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

Whether or not the neuron emits a spike in response to stimulation by an excitatory current pulse is determined by a strength-duration curve (SDC) for the pulse parameters. The SDC is a dependence of the minimal pulse amplitude required to elicit the spiking response on either the pulse duration or its decay time. Excitatory neurons affect the others through pulses of excitatory postsynaptic current. A simple yet plausible approximation for the time course of such a pulse 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 hitherto not known, even for the leaky integrate- and-fire (LIF) neuron, the simplest spiking neuron model used in practice. We have obtained general SDC equations for the LIF neuron. Using the Lambert W function -- a widely-implemented special function, we have found the exact analytical SDC for the spiking response of the LIF neuron stimulated by an excitatory current pulse in the form of the alpha function. To compare results in a unified way, we have also derived the analytical SDCs for (i) rectangular pulse, (ii) ascending ramp pulse, and (iii) instantly rising and exponentially decaying pulse. In the limit of no leakage, we show that the SDC is reduced to the classical hyperbola for all considered cases.