Stochastic Neural Field Equations Research Articles

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.

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.

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.

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.

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 fields as gradient dynamical systems.

Continuous attractor neural networks are used extensively to model a variety of experimentally observed coherent brain states, ranging from cortical waves of activity to stationary activity bumps. The latter are thought to play an important role in various forms of neural information processing, including population coding in primary visual cortex (V1) and working memory in prefrontal cortex. However, one limitation of continuous attractor networks is that the location of the peak of an activity bump (or wave) can diffuse due to intrinsic network noise. This reflects marginal stability of bump solutions with respect to the action of an underlying continuous symmetry group. Previous studies have used perturbation theory to derive an approximate stochastic differential equation for the location of the peak (phase) of the bump. Although this method captures the diffusive wandering of a bump solution, it ignores fluctuations in the amplitude of the bump. In this paper, we show how amplitude fluctuations can be analyzed by reducing the underlying stochastic neural field equation to a finite-dimensional stochastic gradient dynamical system that tracks the stochastic motion of both the amplitude and phase of bump solutions. This allows us to derive exact expressions for the steady-state probability density and its moments, which are then used to investigate two major issues: (i) the input-dependent suppression of neural variability and (ii) noise-induced transitions to bump extinction. We develop the theory by considering the particular example of a ring attractor network with SO(2) symmetry, which is the most common architecture used in attractor models of working memory and population tuning in V1. However, we also extend the analysis to a higher-dimensional spherical attractor network with SO(3) symmetry which has previously been proposed as a model of orientation and spatial frequency tuning in V1. We thus establish how a combination of stochastic analysis and group theoretic methods provides a powerful tool for investigating the effects of noise in continuous attractor networks.

A gradient flow formulation for the stochastic Amari neural field model.

We study stochastic Amari-type neural field equations, which are mean-field models for neural activity in the cortex. We prove that under certain assumptions on the coupling kernel, the neural field model can be viewed as a gradient flow in a nonlocal Hilbert space. This makes all gradient flow methods available for the analysis, which could previously not be used, as it was not known, whether a rigorous gradient flow formulation exists. We show that the equation is well-posed in the nonlocal Hilbert space in the sense that solutions starting in this space also remain in it for all times and space-time regularity results hold for the case of spatially correlated noise. Uniqueness of invariant measures, ergodic properties for the associated Feller semigroups, and several examples of kernels are also discussed.

Well-posedness of the stochastic neural field equation with discontinuous firing rate

We study the existence and uniqueness of mild solutions to the deterministic and the stochastic neural field equation with Heaviside firing rate. Since standard well-posedness results do not apply in case of a discontinuous firing rate, we present a monotone Picard iteration scheme to show the existence of a maximal mild solution. Further, we illustrate that general uniqueness does not hold, and therefore investigate uniqueness under suitable additional properties of the solutions. Here a novel criterion, the so-called absolute continuity condition is introduced. Moreover, we observe regularisation by noise: with a suitable choice of spatially correlated additive noise uniqueness is restored without imposing any additional structural assumptions.

A multiscale-analysis of stochastic bistable reaction–diffusion equations

A multiscale analysis of 1D stochastic bistable reaction–diffusion equations with additive noise is carried out w.r.t. travelling waves within the variational approach to stochastic partial differential equations. It is shown with explicit error estimates on appropriate function spaces that up to lower order w.r.t. the noise amplitude, the solution can be decomposed into the orthogonal sum of a travelling wave moving with random speed and into Gaussian fluctuations. A stochastic differential equation describing the speed of the travelling wave and a linear stochastic partial differential equation describing the fluctuations are derived in terms of the coefficients. Our results extend corresponding results obtained for stochastic neural field equations to the present class of stochastic dynamics.

Finite-Size Effects on Traveling Wave Solutions to Neural Field Equations

Neural field equations are used to describe the spatio-temporal evolution of the activity in a network of synaptically coupled populations of neurons in the continuum limit. Their heuristic derivation involves two approximation steps. Under the assumption that each population in the network is large, the activity is described in terms of a population average. The discrete network is then approximated by a continuum. In this article we make the two approximation steps explicit. Extending a model by Bressloff and Newby, we describe the evolution of the activity in a discrete network of finite populations by a Markov chain. In order to determine finite-size effects—deviations from the mean-field limit due to the finite size of the populations in the network—we analyze the fluctuations of this Markov chain and set up an approximating system of diffusion processes. We show that a well-posed stochastic neural field equation with a noise term accounting for finite-size effects on traveling wave solutions is obtained as the strong continuum limit.

A General Framework for Stochastic Traveling Waves and Patterns, with Application to Neural Field Equations

In this paper we present a general framework in which to rigorously study the effect of spatio-temporal noise on traveling waves and stationary patterns. In particular, the framework can incorporate versions of the stochastic neural field equation that may exhibit traveling fronts, pulses, or stationary patterns. To do this, we first formulate a local SDE that describes the position of the stochastic wave up until a discontinuity time, at which point the position of the wave may jump. We then study the local stability of this stochastic front, obtaining a result that recovers a well-known deterministic result in the small-noise limit. We finish with a study of the long-time behavior of the stochastic wave.

Stochastic neural field equations: a rigorous footing.

We here consider a stochastic version of the classical neural field equation that is currently actively studied in the mathematical neuroscience community. Our goal is to present a well-known rigorous probabilistic framework in which to study these equations in a way that is accessible to practitioners currently working in the area, and thus to bridge some of the cultural/scientific gaps between probability theory and mathematical biology. In this way, the paper is intended to act as a reference that collects together relevant rigorous results about notions of solutions and well-posedness, which although may be straightforward to experts from SPDEs, are largely unknown in the neuroscientific community, and difficult to find in a very large body of literature. Moreover, in the course of our study we provide some new specific conditions on the parameters appearing in the equation (in particular on the neural field kernel) that guarantee the existence of a solution.

Front Propagation in Stochastic Neural Fields: A Rigorous Mathematical Framework

We develop a complete and rigorous mathematical framework for the analysis of stochastic neural field equations under the influence of spatially extended additive noise. By comparing a solution to a fixed deterministic front profile it is possible to realise the difference as strong solution to an $L^2(\mathbb{R})$-valued SDE. A multiscale analysis of this process then allows us to obtain rigorous stability results. Here a new representation formula for stochastic convolutions in the semigroup approach to linear function-valued SDE with adapted random drift is applied. Additionally, we introduce a dynamic phase-adaption process of gradient type.

Large deviations for nonlocal stochastic neural fields.

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers’ law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Mathematics Subject Classification (2000): 60F10, 60H15, 65M60, 92C20.

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.