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Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation

In: Neural Fields

Author

Listed:
  • G. Bard Ermentrout

    (University of Pittsburgh, Department of Mathematics)

  • Stefanos E. Folias

    (University of Alaska Anchorage, Department of Mathematics & Statistics)

  • Zachary P. Kilpatrick

    (University of Houston, Department of Mathematics)

Abstract

We study spatiotemporal patterns of activity that emerge in neural fields in the presence of linear adaptation Adaptation . Using an amplitude equation approach, we show that bifurcations from the homogeneous rest state can lead to a wide variety of stationary and propagating patterns on one- and two-dimensional periodic domains, particularly in the case of lateral-inhibitory synaptic weights. Other typical solutions are stationary and traveling localized activity bumps Bumps ; however, we observe exotic time-periodic localized patterns as well. Using linear stability analysis that perturbs about stationary and traveling bump Bumps solutions, we study conditions for the activity to lock to a stationary or traveling external input on both periodic and infinite one-dimensional spatial domains. Hopf and saddle-node bifurcations can signify the boundary beyond which stationary or traveling bumps Bumps Bumps traveling fail to lock to external inputs. Just beyond a Hopf bifurcation Bifurcation Hopf point, activity bumps often begin to oscillate, becoming breather Breathers or slosher Sloshers solutions.

Suggested Citation

  • G. Bard Ermentrout & Stefanos E. Folias & Zachary P. Kilpatrick, 2014. "Spatiotemporal Pattern Formation in Neural Fields with Linear Adaptation," Springer Books, in: Stephen Coombes & Peter beim Graben & Roland Potthast & James Wright (ed.), Neural Fields, edition 127, chapter 0, pages 119-151, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-54593-1_4
    DOI: 10.1007/978-3-642-54593-1_4
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