Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 4951p(t)sin2tsin 2 t00phase of oscillationFigure 10.37: Synaptic transmission function p(t) typically has an asymmetric shapewith fast rise and slow decay.The phase difference χ = ϕ 2 − ϕ 1 satisfiesχ ′ = ε π(sin 2 (χ + d) − sin 2 (χ − d) ) . =ε sin 2dπsin 2χ .The stability of synchronized states is determined by the sign of the function sin 2d.The in-phase state χ = 0 is unstable when sin 2d > 0, i.e., when the delay is shorterthan the half-period π/2, stable when the delay is longer than half-period but shorterthan one period π, unstable for even longer delays, etc. The stability of the anti-phasestate χ = π/2 is reversed, i.e., it is stable for short delays, unstable for longer delays,then stable again for even longer delays, etc. Finally, when the pulses are inhibitory(ε < 0), the (in)stability character is flipped so that the in-phase state becomes stablefor short delays.Weak synapsesNow suppose that each pulse is not a delta function, but it is smeared in time, i.e., ithas a time course p(t − t i ) with p(0) = p(π) = 0. That is, the synaptic transmissionstarts right after the spike of the pre-synaptic neuron and ends before the onset ofthe next spike. The function p has a typical unimodal shape with fast rise and slowdecay depicted in Fig. 10.37. The discussion below is equally applicable to the caseof p(t, x) = g(t)(E − x) with g > 0 being the synaptic conductance with the shape inthe figure and E being the synaptic reverse potential, positive (negative) for excitatory(inhibitory) synapses.Two weakly synaptically coupled SNIC (Class 1) oscillatorsx ′ 1 = 1 + x 2 1 + εp(t − t 2 ) ,x ′ 2 = 1 + x 2 2 + εp(t − t 1 )can be converted into a general phase model with the connection function (10.16) inthe formH(χ) = 1 π∫ π0sin 2 t p(t + χ) dt ,