Mintchev, Stanislav M

VCU Scholars Compass

Virginia Commonwealth UniversityVCU Scholars CompassBiology and Medicine Through MathematicsConference 2016May 20th, 11:45 AM - 12:15 PMRobust traveling waves in chains of simple neuraloscillatorsStanislav M. MintchevThe Cooper Union for the Advancement of Science and Art, mintchev@cooper.eduFollow this and additional works at: http://scholarscompass.vcu.edu/bammPart of the Dynamical Systems Commons, Non-linear Dynamics Commons, and the OrdinaryDifferential Equations and Applied Dynamics CommonsThis Event is brought to you for free and open access by the Dept. of Mathematics and Applied Mathematics at VCU Scholars Compass. It has beenaccepted for inclusion in Biology and Medicine Through Mathematics Conference by an authorized administrator of VCU Scholars Compass. Formore information, please contact libcompass@vcu.edu.http://scholarscompass.vcu.edu/bamm/2016/May20/6Robust traveling waves in chains of simple neural oscillatorsStanislav M. Mintchev, The Cooper Union, USAmintchev@cooper.eduBastien Fernandez, CNRS - Universite´ Paris 7 Denis Diderot, FRfernandez@math.univ-paris-diderot.frThe generation and persistence of traveling wave solutions (TWs) in neural ensembles is a pri-mary/major topic of mathematical neuroscience, where it finds applications to signal transmissionand propagation of electrical impulses in brain tissues. Here, we report on some progress in ourmathematical understanding of the underlying mechanisms providing robust presence of these regu-lar patterns, in a class of phase oscillator chains modeled by the coupled ODEsdθsdt= ω + ∆(θs)δ(θs−1), ∀s = 1, 2, · · ·where θs ∈ T := R/Z. The functions δ and ∆ represent the stimulus due to coupling to left neighborsand the phase response curve (PRC) [1, 3] that modulates the input, respectively. Previously, suchsystems were shown to exhibit families of stable TWs in the case of smooth pulse-like stimulusand type-I PRC [4]; this was done by way of a combined numerical/analytical methodology thatestablished wave existence via an iterative numerical algorithm and then deduced a stability theoremby way of a suitably-taken, numerically-justified hypothesis. A subsequent, mathematically-rigorousinvestigation of a simplified model [2], where the stimulus and PRC take either the value 1 (“on”) or0 (“off”) on certain sub-intervals of T, has achieved the full spectrum of these results analytically; inthe process, the latter study establishes a one-to-one correspondence between the wave’s period andits velocity of propagation through the chain, a notion predicted by the aforementioned numericalwork. At present, we aim to give an overview of the existence, stability, and wave generation resultsobtained in [2, 4, 5], as well as a discussion of strategies for extending these results to oscillatorensembles featuring type-II phase response and excitable dynamics.References[1] E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and response dynamics of neuraloscillator populations, Neural Comp. 16 (2004) 673-715.[2] B. Fernandez and S.M. Mintchev, Wave generation in unidirectional chains of idealized neuraloscillators, arXiv:1412.0812 [nlin.AO].[3] D. Hansel, G. Mato and C. Meunier, Synchronization in excitatory neural networks, Neural Comp.7 (1995) 307-337.[4] O.E. Lanford III and S.M. Mintchev, Stability of a family of traveling wave solutions in a feed-forward chain of phase oscillators, Nonlinearity 28 (2015) 237-261.[5] S.M. Mintchev and L.-S. Young, Self-organization in predominantly feedforward oscillator chains,Chaos, 19 (2009) 043131.

Robust traveling waves in chains of simple neural oscillators

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Robust traveling waves in chains of simple neural oscillators

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