Stochastic Neural Field Research Articles

Well-posedness and stability of a stochastic neural field in the form of a partial differential equation

A system of partial differential equations representing stochastic neural fields was recently proposed with the aim of modelling the activity of noisy grid cells when a mammal travels through physical space. The system was rigorously derived from a stochastic particle system and its noise-driven pattern-forming bifurcations have been characterised. However, due to its nonlinear and non-local nature, standard well-posedness theory for smooth time-dependent solutions of parabolic equations does not apply. In this article, we transform the problem through a suitable change of variables into a nonlinear Stefan-like free boundary problem and use its representation formulae to construct local-in-time smooth solutions under mild hypotheses. Then, we prove that fast-decaying initial conditions and globally Lipschitz modulation functions imply that solutions are global-in-time. Last, we improve previous linear stability results by showing nonlinear asymptotic stability of stationary solutions under suitable assumptions.

Maximum-correntropy-based sequential method for fast neural population activity reconstruction in the cortex from incomplete abnormally-disturbed noisy measurements

This paper continues to explore the membrane potential reconstruction and pattern recognition problem in a neural tissue modeled by Stochastic Dynamic Neural Field (SDNF) equation. Although recent research has suggested an efficient solution based on the state-space approach through nonlinear Bayesian filtering framework, it is becoming extremely difficult to ignore the existence of non-Gaussian uncertainties in the SDNFs as well as the stability problem of neuronal population dynamics to outliers. Motivated by recent events in signal processing and mathematical neuroscience, this paper explores the SDNFs in a presence of non-Gaussian uncertainties, which is the shot noise case, where the corrupted data might appear due to broken sensors. We derive the “distributionally robust” state estimator for the membrane potential reconstruction process that is the Maximum Correntropy Criterion Extended Kalman Filter (MCC-EKF) as well as its fast and numerically robust (to roundoff) implementation method by using the sequential principle of processing the measurement vectors. The numerical experiments are provided to illustrate the performance of the novel estimation methods.

Dynamical behaviors of an impulsive stochastic neural field lattice model

This paper is concerned with the asymptotic behaviors of the solutions of an impulsive stochastic neural field lattice model driven by nonlinear noise. We first show the existence and uniqueness of weak pullback mean random attractors for the impulsive stochastic systems. Then by the properties of Markov processes, the existence of evolution system of measures for the impulsive stochastic systems is established. To this end, we employ the idea of uniform estimates on the tails of the solutions to show the tightness of a family of distributions of the solutions of the lattice systems.

Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity

Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity

Euler-Maruyama-Based Data-Driven State Restoration and Parameter Adaptation in Stochastic Neural Fields with Finite Signal Transmission Rate

Euler-Maruyama-Based Data-Driven State Restoration and Parameter Adaptation in Stochastic Neural Fields with Finite Signal Transmission Rate

Noise-driven bifurcations in a nonlinear Fokker–Planck system describing stochastic neural fields

The existence and characterisation of noise-driven bifurcations from the spatially homogeneous stationary states of a nonlinear, non-local Fokker–Planck type partial differential equation describing stochastic neural fields is established. The resulting theory is extended to a system of partial differential equations modelling noisy grid cells. It is shown that as the noise level decreases, multiple bifurcations from the homogeneous steady state occur. Furthermore, the shape of the branches at a bifurcation point is characterised locally. The theory is supported by a set of numerical illustrations of the condition leading to bifurcations, the patterns along the corresponding local bifurcation branches, and the stability of the homogeneous state and the most prevalent pattern: the hexagonal one.

Data-driven parameter estimation in stochastic dynamic neural fields by state-space approach and continuous-discrete extended Kalman filtering

This work examines the state and parameter estimation problems for Stochastic Dynamic Neural Field (SDNF) models. Unlike other studies, we suggest a general approach based on the Galerkin method that reduces the SDNF model to a standard state-space representation. This naturally yields nonlinear continuous-discrete stochastic systems with complicated nonlinear dynamics and linear measurements of a high dimension because of a spatial approximation. This in turn paves a systematic way for a strong integration of nonlinear Bayesian filtering methods into mathematical neuroscience. Because of a high resolution utilized to keep an accurate spatial approximation, a key aspect of nonlinear filters to be derived for the SDNF models is their computational budgets. In this paper, we derive the sequential Itô-Taylor-based continuous-discrete Extended Kalman filter for state and parameter estimation of the SDNF models. The novel method processes each measurement vector in a component-like manner and, hence, avoids an inversion of a large matrix. Thus, the newly-derived sequential estimator is faster and, additionally, more numerically stable compared to the previously developed SDNF-oriented estimators. Besides, the novel filter is an adaptive method, i.e. it solves both problems, which are the state and parameter estimation, in parallel. Finally, the proposed method is applied to numerical examples to provide the comparative study and to show the effectiveness of our result.

Distinct Excitatory and Inhibitory Bump Wandering in a Stochastic Neural Field.

Localized persistent cortical neural activity is a validated neural substrate of parametric working memory. Such activity "bumps" represent the continuous location of a cue over several seconds. Pyramidal (excitatory ()) and interneuronal (inhibitory ()) subpopulations exhibit tuned bumps of activity, linking neural dynamics to behavioral inaccuracies observed in memory recall. However, many bump attractor models collapse these subpopulations into a single joint /(lateral inhibitory) population and do not consider the role of interpopulation neural architecture and noise correlations. Both factors have a high potential to impinge upon the stochastic dynamics of these bumps, ultimately shaping behavioral response variance. In our study, we consider a neural field model with separate / populations and leverage asymptotic analysis to derive a nonlinear Langevin system describing / bump interactions. While the bump attracts the bump, the bump stabilizes but can also repel the bump, which can result in prolonged relaxation dynamics when both bumps are perturbed. Furthermore, the structure of noise correlations within and between subpopulations strongly shapes the variance in bump position. Surprisingly, higher interpopulation correlations reduce variance.

Accuracy analysis of numerical simulations and noisy data assimilations in two-dimensional stochastic neural fields with infinite signal transmission speed

This study addresses the accuracy of stochastic simulations performed in Two-Dimensional Stochastic Neural Fields (2D-SNFs) with the infinite signal transmission speed and in the presence of external stimuli input. The numerical method in use belongs to the family of Galerkin-kind spectral approximations to Two-Dimensional Stochastic Neural Field Equations (2D-SNFEs). It translates the partial integro-differential fashion of such models into a large system of ordinary Stochastic Differential Equations (SDEs). Eventually, these SDEs are integrated approximately by the Euler–Maruyama scheme of the strong convergence order 0.5. In this paper, we devise a different-order approximate solution to the SNFE models at hand and look at the difference of such stochastic simulations on average for evaluating the consistency of the Euler–Maruyama-based numerical solution derived. The error committed in the 2D-SNFE-numerical-integration-scheme under study becomes available in our research. The other issue of particular attention and interest is hidden state reconstructions rooted in the 2D-SNFE approximations and incomplete noisy measurements of the membrane potential fulfilled at some user-assigned space positions and time instants. This statement leads to high-dimensional prediction and filtering problems to be solved. Here, we implement the Extended Kalman Filtering (EKF) approach, but accommodate it to our 2D-SNFE-oriented data assimilation scheme of huge size because of the two-dimensional manner of the stochastic process models in use. A sound performance of the newly-devised hidden state estimation technique is observed and exposed on a challenging 2D-SNFE example of computational neuroscience in Matlab.

Self-organized criticality in a mesoscopic model of excitatory-inhibitory neuronal populations by short-term and long-term synaptic plasticity.

Dynamics of an interconnected population of excitatory and inhibitory spiking neurons wandering around a Bogdanov-Takens (BT) bifurcation point can generate the observed scale-free avalanches at the population level and the highly variable spike patterns of individual neurons. These characteristics match experimental findings for spontaneous intrinsic activity in the brain. In this paper, we address the mechanisms causing the system to get and remain near this BT point. We propose an effective stochastic neural field model which captures the dynamics of the mean-field model. We show how the network tunes itself through local long-term synaptic plasticity by STDP and short-term synaptic depression to be close to this bifurcation point. The mesoscopic model that we derive matches the directed percolation model at the absorbing state phase transition.

Numerical simulations of one- and two-dimensional stochastic neural field equations with delay.

Neural Field Equations (NFE) are intended to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind of integro-differential equations proved to be a useful tool to describe the spatiotemporal neuronal activity from a macroscopic point of view, allowing the study of a wide variety of neurobiological phenomena, such as the sensory stimuli processing. The present article aims to study the effects of additive noise in one- and two-dimensional neural fields, while taking into account finite axonal velocity and an external stimulus. A Galerkin-type method is presented, which applies Fast Fourier Transforms to optimise the computational effort required to solve these equations. The explicit Euler-Maruyama scheme is implemented to obtain the stochastic numerical solution. An open-source numerical solver written in Julia was developed to simulate the neural fields in study.

Numerical solution of the stochastic neural field equation with applications to working memory

The main goal of the present work is to investigate the effect of noise in some neural fields, used to simulate working memory processes. The underlying mathematical model is a stochastic integro-differential equation. In order to approximate this equation we apply a numerical scheme which uses the Galerkin method for the space discretization. In this way we obtain a system of stochastic differential equations, which are then approximated in two different ways, using the Euler–Maruyama and the Itô–Taylor methods. We apply this numerical scheme to explain how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Numerical examples are presented and their results are discussed.

Sequential method for fast neural population activity reconstruction in the cortex from incomplete noisy measurements

During recent years there has been a growing interest in stochastic dynamic neural fields employed for modeling and predictions in biomedical and technical systems. In this paper, given some incomplete noisy data available from sensors, we propose and explore a state estimation method for fast restorations of membrane potential in the cortex based on such measurements and the Amari equation used for simulations of neural population activity in a stochastic setting. Our novel technique relies upon a Galerkin-type spectral approximation utilized within the conventional state-space approach. Translating a stochastic system into its state-space form creates a straightforward and fruitful way to the data-driven parameter estimation, filtering, prediction and smoothing. The present study is particularly focused on establishing a nonlinear stochastic Galerkin-spectral-approximation-induced system of large size, which is further estimated by the traditional extended Kalman filter (EKF). The efficiency of calculations is the main purpose of our research. That is why the fast filtering solution devised is based on processing the incoming data incrementally, that is, by processing measurements one at a time, rather than handling them as a unified high-dimensional vector. Such sequential filters suit well for dealing with large data sets as well as with real-time on-line computations. Also, their derivation and substantiation is of great interest in the context of neural network training because of large stochastic systems arisen there. In comparison to the batch filtering, our novel algorithm reduces the computational cost of membrane potential reconstructions in terms of the amount of grid nodes N accepted in the underlying spacial discretization, significantly. Apart from its computation efficiency, this sequential method is more robust to round-off errors committed within a computer-based finite precision arithmetics than the classical EKF because of the (N × N)-matrix inversion elimination from such membrane potential calculations. The superior performance of our technique is examined and confirmed in comparison to the batch one on two known scenarios in the dynamic neural field modeling.

Plastic systemic inhibition controls amplitude while allowing phase pattern in a stochastic neural field model.

We investigate oscillatory phase pattern formation and amplitude control for a linearized stochastic neuron field model by simulating Mexican-hat-coupled stochastic processes. We find, for several choices of parameters, that spatial pattern formation in the temporal phases of the coupled processes occurs if and only if their amplitudes are allowed to grow unrealistically large. Stimulated by recent work on homeostatic inhibitory plasticity, we introduce static and plastic (adaptive) systemic inhibitory mechanisms to keep the amplitudes stochastically bounded. We find that systems with static inhibition exhibited bounded amplitudes but no sustained phase patterns. With plastic systemic inhibition, on the other hand, the resulting systems exhibit both bounded amplitudes and sustained phase patterns. These results demonstrate that plastic inhibitory mechanisms in neural field models can dynamically control amplitudes while allowing patterns of phase synchronization to develop. Similar mechanisms of plastic systemic inhibition could play a role in regulating oscillatory functioning in the brain.

Graph neural fields: A framework for spatiotemporal dynamical models on the human connectome.

Tools from the field of graph signal processing, in particular the graph Laplacian operator, have recently been successfully applied to the investigation of structure-function relationships in the human brain. The eigenvectors of the human connectome graph Laplacian, dubbed "connectome harmonics", have been shown to relate to the functionally relevant resting-state networks. Whole-brain modelling of brain activity combines structural connectivity with local dynamical models to provide insight into the large-scale functional organization of the human brain. In this study, we employ the graph Laplacian and its properties to define and implement a large class of neural activity models directly on the human connectome. These models, consisting of systems of stochastic integrodifferential equations on graphs, are dubbed graph neural fields, in analogy with the well-established continuous neural fields. We obtain analytic predictions for harmonic and temporal power spectra, as well as functional connectivity and coherence matrices, of graph neural fields, with a technique dubbed CHAOSS (shorthand for Connectome-Harmonic Analysis Of Spatiotemporal Spectra). Combining graph neural fields with appropriate observation models allows for estimating model parameters from experimental data as obtained from electroencephalography (EEG), magnetoencephalography (MEG), or functional magnetic resonance imaging (fMRI). As an example application, we study a stochastic Wilson-Cowan graph neural field model on a high-resolution connectome graph constructed from diffusion tensor imaging (DTI) and structural MRI data. We show that the model equilibrium fluctuations can reproduce the empirically observed harmonic power spectrum of resting-state fMRI data, and predict its functional connectivity, with a high level of detail. Graph neural fields natively allow the inclusion of important features of cortical anatomy and fast computations of observable quantities for comparison with multimodal empirical data. They thus appear particularly suitable for modelling whole-brain activity at mesoscopic scales, and opening new potential avenues for connectome-graph-based investigations of structure-function relationships.

Mathematical Modeling of Working Memory in the Presence of Random Disturbance using Neural Field Equations

In this paper, we describe a neural field model which explains how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Moreover, we investigate how noise-induced perturbations may affect the coding process. This is obtained by means of a two-dimensional neural field equation, where one dimension represents the nature of the event (for example, the color of a light signal) and the other represents the moment when the signal has occurred. The additive noise is represented by a Q-Wiener process. Some numerical experiments reported are carried out using a computational algorithm for two-dimensional stochastic neural field equations.

Wandering bumps in a stochastic neural field: A variational approach

We develop a generalized variational method for analyzing wandering bumps in a stochastic neural field model defined on some domain U. For concreteness, we take U=S1 and consider a stochastic ring model. First, we decompose the stochastic neural field into a phase-shifted deterministic bump solution and a small error term, which is assumed to be valid up to some exponentially large stopping time. An exact, implicit stochastic differential equation (SDE) for the phase of the bump is derived by minimizing the error term with respect to a weighted L2(U,ρ) norm. The positive weight ρ is chosen so that the error term consists of fast transverse fluctuations of the bump profile. We then carry out a perturbation series expansion of the exact variational phase equation in powers of the noise strength ϵ to obtain an explicit nonlinear SDE for the phase that decouples from the error term. Solving the corresponding steady-state Fokker–Planck equation up to O(ϵ), we determine a leading-order expression for the long-time distribution of the position of the bump. Finally, we use the variational formulation to obtain rigorous exponential bounds on the error term, demonstrating that with very high probability the system stays in a small neighborhood of the bump for times of order exp(Cϵ−1).

Numerical solution of the neural field equation in the presence of random disturbance

This paper aims at presenting an efficient and accurate numerical method for treating both deterministic- and stochastic-type neural field equations (NFEs) in the presence of external stimuli input (or without it). The devised numerical integration means belongs to the class of Galerkin-type spectral approximations. The particular effort is focused on an efficient practical implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use, which are to be integrated, numerically. Our method is implemented in Matlab. Its practical performance and efficiency are investigated on three variants of an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior. First, we observe only stable one-bump solutions in the deterministic neural field scenario, which, in general, will be preserved in our stochastic NFE scenario if the level of random disturbance is sufficiently small. Second, if the area of the external stimuli is large enough and exceeds the size of the bump, considerably, the stochastic neural field solution’s behavior may change dramatically and expose also two- and three-bump patterns. In addition, we show that strong random disturbances, which may occur in neural fields, fully alter the behavior of the deterministic NFE solution and allow for multi-bump (and even periodic-type) solutions to appear in all variants of the stochastic NFE model studied in this paper.

Noise sharing and Mexican-hat coupling in a stochastic neural field.

A diffusion-type coupling operator that is biologically significant in neuroscience is a difference of Gaussian functions (Mexican-hat operator) used as a spatial-convolution kernel. We are interested in pattern formation by stochastic neural field equations, a class of space-time stochastic differential-integral equations using the Mexican-hat kernel. We explore quantitatively how the parameters that control the shape of the coupling kernel, the coupling strength, and aspects of spatially smoothed space-time noise influence the pattern in the resulting evolving random field. We confirm that a spatial pattern that is damped in time in a deterministic system may be sustained and amplified by stochasticity. We find that spatially smoothed noise alone causes pattern formation even without direct spatial coupling. Our analysis of the interaction between coupling and noise sharing allows us to determine parameter combinations that are optimal for the formation of spatial pattern.

Stochastic neural field model: multiple firing events and correlations.

This paper studies a nonlinear dynamical phenomenon called the multiple firing event (MFE) in a spatially heterogeneous stochastic neural field model, which is extended from that in our previous paper (Li et al. in J Math Biol 78:83-115, 2018). MFEs are a partially synchronized spiking barrages that are believed to be responsible for the Gamma oscillation. Rigorous results about the stochastic stability and the law of large numbers are proved, which further imply the well-definedness and computability of many quantities related to MFEs. Then we devote to study spatial and temporal properties of MFEs. Our key finding is that MFEs are spatially correlated but the spatial correlation decays quickly. Detailed mathematical justifications are made based on our qualitative models that aim to demonstrate the mechanism of MFEs.