# Dynamical System Modeling in Schizophrenia: A Narrative Review of Computational Psychiatry Frameworks

Hélène Soydemir, Christophe Gauld, Mathieu Desroches, Baptiste Pignon, Damien Depannemaecker

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Hélène Soydemir, Christophe Gauld, Mathieu Desroches, Baptiste Pignon, Damien Depannemaecker. Dynamical System Modeling in Schizophrenia: A Narrative Review of Computational Psychiatry Frameworks. 2026. hal-05487358

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# Dynamical System Modeling in Schizophrenia: A
Narrative Review of Computational Psychiatry
## Frameworks

Hélène Soydemir¹, Christophe Gauld², Mathieu Desroches³,⁴,
Baptiste Pignon⁵, and Damien Depannemaecker ¹

¹Aix Marseille Univ, INSERM, INS, Inst Neurosci Syst, Marseille, France

²Service de psychopathologie du développement, Hospices Civils de Lyon, France,
SANPSY, CNRS, UMR 6033, F-33076 Bordeaux, France

³Inria Branch at the University of Montpellier, Montpellier, France

⁴Basque Center for Applied Mathematics, Bilbao, Spain

⁵APHP, Paris, France

<figure><img src="image_2.png" /></figure>

damien.depannemaecker@univ-amu.fr

## Abstract

The symptom dynamics in schizophrenia often exhibit complex, non-linear temporal patterns, alternating between relatively predictable and disorganized states. Conventional diagnostic frameworks often struggle to capture these evolving dynamics, prompting the development of diverse modeling approaches aimed at describing symptom trajectories more faithfully. Among these approaches, dynamical systems theory provides a particularly powerful framework for addressing temporal dependencies, non-linear interactions, and state-dependent transitions. Recent advances in longitudinal data collection and analysis, including improved temporal resolution
and computational methods, now enable both the construction of interpretable models that distinguish distinct trajectories of symptom evolution and the extraction of informative dynamical features. This narrative review synthesizes the growing body of work applying dynamical systems theory to schizophrenia spectrum. It highlights convergent insights across studies while explicitly distinguishing between conceptually motivated models, emerging computational approaches, and empirically validated frameworks. This narrative review also addresses current limitations, including methodological constraints, issues of feasibility, and the need for further longitudinal and clinical validation. The aim is to provide a coherent conceptual framework for understanding schizophrenia as a dynamic process, to promote cross-disciplinary dialogue, and to guide future research in translational computational psychiatry.

**Keywords:** Schizophrenia; Dynamical Systems Modeling; Symptom Trajectories; Translational research; Precision Psychiatry; Computational Psychiatry

# 1 Introduction

Schizophrenia (SZ) is a severe psychiatric disorder that profoundly disrupts brain function and daily life. It is widely recognized as a neurodevelopmental disorder and is characterized by a heterogeneous constellation of symptoms, including hallucinations, delusions, cognitive deficits, disorganized thinking, and motivational impairments (Jiang et al. 2025). With a chronic course typically beginning in early adulthood and a lifetime prevalence of approximately 0.7%, SZ represents a major contributor to global disability-adjusted life years (McGrath et al. 2008), affecting an estimated 21 million individuals worldwide (Vigo et al. 2016). Given that longer durations of untreated psychosis are associated with poorer outcomes, early detection and timely intervention are critical (Salazar de Pablo et al. 2024). Furthermore, diagnostic and therapeutic decisions often depend on subjective judgment, which can lead to substantial rates of misdiagnosis—reported in up to 25% of cases—and to suboptimal treatment strategies (Ayano et al. 2021).

Approximately 20-30% of patients exhibit resistance to conventional antipsychotic
treatments, experiencing persistent symptoms and significant side effects without ade-
quate therapeutic benefit (Siskind et al. 2017; Chand et al. 2020). Treatment resistance is
also associated with elevated rates of psychiatric comorbidity, reflecting overlapping symp-
tom dimensions rather than clearly delineated disease entities (Pizzol et al. 2023). This
high level of multimorbidity underscores the etiological complexity of SZ, arising from the
intricate interplay of biological, environmental, and psychological factors. These observa-
tions highlight the need for more flexible and temporally informed diagnostic frameworks,
particularly when addressing psychiatric comorbidities (Szoke et al. 2024).

SZ diagnosis and management still rely on static, categorical systems (DSM, ICD),
which insufficiently capture the dynamic, heterogeneous, and complex evolution of symp-
toms. Symptom profiles vary markedly across individuals and may evolve throughout the
course of illness (Austin et al. 2015). Advances in AI, large datasets, and computational
power have improved the characterization of psychiatric disorders (Baydili et al. 2025),
supporting biomarker discovery and individualized interventions in SZ (Jiang et al. 2025).
Yet, these approaches mainly describe symptom patterns and provide limited insight into
the mechanisms driving symptom evolution, especially as patients shift between clinical
profiles, limiting their translational and therapeutic utility.

From an ethical and practical perspective, intensive protocols (frequent assessments,
sensors, large-scale digital analyses) raise concerns regarding informed consent, data pri-
vacy, and the transparency of predictive models (Mabire-Yon et al. 2025). Using simula-
tions to virtually evaluate interventions also demands scientific rigor and clear communica-
tion with participants, to avoid giving a misleading impression of deterministic outcomes.
Moreover, the high volatility of symptoms in psychiatric conditions further complicates
model application, and real-world feasibility is constrained by the burden of frequent
symptom monitoring, unequal access to digital tools, and ethical concerns regarding the
prediction of psychotic episodes or other severe outcomes (Deserno et al. 2020).
## 1.1 Dynamics in Schizophrenia

Since Kraepelin's early description of dementia praecox as a chronic degenerative illness (Kraepelin 1913), SZ has been viewed as progressive mental decline. In contrast, Bleuler introduced the term SZ to describe a fundamental 'splitting' of psychic functions—affect, thought, and behavior—reflecting global mental disorganization (Bleuler 1911). He emphasized core structural features, proposing the 'four As' (affective flattening, associative loosening, ambivalence, autism), recognizing both clinical heterogeneity and possible dynamic stability. Focusing on patients' subjective experience, Clérambault described 'mental automatism'—a sequence of passive, anideic phenomena lacking clear ideational content (Clérambault 1920).

Early conceptualizations of neural activity provided a basis for understanding dynamic processes in SZ. For example, du Bois-Reymond and Lapicque described neurons as excitable elements generating ionic currents (Finger 2001), highlighting how neuronal dynamics can produce time-dependent activity patterns. In parallel, Clérambault examined clinical phenomena, how fluctuations in mental states might reflect underlying dynamic processes in the nervous system (Ey 1954). In a way, his work paved the way for modern approaches that relate neural network dynamics to symptom variability.

Importantly, it is useful to distinguish between two types of dynamics. *Neural dynamics* refer to time-dependent changes in neuronal activity across brain networks, which can be mathematically modeled and empirically studied. *Clinical symptom dynamics*, in contrast, describe temporal variations in the intensity and combination of positive, negative, and disorganized symptoms across individuals. These two levels of dynamics are related but not identical: similar neural activity patterns may produce variable symptom trajectories due to environmental influences, internal states, and individual differences. In parallel, apparently similar symptoms manifestation may be associated with different brain activity patterns.

Today, SZ is understood as a dynamical interplay of these symptoms, with combinations and intensities that fluctuate over time and across individuals (Schneider 1959). Symptom dynamics are characterized by intermittent changes and long-range correlations
(Tschacher et al. 1996, 1997), reflecting complex temporal interactions and possible transitions between multiple stable states (Grassberger 1986). Long-term outcome studies, including World Health Organization (WHO) data (Harrison et al. 2001), identify three main SZ trajectories: 20–25% recover, 50% follow a relapsing/remitting course with moderate impairment, and 20–25% develop chronic, severe illness (Häfner et al. 1999).

This heterogeneity reflects neurodevelopmental processes in which early abnormalities combine with dynamic postnatal events, such as apoptosis and synaptic pruning, shaping diverse illness courses without gliosis (Woods 1998). Also, SZ is viewed as a multifactorial, dynamic disorder in which genes, environment, and inflammation interact to shape symptom progression. Psychotic dimensions may share common origins in early psychosocial stress. Transdiagnostic effects of such stressors have been observed in subclinical symptoms, indicating a latent vulnerability (Pignon et al. 2021). Genomic studies reveal pleiotropic variants across multiple psychiatric conditions acting via shared protein networks (Lee et al. 2025), supporting a common biological substrate and a dynamic model of disease progression. SZ is increasingly viewed as a multifactorial, dynamic disorder shaped by gene and environment factors (Rantala et al. 2022).

SZ has been described as often beginning insidiously (in 70 % of cases) rather than as a sudden onset (Fusar-Poli et al. 2013). The prodromal phase often features subtle self-disturbances, hyper-reflexivity, thought interference, and blurred perception-imagination boundaries, marking the early psychosis spectrum (van Os 2014). The basic symptoms paradigm focused on early, self-perceived anomalies like derealization and loss of natural self-evidence as signs of emerging dysfunction (Schultze-Lutter et al. 2007). The Ultra High Risk (UHR) concept was developed to identify prodromal psychosis through clinical markers and risk profiles (McGorry et al. 2014), leading to tools such as Comprehensive Assessment of At Risk Mental States, which assesses attenuated symptoms (CAARMS), which assesses attenuated symptoms, Brief Limited Intermittent Psychotic Symptoms, part of UHR criteria (BLIPS), and genetic risk combined with functional decline (Yung et al. 2005). However, the approach faces major criticisms: most future psychosis cases never meet UHR criteria, transition rates are low (15–30 % in two years), and monitoring
is resource-intensive, limiting clinical utility and raising concerns about false positives, stigma, and preventive efficacy (Fusar-Poli et al. 2012). This has prompted moves toward dynamic, individualized and transdiagnostic risk models (Van Os and Guloksuz 2017).

To address illness progression, clinical staging models were introduced to guide early detection, especially in youth at risk for severe mental illness (McGorry et al. 2014), and show empirical validity in predicting outcomes (Salokangas 2008). SZ is often described across three phases—premorbid, prodromal, and chronic—each linked to distinct neuroimaging, cognitive, and clinical profiles (Ciompi 1989; Agius et al. 2010). These phase-specific insights support multimodal, cross-diagnostic staging models relating dimensional symptoms to neurobiological measures (Shah et al. 2020). Yet most staging frameworks assume linear progression (McGorry et al. 2014), while SZ typically shows variable, non-linear courses with symptom fluctuations, remissions, and relapses that resist static classification or prediction (Ciompi 1997; van Os and Kapur 2009).

## 1.2 From Dynamical Systems to Neuropsychiatry

In this context, approaches based on Dynamical Systems Theory (DST) offer a complementary perspective, focusing on the temporal structure and underlying mechanisms of symptom evolution. DST offers a way to connect biological, cognitive, and clinical levels, supporting a non-linear psychiatry aimed at modeling and predicting psychopathological transitions (West 2012).

DST originates in classical mechanics, notably in Newton's description of systems whose states evolve continuously over time and are governed by differential equations. It also draws on Leibniz's development of calculus, which provided a systematic way to describe continuous change, namely, how small moment-to-moment variations can accumulate into large-scale transformations. Together, these ideas provide a general framework for understanding how complex systems evolve over time, making them particularly relevant for modeling neural activity and its gradual reorganization in cognition and psychopathology (Rabinovich et al. 2006; Breakspear 2017). Poincaré introduced a qualitative approach to nonlinear dynamics, focusing on the geometric organization of trajectories in
phase space rather than on explicit analytical solutions (Poincaré 1893). The phase space
is defined by the system's state variable, with each point representing an instantaneous
state and trajectories describing its temporal evolution.

These state variable—which may include membrane potentials, synaptic currents, or
clinical symptoms in SZ applications—evolve over time according to underlying (bio)physical
laws and fixed parameters, such as synaptic weights, symptom interaction strengths, treat-
ment effects, or Nernst potentials. The full set of possible configurations constitutes the
phase space (here, ‘phase’ refers not to oscillatory phase but more generally to all con-
figurations the system can explore), where each point corresponds to a unique system
state (Nolte 2010). The equations define a flow in phase space, with trajectories tracing
the system’s temporal evolution (Breakspear 2017; Rabinovich et al. 2006). Over time,
these trajectories often converge toward stable geometric structures, or attractors, such
as fixed points, limit cycles, or strange attractors, which characterize the system’s long-
term behavior (e.g., periodic oscillations or chaotic dynamics). Within this framework,
qualitative changes in behavior resulting from parameter variations are captured by bi-
furcations (Izhikevich 2007; Breakspear 2017). Conceptual examples of trajectories for
1D, 2D, and 3D systems are shown in Figure 1, illustrating how time series translate into
geometric structures in phase space, and how interventions can alter system dynamics.

A system is considered structurally stable if small changes in parameters produce only
minor deformations in its attractors, whereas a slight parameter variation that causes a
qualitative change in dynamics corresponds to a bifurcation. A fixed point corresponds to a
stationary state in which all state variable remain constant, whereas a limit cycle describes
a stable, self-sustained oscillatory trajectory in phase space. This perspective allows the
analysis of stability and transitions between distinct dynamical regimes in complex systems
without requiring closed-form solutions of the governing equations.
<figure><img src="image_3.png" /><figcaption>**Figure 1:** Conceptual trajectories illustrating dynamics of 1D, 2D, and 3D systems. Panel A (blue): 1D sine wave time series. Shows $x(t)$ over time. Panel B (red): 2D simplified model of neural-like dynamics, showing Time series with slow (dashed) and fast components. Panel C (green): 3D Lorenz system. Left: time series of $x(t)$. Right: Lorenz attractor in 3D. Panel D: 3D 'Psychiatrist Toy-model' (Gauld & Depannemaecker, 2023). Left: time series of symptoms $x$ and internal state $y$, with treatment indicated. Right: phase space $(x, y, f)$ before (orange) and after (green) treatment.</figcaption></figure>
## 1.3 Self-Organization and Collective Dynamics in Complex Systems

Complexity theory emerged formally in the mid-twentieth century through pioneering work on nonlinear dynamical systems and deterministic chaos (Lorenz 1963). Here, chaos does not mean randomness or disorder, but a regime in which the system remains deterministic while becoming highly sensitive to internal states and external perturbations. Complex systems with many interacting elements can exhibit emergent collective dynamics irreducible to individual components. This phenomenon is formalized in Haken's theory of Synergetics. According to this framework, systems driven far from equilibrium like the brain, can spontaneously self-organize, producing coordinated patterns of activity without the need for a central 'conductor' or controller. In this process, a small number of slow variables, or 'order parameters', dominate the collective dynamics, while the many faster, microscopic processes adapt to these dominant modes. This aligns with the principle of 'disorder creates order' advocated by Prigogine (Prigogine and Nicolis 1977).

Applied to brain function, this principle provides a framework for understanding how coherent cognitive and behavioral states can emerge from distributed neural activity. Fluctuations and ongoing metabolic processes allow the brain to organize and maintain structured activity. When key parameters of neural activity cross critical thresholds, the system can undergo qualitative reorganizations—akin to 'phase Transitions'. Normal cognition appears to rely on neural dynamics operating near regimes where structured patterns can form, supporting flexibility, efficiency, and robustness. In contrast, pathological conditions may reflect either an inability to reach such organized regimes or an excessive sensitivity to fluctuations that destabilizes functional patterns (Kelso 1995; Haken 2006, 2013; van der Maas 2024). Brain function is better described as a continuously evolving dynamical process than as convergence toward static fixed points (Jirsa 2020). A useful analogy is that of a tennis player preparing to return a serve: readiness emerges from ongoing coordinated motion, enabling rapid adaptation with minimal energy expenditure.

Despite the high dimensionality of neural systems, brain activity often evolves along a small number of collective variables, effectively collapsing onto a low-dimensional manifold
(Langdon et al. 2023). In other words, the brain's dynamics behave as if constrained to a smaller space of stable patterns. These patterns are not arbitrary. They reflect forms of organization that maximize structured information while respecting the brain's physical and functional constraints (Haken 2006). Disruptions of these organizing structures can distort continuous attractors into discrete or unstable states, impairing the representation of internal variables and potentially contributing to cognitive dysfunction and psychopathology.

Over subsequent decades, this interdisciplinary framework expanded to encompass self-organization in biological and social systems (Haken 1990), as well as applications to cognition and psychology (Kelso 1995; Scheffer et al. 2024). Biological and cognitive systems can be understood as open systems that continuously exchange energy, matter, and information with their environment. Even subtle variations in neural activity, environment, or cognitive context can produce large differences in outcomes, which helps explain the variability and unpredictability observed in behavior and mental states. Such nonequilibrium dynamics support cognitive flexibility, multistability, and rapid reconfiguration of functional networks, while also increasing sensitivity to perturbations (Nartallo-Kaluarachchi et al. 2025; Deco et al. 2017).

## 1.4 Schizophrenia as a Nonlinear Dynamical System

Early studies suggested that SZ could be conceptualized as a 'dynamical disease' (West 2012). These early ideas laid the conceptual groundwork for modern investigations. Subsequent analyses of EEG signal complexity show that erratic brain activity, characterized by irregular oscillations, abrupt fluctuations, and high temporal variability, is correlated with cognitive and clinical states in SZ (Itil 1977; West 2012).

Non-linear brain dynamics have been suggested to underlie cognitive and perceptual disruptions observed in SZ (An der Heiden 2006; Breakspear 2017). In fact, SZ psychoses may function as systems where affective states and cognitive processes interact in a nonlinear manner (Ciompi 1997). Such interactions can produce sudden changes in the system's
behavior. These changes are known as bifurcations. They can also give rise to multiple
stable states, or attractors. The system can be highly sensitive to small disturbances.
This phenomenon is often called the 'butterfly effect'. The system may retain a form
of memory, meaning that some changes can be irreversible. When the system is pushed
beyond a certain point, for instance by intense anxiety, it can settle into a new stable
psychotic state. This process is similar to how a chaotic systems reorganizes after a bifur-
cation (Scrimali 2018). Even minor shifts in emotions or cognition can lead to large and
unpredictable changes in behavior or experience. This perspective helps us understand
why SZ trajectories are so variable. Gradual fluctuations in emotional states can slowly
shape symptom evolution. They produce distinct courses of the disorder in different in-
dividuals. Figure 2 illustrates patient-derived (synthetic) trajectories of SZ symptoms. It
shows diverse dynamical patterns such as rapid daily fluctuations, seasonal modulations,
irregular chronic fluctuations, and latent clinical dynamics.

From this perspective, SZ can be viewed as a nonlinear disturbance of normal emotion-cognition dynamics. Especially since, SZ arises from dynamic developmental trajectories and critical periods, during which environmental and psychosocial influences, such as social exclusion or substance use, can alter individual risk (Szoke et al. 2020). Neuro-modulators such as dopamine, serotonin, and acetylcholine can act as control parameters, reshaping manifold geometry and altering the trajectories the brain can explore. Neural mass model studies show that increased dopaminergic input expands the repertoire of dynamical patterns and changes their frequency, highlighting neuromodulation's role in regulating brain activity (Depannemaecker et al. 2025).

The aim of the present narrative review was to integrate and synthesize prior work that have mobilized dynamic systems theory to describe, explain, or model SZ. By identifying convergent ideas, validated results, and hypotheses, this review aimed to clarify which dynamical principles have proven robust, which remain exploratory, and how DST-based models now enable a more unified and mechanistically grounded interpretation of SZ.
<figure><img src="image_4.png" /><figcaption>Figure 2: **Patient-derived trajectories illustrating symptom dynamics and phase space.** All data are synthetic, simulated to reproduce dynamics observed in real patients.</figcaption></figure>

**Panel A (blue):** Rapid daily symptom fluctuations. Left: time series of smoothed symptoms. Right: phase space (x vs dx/dt). **Panel B (orange):** Seasonal modulation of SZ symptoms. Left: time series with slow seasonal component (dashed) and fast noise. Right: phase space (x vs dx/dt). Shows increased relapses in spring. **Panel C (green):** Irregular chronic fluctuations (chaos-like). Left: time series. Right: reconstructed 3D phase space (x, dx/dt, d²x/dt²). **Panel D (red):** Data-driven latent clinical dynamics. Left: time series of symptoms and latent vulnerability. Right: 3D phase space (x = symptoms, y = latent state, f = slow clinical history).
# 2 Methods Consideration

This work was conducted in accordance with SANRA guidelines, ensuring transparency in article selection and synthesis. Search terms combined SZ with concepts from DST, nonlinear dynamics, attractors, and bifurcations. Article selection prioritized conceptual relevance, historical significance, and methodological contribution, rather than exhaustive- tiveness (1800–2025). Articles in the Table 1 were preferentially selected if they employed Ordinary Differential Equations (ODEs) to model dynamic systems, whether or not they were originally validated against empirical data, reflecting a methodological focus on this approach.

## 2.1 Towards a DST modeling of Schizophrenia

The reviewed studies are organized chronologically into two main subsections: approaches focusing on brain-level dynamics, and those primarily addressing symptom-level phe- nomenology.

### 2.1.1 Brain-level models

Early applications of DST to the central dopaminergic system showed that, for specific values of postsynaptic dopamine (DA) efficacy, the system can transition into chaotic regimes (King et al. 1984). As discussed earlier, DA plays a central role in SZ, with receptor overactivity and altered metabolite levels correlating with symptom severity and psychotic episodes (Tschacher et al. 1992). Together, these findings suggest that the intrinsic nonlinear structure of the dopaminergic system may give rise to irregular neural dynamics, potentially contributing to the heightened behavioral variability observed in SZ, even in the absence of external perturbations.

Disruption of this attractor-based organization, such as through excessive pruning of long-range cortical connections, can fragment large-scale brain dynamics and disrupt co- ordinated activity(Hoffman and McGlashan 2001). Developmentally reduced synaptic con-
nectivity and progressive pruning degrade global network function, producing hallucination-like errors (McGlashan and Hoffman 2000). Dynamically, this reshapes the attractor landscape, creating new stable minima and disrupting coordinated activity, offering a mechanistic explanation for persistent hallucinations and delusions under ambiguous inputs.

Imbalances between excitation and inhibition, together with modulatory effects of DA, destabilize cortical attractors, facilitating erratic transitions between network states. Gradual changes in DA function act as a control parameter, influencing network stability and transitions between healthy and pathological states in SZ (An der Heiden 2006). Recurrent excitatory-inhibitory cortical circuits and thalamo-cortico-striatal loops can exhibit irregular or chaotic firing, which can be captured computationally using coupled differential equations for excitatory input, inhibitory postsynaptic potentials, and output firing rates, with DA modulating inhibitory efficacy.

At a more local level, reduced N-Méthyl-D-Aspartate (NMDA) receptor conductance flattens cortical attractor basins, increasing susceptibility to noise and impairing working memory, attention, and firing rates in prefrontal, orbitofrontal, and anterior cingulate cortices (Loh et al. 2007; Rolls 2021). Computational models, including integrate-and-fire and stochastic recurrent networks with DA modulation, show that interactions between excitatory (NMDA) and inhibitory Gamma-aminobutyric acid (GABA) populations determine attractor stability. Destabilization of high-activity attractors in dorsolateral prefrontal cortex underlies cognitive symptoms, while shallower basins in orbitofrontal and anterior cingulate regions contribute to negative symptoms. Noise-induced transitions in temporal and auditory association networks, exacerbated by GABA hypofunction, may generate positive symptoms such as hallucinations and delusions. DA modulation can partially restore stability: D1 receptor agonists strengthen high-activity states, improving cognitive and negative symptoms, while D2 antagonists stabilize spontaneous states, reducing positive symptoms (Rolls 2021). Empirical support includes reduced dendritic spine density, decreased GABA-mediated inhibition, and altered functional connectivity and temporal variability observed in fMRI.

Building on these perspectives, a multi-attractor brain network model has been applied
to resting-state fMRI data from healthy controls and patients with SZ, bipolar disorder, and ADHD (Attention-Deficit/Hyperactivity Disorder) (Sun et al. 2023). The model is formulated as a system of coupled ODEs describing the dynamics of excitatory and inhibitory neural populations. By computing all stable attractors, a cross-attractor coordination matrix, and energy gaps, the approach links alterations in global coupling and excitation/inhibition (E/I) balance to psychiatric conditions. Findings indicate lower global coupling and smaller energy gaps, particularly in the Default Mode Network (DMN), somatomotor, and visual networks, in patients, along with a higher E/I ratio in SZ compared with ADHD, suggesting that psychiatric disorders involve modifications in the brain's attractor landscape.

### 2.1.2 Symptom-level models

Early applications of DST to social and clinical contexts, such as modeling family dynamics, showed that crossing bifurcation points can trigger abrupt transitions between distinct stable states or from non-oscillatory to periodic regimes (Elkaïm et al. 1987), with implications for interventions targeting maladaptive cyclical patterns.

The same principles of nonlinear dynamics can also describe individual physiological endophenotype of SZ. Anomalies in smooth pursuit eye movements in SZ, can be interpreted using a deterministic nonlinear framework (Huberman 1987). The oculomotor system can be modeled as a nonlinear oscillator with cubic stiffness, also called a Duffing oscillator. Such a system can shift between different patterns of behavior: stable tracking, where the eyes follow the target smoothly; symmetry breaking, where the movement becomes biased or uneven; chaotic behavior, characterized by irregular and unpredictable trajectories; and hysteresis, where the system's response depends on its past history. These dynamics closely reproduce the irregular eye movements observed in patients. This suggests that SZ may involve inherently unstable eye movement control due to the amplification of nonlinear effects.

Nonlinear dynamical simulations in psychiatry show that small changes in social or
cognitive factors can lead to divergent SZ trajectories (Schiepek et al. 1992). These discrete-time models represent cognition, stress, social withdrawal, expressed emotion, and delusions as interacting variables with feedback and stochastic fluctuations, revealing stable and unstable illness regimes and showing how slow changes in psychological, social, or biochemical parameters can trigger abrupt transitions such as relapse or recovery.

Schizophrenic psychoses can be seen as dissipative structures because their complex symptom dynamics emerge from nonlinear interactions (biological, cognitive, and social processes) operating far from equilibrium, generating self-organized patterns and attractors despite ongoing perturbations (Ciompi 1997). Nonlinear analyses of short- and long-term symptom fluctuations using methods such as the Grassberger–Procaccia algorithm and the Sugihara–May method (Empirical Dynamic Modeling) approach reveal low-dimensional chaotic attractors, indicating that psychotic dynamics are governed by fractal, self-organizing processes rather than random noise (Ciompi 1994). Computational simulations reproduce both rapid symptom changes and long-term trajectories, providing a mechanistic account of SZ's heterogeneous course. Within this framework, affective states act as control parameters that modulate cognitive–affective interactions and shape accessible dynamical states (Ciompi 1976), with emergent patterns, attractors, and bifurcations reflecting multistability. Depending on initial conditions and affective modulation, symptom severity may worsen, fluctuate episodically, stabilize, or remit, illustrating how clinical courses of SZ emerge from a common underlying dynamical system influenced by affect and stochastic perturbations.

Within this perspective, the textit‘Chaos theory of SZ’ extends the view of schizophrenia as a dynamical system by modeling symptom evolution with a few nonlinearly coupled variables, producing complex yet structured temporal patterns (Tschacher et al. 1992). Analyses of daily symptom time series reveal distinct dynamical classes, including chaotic, linear autoregressive, or stochastic, and indicate that many patients follow nonlinear, potentially chaotic trajectories (Tschacher et al. 1997). Although short clinical time series cannot prove deterministic chaos, features such as decaying predictability and bifurcation-like transitions suggest that SZ dynamics are structured but highly sensitive to small
changes.

Finally, a 3+1 dimensional toy-model using a non-linear DST framework has been developed to describe the nonlinear progression of psychiatric symptoms (Gauld and De-pannemaecker 2023). Variables include symptoms (x), internal factors (y), environmental noise (z), and temporal specificity (f). The model simulates different clinical scenarios: healthy, episodic, kindling-like, and highly environment-sensitive disorders, and can test treatment effects. This framework explicitly links symptom dynamics to the concept of SZ as a dynamical disease: rather than being static, symptom profiles emerge from interactions between intrinsic neuronal or cognitive factors and extrinsic environmental influences, evolving over time in complex, sometimes abrupt ways. By mapping these trajectories in phase space, the model illustrates how transitions between clinical states (e.g., remission, relapse, or exacerbation) correspond to shifts in attractor landscapes, providing a mechanistic perspective on the temporal variability and unpredictability of SZ. While theoretical, it offers a concrete tool for predictive, personalized, and intervention-oriented computational psychiatry, bridging the gap between longitudinal symptom observations and mechanistic models.

<table><thead><tr><th>Reference</th><th>Model</th><th>ODE</th><th>Methods</th><th>Variables</th><th>Timescale</th><th>Dynamics</th></tr></thead><tbody><tr><td colspan="7">Brain-level models</td></tr><tr><td>King & Barchas; Huberman 1984</td><td>Nigrostriatal dopaminergic system</td><td>✓</td><td>DDE (ODEs with delay) + nonlinear iterative map: Simulations, spectral and Lyapunov</td><td>Firing rate, DA concentrations, synaptic stores, postsynaptic efficacy</td><td>ms-s</td><td>Multistability, Period-doubling bifurcations, Chaotic transitions, and Erratic fluctuations linked to Symptom variability</td></tr><tr><td>McGlashan & Hoffman, 2000</td><td>Develop-mentally Reduced Synaptic Connectivity [1]</td><td>×</td><td>Neural network simulation of speech perception with synaptic pruning</td><td>S(t) = synaptic density, Cognitive performance (identified words), Simulated hallucinations (misidentified words)</td><td>months-years</td><td>Synaptic loss: &lt;30% optimal → 35-40% cognitive decline → &gt;40% hallucinations → plateau</td></tr><tr><td>An der Heiden, 2006</td><td>Recurrent inhibitory cortical circuit</td><td>✓</td><td>Simulation of delay differential equations, bifurcation analysis</td><td>E(t), I(t); pyramidal cell output; inhibitory cell output; receptor parameters</td><td>ms-s</td><td>Periodic and chaotic firing; bifurcations via inhibitory receptor changes (DA); healthy → pathological transitions.</td></tr><tr><td>Loh et al., 2007</td><td>Recurrent attractor model</td><td>✓</td><td>IF neurons: simulations with Poisson inputs and MF of attractor states</td><td>Firing rates of selective pools+ inhibitory pool; synaptic currents (AMPA, NMDA, GABA)</td><td>ms-s</td><td>Stability of spontaneous/persistent attractors; distractibility; attractor transitions and flow; NMDA/GABA modulation effects.</td></tr></tbody></table>
<table><caption>**Table 1:** Non-exhaustive overview of different approaches using dynamical systems models applied to SZ. It included models that apply DST either directly, by formulating systems of ODEs to model symptom dynamics in SZ, or conceptually, by invoking DST concepts without specifying a system of ODEs.</caption><thead><tr><th>Reference</th><th>Model</th><th>ODE</th><th>Methods</th><th>Variables</th><th>Timescale</th><th>Dynamics</th></tr></thead><tbody><tr><td>Rolls &amp; Deco, 2021</td><td>Stochastic recurrent cortical network IF (SDE)</td><td>✓</td><td>Simulation, attractor stability, exploration of reduced NMDA and GABA effects</td><td>Membrane potential, firing rates, NMDA/AMPA/GABA synaptic currents, mean activity of neuron pools</td><td>ms-s</td><td>Spontaneous (low-firing) and persistent (high-firing) attractor states, noise-driven transitions and destabilization of cortical activity, conceptually linked to negative and positive symptom dimensions.</td></tr><tr><td>Sun et al., 2023</td><td>BNM (Wong-Wang+ Wilson-Cowan)</td><td>✓</td><td>Coupled ODEs (exc./inh. populations linked by SC), RSFC, energy gaps</td><td>Synaptic activity; inputs; global coupling, local parameters</td><td>ms-s</td><td>Stable fixed points (spirals/damped oscillations) with possible transitions (bifurcation-like barriers)</td></tr><tr><td colspan="7" style="text-align: center; font-weight: bold;">Symptom-level models</td></tr><tr><td>Elkaïm et al. 1987</td><td>Mother-child interaction model</td><td>✓</td><td>Steady-state analysis, linear stability, bifurcation analysis</td><td>Aggressiveness of child + Mother's tension</td><td>days</td><td>Trivial and non-trivial steady states, marginally stable oscillations (sustained cycles),damped oscillations</td></tr><tr><td>Huberman 1987</td><td>Deterministic model of smooth pursuit eye movements</td><td>✓</td><td>Forced and damped nonlinear oscillator; Bifurcation, power spectra, Poincaré maps, noise effects</td><td>Eye deviation angle, Angular velocity</td><td>ms-s</td><td>Dynamic regimes: stable tracking, asymmetric oscillations, chaotic/unpredictable behavior, history-dependent transitions, effect of noise on bifurcations</td></tr><tr><td>Schiepek et al. 1992</td><td>Mixed feedback model of SZ [2]</td><td>×</td><td>Discrete nonlinear difference equations, Slow-fast dynamics Simulation, logistic map, Feigenbaum route, random fluctuations</td><td>Fast: cognitive, stress, withdrawal, emotion, delusions; Slow: cognition, neurochemistry, social skills</td><td>days-weeks</td><td>Asymptotic fixed points, periodic oscillations, nonlinear feedbacks, episodic patterns, chaotic behavior</td></tr><tr><td>Ciompi 1976</td><td>Affect-logic [4]</td><td>×</td><td>Conceptual dynamic framework</td><td>Affect, cognition, psychosocial, biological factors</td><td>days-months</td><td>Premorbid vulnerability, Decomensation → psychosis, Long-term evolution with attractors, bifurcations, variability in remission and chronicity</td></tr><tr><td>Tschacher et al. 1997</td><td>Nonlinear dynamics of psychotic symptoms in SZ [3]</td><td>×</td><td>Phase space reconstruction, forecasting algorithms, surrogate data testing to distinguish chaos/noise/linearity</td><td>Daily rating of psychotic derealization (univariate time series)</td><td>days</td><td>Three types: (1) nonlinear/chaotic dynamics (57% of patients), (2) linear (AR/ARMA), (3) random/noise.</td></tr><tr><td>Gauld et al., 2023</td><td>Toy-model 'Psychiatr'</td><td>✓</td><td>4 coupled nonlinear ODEs; dynamical simulation</td><td>Symptoms x, internal factors y, environmental noise z, temporal specificity f</td><td>months-years</td><td>Nonlinear interactions → multistability, slow-fast dynamics, stochastic perturbations, simulation of treatment effects</td></tr></tbody></table>

[1] Conceptual discrete-time neural network inspired by DST, using phase-space representations of $S(t)$ and hallucination-related outputs.

[2] Discrete-time analogues of ODEs, capturing coupled fast-slow dynamics.

[3] Implicit discrete-time formulation, conceptually related to a stochastic differential equation.

[4] Conceptual DST-based framework addressing affect-cognition interactions and qualitative bifurcation concepts.

Together, brain-level models suggest that SZ is fundamentally characterized by alterations
in attractor landscapes—a hypothesis that can be directly examined at the symptom
level using high-frequency longitudinal data. The next steps in DST-based modeling of
SZ critically depend on the availability of longitudinal datasets, which are essential for
empirical validation, model refinement, and the translation of theoretical predictions into
clinically meaningful insights.

## 2.2 Data in Schizophrenia

Until recently, the study of SZ has been limited by the scarcity of high-quality longitudinal datasets. Early reviews already highlighted the 'extreme shortcoming of longitudinal research' in the field (Häfner et al. 1999), a limitation that largely persists today. Ciompi emphasized both decades-long and day-to-day variability in symptom expression (Aebi and Ciompi 1989), illustrating the need for data capturing within-person variations across micro (daily) and macro (yearly) timescales (Ciompi 1989; Nelson et al. 2017). Such temporal richness is necessary to trace how genetic vulnerability and environmental context interact to shape symptom onset and course. However, even existing longitudinal studies face structural and logistical challenges. The lack of standardized, scalable infrastructures for multimodal data collection hinders reproducibility and mechanistic insights, while long-term follow-up is complicated by high dropout rates, heterogeneous clinical courses, asynchronous acquisitions, inconsistent protocols, and sparse or irregular sampling (Hegarty et al. 1994). These limitations highlight the critical need for high-frequency, longitudinal, and multimodal datasets, which are essential both to construct and validate DST-based models of SZ.

The development of Ecological Momentary Assessment (EMA) has opened new avenues for understanding psychiatric disorders through the lens of DST. ODEs provide a powerful framework to model high-frequency longitudinal data from EMA, capturing and predicting complex temporal patterns. By enabling in situ measurements, EMA allows extraction of real-time symptom dynamics in naturalistic settings (Misdrahi et al. 2023; Gauld et al. 2025; Barbosa et al. 2025). These data capture within-subject variability and context-dependent fluctuations often hidden in cross-sectional or retrospective
assessments. For example, conducted on Substance Use Disorders (SUD) with over 200 participants and 8,000+ EMA observations, authors used optimized linear Seasonal Auto-Regressive Integrated Moving Average with exogenous variables (SARIMAX) models to phenotype patients and characterize bidirectional interactions between cues and craving. These empirical findings were replicated by two DST models describing escalation and saturation dynamics of craving and cue exposure (Gauld et al. 2025).

More recently, analyses on FACE-SZ have already revealed robust clinical predictors of relapse (Barbosa et al. 2025), medication adherence (Misdrahi et al. 2023), and quality of life, using established statistical and machine learning tools such as generalized Cox models, random forests, and classification trees. However, these focus on point-in-time predictions and correlations without capturing the latent structural and temporal dynamics of symptom trajectories (Barbosa et al. 2025). This enables identification of new biomarkers, guides real-time interventions, and stratifies patients by dynamic trajectories.

## 2.3 Model validation and model personalization

By fitting model parameters to observed data, model inversion can allows DST to accurately capture individual variability in brain or symptom dynamics. This process not only tests the model's validity against empirical observations but also tailors the model to reflect personalized trajectories, enabling more precise diagnosis and intervention (Zi-aemehr et al. 2025).

# 3 Discussion

The use of dynamical systems models and complexity theory in the study of psychological and psychiatric phenomena significantly enriches behavioral analysis. The models examined here—ranging from brain-level neuronal systems (King et al. 1984; Loh et al. 2007; Sun et al. 2023) to symptom-level dynamics (Ciompi 1976; Huberman 1987; Schiepek et al. 1992; Gauld and Depannemaecker 2023) provide a framework for understanding emergent
processes, multistability, bifurcations, and chaotic or erratic transitions within complex
systems. These properties, observable across different temporal and spatial scales, can
complement classical statistical approaches in psychology, such as linear regression or
ANOVA.

At the brain level, these models illustrate how short-term dynamics, chaos, and erratic
fluctuations, can relate to observable behaviors or symptoms, such as variability in motor,
perceptual, or cognitive responses. More recent models, such as Sun et al. (2023), combine
excitatory and inhibitory populations coupled through structural brain connectivity to
study stability and state transitions, highlighting the relevance of multilayer approaches
for capturing interactions between local and global scales (Sun et al. 2023).

Other approaches such as Dynamic Causal Modeling, energy landscape analyses, net-
work control theory, and whole-brain simulations have revealed how alterations in effective
connectivity, synaptic gain, and functional state stability may underlie cognitive inflexibil-
ity and symptom maintenance (Fogelson et al. 2014; Braun et al. 2021; Li et al. 2023; Tyras
and Wojnar 2025). Recurrent neural networks and reservoir computing capture nonlinear
temporal dependencies in neural and behavioral time series, enabling predictive and dis-
criminative analyses, though these remain largely data-driven and non-mechanistic (Yan
et al. 2019). Mechanistic concepts such as slow-fast dynamics, bifurcations, and critical
transitions help explain symptom volatility, abrupt state changes, and multi-timescale vari-
ability observed in SZ, providing a theoretical bridge from neural dynamics to observable
symptom trajectories (Rabinovich et al. 2006; Pavlidis et al. 2024). While informative for
studying general neural dynamics, these models are primarily theoretical and not tailored
to the explicit modeling of clinical symptoms, which constrains their use for translational
applications.

At the symptom level, different works presented in this narrative review demonstrate
the value of dynamical modeling for understanding nonlinear symptom trajectories, the
occurrence of crises or decompensations, and responses to interventions (Ciompi 1976;
Schiepek et al. 1992; Gauld and Depannemaecker 2023). Models incorporating fast-slow
dynamics, multistability, and stochastic perturbations offer a conceptual approach to link
internal processes (cognitive, emotional) and environmental factors to clinical outcomes. Clinically, this view implies that symptoms are not entirely random and that interventions can exploit the system's short-term determinism, targeting chaotic regimes to stabilize favorable behavior (Elbert et al. 1992; Tschacher et al. 1997). Overall, a parsimonious nonlinear framework may better explain SZ than traditional multicausal models, with implications for both theory and therapeutic strategies.

Also, idiographic DST frameworks and EMA approaches allow reconstruction of patient-specific dynamic landscapes, identifying potential Early Warning Signals (EWS) for clinical transitions (Schiepek 2003; Friston et al. 2012; Van de Leemput et al. 2014; Schreuder et al. 2023). While conceptually compelling, these EWS are not yet validated in real-world settings, and their practical utility is constrained by measurement noise, partial observability, high inter-individual heterogeneity, environmental influences, and the burden of intensive symptom tracking (Nelson et al. 2017; Barbosa et al. 2025). Moreover, translating DST concepts into clinically actionable tools requires longitudinal, high-resolution datasets, ideally in multi-centre cohorts, to test mechanistic hypotheses and confirm predictive markers before routine application can be considered. Thus, DST currently serves more as a theoretical and mechanistic framework than a fully validated clinical instrument.

However, these approaches face several methodological limitations. Clear links between phenomenological descriptors and dynamical classifications remain difficult to establish (Tschacher et al. 1997). Moreover, implementing coupled or multilayer models remains technically challenging and requires high-resolution, long-term data. At the symptom level, identifying EWS or indicators of state transitions remains difficult, and can lead to erroneous conclusions.

# 4 Conclusion

Despite these limitations, DST offers a powerful conceptual language to integrate observations across neural, behavioral, and psychosocial domains, highlighting attractors, bifurca-
tions, and multistability as key properties of SZ dynamics (Breakspear 2017; Mabire-Yon et al. 2025). While many models remain exploratory or retrospective, they provide insights beyond static or categorical frameworks, formalizing latent structures, regime shifts, and multi-scale interactions. Realizing the translational potential of DST will require rigorous validation, longitudinal multimodal datasets, and dynamical markers in real-world patient populations (Barbosa et al. 2025; Leboyer and Llorca 2025).

Finally, it is important to remember that dynamical modeling is not intended to per-
fectly replicate reality, but rather to provide useful approximations of the underlying mech-
anisms driving behavior. Although simplified, such models are valuable for identifying
key dynamical properties, exploring regime shifts, and generating mechanistic hypotheses
without requiring full biological realism. Explaining cognition requires intermediate theo-
retical descriptions that bridge neural activity and behavior, since neural correlations alone
do not provide a mechanistic understanding (Marr 2010; Krakauer et al. 2017). While stud-
ies of neural implementation are necessary for testing causal hypotheses, the challenge lies
in articulating why a purely reductionist approach may be insufficient. Especially given
recent experimental advances, ongoing philosophical debates about explanatory levels, and
questions regarding downward causation. In this sense, the limitations of models are not
a weakness but a prerequisite for understanding complexity—a principle that continues to
guide the careful application of complex models in mental health research and psychiatry
(Epstein 2008; Smaldino 2016).

# 5 Disclosure Statement

No potential competing interest was reported by the authors.

6 Acknowledgment

The project leading to this publication has received funding from the Excellence Initiative of Aix-Marseille Université - A*Midex, a French “Investissements d'Avenir programme”
AMX-21-IET-017. This research has received funding from the European Union's Horizon Europe Programme under the Specific Grant Agreement No. 101147319 (EBRAINS 2.0 Project). It has also received funding from the European Union's Horizon Europe Programme under the Specific Grant Agreement No. 101137289 (Virtual Brain Twin Project). This work has benefited from a government grant managed by the Agence Nationale de la Recherche (ANR) under the France 2030 program, reference ANR-22-PESN-0012.

## Glossary

**bifurcations** A qualitative change in the dynamics of a system caused by a small change in a parameter, leading to the emergence or disappearance of attractors. Bifurcations can correspond to transitions between stable symptom states in computational models of psychiatric disorders. 7

**chaotic systems** Chaotic systems are deterministic nonlinear dynamical systems that exhibit sensitive dependence on initial conditions, leading to exponential divergence of nearby trajectories in phase space. Despite being governed by deterministic equations, their long-term behavior is effectively unpredictable. Chaotic dynamics are characterized by aperiodicity, positive Lyapunov exponents, and complex attractor structures, and they occur in a wide range of physical, biological, and computational systems. 11

**Grassberger–Procaccia algorithm** A nonlinear time-series analysis method used to estimate the correlation dimension of an attractor from scalar observations, allowing the identification of low-dimensional deterministic chaos in empirical data by assessing how the number of state-space point pairs scales with distance. 16

**manifold** A lower-dimensional surface embedded in phase space that organizes the dynamics of a system. In neural and psychiatric models, manifolds can capture corre-
lated patterns of brain activity or constrain the evolution of symptom trajectories.

9

**Neural mass model** Neural mass models are mathematical models that describe the collective dynamics of large populations of neurons by averaging their activity into a small number of state variables. They are commonly used to model mesoscale cortical dynamics and to account for macroscopic electrical and magnetic signals measured with electroencephalography (EEG) and magnetoencephalography (MEG).

11

**phase space** An abstract space in which all possible states of a system are represented, with each axis corresponding to a state variable. Trajectories in phase space describe how the system evolves over time. 7

**phase Transitions** A transition between distinct dynamical regimes of a system, occurring when gradual parameter changes lead to abrupt qualitative shifts in attractor structure, stability, or variability. A sudden reorganization of neural or behavioral dynamics whereby small changes in control variables (e.g., neuromodulation, stress, connectivity) lead to abrupt shifts between functional states. In psychopathology, such transitions may correspond to symptom onset, remission, or relapse. 9

**state variable** A variable representing the current condition or state of a system at a given time, e.g., neural activity, symptom severity, or network connectivity. State variables evolve over time according to the system's dynamics and are used in modeling and simulation of psychiatric disorders. 7

**Sugihara–May method (Empirical Dynamic Modeling)** A nonlinear forecasting approach that tests for state-dependent dynamics by reconstructing the system's attractor from time-series data and evaluating predictive skill, thereby distinguishing deterministic nonlinear processes from stochastic noise. 16
Acronyms

BLIPS Brief Limited Intermittent Psychotic Symptoms, part of UHR criteria. 5

**CAARMS** Comprehensive Assessment of At Risk Mental States, which assesses attenuated symptoms. 5

DST Dynamical Systems Theory. 6, 13, 17–20

EMA Ecological Momentary Assessment. 19, 20, 22

EWS Early Warning Signals. 22

ODEs Ordinary Differential Equations. 13, 15, 18, 19

**SARIMAX** Seasonal Auto-Regressive Integrated Moving Average with exogenous variables. 20

SUD Substance Use Disorders. 20

SZ Schizophrenia. 2–7, 10, 11, 13–19

UHR Ultra High Risk. 5

WHO World Health Organization. 5

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