Stochastic hybrid model of spontaneous dendritic NMDA spikes

Following recent advances in imaging techniques and methods of dendritic stimulation, active voltage spikes have been observed in thin dendritic branches of excitatory pyramidal neurons, where the majority of synapses occur. The generation of these dendritic spikes involves both Na(+) ion channels and M-methyl-D-aspartate receptor (NMDAR) channels. During strong stimulation of a thin dendrite, the resulting high levels of glutamate, the main excitatory neurotransmitter in the central nervous system and an NMDA agonist, modify the current-voltage (I-V) characteristics of an NMDAR so that it behaves like a voltage-gated Na(+) channel. Hence, the NMDARs can fire a regenerative dendritic spike, just as Na(+) channels support the initiation of an action potential following membrane depolarization. However, the duration of the dendritic spike is of the order 100 ms rather than 1 ms, since it involves slow unbinding of glutamate from NMDARs rather than activation of hyperpolarizing K(+) channels. It has been suggested that dendritic NMDA spikes may play an important role in dendritic computations and provide a cellular substrate for short-term memory. In this paper, we consider a stochastic, conductance-based model of dendritic NMDA spikes, in which the noise originates from the stochastic opening and closing of a finite number of Na(+) and NMDA receptor ion channels. The resulting model takes the form of a stochastic hybrid system, in which membrane voltage evolves according to a piecewise deterministic dynamics that is coupled to a jump Markov process describing the opening and closing of the ion channels. We formulate the noise-induced initiation and termination of a dendritic spike in terms of a first-passage time problem, under the assumption that glutamate unbinding is negligible, which we then solve using a combination of WKB methods and singular perturbation theory. Using a stochastic phase-plane analysis we then extend our analysis to take proper account of the combined effects of glutamate unbinding and noise on the termination of a spike.