Relation Between Firing Statistics of Spiking Neuron with Delayed Fast Inhibitory Feedback and Without Feedback

Abstract

We consider a class of spiking neuronal models, defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire or the binding neuron model and also for some artificial neurons. A neuron is fed with a Poisson process. Each output impulse is applied to the neuron itself after a finite delay [Formula: see text]. This impulse acts as being delivered through a fast Cl-type inhibitory synapse. We derive a general relation which allows calculating exactly the probability density function (pdf) [Formula: see text] of output interspike intervals of a neuron with feedback based on known pdf [Formula: see text] for the same neuron without feedback and on the properties of the feedback line (the [Formula: see text] value). Similar relations between corresponding moments are derived.Furthermore, we prove that the initial segment of pdf [Formula: see text] for a neuron with a fixed threshold level is the same for any neuron satisfying the imposed conditions and is completely determined by the input stream. For the Poisson input stream, we calculate that initial segment exactly and, based on it, obtain exactly the initial segment of pdf [Formula: see text] for a neuron with feedback. That is the initial segment of [Formula: see text] is model-independent as well. The obtained expressions are checked by means of Monte Carlo simulation. The course of [Formula: see text] has a pronounced peculiarity, which makes it impossible to approximate [Formula: see text] by Poisson or another simple stochastic process.