Quadratization: From Conductance-Based Models To Caricature Models With Parabolic Nonlinearities

Abstract

DefinitionQuadratization of biophysical (conductance-based) models having a parabolic-like voltage nullcline in the subthreshold voltage regime refers to the process by which these models are substituted by \"caricature\" models having a strictly parabolic voltage nullcline and a linear nullcline for the recovery variable. We refer to the latter as quadratic or parabolic models. The parabolic-like and strictly parabolic voltage nullclines coincide at their extrema (minima or maxima) and are well approximated by each other in vicinities of these extrema whose size depend on the model parameters. Quadratic models are simplified by a change of variables that translates these extrema into the origin of the phase-plane diagram. A further simplification (parameter reduction) can be achieved by nondimensionalizing the quadratic models. This procedure can be extended to three-dimensional models having a parabolic-cylinder-like shaped voltage nullsurface and to models having time-dependent inputs and synaptic currents.