Neuronal Spike Trains as Functional-Analytic Distributions: Representation, Analysis, and Significance

The action potential constitutes the digital component of the signaling dynamics of neurons. But the biophysical nature of the full time course of the action potential associated with changes in membrane potential is fundamentally and mathematically distinct from its representation as a discrete set of events that encode when action potentials triggered in a collection spike trains. In this paper, we rigorously explore from first principles the transition and modeling from the standard biophysical picture of a single action potential to its representation as a spike in a spike train. In particular, we adopt a functional-analytic framework, using Schwartz distribution theory to represent spike trains as generalized Dirac delta functions acting on smooth test functions. We then show how and why this representation transcends a purely descriptive formalism to support deep downstream analysis and modeling of spike train neural dynamics in a mathematically consistent way.