2015/09/28 by Jonathan E. Rubin, Rubin, Jonathan E., Justyna Signerska‐Rynkowska +6 · 1 citation
Computer Science · Neuroscience · Physics and Astronomy · #34K34 #37E05 #37E10 #37E45 #37N25 #92C20 #Dynamical Systems (math.DS) #FOS: Mathematics #Neural dynamics and brain function #Nonlinear Dynamics and Pattern Formation #Photoreceptor and optogenetics research #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1509.08282
openalex publication_date 2015/09/28 · openalex created_date 2017/01/13 · openalex updated_date 2026/07/28
This work continues the analysis of complex dynamics in a class of\nbidimensional nonlinear hybrid dynamical systems with resets modeling neuronal\nvoltage dynamics with adaptation and spike emission. We show that these models\ncan generically display a form of mixed-mode oscillations (MMOs), which are\ntrajectories featuring an alternation of small oscillations with spikes or\nbursts (multiple consecutive spikes). The mechanism by which these are\ngenerated relies fundamentally on the hybrid structure of the flow: invariant\nmanifolds of the continuous dynamics govern small oscillations, while discrete\nresets govern the emission of spikes or bursts, contrasting with classical MMO\nmechanisms in ordinary differential equations involving more than three\ndimensions and generally relying on a timescale separation. The decomposition\nof mechanisms reveals the geometrical origin of MMOs, allowing a relatively\nsimple classification of points on the reset manifold associated to specific\nnumbers of small oscillations. We show that the MMO pattern can be described\nthrough the study of orbits of a discrete adaptation map, which is singular as\nit features discrete discontinuities with unbounded left- and\nright-derivatives. We study orbits of the map via rotation theory for\ndiscontinuous circle maps and elucidate in detail complex behaviors arising in\nthe case where MMOs display at most one small oscillation between each\nconsecutive pair of spikes.\n