Dynamics of synaptically coupled McKean neurons

The work in this thesis uses geometric dynamical systems methods to derive phase equations for networks of weakly connected McKean relaxation oscillators. Importantly, this particular single neuron model, with appropriate modifications, is shown to mimic very closely the behaviour of the more biophysically complicated Hodgkin–Huxley model, whilst remaining analytically tractable (albeit in some singular limit). We consider realistic forms of axo-dendritic synaptic coupling with chemical synapses modelled as the convolution of some input spike train with an appropriate temporal kernel. Using explicit forms for the phase response curves (PRCs), for a range of single neuron models, we are able to derive explicit formulas for the phase interaction function in an arbitrary synaptically interacting network of neural oscillators. The PRC for the McKean model is calculated exactly, whilst those for other models is found numerically. In both cases we make extensive use of Fourier representations for synaptic currents, to investigate the effects of axonal, synaptic and dendritic delays on the existence and stability of phase-locked states. [Continues.]