Abstract: Networks of oscillatory neural mass nodes with delayed interactions are increasingly being used as models for large-scale brain activity, but understanding the influence of the delays on emergent phenomena such as oscillations and waves remains a challenge. One approach is to consider limit cycle oscillator networks with multiple delays that can be reduced to a delay differential equation (DDE) system for the evolution of phases. In this talk we will first discuss how phase reduction techniques can be extended to networks where the node oscillations are induced by delays and where there are also conduction delays. We will include cases where the decay of perturbations to the node limit cycle is slow in some direction necessitating the use of a notion of distance from limit cycle to accurately capture network behaviours. Once we have a delayed phase oscillator network description, we will explore patterning in networks with space-dependent delays, such as those that arise in neuroscience, using tools that include symmetric bifurcation theory, linear stability, and scientific computation.
In brain dynamics, delays are determined by the speed of a communicating signal (action potential) along a physical fibre (axon). Importantly, these are now known to be state dependent since the myelin (white matter) that electrically insulates axons is plastic and can change in response to neuronal activity. A simple phenomenological model of this process in the phase reduced description will be introduced, and we will show that this system with state-dependent delays is amenable to network analysis with a suitable extension of the techniques developed for fixed delays. Interestingly, our analysis suggests that white matter plasticity can stabilise a wide variety of phase-locked states and drive networks to more coherent behaviour.
This is joint work with Stephen Coombes, Robert Allen, Grace Jolly (University of Nottingham) and Gulistan Iskenderoglu (Istanbul Ticaret University)