Abstract
We study synchronization phenomenon of coupled neuronal oscillators using the theory of weakly coupled oscillators. The role of sudden jumps in the phase response curve profiles found in some experimental recordings and models on the ability of coupled neurons to exhibit synchronous and antisynchronous behavior is investigated, when the coupling between the neurons is electrical. The level of jumps in the phase response curve at either end, spike width and frequency of voltage time course of the coupled neurons are parameterized using piecewise linear functional forms, and the conditions for stable synchrony and stable antisynchrony in terms of those parameters are computed analytically. The role of the peak position of the phase response curve on phase-locking is also investigated.
Highlights
A phase response curve (PRC) quantifies temporal deviations of an oscillator in response to an oncoming stimulus [1,2,3]
Except the leaky integrate-and-fire (LIF) model which has an exponential form of PRC, the other three PRCs [quadratic integrate-and-fire (QIF), adaptive exponential integrate-and-fire, and modified Wang-Buzsaki model] presented in Fig. 1 can be approximated by piecewise linear (PWL) formulations in our models
The LIF and QIF models both have sharp jumps in the PRC at both early and late phases, and no adaptation currents are present in either model
Summary
A phase response curve (PRC) quantifies temporal deviations of an oscillator in response to an oncoming stimulus [1,2,3]. Adapted exponential integrate-and-fire neuron model [21,25] is another example that displays sharp PRC jumps In all these cases weakly coupled oscillator theory has been used to predict the stability of synchrony and antisynchrony. The case of zero spike width (W =T~0) presents an enigmatic situation when the unstable synchronous branch and a stable non-synchronous but phase-locked branch converge at the same point [Fig. 2(d)] This situation arises because of the fact that the discontinuity in the voltage due to W ~0 and the discontinuous jump in the PRC occur at the same temporal location. The other spike parameters are derived from the same model
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