Non-weak inhibition and phase resetting at negative values of phase in cells with fast-slow dynamics at hyperpolarized potentials

Phase response is a powerful concept in the analysis of both weakly and non-weakly perturbed oscillators such as regularly spiking neurons, and is applicable if the oscillator returns to its limit cycle trajectory between successive perturbations. When the latter condition is violated, a formal application of the phase return map may yield phase values outside of its definition domain; in particular, strong synaptic inhibition may result in negative values of phase. The effect of a second perturbation arriving close to the first one is undetermined in this case. However, here we show that for a Morris–Lecar model of a spiking cell with strong time scale separation, extending the phase response function definition domain to an additional negative value branch allows to retain the accuracy of the phase response approach in the face of such strong inhibitory coupling. We use the resulting extended phase response function to accurately describe the response of a Morris–Lecar oscillator to consecutive non-weak synaptic inputs. This method is particularly useful when analyzing the dynamics of three or more non-weakly coupled cells, whereby more than one synaptic perturbation arrives per oscillation cycle into each cell. The method of perturbation prediction based on the negative-phase extension of the phase response function may be applicable to other excitable cell models characterized by slow voltage dynamics at hyperpolarized potentials.