Mode locking in a periodically forced integrate-and-fire-or-burst neuron model

The minimal “integrate-and-fire-or-burst” (IFB) neuron model reproduces the salient features of experimentally observed thalamocortical relay neuron response properties, including the temporal tuning of both tonic spiking (i.e., conventional action potentials) and post-inhibitory rebound bursting mediated by the low-threshold ${\mathrm{Ca}}^{2+}$ current, $I_T.$ In previous work focusing on experimental and IFB model responses to sinusoidal current injection, large regions of stimulus parameter space were observed for which the response was entrained to periodic applied current, resulting in repetitive burst, tonic, or mixed (i.e., burst followed by tonic) responses. Here we present an exact analysis of such mode-locking in the integrate-and-fire-or-burst model under the influence of arbitrary periodic forcing that includes sinusoidally driven responses as one case. In this analysis, the instabilities of mode-locked states are identified as both smooth bifurcations of an associated firing time map and nonsmooth bifurcations of the underlying discontinuous flow. The explicit construction of borders in parameter space that define the instabilities of mode-locked zones is used to build up the Arnol’d tongue structure for the model. The zones for mode-locking are shown to be in excellent agreement with numerical simulations and are used to explore the observed stimulus dependence of burst versus tonic response of the IFB neuron model.