Energy efficient neurons with generalized inverse Gaussian interspike interval durations

We develop a linkage between the mathematical analysis of a single neuron and the statistical connection of that neuron to the rest of the brain. The core of a stochastic neuron model is the selection of a conditional probability density, f_T|Λ(·|λ), for the random time T that it takes the neuron´s postsynaptic potential to cross a possibly varying threshold given that the neuron´s random excitation intensity Λ has assumed a particular value λ. For reasons we develop in detail, we have selected a certain subfamily of inverse Gaussian (IG) probability densities to serve in this capacity. We assume the neuron is energy efficient in the sense that it maximizes the Shannon mutual information it conveys to its targets per Joule of energy it expends to generate and propagate its train of neural spikes. Using information theory, calculus of variations and Laplace transforms, we derive and solve a pair of coupled integral equations that describe how Λ must be distributed in order for the neuron to maximize bits transmitted per Joule expended (bpJ). The first equation´s solution establishes that the at-this-point unknown bpj-maximizing probability density fΛ(λ) must induce via f_T|Λ(·|λ) a random ISI duration whose probability density f_T(t) belongs to the generalized inverse Gaussian (GIG) family. The algebraic shape factor of this f_T(t) has the form t^-(3/2+D) where D >; 0, as compared with the standard IG density´s shape factor t^-3/2. This result agrees with work on best matching of experimentally observed ISI durations reported in the literature. The solution of the second integral equation yields the exact form of the bpj-maximizing fΛ(λ). This formula for fΛ(λ) is our principal result in that Λ is created not by the neuron being modeled but by those of the brain´s neurons - hose spike trains are afferent to one or more of the modeled neuron´s excitatory synapses. Accordingly, fΛ(λ) serves as the abovementioned bridge that specifies how an energy efficient brain needs to match the long term statistics of each of its neuron´s inputs to that neuron´s particular design.