The Graded–Spike Continuum Law: A Unified Framework Linking Discrete Spiking and Continuous Learning Dynamics

The investigation originates from an apparent contradiction between the discrete nature of neuronal spiking and the continuity required for gradient-based learning. Drawing an analogy to color synthesis—where a limited set of base channels generates a continuous visual spectrum—the study postulates that spike intensity, rather than spike count, can convey continuous information while preserving event-driven efficiency. From this analogy emerges the Graded–Spike Continuum Law (GSCL), which formalizes spiking activity as a smooth, amplitude-encoded surrogate of binary firing. The law demonstrates that introducing graded amplitudes transforms non-differentiable temporal dynamics into a mathematically continuous system, with provable convergence properties and an O(1/κ)bound on surrogate bias. Verification proceeds through symbolic derivation, and analytical proof, each confirming the theoretical predictions of stability and energy efficiency. The framework therefore establishes a reproducible bridge between biological realism and tractable optimization, extending the interpretive power of analog reasoning into formal neuromorphic theory.