The neuron’s membrane potential build-up allows for refractoriness. In this context, the refractory effect is often the primary reason spike trains are not accurately described by traditional counting distributions. Inspired by this biological phenomenon, this paper introduces a counting distribution that includes a parameter specifying the length of the refractory period in neural spike generation. Specifically, the paper considers a lower bound on the intervals between any two spikes that follow an exponential or Gamma distribution. The proposed distribution is a generalized form of standard counting distributions, such as the Poisson distribution, and is named the refractory Poisson distribution. This study aims to provide estimation methods for the proposed distribution. Additionally, it introduces a novel estimation approach called “one-shot” maximum likelihood estimation. The research also provides guidance on the maximum amount of information that can be reliably transferred through a single neuron due to refractoriness. The optimality conditions for the capacity-achieving distribution in the capacity per unit cost problem are presented, which, to the best of our knowledge, have not been reported in the literature. The proposed counting distribution can be used for numbering any random phenomena where a refractory period exists between occurrences. Additionally, the distribution can be applied in various scenarios. For example, it can replace the Poisson distribution in the Generalized Linear Model as an alternative counting distribution.

Maximizing the information per the expected unit energy cost in biological neurons is vital in understanding the functionality of our nervous system and improving the bio-inspired nano-networks. Hence, toward a better comprehension of neuronal information processing and communication from an information-energy standpoint, this paper presents the following novel results on the capacity-achieving distribution: 1) For the first time, the paper presents the probability mass function of the number of spikes observed in a given time window where inter-spike intervals follow the Inverse Gaussian distribution. To extend this result, the first and second moments of the underlying probability mass function will be determined. 2) The paper proves that the relationship between the underlying probability mass function and the marginal distribution of the intensity parameter is one-to-one when the parameter is a random variable. 3) Proof that a unique capacity-achieving distribution for the stimulus intensity exists in a discrete form with a finite number of mass points that varies with the energy budget. Then, the paper presents an algorithm that numerically computes a trade-off between the maximum transmit bits and the coding-duration energy expenditure. Subsequently, the optimal multiplier along with the optimal distribution and the corresponding capacity will be obtained.