Wilson-Cowan Equations Research Articles

Iterated Crank–Nicolson Runge–Kutta Methods and Their Application to Wilson–Cowan Equations and Electroencephalography Simulations

The Wilson–Cowan model has been widely applied for the simulation of electroencephalography (EEG) waves associated with neural activities in the brain. The Runge–Kutta (RK) method is commonly used to numerically solve the Wilson–Cowan equations. In this paper, we focus on enhancing the accuracy of the numerical method by proposing a strategy to construct a class of fourth-order RK methods using a generalized iterated Crank–Nicolson procedure, where the RK coefficients depend on a free parameter c2. When c2 is set to 0.5, our method becomes a special case of the classical fourth-order RK method. We apply the proposed methods to solve the Wilson–Cowan equations with two and three neuron populations, modeling EEG epileptic dynamics. Our simulations demonstrate that when c2 is set to 0.4, the proposed RK4-04 method yields smaller errors compared to those obtained using the classical fourth-order RK method. This is particularly visible when the spectral radius of the connection matrix or the excitation-inhibition coupling coefficient is relatively large.

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.

Stability of coupled Wilson–Cowan systems with distributed delays

Building upon our previous work on the Wilson-Cowan equations with distributed delays (Kaslik et al., 2022), we study the dynamic behavior in a system of two coupled Wilson-Cowan pairs. We focus in particular on understanding the mechanisms that govern the transitions in and out of oscillatory regimes associated with pathological behavior. We investigate these mechanisms under multiple coupling scenarios, and we compare the effects of using discrete delays versus a weak Gamma delay distribution. We found that, in order to trigger and stop oscillations, each kernel emphasizes different critical combinations of coupling weights and time delay, with the weak Gamma kernel restricting oscillations to a tighter locus of coupling strengths, and to a limited range of time delays. We finally illustrate the general analytical results with simulations for two particular applications: generation of beta-rhythms in the basal ganglia, and alpha oscillations in the prefrontal-limbic system.

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.

Approximating the Effect of Consciousness on Stochastic Brain Structures

There is a large amount of evidence in the Parapsychology literature that indicates thatconsciousness is not an emergent property of neuronal interactions and can exist andfunction independently of a brain. Here we examine mathematical methods that canbe used to derive dynamic equations for the mind-matter interactions occurring in thebrain that these observations imply the existence of. We use the moments method toapproximate the effect that consciousness has on stochastic binary decision-makingneural networks , which we model using biologically realistic Wilson-Cowan equations(Deco et al., 2007). We show that small changes in the variance of the randomness (onthe order of 0.1 Hz2) consumed by the neural networks can bias the networks to selectone binary decision value over the other. Using observations about the interconnectednessof relatively isolated groups of neurons, we argue that biases of the size predictedby the approximation would be sufficient to allow a brain-independent consciousnessto exert significant control over the brain. The results that we present here are relevantto any theory positing that rote computations carried out by neurons are not the solecontributor to consciousness.

Gell-Mann-Low Criticality in Neural Networks.

Criticality is deeply related to optimal computational capacity. The lack of a renormalized theory of critical brain dynamics, however, so far limits insights into this form of biological information processing to mean-field results. These methods neglect a key feature of critical systems: the interaction between degrees of freedom across all length scales, required for complex nonlinear computation. We present a renormalized theory of a prototypical neural field theory, the stochastic Wilson-Cowan equation. We compute the flow of couplings, which parametrize interactions on increasing length scales. Despite similarities with the Kardar-Parisi-Zhang model, the theory is of a Gell-Mann-Low type, the archetypal form of a renormalizable quantum field theory. Here, nonlinear couplings vanish, flowing towards the Gaussian fixed point, but logarithmically slowly, thus remaining effective on most scales. We show this critical structure of interactions to implement a desirable trade-off between linearity, optimal for information storage, and nonlinearity, required for computation.

Before and beyond the Wilson-Cowan equations.

The Wilson-Cowan equations represent a landmark in the history of computational neuroscience. Along with the insights Wilson and Cowan offered for neuroscience, they crystallized an approach to modeling neural dynamics and brain function. Although their iconic equations are used in various guises today, the ideas that led to their formulation and the relationship to other approaches are not well known. Here, we give a little context to some of the biological and theoretical concepts that lead to the Wilson-Cowan equations and discuss how to extend beyond them.

Hyperchaos in Wilson-Cowan oscillator circuits.

The Wilson-Cowan equations were originally shown to produce limit cycle oscillations for a range of parameters. Others subsequently showed that two coupled Wilson-Cowan oscillators could produce chaos, especially if the oscillator coupling was from inhibitory interneurons of one oscillator to excitatory neurons of the other. Here this is extended to show that chains, grids, and sparse networks of Wilson-Cowan oscillators generate hyperchaos with linearly increasing complexity as the number of oscillators increases. As there is now evidence that humans can voluntarily generate hyperchaotic visuomotor sequences, these results are particularly relevant to the unpredictability of a range of human behaviors. These also include incipient senescence in aging, effects of concussive brain injuries, autism, and perhaps also intelligence and creativity.NEW & NOTEWORTHY This paper represents an exploration of hyperchaos in coupled Wilson-Cowan equations. Results show that hyperchaos (number of positive Lyapunov exponents) grows linearly with the number of oscillators in the array and leads to high levels of unpredictability in the neural response.

Visual illusions via neural dynamics: Wilson-Cowan-type models and the efficient representation principle.

We reproduce suprathreshold perception phenomena, specifically visual illusions, by Wilson-Cowan (WC)-type models of neuronal dynamics. Our findings show that the ability to replicate the illusions considered is related to how well the neural activity equations comply with the efficient representation principle. Our first contribution consists in showing that the WC equations can reproduce a number of brightness and orientation-dependent illusions. Then we formally prove that there cannot be an energy functional that the WC dynamics are minimizing. This leads us to consider an alternative, variational modeling, which has been previously employed for local histogram equalization (LHE) tasks. To adapt our model to the architecture of V1, we perform an extension that has an explicit dependence on local image orientation. Finally, we report several numerical experiments showing that LHE provides a better reproduction of visual illusions than the original WC formulation, and that its cortical extension is capable also to reproduce complex orientation-dependent illusions.NEW & NOTEWORTHY We show that the Wilson-Cowan equations can reproduce a number of brightness and orientation-dependent illusions. Then we formally prove that there cannot be an energy functional that the Wilson-Cowan equations are minimizing, making them suboptimal with respect to the efficient representation principle. We thus propose a slight modification that is consistent with such principle and show that this provides a better reproduction of visual illusions than the original Wilson-Cowan formulation. We also consider the cortical extension of both models to deal with more complex orientation-dependent illusions.

Traveling waves in a spatially-distributed Wilson–Cowan model of cortex: From fronts to pulses

Wave propagation in excitable media has been studied in various biological, chemical, and physical systems. Waves are among the most common evoked and spontaneous organized activity seen in cortical networks. In this paper, we study traveling fronts and pulses in a spatially-extended version of the Wilson–Cowan equations, a neural firing rate model of sensory cortex having two population types: Excitatory and inhibitory. We are primarily interested in the case when the local or space-clamped dynamics has three fixed points: (1) a stable down state; (2) a saddle point with stable manifold that acts as a threshold for firing; (3) an up state having stability that depends on the time scale of the inhibition. In the case when the up state is stable, we look for wave fronts, which transition the media from a down to up state, and when the up state is unstable, we are interested in pulses, a transient increase in firing that returns to the down state. We explore the behavior of these waves as the time and space scales of the inhibitory population vary. Some interesting findings include bistability between a traveling front and pulse, fronts that join the down state to an oscillation or spatiotemporal pattern, and pulses which go through an oscillatory instability.

Firing rate equations require a spike synchrony mechanism to correctly describe fast oscillations in inhibitory networks.

Recurrently coupled networks of inhibitory neurons robustly generate oscillations in the gamma band. Nonetheless, the corresponding Wilson-Cowan type firing rate equation for such an inhibitory population does not generate such oscillations without an explicit time delay. We show that this discrepancy is due to a voltage-dependent spike-synchronization mechanism inherent in networks of spiking neurons which is not captured by standard firing rate equations. Here we investigate an exact low-dimensional description for a network of heterogeneous canonical Class 1 inhibitory neurons which includes the sub-threshold dynamics crucial for generating synchronous states. In the limit of slow synaptic kinetics the spike-synchrony mechanism is suppressed and the standard Wilson-Cowan equations are formally recovered as long as external inputs are also slow. However, even in this limit synchronous spiking can be elicited by inputs which fluctuate on a time-scale of the membrane time-constant of the neurons. Our meanfield equations therefore represent an extension of the standard Wilson-Cowan equations in which spike synchrony is also correctly described.

Wilson-Cowan Equations for Neocortical Dynamics.

In 1972–1973 Wilson and Cowan introduced a mathematical model of the population dynamics of synaptically coupled excitatory and inhibitory neurons in the neocortex. The model dealt only with the mean numbers of activated and quiescent excitatory and inhibitory neurons, and said nothing about fluctuations and correlations of such activity. However, in 1997 Ohira and Cowan, and then in 2007–2009 Buice and Cowan introduced Markov models of such activity that included fluctuation and correlation effects. Here we show how both models can be used to provide a quantitative account of the population dynamics of neocortical activity.We first describe how the Markov models account for many recent measurements of the resting or spontaneous activity of the neocortex. In particular we show that the power spectrum of large-scale neocortical activity has a Brownian motion baseline, and that the statistical structure of the random bursts of spiking activity found near the resting state indicates that such a state can be represented as a percolation process on a random graph, called directed percolation.Other data indicate that resting cortex exhibits pair correlations between neighboring populations of cells, the amplitudes of which decay slowly with distance, whereas stimulated cortex exhibits pair correlations which decay rapidly with distance. Here we show how the Markov model can account for the behavior of the pair correlations.Finally we show how the 1972–1973 Wilson–Cowan equations can account for recent data which indicates that there are at least two distinct modes of cortical responses to stimuli. In mode 1 a low intensity stimulus triggers a wave that propagates at a velocity of about 0.3 m/s, with an amplitude that decays exponentially. In mode 2 a high intensity stimulus triggers a larger response that remains local and does not propagate to neighboring regions.

Modeling cortical dynamics with Wilson-Cowan equations

Experimental data collected over the last decade indicates that there exist at least two distinct modes of cortical response to stimuli. In mode 1 a low intensity stimulus triggers a wave that propagates at a velocity of about 0.3 m/sec, with an amplitude that decays exponentially. In mode 2 a high intensity stimulus triggers a larger response that remains local, and does not propagate to neighboring regions. Other data indicate that unstimulated or resting cortex exhibits pair correlations between neighboring cells, the amplitudes of which decay slowly with distance, whereas stimulated cortex exhibits pair correlations whose amplitude falls of rapidly with distance. Here we show how the mean-field Wilson-Cowan equations can account precisely for the two modes of cortical response, and how stochastic Wilson-Cowan equations can account for the behavior of the pair correlations. We will present these results after outlining the basic properties of both the mean-field and stochastic equations.

Modeling lightness induction with Wilson-Cowan equations

Modeling lightness induction with Wilson-Cowan equations

A model of color constancy and efficient coding can predict lightness induction

A recent variational implementation of Land's Retinex theory of color vision can improve the appearance of images without introducing artifacts (Bertalmío et al. 2009). Through anti-symmetrization of the resulting partial differential equation, this methodology emulates color constancy on both under and over exposed images. Remarkably, this formulation of Retinex is formally equivalent to local histogram equalization and has the same form as the Wilson-Cowan equations that model the response of populations of cortical neurons. The model cannot predict the phenomena of color assimilation because it can only increase the contrast of an image. However an addition to the model, designed to allow it to reach a steady state in a constant time regardless of stimulus structure, can predict assimilation. This approach weights the magnitude of each convergence step by computing the ratio of image variance and histogram uniformity. This model can predict the perception of achromatic induction from the data of Rudd (2010) investigating the perceived lightness of a disk-and-ring stimulus. In particular the model predicts when contrast switches to assimilation, and it can also predict how the slope and curvature of such functions vary with ring width and polarity. Meeting abstract presented at VSS 2014

Laws of Large Numbers and Langevin Approximations for Stochastic Neural Field Equations

In this study, we consider limit theorems for microscopic stochastic models of neural fields. We show that the Wilson–Cowan equation can be obtained as the limit in uniform convergence on compacts in probability for a sequence of microscopic models when the number of neuron populations distributed in space and the number of neurons per population tend to infinity. This result also allows to obtain limits for qualitatively different stochastic convergence concepts, e.g., convergence in the mean. Further, we present a central limit theorem for the martingale part of the microscopic models which, suitably re-scaled, converges to a centred Gaussian process with independent increments. These two results provide the basis for presenting the neural field Langevin equation, a stochastic differential equation taking values in a Hilbert space, which is the infinite-dimensional analogue of the chemical Langevin equation in the present setting. On a technical level, we apply recently developed law of large numbers and central limit theorems for piecewise deterministic processes taking values in Hilbert spaces to a master equation formulation of stochastic neuronal network models. These theorems are valid for processes taking values in Hilbert spaces, and by this are able to incorporate spatial structures of the underlying model.Mathematics Subject Classification (2000): 60F05, 60J25, 60J75, 92C20.

Top-Down Influences on Local Networks: Basic Theory with Experimental Implications

The response of a population of cortical neurons to an external stimulus depends not only on the receptive field properties of the neurons, but also the level of arousal and attention or goal-oriented cognitive biases that guide information processing. These top-down effects on cortical neurons bias the output of the neurons and affect behavioral outcomes such as stimulus detection, discrimination, and response time. In any physiological study, neural dynamics are observed in a specific brain state; the background state partly determines neuronal excitability. Experimental studies in humans and animal models have also demonstrated that slow oscillations (typically in the alpha or theta bands) modulate the fast oscillations (gamma band) associated with local networks of neurons. Cross-frequency interaction is of interest as a mechanism for top-down or bottom up interactions between systems at different spatial scales. We develop a generic model of top-down influences on local networks appropriate for comparison with EEG. EEG provides excellent temporal resolution to investigate neuronal oscillations but is space-averaged on the cm scale. Thus, appropriate EEG models are developed in terms of population synaptic activity. We used the Wilson–Cowan population model to investigate fast (gamma band) oscillations generated by a local network of excitatory and inhibitory neurons. We modified the Wilson–Cowan equations to make them more physiologically realistic by explicitly incorporating background state variables into the model. We found that the population response is strongly influenced by the background state. We apply the model to reproduce the modulation of gamma rhythms by theta rhythms as has been observed in animal models and human ECoG and EEG studies. The concept of a dynamic background state presented here using the Wilson–Cowan model can be readily applied to incorporate top-down modulation in more detailed models of specific cortical systems.

A dynamical neural simulation mapping feature-based attention to location with non-linear cortical circuits

Visual attention mechanisms allow the visual cortex todirect cortical processing to salient locations within avisual scene, at the expense of diminished processing ofstimuli at other locations. The deployment of attentionto a given location is known as spatial attention.Attention may also be applied to a known feature in avisual scene, such as a colour or shape, enhancing itsoccurrences across the visual field. In [1], a model ofbinding distributed feature representations throughfeature-based attention predicted that attentional feed-back would act in all positions of the visual scene in lowlevel visual areas. There is now experimental evidencethat this mechanism exists [2].We present here a neural dynamical model of feature-based attention that directs attention to salient locationsinamulti-objectscenethrough interaction of stimulusdriven and cue driven neural activation. This interactionis managed by a disinhibition circuit gating output to aretinotopic neural region of the dorsal stream, thelateral intraparietal area (LIP). When visual stimuli arepresented to primary visual cortex (V1), neural activa-tion spreads into higher ventral stream areas throughincreasing neural receptive fields between layers.Stimulus driven neurons activate excitatory output andinhibitory interneuron populations in the disinhbitioncircuit. When a salient feature is cued by activation ofits associated anterior inferotemporal (AIT) neuronalpopulation, neural activity propagates to lower visualareas across all locations of the visual field. Activity inthis top-down flow inhibits the inhibitory interneuronpopulation allowing excitatory output from the disinhi-bition circuit to LIP in a retinotopic manner whereactivity is correlated in the stimulus and cue drivenpathways. In areas where the two pathways do notmatch, local inhibition in the disinhibition layer preventsneighbouring circuits’ excitatory populations from out-putting to LIP.The model exhibits effects of feature-based attention,such as the feature similarity gain principle and biasedcompetition through interactions of stimulus and cueinitiated neural activity. The dynamical nature of themodel as neural populations of Wilson-Cowan differen-tial equations allows the time-course of neural activityto be investigated.ConclusionsAlthough originally designed as a model for binding, themodel replicates the experimental correlates of feature-based attention, which strongly suggests its functionalrole is actually binding An important prediction of themodel is that local non-matches between cue andstimulus are the most important information projectedfrom the ventral to the dorsal stream and that this inter-action is highly non-linear. We present candidates forneural circuits implementing this interaction.

Modelling gait transition in two-legged animals

The study of locomotor patterns has been a major research goal in the last decades. Understanding how intralimb and interlimb coordination works out so well in animals’ locomotion is a hard and challenging task. Many models have been proposed to model animal’s rhythms. These models have also been applied to the control of rhythmic movements of adaptive legged robots, namely biped, quadruped and other designs. In this paper we study gait transition in a central pattern generator (CPG) model for bipeds, the 4-cells model. This model is proposed by Golubitsky, Stewart, Buono and Collins and is studied further by Pinto and Golubitsky. We briefly resume the work done by Pinto and Golubitsky. We compute numerically gait transition in the 4-cells CPG model for bipeds. We use Morris–Lecar equations and Wilson–Cowan equations as the internal dynamics for each cell. We also consider two types of coupling between the cells: diffusive and synaptic. We obtain secondary gaits by bifurcation of primary gaits, by varying the coupling strengths. Nevertheless, some bifurcating branches could not be obtained, emphasizing the fact that despite analytically those bifurcations exist, finding them is a hard task and requires variation of other parameters of the equations. We note that the type of coupling did not influence the results.

Emergent oscillations in networks of stochastic spiking neurons.

Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.