History of the application of the phase resetting curve to neurons coupled in a pulsatile manner

Abstract

Phase resetting curves (PRCs) keep track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. In this chapter, a history of the application of PRC theory to circuits of pulse-coupled neurons is presented. In these circuits, the interactions between oscillators take the form of brief pulses such that the effect of one input dies out before the next is received. In some approaches, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is measured using a perturbation that approximates the input received in the intact network. In general, this approach requires that a circuit architecture and a specific firing pattern be assumed. This allows the construction of a map from one discrete event, such as the phase at which an input is received, to the next event. The fixed points of these maps correspond to periodic firing modes and are relatively easy to locate and analyze for stability compared to analyzing the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks. The results obtained provide insights regarding synchronization tendencies.