# A one-population Amari model with periodic microstructure

**Nonlinearity**(0951-7715). Vol. 27 (2014), 6, p. 1391-1417.

[Artikel, refereegranskad vetenskaplig]

We review the derivation of the homogenized one- population Amari equation by means of the two-scale convergence technique of Nguetseng in the case of periodic microvariation in the connectivity function. A key point in this derivation is Visintin's theorem for two-scale convergence of convolution integrals. We construct single bump solutions of the resulting homogenized equation using a pinning function technique for the case where the solutions are independent of the local variable and the firing rate function is modelled as a unit step function. The parameter measuring the degree of heterogeneity plays the role of a control parameter. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. A framework for analysing the stability of these structures is formulated. This framework is based on spectral theory for Hilbert-Schmidt integral operators and it deforms to the standard Evans function approach for the translational invariant case in the limit of no heterogeneity. The upper and lower bounds of the growth/decay rates of the perturbations imposed on the bump states can be expressed in terms of the operator norm of the actual Hilbert- Schmidt operator. Intervals for which the pinning function is increasing correspond to unstable bumps, while complementary intervals where the pinning function decreases correspond to stable bumps, just as in the translational invariant case. Examples showing the properties of the bumps are discussed in detail when the connectivity kernels are given in terms of an exponential decaying function, a wizard hat function and a damped oscillating function.

**Nyckelord: **neural field models; homogenization; existence and stability of single bumps

*In the preparation phase of this paper one of the authors, N Svanstedt, sadly passed away (28 April 2012). The paper is dedicated to his memory.*

Denna post skapades 2014-07-14. Senast ändrad 2014-08-26.

CPL Pubid: 200440