Existence and Stability Criteria for Phase-Locked Modes in Ring Networks Using Phase-Resetting Curves and Spike Time Resetting Curves

Abstract

Both phase resetting curves (PRCs) and spike time resetting curves (STRCs) are tools used for predicting the phase locked modes in neural networks and investigating their stability. The PRC tabulates the relative change in the duration of the interspike intervals due to a presynaptic input. In networks of neurons that generate brief action potentials (spiking neurons) with pulsatile couplings, the first order PRC, which accounts only for the change in the duration of the interspike interval that includes the perturbation itself, accurately predicts the phase-locked modes. Although the STRC-based method is equivalent to the first order PRC, it has the advantages of unambiguously defining the temporal intervals in networks of dissimilar neurons and a straightforward geometric interpretation. When using the PRC to predict the phase-locked modes it is generally assumed that (1) the open loop stimulus is identical to the recursive inputs received by the same neuron in closed loop setup, (2) the isolated neural oscillator returns to its unperturbed limit cycle before receiving the next input, and (3) each neuron receives only one presynaptic input and sends its output to only one other neuron in the network. These assumptions limit the applicability of the PRC and the STRC methods to 1:1 phase-locked modes to ring networks. For spiking neurons, the PRC/STRC-based theoretical predictions regarding the phase-locked modes and their stability were in very good agreement with the closed loop numerical simulations. Since neurons that generate sustained and closely spaced trains of action potentials (bursts) maintain their postsynaptic coupling over a significant duration, their influence on the postsynaptic neurons extend beyond the first order PRC. For bursting neurons, the first two assumptions were successfully relaxed by considering higher order PRCs and the burst resetting curves (BRCs) in predicting the phase-locked modes in hybrid networks controlled via dynamic clamp.