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Quasicrystal patterns in a neural field model

Gökçe, Aytül; Coombes, Stephen; Avitabile, Daniele

Authors

Aytül Gökçe

Daniele Avitabile



Abstract

Doubly periodic patterns in planar neural field models have been extensively studied since the 1970s for their role in explaining geometric visual hallucinations. The study of activity patterns that lack translation invariance has received little, if any, attention. Here we show that a scalar neural field model with a translation invariant kernel can support quasicrystal solutions, and that these can be understood using many of the theoretical tools developed previously for materials science. Our approach is constructive in that we consider constraints on the non-local kernel describing interactions in the neural field that lead to the simultaneous excitation of two periodic spatial patterns with incommensurate wavelengths. The resulting kernel has a shape that is a modulation of a Mexican-hat kernel. In the neighbourhood of the degenerate bifurcation of a homogeneous steady state we use a Fourier amplitude approach to determine the value of a Lyapunov functional for various periodic and quasicrystal states. For some values of the parameters defining a translation- invariant synaptic kernel of the model we find that quasicrystal states have the lowest value of the Lyapunov functional. We observe patterns of 12-, 10- and 6- fold rotational symmetry that are stable, but none with 8- fold symmetry. We describe some of the visual hallucination patterns that would be perceived from these quasicrystal cortical patterns, making use of the well known inverse retino-cortical map from visual neuroscience.

Citation

Gökçe, A., Coombes, S., & Avitabile, D. (2020). Quasicrystal patterns in a neural field model. Physical Review Research, 2(1), https://doi.org/10.1103/physrevresearch.2.013234

Journal Article Type Article
Acceptance Date Feb 3, 2020
Online Publication Date Mar 2, 2020
Publication Date 2020-03
Deposit Date Mar 8, 2020
Publicly Available Date Mar 11, 2020
Journal Physical Review Research
Electronic ISSN 2643-1564
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 2
Issue 1
Article Number 013234
DOI https://doi.org/10.1103/physrevresearch.2.013234
Public URL https://nottingham-repository.worktribe.com/output/4106378
Publisher URL https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.2.013234

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