Dynamics of Partial Differential Equations

Volume 2 (2005)

Number 1

Steady-state solutions in nonlocal neuronal networks

Pages: 71 – 100

DOI: http://dx.doi.org/10.4310/DPDE.2005.v2.n1.a4

Author

Linghai Zhang (Department of Mathematics, Lehigh University, Bethlehem, Penn., U.S.A.)

Abstract

We focus on the mathematical analysis ofexistence and nonlinear exponential stability(linear exponential instability, respectively)of steady-state solutions of the scalar nonlocal equation$$\h{\p u}{\p t}+u=(\a-\c u)\i_{\R}K(x-y)H(u(y,t)-\^)dy+(\b-\d u)\i_{\R}K(x-y)H(u(y,t)-\Theta)dy,$$and the nonlinear system of integral-differential equations\1a\h{\p u}{\p t}+u+w&=&(\a-\c u)\i_{\R}K(x-y)H(u(y,t)-\^)dy\\&+&(\b-\d u)\i_{\R}K(x-y)H(u(y,t)-\Theta)dy,\\\h{\p w}{\p t}&=&\e(u-\tau w).\2aThe steady-states may cross the threshold $\^$ or $\Theta$ only.More interesting cases are that they may cross boththresholds $\^$ and $\Theta$.Stable waves represent attractors of the dynamical system.We also investigate bifurcations of solutions of these equations.The kernel is either an even probability function with exponential decay at infinityor a Mexican hat type function, e.g. $K(x)=A\exp(-a|x|)-B\exp(-b|x|)$ and$K(x)=A\exp(-ax^2)-B\exp(-bx^2)$, where $A>B>0$ and $a>b>0$ are constants.The firing rate $H$ is the Heaviside step function.

Keywords

nonlocal neuronal networks, integral-differential equations, steady state solutions, existence and stability, bifurcation

2010 Mathematics Subject Classification

35B35, 35R10, 45K05, 92C20

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