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Spatiotemporal canards in neural field equations

D. Avitabile1,*, M. Desroches2, and E. Knobloch3

  • 1Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG9 7RD, United Kingdom
  • 2Inria Sophia Antipolis Méditerranée Research Centre, MathNeuro Team, 2004 route des Lucioles–Boîte Postale 93 06902 Sophia Antipolis, Cedex, France
  • 3Department of Physics, University of California, Berkeley, California 94720, USA

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Phys. Rev. E 95, 042205 – Published 12 April, 2017

DOI: https://doi.org/10.1103/PhysRevE.95.042205

Abstract

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.

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Outline

Information

Phys. Rev. E 95, 042205– Published 12 April, 2017
  • Received 14 October 2016
  • Revised 7 March 2017
Article part of CHORUS

©2017 American Physical Society

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