Inhomogeneity-induced bifurcation of stationary and oscillatory pulses
This paper discusses the existence and stability of stationary and periodic pulse solutions to two different mathematical models for the electrical potential in one-dimensional neural networks. One of the models is a Fitzhugh-Nagumo reaction-diffusion system whereas in the other one the diffusion term is substituted by a convolution integral. For a special form of the kernel, the integro-differential equation can be reduced to a partial differential equation which has stationary solutions identical to those of the reaction-diffusion model. In both cases two forms of the nonlinearity are considered: a cubic one and a piecewise linear one. Moreover a spatial inhomogeneity is assumed taking the form of an additive term which has the shape of a Gaussian or Mexican-hat function. When there is no inhomogeneity no stable stationary pulse solution can exist, but the introduction of inhomogeneity makes the existence of stable stationary pulse solutions possible which are not Turing patterns. Mainly taking as parameters the size of the inhomogeneity and the time constant of the slow negative feedback component, an exhaustive construction of the bifurcation diagrams is performed both analytically and numerically in the case of the PDE model giving rise to regions of stability of the stationary pulses, to super- and sub-critical Hopf bifurcations (stable and unstable periodic pulses) and to pulse generators (an oscillatory pulse emitting traveling pulses). In the case of the integro-differential model the stability of the stationary pulse solutions is shown to be qualitatively the same as the one of the PDE system both in the case of piecewise linear nonlinearity and in the case of the cubic one.
super- and sub-critical Hopf bifurcations
scientific article; zbMATH DE number 2173558
scientific article