Controllability of excitable systems

Mathematical models of cell electrical activity typically consist of a current balance equation, channel activation (or inactivation) variables and concentrations of regulatory agents. These models can be thought of as nonlinear filters whose input is some applied current I (possibly zero) and output is a membrane potential V. A natural question to ask is if the applied current I can be deduced from the potential V. For a surprisingly large class of models the answer to this question is yes. To show this, we first demonstrate how many models can be embedded into higher dimensional quasilinear systems. For quasilinear models, a procedure for determining the inverse of the nonlinear filter is then described and demonstrated on two models: (1) the FitzHugh-Nagumo model and (2) the Sherman-Rinzel-Keizer (SRK) [Sherman et al., (1988, Biophysics Journal, 54, 411-425)] model of bursting electrical activity in pancreatic beta-cells. For the latter example, the inverse problem is then used to deduce model parameter values for which the model and experimental data agree in some measure. An advantage of the correlation technique is that experimental values for activation (and/or regulatory) variables need not be known to make the estimates for these parameter values.