Dynamics of neural fields with exponential temporal kernel

Various experimental methods of recording the activity of brain tissue in vitro and in vivo demonstrate the existence of traveling waves. Neural field theory offers a theoretical framework within which such phenomena can be studied. The question then is to identify the structural assumptions and the parameter regimes for the emergence of traveling waves in neural fields. 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 the finite memory of past activities of neurons, which the Green's function does not. Through a dynamic bifurcation analysis, we give explicit Hopf (temporally non-constant, but spatially constant solutions) and Turing-Hopf (spatially and temporally non-constant solutions, in particular traveling waves) bifurcation conditions on the parameter space which consists of the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.Comment: 25 pages, 8 Figures, 44 Reference