Abstract
We study pattern formation in a 2-population homogenized neural field model of the Hopfield type in one spatial dimension with periodic microstructure. The connectivity functions are periodically modulated in both the synaptic footprint and in the spatial scale. It is shown that the nonlocal synaptic interactions promote a finite band width instability. The stability method relies on a sequence of wave-number dependent invariants of 2times 2-stability matrices representing the sequence of Fourier-transformed linearized evolution equations for the perturbation imposed on the homogeneous background. The generic picture of the instability structure consists of a finite set of well-separated gain bands. In the shallow firing rate regime the nonlinear development of the instability is determined by means of the translational invariant model with connectivity kernels replaced with the corresponding period averaged connectivity functions. In the steep firing rate regime the pattern formation process depends sensitively on the spatial localization of the connectivity kernels: For strongly localized kernels this process is determined by the translational invariant model with period averaged connectivity kernels, whereas in the complementary regime of weak and moderate localization requires the homogenized model as a starting point for the analysis. We follow the development of the instability numerically into the nonlinear regime for both steep and shallow firing rate functions when the connectivity kernels are modeled by means of an exponentially decaying function. We also study the pattern forming process numerically as a function of the heterogeneity parameters in four different regimes ranging from the weakly modulated case to the strongly heterogeneous case. For the weakly modulated regime, we observe that stable spatial oscillations are formed in the steep firing rate regime, whereas we get spatiotemporal oscillations in the shallow regime of the firing rate functions.
Highlights
It is common to investigate large-scale activity of neural tissue by means of nonlocal models
In the present paper we have investigated the effect of periodic microstructure on the pattern formation mechanism in a 2-population neural field model
This work presents an extension of the previous paper by Wyller et al [15] on Turing type of instability and pattern formation within the framework of a 2-population neural field model with homogeneous and isotropic connectivity strengths
Summary
It is common to investigate large-scale activity of neural tissue by means of nonlocal models. Since the seminal works of Amari [1, 2] and Wilson and Cowan [3, 4] such models have been subject to a vast number of investigations, e.g., [5] and the references therein. 1- and 2-population neural field models have been used to understand spatiotemporal dynamics of the cortex of the brain. Stationary spatially-extended patterns are related to visual hal-. Kolodina et al Journal of Mathematical Neuroscience (2021) 11:9 lucinations [6,7,8], while stationary localized structures (bumps) are related to short term memory [9,10,11]. The 2-population neural field model of the Hopfield type ∂ ∂t ue = –ue + ωee ⊗ Pe(ue θe) ωie Pi Traveling waves (fronts, pulses, target waves and spirals) are connected to information processing [12, 13].
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