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Single-pulse solutions for oscillatory coupling functions in neural networks. (English) Zbl 1153.45010

The paper deals with the existence and linear stability of stationary pulse solutions of an integro-differential equation
\[ \frac{\partial u(x,t)}{\partial t} = -u(x,t) + \int_{-\infty}^{\infty} u(x-y)f[u(y,t)]\,dt, \]
which is a model of the coarse-grained averaged activity of a single layer of interconnected neurons. The neuronal connections considered feature lateral oscillations with an exponential rate of decay and variable period. The authors identify regions in the parameter space where the solutions exhibit areas of excitation with single- and dimpled-pulses. When the gain function reduces to the Heaviside function, they establish existence of single-pulse solutions analytically. For a more general gain function, the authors include a numerical support of the existence of pulse-like solutions. Then the linear stability behavior of these solutions is investigated.

MSC:

45K05 Integro-partial differential equations
45M10 Stability theory for integral equations
45G05 Singular nonlinear integral equations
92B20 Neural networks for/in biological studies, artificial life and related topics
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