Spiking Activity of a LIF Neuron in Distributed Delay Framework

Abstract

-Evolution of membrane potential and spiking activity for a single leaky integrate-and-fire (LIF) neuron in distributed delay framework (DDF) is investigated. DDF provides a mechanism to incorporate memory element in terms of delay (kernel) function into a single neuron models. This investigation includes LIF neuron model with two different kinds of delay kernel functions, namely, gamma distributed delay kernel function and hypo-exponential distributed delay kernel function. Evolution of membrane potential for considered models is studied in terms of stationary state probability distribution (SPD). Stationary state probability distribution of membrane potential (SPDV) for considered neuron models are found asymptotically similar which is Gaussian distributed. In order to investigate the effect of membrane potential delay, rate code scheme for neuronal information processing is applied. Firing rate and Fano-factor for considered neuron models are calculated and standard LIF model is used for comparative study. It is noticed that distributed delay increases the spiking activity of a neuron. Increase in spiking activity of neuron in DDF is larger for hypo-exponential distributed delay function than gamma distributed delay function. Moreover, in case of hypo-exponential delay function, a LIF neuron generates spikes with Fano-factor less than 1.Keywords -Distributed Delay Framework, Fokker-Planck Equation, Gamma Distribution, Hypo-exponential Distribution, Stationary Probability Distribution, Spiking Activity. I. InTRoducTIonT heRe are a number of neuron models depending on biophysical and electrical properties of a neuron suggested in literature [1][2][3][4][5]. Among these neuron models, Leaky integrate-and-fire (LIF) model has become a backbone for theoretical as well as experimental investigation of neuronal dynamics due to its simplicity and analytical solvable capability [5,6,7]. This model is an RC-circuit equivalent representation of a neuron with additional spiking constraint and is widely used for mathematical explanation of bio-physical mechanism and information processing of neurons [1,2,4,8,9]. Based on some specific properties, many variants of LIF model are suggested in literature [1,5,7,8]. These neuron models explain neuronal dynamics adequately, but when we talk about memory, we have to rely on group of few neurons or neural networks. None of these single neuron models capture memory element. Recently, Karmeshu et. al. [10] have suggested a distributed delay framework for incorporating the effect of previous values of membrane potential (memory) on neuronal dynamics, in which, a kernel function is included in LIF model to capture the aggregate effect of previous values of membrane potential on its further evolution. It is a challenging problem to find an appropriate kernel function so that the resulting model can explain most of the variability in neuronal responses. To this end, Karmeshu et. al. [10] have investigated their proposed framework with exponential distributed delay kernel and...

Information Processing in Neuron with Exponential Distributed Delay

“… is taken ranging from 0.01 to 0.3 with step-size of 0.01 [7], [8]. Spike rate for LIF model in EDD is similar to that of LIF model with stochastic input at small  and reduces in LIF model in EDD is slowly than LIF model with stochastic input for increment in  as shown in Fig.…”

“…The aggregated effect of these ions and molecules can be captured in term of stochastic current. Thus, the rate of change of membrane potential ( () Vt) for LIF model with stochastic input stimulus is defined as [5], [6], [7], [10].…”

“…. DDF for the LIF model is defined as [7], [20] V t t  into a non-Markovian process. Analytical analysis of a non-Markovian process is a too complex and has to look on some other methods.…”

“…Neural networks are artificial human-brain like computing structure which incorporates Lapicque (1907) has proposed the integrate-and-fire neuron model (IF model), first neuron model, equivalent to an electrical RC-circuit with an additional threshold constraint, i.e. neuron potential resets a lower value as soon as it reaches to a certain value (threshold) [1], [7], [9]. Under the influence of noisy input stimulus and noisy environment, neuron reflects the highest level of variation in spiking pattern [25].…”

Temporal Information Processing and Stability Analysis of the MHSN Neuron Model in DDF

“…al. [30] has shown that "the distributed delay has no effect on stationary state membrane potential distribution of a LIF neuron". In addition, we say that the distributed delay has no effect on steady state solution of the threshold based linear neuron models.…”