affective states) or how interactions between certain symptoms occur (e.g., low mood and overeating or anorexia). Clinical inference from this kind of model should be very careful. By extension, it will be necessary to ensure that these individual-level models are not naively transferred to group-level models.
In the history of clinical psychology and psychiatry, predicting the occurrence of disorders and symptoms has focused on the evaluation of a spectrum of variables – ranging from genetics to the environment, including neurocognitive measurements or subjective feelings. These conditions are particularly difficult to model, and this difficulty is largely due to the lack of dynamic modeling to model them, despite a growing theoretical literature advancing such promises for at least several decades (Nelson et al.,
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s. An example code of an implementation of the model presented in this paper is available at
CG and DD were the principal investigators and study supervisors and interpreted the results and wrote the manuscript. CG conceptualized and supervised the analysis. DD made the mathematical equation formalization and run the simulations. Both authors approved the final version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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