Abstract
The authors show that a modulation of this connectivity structure can lead to the formation of quasicrystal patterns, thus expanding the repertoire of drug induced geometric hallucinations that can be explained by a Turing mechanism.
Highlights
One of the most well known success stories in mathematical neuroscience is the theory of geometric visual hallucinations, developed by Ermentrout and Cowan [1] and subsequently extended by Bressloff et al [2]
II we describe the simple scalar two-dimensional neural field model that is the subject of this paper
We present a Lyapunov functional for the model
Summary
One of the most well known success stories in mathematical neuroscience is the theory of geometric visual hallucinations, developed by Ermentrout and Cowan [1] and subsequently extended by Bressloff et al [2]. At first sight the spatially nonlocal evolution equation for a neural field is somewhat different from local partial differential equation (PDE) models of pattern-forming systems, such as the Swift-Hohenberg equation, it can be analyzed with many of the same tools These include linear Turing instability analysis, weakly nonlinear analysis, and symmetric bifurcation theory to name just the most common ones. We introduce a set of conditions for a translationally invariant kernel to support a Turing instability that will excite two incommensurate spatial scales, whose nonlinear interaction may lead to a quasicrystal pattern This nonlinearity is represented in the model with a two-parameter sigmoidal firing rate function. VI we discuss the shape of the kernel that gives rise to quasicrystal neural activity patterns and suggest that it may be realized in V1 via a spatially varying modulation of anatomical connection strengths
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