Cowan Model 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.

Extension of the SpK atomic physics code to generate global equation of state data

Global microphysics models are required for the modelling of high-energy-density physics (HEDP) experiments, the improvement of which are critical to the path to inertial fusion energy. This work presents further developments to the atomic and microphysics code, SpK, part of the numerical modelling suite of Imperial College London and First Light Fusion. We extend the capabilities of SpK to allow the calculation of the equation of state (EoS). The detailed configuration accounting calculations are interpolated into finite-temperature Thomas–Fermi calculations at high coupling to form the electronic component of the model. The Cowan model provides the ionic contribution, modified to approximate the physics of diatomic molecular dissociation. By utilising bonding corrections and performing a Maxwell construction, SpK captures the EoS from states ranging from the zero-pressure solid, through the liquid–vapour coexistence region and into plasma states. This global approach offers the benefit of capturing electronic shell structure over large regions of parameter space, building highly-resolved tables in minutes on a simple desktop. We present shock Hugoniot and off-Hugoniot calculations for a number of materials, comparing SpK to other models and experimental data. We also apply EoS and opacity data generated by SpK in integrated simulations of indirectly-driven capsule implosions, highlighting physical sensitivities to the choice of EoS models.

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.

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.

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.

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.

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.

The nonequilibrium potential today: A short review

A brief review is made of the birth and evolution of the “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson–Cowan model (a population-competition model of the neocortex) and a “thermodynamic uncertainty relation” (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson–Cowan model and relevant aspects of KPZ phenomenology like the EW–KPZ crossover and the memory of initial conditions.

Оценка состояния пилота по данным видеоокулографии и параметрам альфа-ритмов головного мозга путем моделирования его нейронной активности

The H.R. Wilson and J.D. Cowan model of human brain neural activity is considered, which connects the latent (non-measurable) parameters of external excitation with the brain neuronal activity parameters which are available for observation, with alpha rhythm frequencies being in use in this capacity. By means of computational experiments, revealed are the ranges of latent parameters, which ensure the occurrence of the Poincaré—Andronov—Hopf bifurcation (transforming the spiral sink to the spiral source equilibrium point with a limit cycle, or vice versa) that leads, in the first approximation, to the appearance of alpha rhythms. Relative changes of the latent parameters representing external excitation are estimated through the primary indices of oculomotor activity (gaze movement entropies and gaze fixation durations), which makes it possible to assess the pilot proximity to falling into a low efficiency state, including the real time mode.

Brain rhythm bursts are enhanced by multiplicative noise.

Many healthy and pathological brain rhythms, including beta and gamma rhythms and essential tremor, are suspected to be induced by noise. This yields randomly occurring, brief epochs of higher amplitude oscillatory activity known as "bursts," the statistics of which are important for proper neural function. Here, we consider a more realistic model with both multiplicative and additive noise instead of only additive noise, to understand how state-dependent fluctuations further affect rhythm induction. For illustrative purposes, we calibrate the model at the lower end of the beta band that relates to movement; parameter tuning can extend the relevance of our analysis to the higher frequency gamma band or to lower frequency essential tremors. A stochastic Wilson-Cowan model for reciprocally as well as self-coupled excitatory (E) and inhibitory (I) populations is analyzed in the parameter regime where the noise-free dynamics spiral in to a fixed point. Noisy oscillations known as quasi-cycles are then generated by stochastic synaptic inputs. The corresponding dynamics of E and I local field potentials can be studied using linear stochastic differential equations subject to both additive and multiplicative noises. As the prevalence of bursts is proportional to the slow envelope of the E and I firing activities, we perform an envelope-phase decomposition using the stochastic averaging method. The resulting envelope dynamics are uni-directionally coupled to the phase dynamics as in the case of additive noise alone but both dynamics involve new noise-dependent terms. We derive the stationary probability and compute power spectral densities of envelope fluctuations. We find that multiplicative noise can enhance network synchronization by reducing the magnitude of the negative real part of the complex conjugate eigenvalues. Higher noise can lead to a "virtual limit cycle," where the deterministically stable eigenvalues around the fixed point acquire a positive real part, making the system act more like a noisy limit cycle rather than a quasi-cycle. Multiplicative noise can thus exacerbate synchronization and possibly contribute to the onset of symptoms in certain motor diseases.

Effect of radiation on compressibility of hot dense sodium and iron plasma using improved screened hydrogenic model with l splitting

High pressure investigations of matter involve the study of strong shock wave dynamics within the materials which gives rise to many thermal effects leading to dissociation of molecules, ionization of atoms, and radiation emission, etc. The response of materials experiencing a strong shock can be determined by its shock Hugoniot calculations which are frequently applied in numerical and experimental studies in inertial confinement fusion, laboratory astrophysical plasma, etc. These studies involve high energy density plasmas in which the radiation plays an important role in determining the energy deposition and maximum compressibility achieved by the shock within material. In this study, we present an investigation for the effect of radiation pressure on the maximum compressibility of the material using shock Hugoniot calculations. In shock Hugoniot calculations, an equation of state (EOS) is developed in which electronic contributions for EOS calculations are taken from an improved screened hydrogenic model with –l splitting (I-SHML) [High Energy Density Physics (2018) 26 48] under local thermodynamic equilibrium (LTE) conditions. The thermal ionic part calculations are adopted from the state of the art Cowan model while the cold ionic contributions are adopted from the scaled binding energy model. The Shock Hugoniot calculations are carried out for sodium and iron plasmas and our calculated results show excellent agreement with published results obtained by using either sophisticated self-consistent models or the first principle study.

Progressive alignment of inhibitory and excitatory delay may drive a rapid developmental switch in cortical network dynamics.

Nervous system maturation occurs on multiple levels-synaptic, circuit, and network-at divergent timescales. For example, many synaptic properties mature gradually, whereas emergent network dynamics can change abruptly. Here we combine experimental and theoretical approaches to investigate a sudden transition in spontaneous and sensory evoked thalamocortical activity necessary for the development of vision. Inspired by in vivo measurements of timescales and amplitudes of synaptic currents, we extend the Wilson and Cowan model to take into account the relative onset timing and amplitudes of inhibitory and excitatory neural population responses. We study this system as these parameters are varied within amplitudes and timescales consistent with developmental observations to identify the bifurcations of the dynamics that might explain the network behaviors in vivo. Our findings indicate that the inhibitory timing is a critical determinant of thalamocortical activity maturation; a gradual decay of the ratio of inhibitory to excitatory onset time drives the system through a bifurcation that leads to a sudden switch of the network spontaneous activity from high-amplitude oscillations to a nonoscillatory active state. This switch also drives a change from a threshold bursting to linear response to transient stimuli, also consistent with in vivo observation. Thus we show that inhibitory timing is likely critical to the development of network dynamics and may underlie rapid changes in activity without similarly rapid changes in the underlying synaptic and cellular parameters.NEW & NOTEWORTHY Relying on a generalization of the Wilson-Cowan model, which allows a solid analytic foundation for the understanding of the link between maturation of inhibition and network dynamics, we propose a potential explanation for the role of developing excitatory/inhibitory synaptic delays in mediating a sudden switch in thalamocortical visual activity preceding vision onset.

The role of network structure and time delay in a metapopulation Wilson--Cowan model

We study the dynamics of a network Wilson--Cowan model (a system of connected Wilson--Cowan oscillators) for interacting excitatory and inhibitory neuron populations with time delays. Each node in this model corresponds to a population of neurons, including excitatory and inhibitory subpopulations, and hence it can be viewed as a metapopulation model. It is known that information transfer within each cortical area is not instantaneous, and therefore we consider a system of delay differential equations with two different kinds of discrete delays. We account for the time delay in information propagation between individual excitatory and inhibitory subpopulations at each node via intra-node time delays, and we account for time delay in information propagation between neuron populations at different nodes with inter-node time delays. The biologically relevant resting state solutions are oscillatory (stable limit cycles). After determining the influence of the coupling parameters between nodes, the intra-node delays, and the inter-node delays on the dynamics of the two coupled Wilson--Cowan oscillators, we then explore a variety of larger networks of 16 and 100 nodes, in order to determine how the network topology will influence time delayed Wilson--Cowan dynamics. We find that network structure can regularize or deregularize the dynamics, with networks of higher mean degree permitting stable limit cycles and networks with smaller mean degree yielding less regular dynamics (which may range from chaotic solutions, to solutions for which limit cycles collapse into steady states, which are biologically undesirable compared with the preferred stable limit cycles). Furthermore, heterogeneity in the degree distribution of the network (resulting from networks with nodes of varying degree) can result in asynchronous dynamics, even if at each node the local dynamics are that of a limit cycle, in contrast to the synchronization of dynamics between nodes seen when the degree of all nodes is equal. This suggests that homogeneous and well-connected networks permit robust limit cycles under time-delayed Wilson--Cowan dynamics, whereas heterogeneous or poorly connected networks may fail to provide such desirable dynamics, a phenomena akin to structural loss of neuron connections in neurodegenerative diseases.

Digital Hardware Implementation of Gaussian Wilson–Cowan Neocortex Model

Hardware implementation of biological neural models can help in better understanding of the brain functionality, implementing cognitive tasks, and also studying the brain diseases. Gaussian Wilson–Cowan model as one of the well-known population-based models represents neuronal functionality in neocortex. In this paper, Gaussian Wilson–Cowan model is investigated in terms of its digital implementation feasibility. Digital model is proposed for the Gaussian Wilson–Cowan and examined from dynamical and timing behavior point of view. The evaluations indicate that the digitized model is able to reproduce the dynamical bifurcations as the original model is capable of. An efficient digital hardware system is given for the proposed model with minimum required resources using Verilog Hardware Description Language. Digital architectures are physically implemented on an Altera FPGA board. Experimental results show that the proposed circuits take maximum 2% of the available resources of a Stratix Altera board. In addition, static timing analysis indicates that the circuits can work in a maximum frequency of 244 MHz.

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.

Comparison of Three Models Dealing with Working Memory and Its Dimensions in Second Language Acquisition

The current status of research on working memory (WM) and its components in second language acquisition (SLA) was examined in this review. Literature search was done on four aspects using search terms in Google Scholar. Hence, the review results are given and introduced. 1. In the definition of WM, some confusion exists on whether short term memory (STM) or recent memory is the same as WM or different. 2. In this review, three main models have been discussed elaborately, as they are the only ones discussed in literature. They are: multicomponent model of Baddeley (2000), embedded process model of Cowan (2005) and attention control model of Engle and Kane (2003). 3. The phonological and executive components of WM were examined in more detail, as these determine the two basic aspects of language acquisition: language characteristics and acquisition methods (Wen, 2012). Overall, the variables related to phonological and executive working memories are evident from published research, but their interactive relationships and affecting factors are not entirely clear. 4. Admittedly, several diverse internal and external factors affect WM in relation to SLA. Some practically useful interventions are indicated by certain findings.

Diffusion approximation of the stochastic Wilson–Cowan model

We consider a stochastic version of the Wilson–Cowan model which accommodates for discrete populations of excitatory and inhibitory neurons. The model assumes a finite carrying capacity with the two populations being constant in size. The master equation that governs the dynamics of the stochastic model is analyzed by an expansion in powers of the inverse population size, yielding a coupled pair of non-linear Langevin equations with multiplicative noise. Gillespie simulations show the validity of the obtained approximation, for the parameter region where the system exhibits dynamical bistability. We report analytical progress by silencing the retroaction of excitatory neurons on inhibitory neurons, while still assigning the parameters so to fall in the region of deterministic bistability for the excitatory species. The proposed approach forms the basis of a perturbative generalization which applies to the case where a modest degree of coupling is restored.

Formal Derivation of Lotka–Volterra–Haken Amplitude Equations of Task-Related Brain Activity in Multiple, Consecutively Performed Tasks

The Lotka–Volterra–Haken equations have been frequently used in ecology and pattern formation. Recently, the equations have been proposed by several research groups as amplitude equations for task-related patterns of brain activity. In this theoretical study, the focus is on the circular causality aspect of pattern formation systems as formulated within the framework of synergetics. Accordingly, the stable modes of a pattern formation system inhibit the unstable modes, whereas the unstable modes excite the stable modes. Using this circular causality principle it is shown that under certain conditions the Lotka–Volterra–Haken amplitude equations can be derived from a general model of brain activity akin to the Wilson–Cowan model. The model captures the amplitude dynamics for brain activity patterns in experiments involving several consecutively performed multiple-choice tasks. This is explicitly demonstrated for two-choice tasks involving grasping and walking. A comment on the relevance of the theoretical framework for clinical psychology and schizophrenia is given as well.