Abstract
Consider the following nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks. These model equations generalize many important nonlinear scalar integral differential equations aris ing from synaptically coupled neuronal networks. The kernel functions K and W represent synaptic couplings between neurons in synaptically coupled neuronal networks. The synaptic couplings can be very general, including not only pure excitations (modeled with nonnegative kernel functions), lateral inhibitions (modeled with Mexican hat kernel functions), lateral excitations (modeled with upside down Mexican hat kernel functions), but also synaptic couplings which may change sign for finitely many times or even infinitely many times. The function H = H(u − ) represents the Heaviside step function, which is defined by H(u − ) = 0 for all u . The functions and represent probability density functions defined on (0,1). The parameter c > 0 represents the speed of an action potential and the parameter > 0 represents a constant delay. In these equations, u = u(x, t) stands for the membrane potential of a neuron at position x and time t. The positive constants > 0 and > 0 represent synaptic rates. The positive constants > 0 and > 0 represent thresholds for excitation of neurons. The function f = f(u) represents the sodium currents in neuronal networks. The positive constant w 0 > 0 is to be given. The authors will establish the existence and stability of traveling wave solutions of these nonlinear scalar integral differential equations by coupling together speed index functions, stability index func tions (often called Evans functions, that is, complex analytic functions), implicit function theorem, intermediate value theorem, mean value theorem, global strong maximum principle for Evans func tions, linearized stability criterion and many other important techniques in dynamical systems. They will find sufficient conditions satisfied by the synaptic couplings, by the probability density functions, by the synaptic rate constants and by the thresholds so that the traveling wave solutions and their wave speeds exist, and the stability of the traveling wave solutions is true. The main results obtained in this paper greatly improve many previous results.
Highlights
The spatial temporal dynamics of nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks have attracted much attention in recent years, see [3], [6], [7], [9], [10], [11], [14], [16], [17], [18], [22], [24], [35], [37], [38], [39], [40], [41], [43], [46], [47], [48], [50], [51], [52], [56], [60], [61], [68], [77], [78], [79], [80]
We will accomplish the existence of the traveling wave solutions of the nonlinear scalar integral differential equations (1.1) and (1.2)
We study the following nonlinear scalar integral differential equations
Summary
The spatial temporal dynamics of nonlinear scalar integral differential equations arising from synaptically coupled neuronal networks have attracted much attention in recent years, see [3], [6], [7], [9], [10], [11], [14], [16], [17], [18], [22], [24], [35], [37], [38], [39], [40], [41], [43], [46], [47], [48], [50], [51], [52], [56], [60], [61], [68], [77], [78], [79], [80]. The resulting transmission delay between two spatial positions depends on the axon and changes between one half millisecond and one hundred millisecond Since these delay time are in the same range as time constants of synaptic responses, effects of finite transmission delays on the spatial temporal evolution of activity take place. The corresponding spatial domain is coarse-grained and exhibits spatial patches which are motivated by the macrocolumnar structure in primary sensory areas [32], [34] Such neural field model equations allow the successful reproduction of electroencephalographic activity on the head [36] and the successful description of spiral waves in neural tissue [33] and [45]
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