We consider an idealized network, formed by N neurons individually described by the FitzHugh-Nagumo equations and connected by electrical synapses. The limit for N → ∞ of the resulting discrete model is thoroughly investigated, with the aim of identifying a model for a continuum of neurons having an equivalent behaviour. Two strategies for passing to the limit are analysed: i) a more convention...

We present for the #rst time an analytical approach for determining the time of #ring of nonlinear stochastic neuronal models. The theory of Markov processes is applied to the Fitzhugh–Nagumo system with a constant mean, Gaussian white noise input, representing stochastic excitation and inhibition. An equation obtained for the mean of the time to #rst spike is solved by means of an accurate one...

The Fitzhugh-Nagumo (fn) mathematical model characterizes the action potential of the membrane. The dynamics of the Fitzhugh-Nagumo model have been extensively studied both with a view to their biological implications and as a test bed for numerical methods, which can be applied to more complex models. This paper deals with the dynamics in the (FH) model. Here, the dynamics are analyzed, qualit...

This paper proposes an identification method to estimate the parameters of the FitzHugh-Nagumo (FHN) model for a neuron using noisy measurements available from a voltage-clamp experiment. By eliminating an unmeasurable recovery variable from the FHN model, a parametric second order ordinary differential equation for the only measurable membrane potential variable can be obtained. In the presenc...

We report on experimental and theoretical findings of front destabilization that causes spontaneous spiral–vortex nucleation and produces a state of spatio-temporal disorder. The experiments were carried out on an oscillatory photosensitive Belousov–Zhabotinsky reaction that is periodically forced in time. Numerical studies were carried out on a modified Complex Ginzburg–Landau equation and on ...

We investigate pattern formation in self-oscillating systems forced by an external periodic perturbation. Experimental observations and numerical studies of reaction-diffusion systems and an analysis of an amplitude equation are presented. The oscillations in each of these systems entrain to rational multiples of the perturbation frequency for certain values of the forcing frequency and amplitu...

Abstract. Optimal sparse control problems are considered for the FitzHugh-Nagumo system including the so-called Schlögl model. The non-differentiable objective functional of tracking type includes a quadratic Tikhonov regularization term and the L1-norm of the control that accounts for the sparsity. Though the objective functional is not differentiable, a theory of second order sufficient optim...