Dynamics of Coupled Noisy Neural Oscillators with Heterogeneous Phase Resetting Curves

Abstract

Pulse-coupled phase oscillators have been utilized in a variety of contexts. Motivated by neuroscience, we study a network of pulse-coupled phase oscillators receiving independent and correlated noise. An additional physiological attribute, heterogeneity, is incorporated in the phase-resetting curve (PRC), which is a vital entity for modeling the biophysical dynamics of oscillators. An accurate probability density or mean field description is large-dimensional, requiring reduction methods for tractability. We present a reduction method to capture the pairwise synchrony via the probability density of the phase differences, and we explore the robustness of the method. We find the reduced methods can capture some of the synchronous dynamics in these networks. The variance of the noisy period (or spike times) in this network is also considered. In particular, we find that phase oscillators with predominately positive PRCs (type I) have larger variance with inhibitory pulse-coupling than PRCs with a larger negative region (type II), but with excitatory pulse-coupling the opposite happens-type I oscillators have lower variability than type II. Analysis of this phenomenon is provided via an asymptotic approximation with weak noise and weak coupling, where we demonstrate how the individual PRC alters variability with pulse-coupling. We compare the phase oscillators to full oscillator networks and discuss utility and shortcomings.

Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

“…Without recourse to a phase reduction it is well to mention that Medvedev has been pioneering a phase–amplitude approach to studying the effects of noise on the synchronisation of coupled stochastic limit-cycle oscillators [ 194 , 279 ], and that Newhall et al have developed a Fokker–Planck approach to understanding cascade-induced synchrony in stochastically driven IF networks with pulsatile coupling and Poisson spike-train external drive [ 280 ]. More recent work on pairwise synchrony in network of heterogeneous coupled noisy phase oscillators receiving correlated and independent noise can be found in [ 281 ]. However, note that even in the absence of synaptic coupling, two or more neural oscillators may become synchronised by virtue of the statistical correlations in their noisy input streams [ 282 – 284 ].…”

Oscillating systems with cointegrated phase processes

“…( 2016 ), Burton et al. ( 2012 ), Fernandez and Tsimring ( 2014 ), Ly ( 2014 ), Ly and Ermentrout ( 2011 ).…”

The Winfree model with heterogeneous phase-response curves: analytical results

“…Indeed, broad cell-to-cell differences in PRCs have been recently measured in the olfactory bulb mitral cells (Burton et al 2012), and given that the collective phase dynamics of a synchronized ensemble of oscillators depends crucially on the level of PRC heterogeneity (Nakao et al 2018), it is desirable to deepen our understanding on the effects of heterogeneous PRCs on collective synchronization. However, due to its mathematical complexity, previous attempts to tackle oscillator ensembles with heterogeneous PRCs are scarce, and rely on approximate methods (Tsubo, Teramae & Fukai 2007, Ly 2014.…”