Spiking Neural Networks: Mathematical Foundations

This article presents the mathematical foundations of spiking neural networks (SNNs) in a unified formalism, with a deliberate emphasis on derivational provenance. The same neuron model is written one way in computational neuroscience textbooks, another way in machine learning papers, and a third way in the stochastic process literature. Even within a single line of work, papers absorb constants into other constants until two equations from two sources cannot be compared by inspection. We collect the core mathematics in one place, and we attach a status label to every major equation so that the reader sees at a glance whether a given step is a mathematical identity, a parameter limit, a formal approximation under stated conditions, or a useful but unproven heuristic. The labels are exact, reduction, approximation, and heuristic. The substantive content is the following. The reduction chain from Hodgkin-Huxley dynamics through the adaptive exponential integrate-and-fire model down to leaky integrate-and-fire (LIF) is given with status labels at every step, including the spike response model as an exact reformulation under linear subthreshold dynamics. Reset semantics are analyzed in three forms (hard, soft, no reset), with implications for both spike statistics and gradient flow. Network dynamics are written down in a coupled form, and the analytical theory of recurrent SNNs (liquid state machines, the echo state property, balanced excitatory-inhibitory networks) is reviewed with explicit conditions on time constants and weight matrices. The full point process formulation is developed: counting processes, conditional intensities, the time-rescaling theorem, the likelihood for general history-dependent point processes, and the canonical model classes (homogeneous Poisson, inhomogeneous Poisson, Hawkes, point-process generalized linear models). The bridge between state-space SNNs and intensity-based formulations is made explicit, including conditions under which a generalized linear model can be embedded in a finite-dimensional spiking state space. Information-theoretic aspects of spike coding are presented through Fisher information, with a quantitative comparison of rate and time-to-first-spike codes. Computational capacity is treated through three lines of results: the Maass third-generation argument and its noisy temporal-coding strengthening, the Stanojevic exact mapping from feedforward ReLU networks to time-to-first-spike SNNs, and the Date-Schuman Turing-completeness construction. The article closes with a status-labeled taxonomy of the hazard-based H-LIF family and its Liquid extension, drawn from a public, patent-scoped reference implementation with custom CUDA kernel and FPGA validation; the other LIF variant families (multi-spectral, wavelet, fractional, control-theoretic, information-theoretic, and domain-specific gating) are deferred to a companion v2.This article is the second installment in a series on spiking neural networks. The first installment, Spiking Neural Networks: A Tutorial on Models, Coding, and Training [1], introduces the practical side at a tutorial level; the present article develops the underlying mathematics in depth. The two share notation, and a reader who has followed the first installment can read this one essentially in any order; the cross-references between them are explicit. The intended audience is the graduate student or researcher who needs the mathematical underpinnings of SNNs in a single document, rather than reconstructed from a dozen textbooks and review papers.