Dynamics of neural fields with exponential temporal kernel

Abstract

In this paper, we consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatio-temporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations, in contrast to the Green's function used by Atay and Hutt (SIAM J. Appl. Math. 65: 644-666, 2004). However, the exponential temporal kernel possesses the important property that it takes into account finite memory of past activities of neurons, which the Green's function does not. Through a dynamic bifurcation analysis we give explicit Hopf (temporally nonconstant, but spatially constant solutions) and Turing-Hopf (spatially and temporally non-constant solutions) bifurcation conditions on the parameter space which consists of the internal input current, the time delay rate of synapses, the ratio of excitatory to inhibitory synaptic weights, the coefficient of the exponential temporal kernel, and the transmission speed of neural signals.