The Three Symmetries of Ion Channel Dynamics

Each symmetry independently constrains the mathematical structure of the Hodgkin–Huxley equations

Panel 1  ·  SO(2)
Rotational symmetry
Closed Closing Open Opening
Gating variables confined to [0,1]
by SO(2) compactness
Panel 2  ·  ℝ
Conductance scaling
g₀/4 g₀ 4g₀ conductance g × λ ÷ λ Multiplicative structure g → λg for any λ > 0
Conductance voltage-dependence
exp(±V/VT) from ℝ scale invariance
Panel 3  ·  ℝ
Time translation
t Time-translation invariance t → t + τ leaves dynamics unchanged τ = 0
First-order kinetics dm/dt = α(1−m) − βm
arise from Lie algebra flows
SO(2) ⋉ ℝ × ℝ = SO(2) ⋉ ℝ²
These three symmetries combine into a semidirect product group structure
→ Representation theory then requires integer exponents: m³, h, n⁴
Melendy & Blue · Lie Group Basis of Neuronal Membrane Architecture