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Traveling Waves and Synchrony in an Excitable Large-Scale Neuronal Network with Asymmetric Connections

William C. Troy

2008
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SIAM Journal on Applied Dynamical Systems
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We study (i) traveling wave solutions, (ii) the formation and spatial spread of synchronous oscillations, and (iii) the effects of variations of threshold in a system of integro-differential equations which describe the activity of large scale networks of excitatory neurons on spatially extended domains. The independent variables are the activity level u of a population of excitatory neurons which have long range connections, and a recovery variable v. In the integral component of the equation
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... nt of the equation for u the firing rate function is the heaviside function, and the coupling function w is positive. Thus, there is no inhibition in the system. There is a critical value of the parameter β (β * > 0) that appears in the equation for v, at which the eigenvalues µ ± of the linearization of the system around the rest state (u, v) = (0, 0) change from real to complex. We focus on the range β > β * where µ ± are complex, and analyze properties of wave fronts, 1-pulse and 2-pulse waves when the connection function w is asymmetric. For wave fronts we demonstarte how an initial stimulus evolves into two solutions which propagate in opposite directions with different speeds and shapes. For 1-pulse waves our main theoretical result (Theorem 4.2) shows that there is a range of β > β * where two families of waves exist, each consisting of infinitely many solutions. The waves in these two families also propagate in opposite directions with different speeds and shapes. There is a critical value θ * > 0 such that if θ > θ * then 1-pulse waves can only propagate in one direction. In addition, there is a second critical β value, β * > β * , where bulk oscillations come into existence and the system becomes bistable. When β ≥ β * we show how an initial stimulus evolves into a solution with large amplitude oscillations that spread out uniformly from the point of stimulus. The asymmetry in w causes the rate of spread of the 'region of synchrony' to be more rapid to the right of the point of stimulus than to the left. When θ > θ * we construct a 'uni-directional' circuit where synchronization in one region can trigger synchronization in a distant, second region. However, when synchronization is initially triggered in the second region, it cannot spread to the first region.

doi:10.1137/070709888
fatcat:hhhex3h73jgxpgv455th23mzdy