Using constraints on neuronal activity to reveal compensatory changes in neuronal parameters

In this study, we developed a general description of parameter combinations for which specified characteristics of neuronal or network activity are constant. Our approach is based on the implicit function theorem and is applicable to activity characteristics that smoothly depend on parameters. Such smoothness is often intrinsic to neuronal systems when they are in stable functional states. The conclusions about how parameters compensate each other, developed in this study, can thus be used even without regard to the specific mathematical model describing a particular neuron or neuronal network. We showed that near a generic point in the parameter space there are infinitely many other points, or parameter combinations, for which specified characteristics of activity are the same as in the original point. These parameter combinations form a smooth manifold. This manifold can be extended as long as the gradients of characteristics are defined and independent. All possible variations of parameters compensating each other are simply all possible charts of the same manifold. The number of compensating parameters (but not parameters themselves) is fixed and equal to the number of the independent characteristics maintained. The algorithm that we developed shows how to find compensatory functional dependencies between parameters numerically. Our method can be used in the analysis of the homeostatic regulation, neuronal database search, model tuning and other applications.