Stability of Bumps in a Two Population Neural Field Model

Abstract

a b s t r a c tNeural-field models describing the spatio-temporal dynamics of the average neural activity are frequently formulated in terms of partial differential equations, Volterra equations or integro-differential equations. We develop a stability analysis for spatially symmetric bumps in a two-population neural-field model of Volterra form for a large class of temporal kernels. We find that the corresponding Evans matrix can be block-diagonalized, where one block corresponds to the symmetric part of the perturbations while the other block takes care of the antisymmetric part of these perturbations including the translational invariance of the bumps. For the class of quasi-power temporal kernels ∼ t k exp(−t) we show that the system of governing equations can be converted to a system of rate equations. We prove that for this class of temporal kernels the stability analysis based on the Evans function approach is equivalent to the phase-space reduction technique termed the generalized Amari approach. We illuminate these results by carrying out numerical simulations based on a fourth-order Runge-Kutta numerical scheme in time for the special mixed case modeled by α-functions in the excitatory equation and exponentially decaying functions in the inhibitory equation. Excellent agreement between analytical predictions from the stability analysis and numerical simulations is obtained. The generic picture consists of an unstable narrow bump pair and a broad bump pair. The broad bump pair is stable for small and moderate values of the relative inhibition time τ , is converted to a stable breather at a critical time constant, τ = τ cr (which is identified as a Hopf-bifurcation point), and becomes unstable as τ exceeds τ cr .

Existence and stability of periodic solutions in a neural field equation

“…We conjecture that the existence and stability results hold for steep sigmoid like functions f , see (1.6). To prove this conjecture we plan to proceed in the way similar to [10] and [23]. It is not possible to apply the results from the mentioned papers directly here since the eigenvalue λ = 1 of H (u p ) is not isolated.…”

“…The first condition, however, is never satisfied. Thus one must employ more detailed analysis of spectral convergence than in the case of bump solutions [23]. However, this is out of the scope of this paper.…”

“…The approximation of f = H with f = S(βu) then must be properly justified. This has been successfully done for bumps solutions in [10,22] and [23].…”