A Mathematical Theory of Energy Efficient Neural Computation and Communication

A neuroscience-based mathematical model of how a neuron stochastically processes data and communicates information is introduced and analyzed. Call the neuron in question \´neuron j", or just "j". The information j transmits approximately describes the time-varying intensity of the excitation j is continuously experiencing from neural spike trains delivered to its synapses by thousands of other neurons. Neuron j "encodes" this excitation history into a sequence of time instants at which it generates neural spikes of its own. By propagating these spikes along its axon, j acts as a multiaccess, partially degraded broadcast channel with thousands of input and output terminals that employs a time-continuous version of pulse position modulation. The mathematical model features three parameters, m, ¿, and b, which largely characterize j as an engine of computation and communication. Each set of values of these parameters corresponds to a long term maximization of the bits j conveys to its targets per joule it expends doing so, which is achieved by distributing the random duration between successive spikes j generates according to a gamma pdf with parameters ¿ and b and distributing b/A according to a beta probability density with parameters ¿ and m-¿, where A is the random intensity of the effectively Poisson process of spikes that arrive to the union of all of j\´s synapses at a randomly chosen time instant.