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 mathematically distinct from its representation as a discrete set of events that encode when action potentials are triggered in a collection of spike trains. In this paper, we develop from first principles a unified functional-analytic framework for neuronal spike trains, grounded in Schwartz distribution theory. We show how this representation provides an exact operational calculus for convolution, distributional differentiation, and distributional support, which enables closed-form analysis of spike train dynamics without discretization, rate approximation, or smoothing. We then analyze the framework in the context of a two-neuron reciprocal circuit with propagation latencies and refractoriness, deriving exact results for synaptic drive, spike timing sensitivity, and causal admissibility of inputs, quantities that are either ill-defined or require approximation in conventional treatments.