### Dynamics of Morris-Lecar networks

Stephen Coombes

Abstract.
The Morris-Lecar (ML) neuron model is a two dimensional conductance
based model that is often used as an idealised fast-spiking pyramidal
neuron. Its planar nature has encouraged much analysis of the single
neuron model using tools from phase-plane analysis and the "geometry
of excitability". When treating synaptic or gap junction coupled
networks of oscillating ML neurons these techniques are the natural
basis for developing a weakly-coupled oscillator theory. However, to
probe network dynamics in the strong coupling regime requires an
alternative approach. I will show how results in this area can be
obtained by using a piece-wise linear caricature of the ML model. In
illustration of the usefulness of such an approach I will first
consider gap junction coupling and show how to analyse emergent
fluctuations in the mean membrane potential (as instabilities of an
asynchronous network state). Next I will treat synaptically coupled
networks with a phenomenological form of retrograde (cannabinoid)
second messenger signalling that can support depolarisation induced
suppression of excitation. In this case I will describe a mechanism
for the emergence of ultra-low frequency (0.01-0.1 Hz) synchronized
oscillations - a hallmark rhythm of the resting brain.