Period doubling of calcium spike firing in a model of a Purkinje cell dendrite

Recordings from cerebellar Purkinje cell dendrites have revealed that in re sponse to sustained current injection, the cell firing pattern can move fro m tonic firing of Ca2+ spikes to doublet firing and even to quadruplet firi ng or more complex firing. These firing patterns are not modified substanti ally if Na+ currents are blocked. We show that the experimental results can be viewed as a slow transition of the neuronal dynamics through a period-d oubling bifurcation. To further support this conclusion and to understand t he underlying mechanism that leads to doublet firing, we develop and study a simple, one-compartment model of Purkinje cell dendrite. The neuron can a lso exhibit quadruplet and chaotic firing patterns that are similar to the firing patterns that some of the Purkinje cells exhibit experimentally. The effects of parameters such as temperature, applied current, and potassium reversal potential in the model resemble their effects in experiments. The model dynamics involve three time scales. Ca2+-dependent K+ currents, with intermediate time scales, are responsible for the appearance of doublet fir ing, whereas a very slow hyperpolarizing current transfers the neuron from tonic to doublet firing. We use the fast-slow analysis to separate the effe cts of the three time scales. Fast-slow analysis of the neuronal dynamics, with the activation variable of the very slow, hyperpolarizing current cons idered as a parameter, reveals that the transitions occurs via a cascade of period-doubling bifurcations of the fast and intermediate subsystem as thi s slow variable increases. We carry out another analysis, with the Ca2+ con centration considered as a parameter, to investigate the conditions for the generation of doublet firing in systems with one effective variable with i ntermediate time scale, in which the rest state of the fast subsystem is te rminated by a saddle-node bifurcation. We find that the scenario of period doubling in these systems can occur only if (1) the time scale of the inter mediate variable (here, the decay rate of the calcium concentration) is slo w enough in comparison with the interspike interval of the tonic firing at the transition but is not too slow and (2) there is a bistability of the fa st subsystem of the spike-generating variables.