The variability and mean of the firing rate of Hodgkin-Huxley and FitzHugh-Nagumo neurons subjected to random synaptic input are only weakly dependent on the level of inhibitory input, unlike integrate-and-fire neurons. For the latter model, substantial inhibitory input is essential to ensure output variability close to Poissonian firing. It cannot therefore be used routinely in stochastic netw...

This thesis considers a stochastic dynamical system that has multiple time scales. The focus lies on the planar slow-fast stochastic differential equation (a singularly perturbed stochastic system) modeling the spiking activity of an isolated FitzHugh-Nagumo neural oscillator. We analyze the model equation on the slow time scale τ (Eq.(1)) and on the fast time scale t (Eq.(2)). { dvτ = ε f(vτ ,...

There have been several existence results for the standing waves of FitzHugh-Nagumo equations. Such are connecting orbits an autonomous second-order Lagrangian system and corresponding kinetic energy is indefinite quadratic form in velocity terms. When has two stable hyperbolic equilibria, there exist fronts, which will be used this paper as building blocks, to construct with multiple fronts ca...

We investigate smooth and sparse optimal control problems for convective FitzHughNagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several ...

During present investigation, a direct algebraic method based on complex solutions of nonlinear partial differential equations is developed and tested in the case of generalised Burgers–Huxley equation. The proposed scheme can be used in a wide class of nonlinear reaction–diffusion equations. These calculations demonstrate that the accuracy of the direct algebraic solutions is quite high even i...