Thursday 14 Jan 2016: Dynamics seminar: Synchrony in neural networks: from Ott-Antonsen reductions to the master stability function
Stephen Coombes - University of Nottingham
Part i) In electrophysiological recordings of the brain, transition from high amplitude to low amplitude signals are likely caused by an underlying change in the synchrony of underlying neuronal population firing patterns. A classic example of such modulation is the strong stimulus-related oscillatory phenomena known as the movement related beta decrease (MRBD) and post-movement beta rebound (PMBR). Here a sharp decrease in neural oscillatory power is observed during movement (MRBD) followed by an increase above baseline on movement cessation (PMBR). These represent important neuroscientific phenomena which have been shown to have clinical relevance. A related phenomenon is movement related beta decrease (MRBD), whereby beta rhythms are suppressed during voluntary movement, typifying event-related desynchronisation. We present a parsimonious model for the dynamics of synchrony within a synaptically coupled spiking network that is able to replicate a human MEG power spectrogram showing the evolution from MRBD to PMBR. Importantly the high-dimensional spiking model has an exact mean field description in terms of four ordinary differential equations that allows considerable insight to be obtained into the cause of the experimentally observed time-lag from movement termination to the onset of PMBR (~ 0.5 s), as well as the subsequent long duration of PMBR (~1-10 s). Our model represents the first to accurately predict these commonly observed and robust phenomena and will represent a key step in their understanding, in health and disease.
Part ii) The master stability function is a powerful tool for determining synchrony in high dimensional networks of coupled limit cycle oscillators. In part this approach relies on the analysis of a low dimensional variational equation around a periodic orbit. For smooth dynamical systems this orbit is not generically available in closed form. However, many models in physics, engineering, and biology admit to piece-wise linear (pwl) caricatures which are also often nonsmooth, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably the master stability function cannot be immediately applied to networks of such elements if they are non-smooth. Here we show how to extend the master stability function to nonsmooth planar pwl systems, and in the process demonstrate that considerable insight into network dynamics can be obtained when choosing the dynamics of the nodes to be pwl. In illustration we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. We contrast this with node dynamics poised near a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.