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Hence, after at most one depolarization, it enters a quiescent state. Thereafter a discrete synaptic input, such as due to an excitatory post-synaptic potential (EPSP), will trigger a further discharge only if it is of sufficient strength to ‘knock’ the system over the inset of the saddle point. This will hence determine whether the resulting neural response is of the green or red waveform as in Fig. 16. 1 -80 -80 -60 -40 -20 V (mV) 0 20 2 4 6 8 10 time (ms) Fig. 16. (a) Representative sub- (green) and supra-threshold orbits (red) and (b) their temporal evolution Neuronal Dynamics and Brain Connectivity 31 Bifurcations and Neuronal Firing in Planar Models Saddle-node bifurcation A further examination of the equation for the V -nullcline (37) shows that the synaptic current is a purely additive term.

Also, setting r small has the effect of ensuring that z evolves on an intrinsically slower time scale than x and y. Together, these constructions have the effect of ensuring that z acts like a slowly varying synaptic current, albeit one which, due to the x term in (45), is also state dependent. Hence as z becomes more negative, it acts like the bifurcation parameter in the FitzHugh Nagumo model and precipitates – via a subcritical Hopf bifurcation - a run of depolarizations. However, due to the x term in (43), these depolarisations have the (relatively slow) effect of increasing z.

A Fig. 12. Saddle-node bifurcation. Solid line denotes fixed point attractor. Dashed line denote fixed point repellors. 4 z1 z1 Fig. 13. Blowout bifurcation in system of two coupled logistic maps. (a) When c < ck the system explores R2 . (b) When c > ck the system contracts onto the diagonal after a brief transient. Arrow points to initial condition. A “blowout bifurcation corresponds to a transition from (b) to (a). Adapted from Breakspear (2004) Other global bifurcations involve intersections of homoclinic or heteroclinic orbits either with themselves, or with fixed point attractors.

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