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We consider the standard neural field equation with an exponential temporalkernel. We analyze the time-independent (static) and time-dependent (dynamic)bifurcations of the equilibrium solution and the emerging spatio-temporal wavepatterns. We show that an exponential temporal kernel does not allow staticbifurcations such as saddle-node, pitchfork, and in particular, static Turingbifurcations, in contrast to the Green's function used by Atay and Hutt (SIAMJ. Appl. Math. 65: 644-666, 2004). However, the exponential temporal kernelpossesses the important property that it takes into account the finite memoryof past activities of neurons, which the Green's function does not. Through adynamic bifurcation analysis, we give explicit Hopf (temporally non-constant,but spatially constant solutions) and Turing-Hopf (spatially and temporallynon-constant solutions, in particular traveling waves) bifurcation conditionson the parameter space which consists of the coefficient of the exponentialtemporal kernel, the transmission speed of neural signals, the time delay rateof synapses, and the ratio of excitatory to inhibitory synaptic weights.

Dynamics of neural fields with exponential temporal kernel