LAUSR

Reduced models of single-neuron dynamics are widely used to characterize neuronal properties, encoding, and a cell’s contribution to network function. Low-dimensional phase oscillator models based on phaseresponse curves, in particular, permit to capture the essence of a neuron’s input-output dynamics at the level of spike times from high-dimensional conductance-based neuron models. In principle, these phase models allow for a direct translation of perturbations in the state variables (such as changes in specific ion channel conductances and kinetics) to shifts in the timing of spikes. Previous mathematical work, however, has predominantly focused on voltage and current perturbations and less on perturbations occuring in any of the involved kinetic equations of ionic channels. Here, we provide the means to mathematically relate properties in these variables to the phase-dependent voltage dynamics. Specifically, we derive the vector of phase-response curves near two different onset bifurcations: the physiologically prominent saddle-node on invariant cycle bifurcation as well as the saddle-node loop bifurcation. We demonstrate that, locally around the saddle-node, the tangent space of the isochrons is spanned by the strongly-stable manifold of the saddle-node. This finding enables us to quantitatively relate perturbations in the gating kinetics of ion channels to the resulting changes in spike timing. These results lay the methodological foundation for future quantitative analyses of the impact of specific biophysical channel properties on spike timing, including but not limited to channel noise. Copyright © 0000 John Wiley & Sons, Ltd.