Action potential propagation properties of 4D, 3D and 2D Hodgkin-Huxley type models

We explore the relationship between the sodium and potassium gating variables in the Hodgkin-Huxley electrophysiology model, reducing the dimension of the original 4-dimensional (4D) HH model and decreasing the complexity of the model equations. New 3D and 2D model equations have been derived. The new 3D and 2D models result from the relationship h≃ c-n, where c is a constant, of the gate variables h and n of the 4D HH model, suggesting an interdependence between the dynamics of the Na^+ and K^+ transmembrane pumps. We have derived the main properties of the propagation speed and width of action potentials for axons with Na^+ and K^+ active channels as a function of the transmembrane capacity C_m and resistivity R of the conducting axon. For the three HH type models, we show that the action potential propagates along the axon with speed well described by v(R, C_m)=α /(C_m R^β)=γ D^β, where α>0, 0<β <1 and γ are constants independent of the local intensity stimulus, and D is the diffusion coefficient of the axon. The width w of the action potential spikes depends on the resistivity of the axon with w = α_2 /R^β_2, where α_2 and β_2 are positive constants.