Stochastic Resonance in an Extended FitzHugh-Nagumo System: the Role of Selective Coupling

Motivated by recent experiments on intracellular calcium dynamics, we study the general issue of fluctuation-induced nucleation of waves in excitable media. We utilize a stochastic Fitzhugh-Nagumo model for this study, a spatially-extended non-potential pair of equations driven by thermal (i.e. white) noise. The nucleation rate is determined by finding the most probable escape path via minimization of an action related to the deviation of the fields from their deterministic t...

This article aims to provide insights into the qualitative analysis of some nonlinear Reaction-Diffusion (RD) systems arising in Neuroscience. We first introduce a non-homogeneous FitzHugh-Nagumo (nhFHN) featuring excitability and oscillatory properties. Then, we discuss the qualitative analysis of a toy model related to nhFHN. In particular, we focus on the convergence of solutions of the toy model toward different solutions (fixed point, periodic) and show the existence of ...

Nonlinear dynamical stochastic models are ubiquitous in different areas. Excitable media models are typical examples with large state dimensions. Their statistical properties are often of great interest but are also very challenging to compute. In this article, a theoretical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous mathematical theories show the covariance decay beh...

We propose theoretical methods to infer coupling strength and noise intensity simultaneously through an observation of spike timing in two well-synchronized noisy oscillators. A phase oscillator model is applied to derive formulae relating each of the parameters to some statistics from spike-time data. Using these formulae, each parameter is inferred from a specific set of statistics. We demonstrate the methods with the FitzHugh-Nagumo model as well as the phase model. Our me...

We investigate the dynamics of a limit of interacting FitzHugh-Nagumo neurons in the regime of large interaction coefficients. We consider the dynamics described by a mean-field model given by a nonlinear evolution partial differential equation representing the probability distribution of one given neuron in a large network. The case of weak connectivity previously studied displays a unique, exponentially stable, stationary solution. Here, we consider the case of strong conne...

The FitzHugh-Nagumo equation, originally conceived in neuroscience during the 1960s, became a key model providing a simplified view of excitable neuron cell behavior. Its applicability, however, extends beyond neuroscience into fields like cardiac physiology, cell division, population dynamics, electronics, and other natural phenomena. In this review spanning six decades of research, we discuss the diverse spatio-temporal dynamical behaviors described by the FitzHugh-Nagumo e...

We use a biophysical model of a local neuronal circuit to study the implications of synaptic plasticity for the detection of weak sensory stimuli. Networks with fast plastic coupling show behavior consistent with stochastic resonance. Addition of an additional slow coupling that accounts for the asynchronous release of neurotransmitter results in qualitatively different properties of signal detection, and also leads to the appearance of transient post-stimulus bistability. Ou...

Mean field approximation of a large collection of FitzHugh-Nagumo excitable neurons with noise and all-to-all coupling with explicit time-delays, modelled by $N\gg 1$ stochastic delay-differential equations is derived. The resulting approximation contains only two deterministic delay-differential equations but provides excellent predictions concerning the stability and bifurcations of the averaged global variables of the exact large system.

We study deterministic systems, composed of excitable units of FitzHugh-Nagumo type, that are capable of self-generating and self-terminating strong deviations from their regular dynamics without the influence of noise or parameter change. These deviations are rare, short-lasting, and recurrent and can therefore be regarded as extreme events. Employing a range of methods we analyze dynamical properties of the systems, identifying features in the systems' dynamics that may qua...

We investigate heterogeneous coupling delays in complex networks of excitable elements described by the FitzHugh-Nagumo model. The effects of discrete as well as of uni- and bimodal continuous distributions are studied with a focus on different topologies, i.e., regular, small-world, and random networks. In the case of two discrete delay times resonance effects play a major role: Depending on the ratio of the delay times, various characteristic spiking scenarios, such as cohe...