A gradient particle method is used to compute probability densities for processes with delay bifurcations that are sensitive to very small noise. The method is particularly useful for computing the probability density in the transition region, where asymptotic approximations may not be valid. For a one dimensional steady bifurcation problem and a two-dimensional FitzHugh-Nagumo model we solve t...

Stochastic Resonance in a coupled FitzHugh-Nagumo equation is investigated. The optimal noise intensity and the optimal input frequency, which maximize the signal to noise ratio of the output signal, are studied numerically, and their dependence on system parameters and connection coe cients is examined. It is found that a network composed of six elements can separate a superimposed periodic pu...

We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involv...

In this paper, we implement the radial basis functions for solving a classical type of time-fractional telegraph equation defined by Caputo sense for ð1oαr2Þ. The presented method which is coupled of the radial basis functions and finite difference scheme achieves the semi-discrete solution. We investigate the stability, convergence and theoretical analysis of the scheme which verify the validi...

The paper deals with a semilinear integrodifferential equation that characterizes several dissipative models of Viscoelasticity, Biology and Superconductivity. The initial boundary problem with Neumann conditions is analyzed. When the source term F is a linear function, then the explicit solution is obtained. When F is non linear, some results on existence, uniqueness and a priori estimates are...

Viability and invariance problems related to a stochastic equation in a Hilbert space H are studied. Finite dimensional invariant C submanifolds of H are characterized. We derive Nagumo type conditions and prove a regularity result: Any weak solution, which is viable in a finite dimensional C submanifold, is a strong solution. These results are related to finding finite dimensional realizations...

We focus on the qualitative analysis of a reaction-diffusion with spatial heterogeneity near a bifurcation. The system is a generalization of the well known FitzHugh-Nagumo system in which the excitability parameter is space dependent. This heterogeneity allows to exhibit concomitant stationary and oscillatory phenomena. We prove the existence of an Hopf bifurcation and determine an equation of...

A space clamped neural axon along which flows a constant current exhibits current dependent oscillations of the action potential. With increasing current a characteristic Hopf bifurcation scenario can be observed: approximately harmonic small amplitude oscillations arise and bifurcate subsequently into relaxation oscillations of relatively large amplitude. Amplitude and period of the oscillatio...