A Mean Field Theory for Pulse-Coupled Oscillators based on the Spike Time Response Curve

A mean field method for pulse-coupled oscillators with delays used a self-connected oscillator to represent a synchronous cluster of N-1 oscillators and a single oscillator assumed to be perturbed from the cluster. A periodic train of biexponential conductance input was divided into a tonic and a phasic component representing the mean field input. A single cycle of the phasic conductance from the cluster was applied to the single oscillator embedded in the tonic component at different phases to measure the change in the cycle length which the perturbation was initiated, that is, the first order phase response curve (PRC), and the second order PRC in the following cycle. A homogeneous network of 100 biophysically calibrated inhibitory interneurons with either shunting or hyperpolarizing inhibition tested the predictive power of the method. A self-consistency criterion predicted the oscillation frequency of the network from the PRCs as a function of the synaptic delay. The major determinant of the stability of synchrony was the sign of the slope of the first order PRC of the single oscillator in response to an input from the self-connected cluster at a phase corresponding to the delay value. For most short delays, first order PRCs correctly predicted the frequency and stability of simulated network activity. However, considering the second order PRC improved the frequency prediction and resolved an incorrect prediction of stability of global synchrony at delays close to the free running period of single neurons in which a discontinuity in the PRC precluded existence of 1:1 self-locking.