Phase oscillator networks with multiple and state-dependent delays: A framework for exploring white matter plasticity in neurodynamics

Abstract

Network science is increasingly focused on how node dynamics influence emergent phenomena such as oscillations, waves, chimeras, and turbulence. In oscillatory systems modeled by networks of coupled ordinary differential equations (ODEs), a common approach is to reduce the system to phase variables. When understanding how network delays shape emergent properties, the delays are often absorbed in the reduced description as a phase shift. The reduced system is a set of ODEs that loses some information about the full delay differential equation (DDE) system. We adopt a less restrictive approach and consider limit cycle oscillator networks with multiple delays that can be reduced to a DDE system. This captures the effects of delayed couplings, including coexistence and multistability of states. We analyze relative equilibria in the form of phase-locked states with tools previously developed for more general DDE settings. This allows us to explore patterning in networks with space-dependent delays, including in neuroscience, using symmetric bifurcation theory, linear stability analysis, and numerical simulations and continuation. In brain dynamics, time delays are determined by the speed of a communicating signal (action potential) along a fiber (axon). Importantly, these are now known to be state-dependent since the myelin (white matter) that insulates axons is plastic and can change in response to neuronal activity. A simple phenomenological model of this process (in the phase reduced description) is introduced and analyzed by extending techniques developed for fixed delays. Our analysis suggests that white matter plasticity can drive networks to more coherent behavior.