A system of ODE’s is used to attempt an approximation of the dynamics of two delayed coupled FitzHugh-Nagumo excitable units, described by delay-differential equations. It is shown that the codimension 2 generalized Hopf bifurcation acts as the organizing center for the dynamics of ODE’s for small time-lags. Furthermore, this is used to explain important qualitative properties of the exact dyna...

An application of the Exp-function method to search for exact solutions of nonlinear differential equations is analyzed. Typical mistakes of application of the Exp-function method are demonstrated. We show it is often required to simplify the exact solutions obtained. Possibilities of the Exp-function method and other approaches in mathematical physics are discussed. The application of the sing...

Abstract. The purpose of the paper is to show that analogue electronic modelling is an extremely fast technique compared to conventional digital computing, especially when solving large sets of coupled nonlinear differential equations. An array of thirty FitzHugh–Nagumo type electronic oscillators, modelling dynamics of the brain neurons, is considered. The decrease of time consumption by a fac...

For a linearized version of the FitzHugh–Nagumo model, firing and resetting conditions are defined by analogy with the Integrate and Fire model. The dynamical behavior of the model, subject to periodic synaptic inputs, is investigated both theoretically and numerically, with particular emphasis on the synchronization properties. The study is then extended to a network of two synaptically couple...

The simplest models used to describe physical systems are often linear. However, several natural phenomena can only be accurately modeled by relatively complex nonlinear systems of equations. There is a wealth of interesting dynamics displayed by systems consisting of nonlinear elements, for example systems in which the dynamics depend delicately on the initial conditions, viz, chaos, and syste...

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall be mainly concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semi...

We investigate the effects of reflecting connectivity in a network composed FitzHugh–Nagumo (FHN) elements linked ring topology. Reflecting is inspired by linking between opposite hemispheres mammalian brain. To study synchronization phenomena and coexistence synchronous asynchronous domains (chimera states) under influence this connectivity, we use two versions FHN model: Version I where membr...