Abstract

The presence of myelin is a powerful structural factor that controls the conduction velocity of mammalian axons. It is the combination of local synaptic activity and non-local delayed axonal interactions within the cortex that is believed to be the major source of large-scale brain signals that are seen in EEG/MEG brain recordings. Here, we present perspectives from neural mass and network modelling and develop a new set of mathematical tools able to unravel the contributions of space-dependent axonal delays to large-scale spatiotemporal patterning of brain activity.  We first analyse a single neuronal population Wilson-Cowan neural mass model with self-feedback and a fixed delay and show how to construct periodic orbits for a Heaviside firing rate.  For this nonsmooth model we perform linear stability analysis by augmenting Floquet theory with saltation operations. Building on this example, we then show how to treat the synchronous oscillatory state in networks of nonsmooth neural masses with multiple and heterogeneous delays.  To complement this advance in the understanding of synchronous network states and their destabilisation to more novel functional connectivity patterns, we also present numerical simulations (developed in Julia) for both Heaviside and sigmoidal firing rate functions.  We use this to highlight the predictive power of the mathematical approach, as well as uncover possible routes to chaos beyond bifurcation.  Finally, we discuss outstanding challenges for when the delays are plastic and state dependent, and present preliminary results for a new form of biologically motivated white matter plasticity rule.