Stochastic FitzHugh-Nagumo Research Articles

An approximate solution for stochastic Fitzhugh–Nagumo partial differential equations arising in neurobiology models

In this paper, approximate solutions for stochastic Fitzhugh–Nagumo partial differential equations are obtained using two‐dimensional shifted Legendre polynomial (2DSLP) approximation. The problem's suitability and solvability are confirmed. The convergence analysis for the proposed methodology and the error analysis in the norm are carried out. Using Maple software, an algorithm is created and implemented to arrive at the numerical solution. The solution thus obtained is compared with the exact solution and the solution obtained using the explicit order RK1.5 method.

Soliton solutions of nonlinear stochastic Fitz-Hugh Nagumo equation

This study deals with the stochastic Fitz-Hugh Nagumo (FHN) equation and its multiple soliton solutions. The underlying model has numerous applications in neuroscience that express the pulse behavior of neurons. In general, different kinds of noise affect neurons, e.g., oscillations in the opening and closing of ion stations within cell membranes and the fluctuations of the different conductivities in the system. This fluctuation creates a sequence of stochastic excitation. Various applications are branching Brownian motion process, flame propagation, nuclear reactor theory, autocatalytic chemical reaction, mobility in neurons, population growth in the open environment, and liquid environment. So, need of the hour to consider the FHN equation under the impact of noise. The ϕ6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi ^6$$\\end{document}-model expansion method is used to extract the analytical solutions that give a dynamic attitude of the transmission for the nerve impulses of a nervous system. The different constraint conditions for the existence of these solutions are also discussed. The solutions of this model are represented in hyperbolic, trigonometric, and rational forms. The 2 and 3-dimensional behavior of these solutions are depicted by choosing the different values of parameters. The impact of noise on the physical system is analyzed and its real-world applications are discussed. The spikes in the solutions are controlled through the Borel function. These important results will open a new horizon of research for the young researchers.

Large deviation principle for stochastic FitzHugh–Nagumo lattice systems

This paper is devoted to the large deviation principle of stochastic FitzHugh–Nagumo lattice system, where the nonlinear drift and diffusion terms are locally Lipschitz continuous. Based on the well-posedness and the convergence of the solutions of the controlled system associated with the considered system, by the weak convergence method, we establish the large deviation principle in C([0,T],l2×l2) for such infinite dimensional system.

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Bi-spatial random attractor for stochastic FitzHugh-Nagumo systems on unbounded thin domain

Random uniform attractors for fractional stochastic FitzHugh-Nagumo lattice systems

<p>The present study focuses on the asymptotic behavior of fractional stochastic FitzHugh-Nagumo lattice systems with multiplicative noise. First, we investigate the well-posedness of solutions for these stochastic systems and subsequently establish the existence and uniqueness of tempered random uniform attractors.</p>

Forecasting coherence resonance in a stochastic Fitzhugh–Nagumo neuron model using reservoir computing

We delve into the intriguing realm of reservoir computing to predict the intricate dynamics of a stochastic FitzHugh–Nagumo neuron model subjected to external noise. Through innovative reservoir design and training, we unveil the remarkable capacity of a reservoir computer to forecast the behavior of this stochastic system across a wide spectrum of noise intensity variations. Notably, our reservoir computer astutely replicates the intricate phenomenon of coherence resonance in the stochastic FitzHugh–Nagumo neuron, signifying the superior modeling capabilities of this approach. A detailed examination of the microscopic dynamics within the reservoir’s hidden layer reveals the emergence of distinct neuronal clusters, each displaying unique behaviors. Certain neurons within the reservoir are adept at faithfully reproducing the dynamical traits of the neuron, particularly the spike generation mechanism. In contrast, the remaining neurons within the reservoir seem to emulate stochastic influences with remarkable precision, accurately capturing the moments of spike generation in the neuron under the sway of noise. This innovative reservoir design proves to be highly effective across a diverse range of noise control parameters, faithfully replicating the essential characteristics of the original stochastic FitzHugh–Nagumo neuron. These findings illuminate the potential of reservoir computing to model and predict the dynamics of complex stochastic systems, showcasing its adaptability and versatility in understanding and simulating natural phenomena.

Well-posedness and dynamics of stochastic retarded FitzHugh-Nagumo lattice systems

Well-posedness and dynamics of stochastic retarded FitzHugh-Nagumo lattice systems

Propagation of chaos in mean field networks of FitzHugh-Nagumo neurons

In this article, we are interested in the behavior of a fully connected network of $N$ neurons, where $N$ tends to infinity. We assume that the neurons follow the stochastic FitzHugh-Nagumo model, whose specificity is the non-linearity with a cubic term. We prove a result of uniform in time propagation of chaos of this model in a mean-field framework. We also exhibit explicit bounds. We use a coupling method initially suggested by A. Eberle (arXiv:1305.1233), and recently extended in (1805.11387), known as the reflection coupling. We simultaneously construct a solution of the $N$-particle system and $N$ independent copies of the non-linear McKean-Vlasov limit in such a way that, considering an appropriate semi-metric that takes into account the various possible behaviors of the processes, the two solutions tend to get closer together as $N$ increases, uniformly in time. The reflection coupling allows us to deal with the non-convexity of the underlying potential in the dynamics of the quantities defining our network, and show independence at the limit for the system in mean field interaction with sufficiently small Lipschitz continuous interactions.

Delay-induced self-oscillation excitation in the Fitzhugh–Nagumo model: Regular and chaotic dynamics

The stochastic FitzHugh–Nagumo model with time delayed-feedback is often studied in excitable regime to demonstrate the time-delayed control of coherence resonance. Here, we show that the impact of time-delayed feedback in the FitzHugh–Nagumo neuron is not limited by control of noise-induced oscillation regularity (coherence), but also results in excitation of the regular and chaotic self-oscillatory dynamics in the deterministic model. We demonstrate this numerically by means of simulations, linear stability analysis, the study of Lyapunov exponents and basins of attraction for both positive and negative delayed-feedback strengths. It has been established that one can implement a route to chaos in the explored model, where the intrinsic peculiarities of the Feigenbaum scenario are exhibited. For large time delay, we complement the study of temporal evolution by the interpretation of the dynamics as patterns in virtual space.

Dynamics of stochastic FitzHugh–Nagumo system on unbounded domains with memory

In this paper, we consider the non-autonomous stochastic FitzHugh–Nagumo system with hereditary memory and a very large class of nonlinearities, which has no restriction on the upper growth of the nonlinearity. The existence of a random pullback attractor is established for this system in all N-dimensional space.

Non‐instantaneous impulsive stochastic FitzHugh–Nagumo equation with fractional Brownian motion

This article is devoted to examine the exponential behavior of stochastic FitzHugh–Nagumo equation in Hilbert space. Initially, the proposed model is reformulated into an abstract space, and the existence of mild solution is constructed by employing Mönch fixed‐point theorem and Hausdorff measure of non‐compactness. Furthermore, suitable conditions are developed to prove the proposed model's exponential behavior which reflects the stable generation and transmission of electric signal in neurons. At last, an example is presented to verify the established theoretical concepts.

Multiscale analysis for traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHughNagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Kr\"uger and Stannat for scalar stochastic bistable reaction-diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.

Backward regularity of attractors for lattice FitzHugh-Nagumo system with double random coefficients

In this paper, we investigate the backward compact dynamical behavior for the stochastic lattice FitzHugh-Nagumo system with double random coefficients and multiplicative noise. It is proved, under some suitable conditions on the nonlinear term and the body force, that the system has a unique backward compact bi-spatial random (ℓσ2×ℓσ2,ℓσp×ℓσ2)-attractor for any p>2. By such a bi-spatial attractor we mean an invariant backward compact set in ℓσp×ℓσ2 that pullback attracts all nonempty subsets of ℓσ2×ℓσ2 under another topology of ℓσp×ℓσ2. The time-dependent uniform pullback asymptotical compactness of the solution operators in ℓσp×ℓσ2 is proved by virtue of a cut-off technique and considering two different attracting universes to overcome several difficulties caused by the lack of compact Sobolev embeddings in the case of a lattice system and the unknown measurability of random attractor.

A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model

In this article, we construct and analyse an explicit numerical splitting method for a class of semi-linear stochastic differential equations (SDEs) with additive noise, where the drift is allowed to grow polynomially and satisfies a global one-sided Lipschitz condition. The method is proved to be mean-square convergent of order 1 and to preserve important structural properties of the SDE. First, it is hypoelliptic in every iteration step. Second, it is geometrically ergodic and has an asymptotically bounded second moment. Third, it preserves oscillatory dynamics, such as amplitudes, frequencies and phases of oscillations, even for large time steps. Our results are illustrated on the stochastic FitzHugh-Nagumo model and compared with known mean-square convergent tamed/truncated variants of the Euler-Maruyama method. The capability of the proposed splitting method to preserve the aforementioned properties may make it applicable within different statistical inference procedures. In contrast, known Euler-Maruyama type methods commonly fail in preserving such properties, yielding ill-conditioned likelihood-based estimation tools or computationally infeasible simulation-based inference algorithms.

Analysis of stochastic fitzhugh–nagumo equation for wave propagation in a neuron arising in certain neurobiology models

In this paper, the numerical solution of stochastic FitzHugh–Nagumo equation (SFNE) has been obtained by Chebyshev spectral collocation. Semi-implicit Euler–Maruyama scheme has been used for temporal variable to discretize the stochastic FitzHugh–Nagumo equation. A detailed stability analysis for stochastic FitzHugh–Nagumo equation has also been discussed. Graphical representations of obtained results of the stochastic FitzHugh–Nagumo equation have been discussed to provide a clear idea about the behavior of the solutions.

Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces

Limiting Dynamics for Stochastic FitzHugh–Nagumo Lattice Systems in Weighted Spaces

Limiting behavior of FitzHugh–Nagumo equations driven by colored noise on unbounded thin domains

We investigate the limiting behavior of dynamics of non-autonomous stochastic FitzHugh–Nagumo equations driven by a nonlinear multiplicative colored noise on unbounded thin domains. We first establish the existence and uniqueness of random attractors for the equations on the thin domains and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain.

Numerical scheme and stability analysis of stochastic Fitzhugh–Nagumo model

This article deals with the Fitzhugh–Nagumo equation in the presence of stochastic function. A numerical scheme has been developed for the solution of such equations which preserves the certain structure of the unknown functions, also we have given the stability analysis, consistency of the problem, and explicitly optimal a priori estimates for the existence of solutions of such equations. A unique solution has been guaranteed. The corresponding explicit estimates in the function spaces are formulated in the form of theorems. Lastly, one important feature of the article is the simulation of the proposed numerical scheme in the form of the 2D and 3D plots which shows the efficacy of the stochastic analysis of such nonlinear partial differential equations.

Wong-Zakai approximations and attractors for non-autonomous stochastic FitzHugh-Nagumo system on unbounded domains

In this paper, we study the Wong-Zakai approximations and long term behavior of the stochastic FitzHugh-Nagumo system driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation system. Then we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic FitzHugh-Nagumo system for both additive and multiplicative noise on unbounded domains. The tail estimates and decomposition method are employed to derive the pullback asymptotic compactness of solutions in order to overcome the obstacles caused by the non-compactness of Sobolev embeddings on unbounded domains as well as the absence of regularity of one component of the solutions.

Wong–Zakai approximations of the non-autonomous stochastic FitzHugh–Nagumo system on RN in higher regular spaces

In this paper, we consider the Wong–Zakai approximations of a non-autonomous stochastic FitzHugh–Nagumo system driven by a multiplicative white noise with an arbitrary intensity. The convergence of solutions of the path-wise deterministic system to that of the corresponding stochastic system is established in higher regular spaces by means of a new iteration technique and an optimal multiplier at different stages. Furthermore, we prove that the random attractor of the path-wise deterministic system converges to that of the non-autonomous stochastic FitzHugh–Nagumo system in higher regular spaces when the size of approximation vanishes, with much looser conditions on the nonlinearity.