Distributed delay framework (DDF) has suggested a mechanism to incorporate the delay factor in the evolution of the membrane potential of a neuron model in terms of distributed delay kernel functions. Orejas et. al. , Shoaib and Liu  have applied delay in neural networks to obtain comparatively more efficient output. Depending on the parameter of investigation, there exist a number of choices of delay kernel function for a neuron model. We investigate the Leaky integrate-and-fire (LIF) neuron model in DDF with hypo-exponential delay kernel. LIF neuron with hypo-exponential distributed delay (LIFH) model is capable to regenerate almost all possible empirically observed spiking patterns. In this article, we perform the detailed analytical and simulation based study of the LIFH model. We compute the explicit expressions for the membrane potential and its first two moment viz. mean and variance, in analytical study. Temporal information processing functionality of the LIFH model is investigated during simulation based study. We find that the LIFH model is capable to reproduce unimodal, bimodal and multimodal inter-spike- interval distributions which are qualitatively similar with the experimentally observed ISI distributions. We also notice the neurotransmitter imbalance situation, where a noisy neuron exhibits long tail behavior in aforementioned ISI distributions which can be characterized by power law behavior.
Keywords: Fokker-Planck Equation, Hypo-exponential Distribution, LIF Neuron, Multimodal Distribution, Power Law
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