No abstract

Activity of inhibitory neuron with delayed feedback is considered in the framework of point stochastic processes. The neuron receives excitatory input impulses from a Poisson stream, and inhibitory impulses from the feedback line with a delay. We investigate here, how does the presence of inhibitory feedback affect the output firing statistics. Using binding neuron (BN) as a model, we derive analytically the exact expressions for the output interspike intervals (ISI) probability density, mean output ISI and coefficient of variation as functions of model's parameters for the case of threshold 2. Using the leaky integrate-and-fire (LIF) model, as well as the BN model with higher thresholds, these statistical quantities are found numerically. In contrast to the previously studied situation of no feedback, the ISI probability densities found here both for BN and LIF neuron become bimodal and have discontinuity of jump type. Nevertheless, the presence of inhibitory delayed feedback was not found to affect substantially the output ISI coefficient of variation. The ISI coefficient of variation found ranges between 0.5 and 1. It is concluded that introduction of delayed inhibitory feedback can radically change neuronal output firing statistics. This statistics is as well distinct from what was found previously (Vidybida and Kravchuk, 2009) by a similar method for excitatory neuron with delayed feedback.

The instantaneous state of a neural network consists of both the degree of excitation of each neuron the network is composed of and positions of impulses in communication lines between the neurons. In neurophysiological experiments, the neuronal firing moments are registered, but not the state of communication lines. But future spiking moments depend essentially on the past positions of impulses in the lines. This suggests, that the sequence of intervals between firing moments (inter-spike intervals, ISIs) in the network could be non-Markovian.In this paper, we address this question for a simplest possible neural "net", namely, a single inhibitory neuron with delayed feedback. The neuron receives excitatory input from the driving Poisson stream and inhibitory impulses from its own output through the feedback line. We obtain analytic expressions for conditional probability density P (t n+1 | t n , . . . , t 1 , t 0 ), which gives the probability to get an output ISI of duration t n+1 provided the previous (n + 1) output ISIs had durations t n , . . . , t 1 , t 0 . It is proven exactly, that P (t n+1 | t n , . . . , t 1 , t 0 ) does not reduce to P (t n+1 | t n , . . . , t 1 ) for any n ≥ 0. This means that the output ISIs stream cannot be represented as a Markov chain of any finite order.

The instantaneous state of a neural network consists of both the degree of excitation of each neuron and the positions of impulses in communication lines between the neurons. In neurophysiological experiments, the neuronal firing moments are registered, but not the state of communication lines. However, future spiking moments depend substantially on the past positions of impulses in the lines. This suggests that the sequence of intervals between firing moments (interspike intervals, ISI) in the network can be non-Markovian. In this paper, we address this question for a simplest possible neural "network", namely, a single neuron with delayed feedback. The neuron receives excitatory input both from the input Poisson process and from its own output through the feedback line. We obtain exact expressions for the conditional probability density P (tn+1 | tn,. .. , t1, t0)dtn+1 and prove that P (tn+1 | tn,. .. , t1, t0) does not reduce to P (tn+1 | tn,. .. , t1) for any n ≥ 0. This means that the output ISI stream cannot be represented as a Markov chain of any finite order. Стан нейронної мережi складається як з величини збудження в кожному з нейронiв, так i зi значень положення iмпульсiв у лiнiях зв'язку. В нейрофiзiологiчних експериментах реєструються моменти пострiлiв окремих нейронiв, а не стани лiнiй зв'язку. Але моменти наступних пострiлiв iстотним чином залежать вiд положення iмпульсiв у лiнiях зв'язку в попереднi моменти. Це наводить на думку, що послiдовнiсть iнтервалiв мiж послiдовними пострiлами окремого нейрона в мережi (мiжспайковi iнтервали, МСI) може складати немарковський точковий стохастичний процес. У цiй роботi дослiджується така можливiсть для найпростiшої з можливих нейронної "мережi", а саме, поодинокого нейрона з затриманим зворотним зв'язком. Нейрон отримує в якостi стимулу збуджувальнi iмпульси вiд пуассонiвського вхiдного процесу i власнi вихiднi збуджувальнi iмпульси через лiнiю зворотного зв'язку. Одержано точнi вирази для щiльностi умовної ймовiрностi P (tn+1 | tn,. .. , t1, t0)dtn+1 i доведено, що P (tn+1 | tn,. .. , t1, t0) не зводиться до P (tn+1 | tn,. .. , t1) для будь-якого n ≥ 0. Це означає, що вихiдний потiк МСI неможливо подати як марковський ланцюг скiнченного порядку.

The binding neuron model (Vidybida 1998) is inspired by numerical simulation of Hodgkin-Huxley-type point neuron (Vidybida 1996), as well as by the leaky integrate-and-fire model (Segundo at all. 1968). In the binding neuron, the trace of an input is remembered for a fixed period of time after which it disappears completely. This is in the contrast with the above two models, where the postsynaptic potentials decay exponentially and can be forgotten only after triggering.