The neurotransmitter acetylcholine has been shown to modulate the firing properties of several types of neurons through the down-regulation of voltage dependent potassium currents such as the muscarine-sensitive M-current. In particular, experimental work has shown that this current can switch the phase resetting curves from type I to type II and computational models have studied the resulting change in the synchronization of networks of such neurons. In the brain, levels of acetylcholine change with activity. For example, acetylcholine is higher during waking and REM sleep and low during slow wave sleep. Thus an accurate model of the effects of acetylcholine should include slow variation of this neurotransmitter. In the first part of the talk we use normal form theory to show how the MMM-current affects the bifurcation structure of any conductance-based neuron model. In particular, we show that the MMM-current induces two co-dimension two bifurcation which cause the model to transition from a class I to class II oscillator. In the second part of the talk, we use a phase model reduction to study the effect of a slowly varying M-current on the synchronization properties of the neural model. We show that as the current is downregulated or upregulated the phase model passes through two pitchfork bifurcations, which are associated in the full model with the transition between synchronous and asynchronous behaviour. The criticality of the pitchfork bifurcations depends on the neural model and whether the coupling is inhibitory or excitatory. We show that periodic slow passage through these pitchfork bifurcation leads to a hysteresis loop and study how different properties of the model affect this loop and the transitions between synchronous and asynchronous behaviour. Numerical simulations confirm the results of the phase model analysis. This is joint work with Victoria Booth, Xueying Wang and Isam Al-Darbasah.

This seminar is jointly run by CMM, CAMBAM and the University of Waterloo.

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