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, or just j. The information transmits approximately describes the time-varying intensity of the excitation is continuously experiencing from neural spike trains delivered to its synapses by thousands of other neurons. Neuron 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, 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, kappa, and b, which largely characterize 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 kappa and b and distributing b/A according to a beta probability density with parameters k and m - kappa, 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.