Wilson-Cowan Model Research Articles

Criticality and partial synchronization analysis in Wilson-Cowan and Jansen-Rit neural mass models.

Synchronization is a phenomenon observed in neuronal networks involved in diverse brain activities. Neural mass models such as Wilson-Cowan (WC) and Jansen-Rit (JR) manifest synchronized states. Despite extensive research on these models over the past several decades, their potential of manifesting second-order phase transitions (SOPT) and criticality has not been sufficiently acknowledged. In this study, two networks of coupled WC and JR nodes with small-world topologies were constructed and Kuramoto order parameter (KOP) was used to quantify the amount of synchronization. In addition, we investigated the presence of SOPT using the synchronization coefficient of variation. Both networks reached high synchrony by changing the coupling weight between their nodes. Moreover, they exhibited abrupt changes in the synchronization at certain values of the control parameter not necessarily related to a phase transition. While SOPT was observed only in JR model, neither WC nor JR model showed power-law behavior. Our study further investigated the global synchronization phenomenon that is known to exist in pathological brain states, such as seizure. JR model showed global synchronization, while WC model seemed to be more suitable in producing partially synchronized patterns.

The dance of neurons: Exploring nonlinear dynamics in brain networks

The brain is a complex, nonlinear system, exhibiting ever-evolving patterns of activities, whether in the presence or absence of external stimuli or task demands. Nonlinearity can notably obscure the link between structural constraints enforced on the interaction and its dynamical consequences. Suitable nonlinear dynamical models and their analysis serve as essential tools not only for bridging structural and functional understanding of the brain but also for predictably altering the complex dynamical organization of the brain. Here, starting from a large-scale network of threshold Hodgkin–Huxley style neurons, we formulate the average nonlinear dynamics implicitly following from the Wilson–Cowan assumptions. We investigate the influence of biophysical and structural properties on the complexity of neural dynamics at the microscale level and its relationship with the macroscopic Wilson–Cowan model. Incorporating the elements in the model can help identify more realistic regimes of activity and connect the mathematical prediction of increasing nonlinearity to physical manipulations. Our simulations of the temporal profiles reveal dependency on the binary state of interacting subpopulations and the random property of structural network at the transition points, when different synaptic weights are considered. For substantial configurations of stimulus intensity, our model provides further estimates of the neural population’s dynamics, ranging from simple-periodic to aperiodic patterns and phase transition regimes. This reflects the potential contribution of the microscopic nonlinear scheme to the mean-field approximation in studying the collective behaviour of individual neurons with particularly concentrating on the occurrence of critical phenomena. We show that finite-size effects kick the system in a state of irregular modes to evolve differently from predictions of the original Wilson–Cowan reference. Additionally, we report that the complexity and temporal diversity of neural dynamics, especially in terms of limit cycle trajectory, and synchronization can be induced by either small heterogeneity in the degree of various types of local excitatory connectivity or considerable diversity in the external drive to the excitatory pool.

Controlling neural activity: LPV modelling of optogenetically actuated Wilson–Cowan model * *This work was supported by ANPCyT, Ministry of Science & Technology, Argentina Grant PICT2017-2417. Demián García-Violini is supported by the Agencia I+D+i, Ministry of Science & Technology, Argentina, under PICT-2021-I-INVI-00190.

Objective. This paper aims to bridge the gap between neurophysiology and automatic control methodologies by redefining the Wilson–Cowan (WC) model as a control-oriented linear parameter-varying (LPV) system. A novel approach is presented that allows for the application of a control strategy to modulate and track neural activity. Approach. The WC model is redefined as a control-oriented LPV system in this study. The LPV modelling framework is leveraged to design an LPV controller, which is used to regulate and manipulate neural dynamics. Main results. Promising outcomes, in understanding and controlling neural processes through the synergistic combination of control-oriented modelling and estimation, are obtained in this study. An LPV controller demonstrates to be effective in regulating neural activity. Significance. The presented methodology effectively induces neural patterns, taking into account optogenetic actuation. The combination of control strategies with neurophysiology provides valuable insights into neural dynamics. The proposed approach opens up new possibilities for using control techniques to study and influence brain functions, which can have key implications in neuroscience and medicine. By means of a model-based controller which accounts for non-linearities, noise and uncertainty, neural signals can be induced on brain structures.

Coarse graining and criticality in the human connectome.

In the face of the stupefying complexity of the human brain, network analysis is a most useful tool that allows one to greatly simplify the problem, typically by approximating the billions of neurons making up the brain by means of a coarse-grained picture with a practicable number of nodes. But even such relatively small and coarse networks, such as the human connectome with its 100-1000 nodes, may present challenges for some computationally demanding analyses that are incapable of handling networks with more than a handful of nodes. With such applications in mind, we set out to study the extent to which dynamical behavior and critical phenomena in the brain may be preserved following a severe coarse-graining procedure. Thus we proceeded to further coarse grain the human connectome by taking a modularity-based approach, the goal being to produce a network of a relatively small number of modules. After finding that the qualitative dynamical behavior of the coarse-grained networks reflected that of the original networks, albeit to a less pronounced extent, we then formulated a hypothesis based on the coarse-grained networks in the context of criticality in the Wilson-Cowan and Ising models, and we verified the hypothesis, which connected a transition value of the former with the critical temperature of the latter, using the original networks. This preservation of dynamical and critical behavior following a severe coarse-graining procedure, in principle, allows for the drawing of similar qualitative conclusions by analyzing much smaller networks, which opens the door for studying the human connectome in contexts typically regarded as computationally intractable, such as Integrated Information Theory and quantum models of the human brain.

First- and second-order phase transitions in electronic excitable units and neural dynamics under global inhibitory feedback

The diverse roles of inhibition in neural circuits and other dynamical networks are receiving renewed interest. Here, it is shown that increasing global inhibitory feedback leads to gradual rounding of first-order transition between dynamical phases, turning it into second-order transition. The effect is initially observed in an electronic model consisting of a bi-dimensional array of neon glow lamps, where global inhibition can be simply introduced through a resistor in series with the supply voltage. The experimental findings are confirmed using both an extended numerical model and a mean-field approximation, then replicated across different models of neural dynamics, namely, the Wilson–Cowan model and a network of leaky integrate-and-fire neurons. Across all these systems, a critical point is always found as a function of a pair of parameters controlling local excitability and global inhibition strength, and a general explanation revealing the roles of the shape of the activation function and voltage fluctuations versus the extinction time-scale is provided. It is speculated that the brain could use global inhibition as a versatile means of shifting between first- and second-order dynamics, addressing the conundrum regarding the coexistence in neural dynamics of phenomena stemming from both. Some reflections regarding the comparison with other physical systems and the possible physiological significance are offered, and a hypothetical setup for an optogenetics experiment on cultured neurons is put forward.

Cortical Divisive Normalization from Wilson–Cowan Neural Dynamics

Divisive Normalization and the Wilson–Cowan equations are well-known influential models of nonlinear neural interaction (Carandini and Heeger in Nat Rev Neurosci 13(1):51, 2012; Wilson and Cowan in Kybernetik 13(2):55, 1973). However, they have been always treated as different approaches and have not been analytically related yet. In this work, we show that Divisive Normalization can be derived from the Wilson–Cowan dynamics. Specifically, assuming that Divisive Normalization is the steady state of the Wilson–Cowan differential equations, we find that the kernel that controls neural interactions in Divisive Normalization depends on the Wilson–Cowan kernel but also depends on the signal. A standard stability analysis of a Wilson–Cowan model with the parameters obtained from our relation shows that the Divisive Normalization solution is a stable node. This stability suggests the appropriateness of our steady state assumption. The proposed theory provides a mechanistic foundation for the suggestions that have been done on the need of signal-dependent Divisive Normalization in Coen-Cagli et al. (PLoS Comput Biol 8(3):e1002405, 2012). Moreover, this theory explains the modifications that had to be introduced ad hoc in Gaussian kernels of Divisive Normalization in Martinez-Garcia et al. (Front Neurosci 13:8, 2019) to reproduce contrast responses in V1 cortex. Finally, the derived relation implies that the Wilson–Cowan dynamics also reproduce visual masking and subjective image distortion, which up to now had been explained mainly via Divisive Normalization.

Metabolic energetics underlying attractors in neural models.

Neural population modeling, including the role of neural attractors, is a promising tool for understanding many aspects of brain function. We propose a modeling framework to connect the abstract variables used in modeling to recent cellular-level estimates of the bioenergetic costs of different aspects of neural activity, measured in ATP consumed per second per neuron. Based on recent work, an empirical reference for brain ATP use for the awake resting brain was estimated as ∼2 × 109 ATP/s-neuron across several mammalian species. The energetics framework was applied to the Wilson-Cowan (WC) model of two interacting populations of neurons, one excitatory (E) and one inhibitory (I). Attractors were considered to exhibit steady-state behavior and limit cycle behavior, both of which end when the excitatory stimulus ends, and sustained activity that persists after the stimulus ends. The energy cost of limit cycles, with oscillations much faster than the average neuronal firing rate of the population, is tracked more closely with the firing rate than the limit cycle frequency. Self-sustained firing driven by recurrent excitation, though, involves higher firing rates and a higher energy cost. As an example of a simple network in which each node is a WC model, a combination of three nodes can serve as a flexible circuit element that turns on with an oscillating output when input passes a threshold and then persists after the input ends (an "on-switch"), with moderate overall ATP use. The proposed framework can serve as a guide for anchoring neural population models to plausible bioenergetics requirements.NEW & NOTEWORTHY This work bridges two approaches for understanding brain function: cellular-level studies of the metabolic energy costs of different aspects of neural activity and neural population modeling, including the role of neural attractors. The proposed modeling framework connects energetic costs, in ATP consumed per second per neuron, to the more abstract variables used in neural population modeling. In particular, this work anchors potential neural attractors to physiologically plausible bioenergetics requirements.

A fractional-order Wilson-Cowan formulation of cortical disinhibition.

This work presents a fractional-order Wilson-Cowan model derivation under Caputo's formalism, considering an order of . To that end, we propose memory-dependent response functions and average neuronal excitation functions that permit us to naturally arrive at a fractional-order model that incorporates past dynamics into the description of synaptically coupled neuronal populations' activity. We then shift ourfocus on a particular example, aiming to analyze the fractional-order dynamics of the disinhibited cortex. This system mimics cortical activity observedduring neurological disorders such as epileptic seizures, where an imbalance between excitation and inhibition is present, which allows brain dynamics to transition to a hyperexcited activity state. In thecontext of the first-order mathematical model, we recover traditional results showing a transition from a low-level activity state to a potentially pathological high-level activity state as an external factor modifies cortical inhibition. On the other hand, under the fractional-order formulation, we establish novel results showing that the system resists such transition as the order is decreased, permitting the possibility of staying in the low-activity state even with increased disinhibition. Furthermore, considering the memory index interpretation of the fractional-order model motivation here developed, our results establish that by increasing the memory index, the system becomes more resistant to transitioning towards the high-level activity state. That is, one possible effect of the memory index is to stabilize neuronal activity. Noticeably, this neuronal stabilizing effect is similar to homeostatic plasticity mechanisms. To summarize our results, we present a two-parameter structural portrait describing the system's dynamics dependent on a proposed disinhibition parameter and the order. We also explore numerical model simulations to validate our results.

Unconventional criticality, scaling breakdown, and diverse universality classes in the Wilson-Cowan model of neural dynamics.

The Wilson-Cowan model constitutes a paradigmatic approach to understanding the collective dynamics of networks of excitatory and inhibitory units. It has been profusely used in the literature to analyze the possible phases of neural networks at a mean-field level, e.g., assuming large fully connected networks. Moreover, its stochastic counterpart allows one to study fluctuation-induced phenomena, such as avalanches. Here we revisit the stochastic Wilson-Cowan model paying special attention to the possible phase transitions between quiescent and active phases. We unveil eight possible types of such transitions, including continuous ones with scaling behavior belonging to known universality classes-such as directed percolation and tricritical directed percolation-as well as six distinct ones. In particular, we show that under some special circumstances, at a so-called "Hopf tricritical directed percolation" transition, rather unconventional behavior is observed, including the emergence of scaling breakdown. Other transitions are discontinuous and show different types of anomalies in scaling and/or exhibit mixed features of continuous and discontinuous transitions. These results broaden our knowledge of the possible types of critical behavior in networks of excitatory and inhibitory units and are, thus, of relevance to understanding avalanche dynamics in actual neuronal recordings. From a more general perspective, these results help extend the theory of nonequilibrium phase transitions into quiescent or absorbing states.

Effects of optogenetic and visual stimulation on gamma activity in the visual cortex

Studying brain functions and activity during gamma oscillations can be a challenge because it requires careful planning to create the necessary conditions for a controlled experiment. Such an experiment consists of placing the brain into a gamma state and investigating cognitive processing with a careful design. Cortical oscillations in the gamma frequency range (30–80 Hz) play an essential role in a variety of cognitive processes, including visual processing and cognition. The present study aims to investigate the effects of a visual stimulus on the primary visual cortex under gamma oscillations. Specifically, we sought to explore the behavior of gamma oscillations triggered by optogenetic stimulation in the II and IV layers of the visual cortex, both with and without concurrent visual stimulation. Our results show that optogenetic stimulation increases the power of gamma oscillation in both layers of the visual cortex. However, the combined stimuli resulted in a reduction of gamma power in layer II and an increase and reinforcement in gamma power in layer IV. Modelling the results with the Wilson-Cowan model suggests changes in the input of the excitatory population due to the combined stimuli. In addition, our analysis of the data using the Lempel-Ziv complexity method supports our interpretations from the modeling. Thus, our results suggest that optogenetic stimulation enhances low gamma power in both layers of the visual cortex, while simultaneous visual stimulation has differing effects on the two layers, reducing gamma power in layer II and increasing it in layer IV.

Temporal characteristics of gamma rhythm constrain properties of noise in an inhibition-stabilized network model.

Gamma rhythm refers to oscillatory neural activity between 30 and 80Hz, induced in visual cortex by stimuli such as iso-luminant hues or gratings. The power and peak frequency of gamma depend on the properties of the stimulus such as size and contrast. Gamma waveform is typically arch-shaped, with narrow troughs and broad peaks, and can be replicated in a self-oscillating Wilson-Cowan (WC) model operating in an appropriate regime. However, oscillations in this model are infinitely long, unlike physiological gamma that occurs in short bursts. Further, unlike the model, gamma is faster after stimulus onset and slows down over time. Here, we first characterized gamma burst duration in local field potential data recorded from two monkeys as they viewed full screen iso-luminant hues. We then added different types of noise in the inputs to the WC model and tested how that affected duration and temporal dynamics of gamma. While the model failed with the often-used Poisson noise, Ornstein-Uhlenbeck noise applied to both the excitatory and the inhibitory populations replicated the duration and slowing of gamma and replicated the shape and stimulus dependencies. Thus, the temporal dynamics of gamma oscillations put constraints on the type and properties of underlying neural noise.

Hierarchical Wilson-Cowan Models and Connection Matrices.

This work aims to study the interplay between the Wilson-Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson-Cowan equations provide a dynamical description of neural interaction. We formulate Wilson-Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson-Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson-Cowan equations be formulated on a compact group. We propose a p-adic version of the Wilson-Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the p-adic version matches the predictions of the classical version in relevant experiments. The p-adic version allows the incorporation of the connection matrices into the Wilson-Cowan model. We present several numerical simulations using a neural network model that incorporates a p-adic approximation of the connection matrix of the cat cortex.

Fluctuation-dissipation relations in the imbalanced Wilson-Cowan model.

The relation between spontaneous and stimulated brain activity is a fundamental question in neuroscience which has received wide attention in experimental studies. Recently, it has been suggested that the evoked response to external stimuli can be predicted from temporal correlations of spontaneous activity. Previous theoretical results, confirmed by the comparison with magnetoencephalography data for human brains, were obtained for the Wilson-Cowan model in the condition of balance of excitation and inhibition, a signature of a healthy brain. Here we extend previous studies to imbalanced conditions by examining a region of parameter space around the balanced fixed point. Analytical results are compared to numerical simulations of Wilson-Cowan networks. We evidence that in imbalanced conditions the functional form of the time correlation and response functions can show several behaviors, exhibiting also an oscillating regime caused by the emergence of complex eigenvalues. The analytical predictions are fully in agreement with numerical simulations, validating the role of cross-correlations in the response function. Furthermore, we identify the leading role of inhibitory neurons in controlling the overall activity of the system, tuning the level of excitability and imbalance.

Optimal control of a Wilson-Cowan model of neural population dynamics.

Nonlinear dynamical systems describe neural activity at various scales and are frequently used to study brain functions and the impact of external perturbations. Here, we explore methods from optimal control theory (OCT) to study efficient, stimulating "control" signals designed to make the neural activity match desired targets. Efficiency is quantified by a cost functional, which trades control strength against closeness to the target activity. Pontryagin's principle then enables to compute the cost-minimizing control signal. We then apply OCT to a Wilson-Cowan model of coupled excitatory and inhibitory neural populations. The model exhibits an oscillatory regime, low- and high-activity fixed points, and a bistable regime where low- and high-activity states coexist. We compute an optimal control for a state-switching (bistable regime) and a phase-shifting task (oscillatory regime) and allow for a finite transition period before penalizing the deviation from the target state. For the state-switching task, pulses of limited input strength push the activity minimally into the target basin of attraction. Pulse shapes do not change qualitatively when varying the duration of the transition period. For the phase-shifting task, periodic control signals cover the whole transition period. Amplitudes decrease when transition periods are extended, and their shapes are related to the phase sensitivity profile of the model to pulsed perturbations. Penalizing control strength via the integrated 1-norm yields control inputs targeting only one population for both tasks. Whether control inputs drive the excitatory or inhibitory population depends on the state-space location.

Modelling homeostatic plasticity in the auditory cortex results in neural signatures of tinnitus

Tinnitus is a clinical condition where a sound is perceived without an external sound source. Homeostatic plasticity (HSP), serving to increase neural activity as compensation for the reduced input to the auditory pathway after hearing loss, has been proposed as a mechanism underlying tinnitus. In support, animal models of tinnitus show evidence of increased neural activity after hearing loss, including increased spontaneous and sound-driven firing rate, as well as increased neural noise throughout the auditory processing pathway. Bridging these findings to human tinnitus, however, has proven to be challenging. Here we implement hearing loss-induced HSP in a Wilson-Cowan Cortical Model of the auditory cortex to predict how homeostatic principles operating at the microscale translate to the meso- to macroscale accessible through human neuroimaging. We observed HSP-induced response changes in the model that were previously proposed as neural signatures of tinnitus, but that have also been reported as correlates of hearing loss and hyperacusis. As expected, HSP increased spontaneous and sound-driven responsiveness in hearing-loss affected frequency channels of the model. We furthermore observed evidence of increased neural noise and the appearance of spatiotemporal modulations in neural activity, which we discuss in light of recent human neuroimaging findings. Our computational model makes quantitative predictions that require experimental validation, and may thereby serve as the basis of future human studies of hearing loss, tinnitus, and hyperacusis.

Bistability and criticality in the stochastic Wilson-Cowan model.

We study a stochastic version of the Wilson-Cowan model of neural dynamics, where the response function of neurons grows faster than linearly above the threshold. The model shows a region of parameters where two attractive fixed points of the dynamics exist simultaneously. One fixed point is characterized by lower activity and scale-free critical behavior, while the second fixed point corresponds to a higher (supercritical) persistent activity, with small fluctuations around a mean value. When the number of neurons is not too large, the system can switch between these two different states with a probability depending on the parameters of the network. Along with alternation of states, the model displays a bimodal distribution of the avalanches of activity, with a power-law behavior corresponding to the critical state, and a bump of very large avalanches due to the high-activity supercritical state. The bistability is due to the presence of a first-order (discontinuous) transition in the phase diagram, and the observed critical behavior is connected with the line where the low-activity state becomes unstable (spinodal line).

Noise-modulated multistable synapses in a Wilson-Cowan-based model of plasticity.

Frequency-dependent plasticity refers to changes in synaptic strength in response to different stimulation frequencies. Resonance is a factor known to be of importance in such frequency dependence, however, the role of neural noise in the process remains elusive. Considering the brain is an inherently noisy system, understanding its effects may prove beneficial in shaping therapeutic interventions based on non-invasive brain stimulation protocols. The Wilson-Cowan (WC) model is a well-established model to describe the average dynamics of neural populations and has been shown to exhibit bistability in the presence of noise. However, the important question of how the different stable regimes in the WC model can affect synaptic plasticity when cortical populations interact has not yet been addressed. Therefore, we investigated plasticity dynamics in a WC-based model of interacting neural populations coupled with activity-dependent synapses in which a periodic stimulation was applied in the presence of noise of controlled intensity. The results indicate that for a narrow range of the noise variance, synaptic strength can be optimized. In particular, there is a regime of noise intensity for which synaptic strength presents a triple-stable state. Regulating noise intensity affects the probability that the system chooses one of the stable states, thereby controlling plasticity. These results suggest that noise is a highly influential factor in determining the outcome of plasticity induced by stimulation.

Power spectrum and critical exponents in the 2D stochastic Wilson–Cowan model

The power spectrum of brain activity is composed by peaks at characteristic frequencies superimposed to a background that decays as a power law of the frequency, f^{-beta }, with an exponent beta close to 1 (pink noise). This exponent is predicted to be connected with the exponent gamma related to the scaling of the average size with the duration of avalanches of activity. “Mean field” models of neural dynamics predict exponents beta and gamma equal or near 2 at criticality (brown noise), including the simple branching model and the fully-connected stochastic Wilson–Cowan model. We here show that a 2D version of the stochastic Wilson–Cowan model, where neuron connections decay exponentially with the distance, is characterized by exponents beta and gamma markedly different from those of mean field, respectively around 1 and 1.3. The exponents alpha and tau of avalanche size and duration distributions, equal to 1.5 and 2 in mean field, decrease respectively to 1.29pm 0.01 and 1.37pm 0.01. This seems to suggest the possibility of a different universality class for the model in finite dimension.

Mechanistic modeling as an explanatory tool for clinical treatment of chronic catatonia.

Mathematical modeling of neural systems is an effective means to integrate complex information about the brain into a numerical tool that can help explain observations. However, the use of neural models to inform clinical decisions has been limited. In this study, we use a simple model of brain circuitry, the Wilson-Cowan model, to predict changes in a clinical measure for catatonia, the Bush-Francis Catatonia Rating Scale, for use in clinical treatment of schizophrenia. This computational tool can then be used to better understand mechanisms of action for pharmaceutical treatments, and to fine-tune dosage in individual cases. We present the conditions of clinical care for a residential patient cohort, and describe methods for synthesizing data to demonstrated the functioning of the model. We then show that the model can be used to explain effect sizes of treatments and estimate outcomes for combinations of medications. We conclude with a demonstration of how this model could be personalized for individual patients to inform ongoing treatment protocols.

Dynamical Mechanism Analysis of Three Neuroregulatory Strategies on the Modulation of Seizures.

This paper attempts to explore and compare the regulatory mechanisms of optogenetic stimulation (OS), deep brain stimulation (DBS) and electromagnetic induction on epilepsy. Based on the Wilson-Cowan model, we first demonstrate that the external input received by excitatory and inhibitory neural populations can induce rich dynamic bifurcation behaviors such as Hopf bifurcation, and make the system exhibit epileptic and normal states. Then, both OS and DBS are shown to be effective in controlling the epileptic state to a normal low-level state, and the stimulus parameters have a broad effective range. However, electromagnetic induction cannot directly control epilepsy to this desired state, even if it can significantly reduce the oscillation frequency of neural populations. One main difference worth noting is that the high spatiotemporal specificity of OS allows it to target inhibitory neuronal populations, whereas DBS and electromagnetic induction can only stimulate excitatory as well as inhibitory neuronal populations together. Next, the propagation behavior of epilepsy is explored under a typical three-node feedback loop structure. An increase in coupling strength accelerates and exacerbates epileptic activity in other brain regions. Finally, OS and DBS applied to the epileptic focus play similar positive roles in controlling the behavior of the area of seizure propagation, while electromagnetic induction still only achieves unsatisfactory effects. It is hoped that these dynamical results can provide insights into the treatment of epilepsy as well as other neurological disorders.