In the first part of this paper [1], a complete description of Q-conditional symmetries for two classes of reaction-diffusion-convection equations with power diffusivities is derived. It was shown that all the known results for reaction-diffusion equations with power diffusivities follow as particular cases from those obtained in [1] but not vise versa. In the second part the symmetries obtaine...

This article presents the reduction of neuronal models from the classic four-dimensional differential model of Hodgkin and Huxley [7] to discrete binary automata which keep the main properties of more complex models. A reduction of Fitzhugh and Nagumo (FHN) model is performed using a numerical strategy introduced in [3] completed by a linearization in the spirit of McKean model [14]. The result...

We study the interplay between diversity and noise in a 3D network of FitzHugh-Nagumo elements, with topology dimensions chosen to model pancreatic beta-cell cluster, as an example excitable cell network. Our results show that are not equivalent sources disorder but have different effects on dynamics. Their synchronization mechanisms may act independently one another or synergistically, dependi...

We consider a general nonlinear diffusion, typified by those deriving from Fitzhugh–Nagumo or Hindmarsh–Rose models of nerve-cell dynamics, perturbed also by 2-parameter white noise. In order to investigate the effects of the nonlinearity, we find for general boundary conditions the mean to order 2 and the four-point covariance to order 3. The derivations involve multiple stochastic integrals i...

A FitzHugh–Nagumo (FHN) model with delayed coupling is considered to investigate the steady state bifurcation due to the coupling strength and delay. The center manifold reduction and normal form method are employed to study the bifurcation from zero singularity. We show that the coupling strength can induce both the pitchfork and transcritical bifurcations and the coupled delay has the most in...

By extending a semi-analytical, dynamical mean-field approximation (DMA) previously proposed by the author [H. Hasegawa, Phys. Rev. E 67, 41903 (2003)], we have developed a generalized theory which takes into account a wide range of couplings in small-world networks consisting of noisy N -unit FitzHugh-Nagumo (FN) neurons: the coordination number on the average Z is allowed to vary from local (...