Extending the Conditions of Application of an Inversion of the Hodgkin-Huxley Gating Model

We present an inversion of the Hodgkin-Huxley formalism to estimate initial conditions and model parameters, including functions of voltage, from the solutions of the underlying ordinary differential equation (ODE) subjected to multiple voltage step stimulations. As such, the procedure constitutes a means to estimate the parameters including functions of voltage of an Hodgkin-Huxley formalism from experimental data. The basic idea was developed in a previous communication (SIAM J. Appl. Math. 64:1264-1274, 2009). The inversion in question applies to currents exhibiting activation and inactivation, but the version, as published previously, cannot estimate the unknowns for channels that rapidly inactivate just after a brief opening. In such cases, the amplitude of the current, in a given voltage range, is too small to be detectable by the instrumentation using previously applied experimental protocols. This is, for example, the case for the sodium channels in a number of excitable tissue for potential in the vicinity of the cell resting potential. The current communication extends the inversion procedure in a manner to overcome this limitation. Furthermore, within the inversion framework, we can determine whether the data at the basis of the estimation sufficiently constrains the estimation problem, i.e., whether it is complete. We exploit this element of our method to document a set of stimulation protocols that constitute a complete data set for the purpose of inverting the Hodgkin-Huxley formalism.