Information transfer through a single neuron is a fundamental component of information processing in the brain, and computing the information channel capacity is important to understand this information processing. The problem is difficult since the capacity depends on coding, characteristics of the communication channel, and optimization over input distributions, among other issues. In this letter, we consider two models. The temporal coding model of a neuron as a communication channel assumes the output is τ where τ is a gamma-distributed random variable corresponding to the interspike interval, that is, the time it takes for the neuron to fire once. The rate coding model is similar; the output is the actual rate of firing over a fixed period of time. Theoretical studies prove that the distribution of inputs, which achieves channel capacity, is a discrete distribution with finite mass points for temporal and rate coding under a reasonable assumption. This allows us to compute numerically the capacity of a neuron. Numerical results are in a plausible range based on biological evidence to date.

Belief propagation (BP) is a universal method of stochastic reasoning. It gives exact inference for stochastic models with tree interactions and works surprisingly well even if the models have loopy interactions. Its performance has been analyzed separately in many fields, such as AI, statistical physics, information theory, and information geometry. This article gives a unified framework for understanding BP and related methods and summarizes the results obtained in many fields. In particular, BP and its variants, including tree reparameterization and concave-convex procedure, are reformulated with information-geometrical terms, and their relations to the free energy function are elucidated from an information-geometrical viewpoint. We then propose a family of new algorithms. The stabilities of the algorithms are analyzed, and methods to accelerate them are investigated.