Ensemble of LIF Neurons With Random Membrane Decay Constant: Emergence of Power-Law Behavior in ISI Distribution

A theoretical framework is proposed to explain the emergence of power law behavior in the spiking neurons where the membrane decay constant is assumed to vary randomly across the population of neurons. The proposed approach, akin to superstatistics, provides a plausible mechanism for generating power law behavior in non-equilibrium systems with fluctuations in intensive quantity. This approach has led to formulation of a hypothesis that power law behavior in inter-spike interval (ISI) distribution results when several neurons group together and fire together. In presence of fluctuations in membrane decay constant governed by gamma distribution, the asymptotic analysis of ISI distribution yields a power law. This finding has been corroborated by simulation study when different types of probability distributions for membrane decay constant are considered. Our results are in agreement with the empirical findings due to Kim (2004) where the spiking trains of Suprachiasmatic nucleus (SCN) neurons display power law behavior. Further investigations in subthreshold regime reveals that the averaging of the membrane potential over a large number of neurons also yields power law.