Nonlinear Thermodynamic Models of Voltage-Dependent Currents

Hodgkin and Huxley provided the first quantitative description of voltage-dependent currents and adjusted their model to experimental data using empirical functions of voltage. A physically plausible formalism was proposed later by assuming that transition rates depend exponentially on a free-energy barrier, by analogy with the theory of reaction rates. It was also assumed that the free energy depends linearly on voltage. This thermodynamic formalism can accurately describe many processes, but the resulting time constants can be arbitrarily fast, which may also lead to aberrant behavior. We considered here a physically plausible solution to this problem by including nonlinear effects of the electrical field on the free energy. We show that including effects such as mechanical constraints, inherent to the structure of the ion channel protein, leads to more accurate thermodynamic models. These models can account for voltage-dependent transitions that are rate-limited in a given voltage range, without invoking additional states. We illustrate their applicability to fit experimental data by considering the case of the T-type calcium current in thalamic neurons.