The Hodgkin-Huxley (HH) model is the cornerstone of computational neuroscience, yet it does not provide a closed-form expression for the spike latency. We show that near the excitation threshold, the HH dynamics reduces to u' = u exp (lambda u), where u = V - Vₜh is the membrane voltage relative to threshold and lambda encodes the sodium channel sensitivity. Using the exact linearization W = E1 (lambda u), we obtain the spike latency: T*ₛpike = E1 (lambda u0), where E1 is the exponential integral. This yields several novel predictions: exact spike latency as a function of initial depolarization, universal scaling T* ~ -ln (lambda) as lambda -> 0, dose-response of a sodium channel blocker: Delta T* ≈ -ln (alpha), independent of u0, and a topological invariant Omega = pi/2 - Si (mu u0) conserved during spike trains. We validate these predictions through numerical simulations of the full HH model and discuss their experimental testability.
Judicael Brindel (Thu,) studied this question.