Resumen
The Hodgkin--Huxley (HH) gating model has been extensively employed over the last half century to describe bioelectricity phenomena related to normal and impaired electrophysiological functions. Since the HH gating model is relatively empirical, the associated modelling methodology requires estimating model parameters (including functions of membrane voltage) from experimental data. Until now, as is the case for most nonlinear models, parameter estimation has been carried out through nonlinear least square fitting, which presents important limitations for the modelling methodology. Here we pursue a different approach to the estimation problem, which allows us to overcome all the limitations inherent to nonlinear fitting. As initially introduced by Beaumont, Roberge, and Leon [Math. Biosci., 115 (1993), pp. 65--101], instead of fitting we invert the solution. Specifically, model parameters (including functions of membrane voltage) are obtained from multiple transformations (or modals) applied to the solution, or equivalently, from an experimental data set. Such transformations enable one to deduce, for a given data point, disjoint ranges of parameter values which allow the model to exactly reproduce the solution. Using sufficiently large data sets and continuity criteria, it is possible to narrow down estimates to a specific value. Our main results are (i) a more accurate estimation procedure; (ii) the ability to determine whether a data set sufficiently constrains the model, i.e., whether it is complete; (iii) if it is not, the possibility to identify a model family capable of reproducing the entire data set; and (iv) stimulation protocols which can produce complete data sets.