Neuronal model with distributed delay: analysis and simulation study for gamma distribution memory kernel

Abstract

A single neuronal model incorporating distributed delay (memory)is proposed. The stochastic model has been formulated as a Stochastic Integro-Differential Equation (SIDE) which results in the underlying process being non-Markovian. A detailed analysis of the model when the distributed delay kernel has exponential form (weak delay) has been carried out. The selection of exponential kernel has enabled the transformation of the non-Markovian model to a Markovian model in an extended state space. For the study of First Passage Time (FPT) with exponential delay kernel, the model has been transformed to a system of coupled Stochastic Differential Equations (SDEs) in two-dimensional state space. Simulation studies of the SDEs provide insight into the effect of weak delay kernel on the Inter-Spike Interval(ISI) distribution. A measure based on Jensen-Shannon divergence is proposed which can be used to make a choice between two competing models viz. distributed delay model vis-á-vis LIF model. An interesting feature of the model is that the behavior of (CV(t))((ISI)) (Coefficient of Variation) of the ISI distribution with respect to memory kernel time constant parameter η reveals that neuron can switch from a bursting state to non-bursting state as the noise intensity parameter changes. The membrane potential exhibits decaying auto-correlation structure with or without damped oscillatory behavior depending on the choice of parameters. This behavior is in agreement with empirically observed pattern of spike count in a fixed time window. The power spectral density derived from the auto-correlation function is found to exhibit single and double peaks. The model is also examined for the case of strong delay with memory kernel having the form of Gamma distribution. In contrast to fast decay of damped oscillations of the ISI distribution for the model with weak delay kernel, the decay of damped oscillations is found to be slower for the model with strong delay kernel.

Information Processing in Neuron with Exponential Distributed Delay

“…. 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.…”

“…Analytical analysis of a non-Markovian process is a too complex and has to look on some other methods. The present non-Markovian process can be extended into higher dimensions to transform membrane potential evolution into Markovian process [20]. LIF model with EDD in extended space takes the following form [20].…”

“…al. [20] has proposed a distributed delay framework and has made an enlarged study of LIF model for exponential distributed delay kernel function and noticed oscillatory spiking patterns in inter-spike-interval distribution [20]. In this article, LIF model with exponentially distributed kernel function is further investigated in terms of spiking activity and SVD.…”

Spiking Activity of a LIF Neuron in Distributed Delay Framework

“…[12] have studied LIF model in DDF with hypo-exponential distributed delay kernel and observed huge variability in spiking pattern of a LIF neuron as obtained in empirical data. These investigations [10,11,12] reflect the novelty and robustness of distributed delay framework for neuronal dynamics, but none of them suggests about the actual effect of distributed delay (memory) on membrane potential evolution and spiking activity of a neuron.Ermentrout and Terman [2], Rudolph and Destexhe [13] have suggested methods for estimating the moments of excitatory and inhibitory conductances from stationary probability distribution of membrane potential (SPDV) for a neuron. Ruan and Filfil [14] have suggested about stability of steady state distribution of membrane potential for discrete delay case and calculated the critical values of discrete delay for stable steady state distribution of membrane potential.…”

“…al. [10], LIF model including gamma distributed delay reads as: (4) and further simplification results a system of coupled SDE …”