Analytical strength-duration curve for spiking response of the LIF neuron to an alpha-function shaped excitatory current pulse

Abstract

AbstractWhether or not the neuron emits a spike in response to stimulation by an excitatory current impulse is determined by a strength-duration curve (SDC) for the impulse parameters. Excitatory neurons affect the others through impulses of excitatory postsynaptic current. A simple yet plausible approximation for the time course of such an impulse is the alpha function, with linear rise at the start and exponential decay at the end. However, an exact analytical SDC for this case is unknown even for the leaky integrate-and-fire (LIF) neuron, the simplest spiking neuron model used in practice. This paper contains the solution: the exact analytical SDC has been found for the spiking response of the LIF neuron stimulated by an excitatory current pulse in the form of alpha function. In addition, we have derived general SDC equations for an arbitrary shape of the stimulating pulse. For comparison, we have also re-derived SDCs in a unified way for (i) a standard rectangular pulse, (ii) ascending ramp pulse, and (iii) instantly rising and exponentially decaying pulse. In the limit of no leakage, for all considered cases the SDC is reduced to the classical hyperbola. The obtained SDC formulas can be of use for theoretical and computational neuroscience studies, as well as for patch-clamp experiments in neuronal electrophysiology.