Geometry and dynamics of activity-dependent homeostatic regulation in neurons

To maintain activity in a functional range, neurons constantly adjust membrane excitability to changing intra- and extracellular conditions. Such activity-dependent homeostatic regulation (ADHR) is critical for normal processing of the nervous system and avoiding pathological conditions. Here, we posed a homeostatic regulation problem for the classical Morris-Lecar (ML) model. The problem was motivated by the phenomenon of the functional recovery of stomatogastric neurons in crustaceans in the absence of neuromodulation. In our study, the regulation of the ionic conductances in the ML model depended on the calcium current or the intracellular calcium concentration. We found an asymptotic solution to the problem under the assumption of slow regulation. The solution provides a full account of the regulation in the case of correlated or anticorrelated changes of the maximal conductances of the calcium and potassium currents. In particular, the solution shows how the target and parameters of the regulation determine which perturbations of the conductances can be compensated by the ADHR. In some cases, the sets of compensated initial perturbations are not convex. On the basis of our analysis we formulated specific questions for subsequent experimental and theoretical studies of ADHR.