Mathematical frameworks for oscillatory network dynamics in neuroscience

TL;DR

This paper reviews mathematical tools for analyzing oscillatory neural networks, extending beyond phase oscillator theory to address complex dynamics like strong coupling and stochastic effects.

Contribution

It introduces a broad set of mathematical frameworks for neural network dynamics, expanding the traditional phase oscillator approach to handle more complex and realistic scenarios.

Findings

Provides insights into network behaviors like synchronization and chimeras.

Highlights limitations of phase oscillator theory in certain conditions.

Offers practical mathematical tools for future neuroscience research.

Abstract

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances when this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear - for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.