Envelope-Phase Dynamics in Stochastic Neural Models with Delayed

Andre Longtin
Department of Physics and Cellular and Molecular Medicine
Centre for Neural Dynamics
University of Ottawa

Brain rhythms typically occur in random epochs of higher amplitude known as “bursts”. Such bursts, in the gamma or beta range frequency ranges, are thought to contribute to the efficiency of working memory, communication and movement tasks. Abnormalities in bursts have also been associated with motor and psychiatric disorders. We present a theory for the generic features of burst generation, based on single cell dynamics and network connectivity. We first present a generic mathematical model for burst generation in an excitatory-inhibitory (EI) network with self-couplings. The resulting equations for the stochastic phase and envelope of the rhythm’s fluctuations, derived using the stochastic averaging method, depend on only one meta-parameter that combines all the network parameters. This allows us to identify different regimes of amplitude excursions, and to highlight the supportive role that noise can have. We discuss how burst attributes, such as their durations and peak frequency content, depend on the network parameters. We also show how to extend this formalism to the delayed coupling of brain rhythms from different areas, and the importance of noise and delay for determining the range of phase differences. Finally we present results on multi-delayed nonlinear feedback systems, studying how their complexity depends on the number of delays in the feedback loops. The goal is to reconcile the notions of increased complexity for larger numbers of delays, and simpler complexity in the limit of a large number of delays. A new method for estimating Lyapunov exponents for such systems is presented, which reveals a novel complexity collapse when the number of delays is sufficiently large. This implies that multiple delayed feedback interactions do not necessarily lead to complex high-dimensional dynamics, a property of interest for understanding the possible range of dynamics exhibited by neural, metabolic and other biological networks.

Arthur Powanwe and Andre Longtin, Scientific Reports 2019
Arthur Powanwe and Andre Longtin, Phys. Rev. Res. 2020
Kamyar Tavakoli and Andre Longtin, Phys. Rev. Res. 2020