Georgi Medvedev
          Program in Computational and Applied Mathematics,
    Princeton University

Based on recent calcium imaging studies, Wilson and Callaway
proposed to model an isolated dopaminergic cell using chains
of coupled oscillators. Unlike in other models, they assume
that dendritic as well as somatic compartments are capable
of autonomous oscillations. All compartments are strongly
electrically coupled. The variation of a cross section diameter
along the cell yields the variation in natural frequencies of
oscillators in the chain. The model was shown to reproduce
many of the experimental observations. In particular, it captures
well the basic features of the rhythmic spiking, the most
common firing pattern in vitro, and displays a spike frequency
adaptation. Both experimental data and simulations show pronounced
transient dynamics following a perturbation of steady state
oscillations.

Our analysis explains the mechanism of synchronous oscillations
of the membrane potential and calcium concentrations in the somatic
and all dendritic compartments. We also give a detailed analytical
description of the transient dynamics. In particular, we show that the
duration of transients is directly proportional to the strength
of coupling. Our results are based on the geometric theory for
singularly perturbed systems, asymptotic expansions, and the
Lyapunov's method. It is the latter, that reveals the nature
of transients and characterizes the steady state oscillations.
This is joint work with N. Kopell.