Abstract
We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.
Highlights
There are a growing number of problems in cell biology that involve the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space Γ, resulting in a stochastic hybrid system, known as a piecewise deterministic Markov process (PDMP) [37]
(ii) Escape trajectories all pass through a narrow region of state space so that, there is no well-defined separatrix for an excitable system, it is possible to formulate an escape problem by determining the mean first passage times (MFPTs) to reach the bottleneck from the resting state
A number of recent experimental studies of intracellular transport in axons of C. elegans and Drosophila have shown that (i) motor-driven vesicular cargo exhibits “stop and go” behavior, in which periods of ballistic anterograde or retrograde transport are interspersed by long pauses at presynaptic sites, and (ii) the capture of vesicles by synapses during the pauses is reversible in the sense that the aggregation of vesicles can be inhibited by signaling molecules resulting in dissociation from the target [96, 148]
Summary
There are a growing number of problems in cell biology that involve the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space Γ , resulting in a stochastic hybrid system, known as a piecewise deterministic Markov process (PDMP) [37]. Finite-size effects can result in the noise-induced spontaneous firing of a neuron due to channel fluctuations Another important example is a gene regulatory network, where the continuous variable is the concentration of a protein product, and the discrete variable represents the activation state of the gene [79, 83, 108, 110, 131]. One example concerns molecular diffusion in cellular and subcellular domains with randomly switching exterior or interior boundaries [12, 17–19, 92] The latter are generated by the random opening and closing of gates (ion channels or gap junctions) within the plasma membrane. Diffusion in randomly switching environments has applications to the branched network of tracheal tubes forming the passive respiration system in insects [18, 92] and volume neurotransmission [90] This tutorial review develops the theory and applications of stochastic hybrid systems within the context of cellular neuroscience.
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