Symmetry-based derivation of neuronal membrane dynamics: a Lie group approach to action potential generation

We derive the nonlinear dynamics of neuronal membrane potential from fundamental physical symmetries using Lie group theory. Starting from three experimentally verified symmetries [rotational invariance of paramagnetic ion (Ca^{2 +}, Na⁺) orientation producing SO(2), scale invariance of membrane conductance ratios producing ℝ, and time-translation symmetry producing ℝ], we prove that these independent physical observations uniquely determine the semidirect product group structure G = SO(2) ⋉ℝ², as established in Theorem 1.4. The structure constants (γ₁ = 1.58 V/s, γ₂ = 0.025 V^-1, γ₃ = 0.003) derive from independent measurements in cable theory, Hodgkin-Huxley voltage gating, and Faraday induction, establishing the complete Lie algebra without reference to any phenomenological action potential model. Despite the presence of these physical symmetries, existing neuromorphic models lack a rigorous mathematical framework that systematically predicts functional forms from first principles, relying instead on empirical fitting or stochastic approximations that obscure the underlying deterministic structure. From the group structure G = SO(2) ⋉ℝ², we derive three mathematical constraints on membrane voltage: SO(2) compactness requires bounded functions (tanh, 1/cosh), scale invariance mandates power-law terms (x ^ξ), and temporal symmetry requires oscillatory behavior [sin(πx/τ₀)]. These constraints predict a unique functional form V(x) ∼ x ^ξ ⋅sin(πx/τ₀)/[G(x)⋅cosh(πx/τ₀)⋅f(x)]. Remarkably, our previously published phenomenological equation (Melendy, 2018) (discovered through empirical fitting before developing the present theoretical framework) exhibits exactly this predicted structure, validating that the Lie group approach captures the fundamental physics governing neuronal excitability. This represents theory predicting phenomenology, not post-hoc rationalization. The symmetries were identified from independent physical principles, the group structure was mathematically derived, and the functional form was predicted: all without reference to the 2018 empirical equation. The subsequent agreement demonstrates that the 2018 equation was not arbitrary phenomenological fitting but rather discovered the unique form required by underlying symmetry principles. We present a deterministic algorithm for neuromorphic circuit synthesis wherein Lie algebra structure maps to circuit topology, with components as physical realizations of group exponentials and Kirchhoff's laws emerging as Casimir constraints. The framework determines concrete physical predictions: the power-law exponent ξ≈ e/2 ≈ 1.359, obtained from Lyapunov trajectory separation analysis, characterizes depolarization growth rate and emerges naturally from the algebraic structure. Comparison with classical Hodgkin-Huxley dynamics demonstrates that the simplified Lie group equation captures physiologically realistic action potential waveforms while revealing why the functional form takes its specific mathematical structure. All transcendental functions employ proper dimensional scaling through characteristic time τ₀, ensuring mathematical rigor. This framework offers a principled foundation for neuromorphic circuit design, bridging mathematical physics and engineering to enable reproducible, scalable, and symmetry-preserving models of neuronal excitability. By grounding neuromorphic systems in fundamental symmetry principles rather than phenomenological approximations, this approach enables systematic circuit synthesis with physical consistency guaranteed by the algebraic structure.