Application of the Hodgkin-Huxley equations to an electric field model for interaction between excitable cells

We previously described a model for the electrical transfer of excitation from one cell to the next which utilized the electric potential generated in the junctional cleft between the cells. Low-resistance connections between the cells were not used in the model, and it was assumed that the junctional membranes were excitable. This model was analyzed for the static case without capacitances and for the dynamic case in which capacitances were part of the circuit elements. For simplicity, the Na⁺ resistance (R_Na), after a threshold potential was exceeded, was allowed to decrease exponentially (to 1% of its initial value) within 0·25–1·0 ms, and possible changes in the K⁺ resistance were ignored. In this paper, we have incorporated the Hodgkin-Huxley equations into the operation of the lumped membrane units for the electrical equivalent circuit of the cell membrane. The parameters varied are the membrane capacitances, resistances, maximum Na⁺ conductance ($\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$), and the radial cleft resistance (R_jc). We demonstrated that our model worked very well, i.e. the successful transfer of action potentials was achieved, with the membrane units following Hodgkin-Huxley dynamics for changes in g_Na and g_K. The calculations indicate that transmission is facilitated when the junctional units have a higher $\text{g}{}_{\mathrm{<mtext>Na</mtext>}}$ and a lower capacitance and when R_jc is elevated. Lowering the resistance of the junctional membrane units several fold, relative to the surface membrane units, also facilitated transmission; however, the absolute resistance of the junctional membrane was still well above the maximum value that would allow sufficient local-circuit current to flow to effect transmission. Thus, the electric field model provides an alternative means of cell-to-cell propagation between myocardial cells which is electrical in nature but does not require the presence of low-resistance connections between cells.