Dynamics of McKean relaxation oscillator networks

The understanding of how an excitable neuron generates output in response to a train of electrical spikes is important in determining the nature of neural coding strategies. The McKean neural model is a planar relaxation oscillator that captures many of the features of a real neuron, and also admits some exact mathematical analysis (although typically in some singular limit). The bifurcation structure of the system under periodic pulsatile stimulation shows that period-adding bifurcations separated by windows of both chaos and periodicity are to be expected.

Geometric dynamical systems methods may also be used to derive phase equations for weakly connected networks. This in turn allows a study of the role that fast and slow synapses, of excitatory and inhibitory type, can play in producing stable phase-locked rhythms.

Liapunov exponent

Relevant publications:

S Coombes and A H Osbaldestin 2000 Period adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator, Physical Review E, Vol 62, 4057-4066
S Coombes 2001 Phase-locking in networks of pulse-coupled McKean relaxation oscillators, Physica D, Vol 2820, 1-16
M Denman-Johnson and S Coombes 2003 A continuum of weakly coupled McKean neurons, Physical Review E, Vol 67, 051903
D T W Chik, S Coombes and Z D Wang 2004 Clustering through post inhibitory rebound in synaptically coupled neurons, Physical Review E, Vol 70, 011908
S Coombes 2008 Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models, SIAM Journal on Applied Dynamical Systems, Vol 7, 1101-1129
S Coombes and R Thul 2016 Synchrony in networks of coupled nonsmooth dynamical systems: Extending the master stability function, European Journal of Applied Mathematics, Vol 27(6), 904–922