Stochastic Hybrid Systems in Cellular Neuroscience

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.

Special Issue from the 2017 International Conference on Mathematical Neuroscience

“…To understand how stochasticity emerges at the macroscopic level, it is important to scale up microscopic models of such fluctuations. Bressloff and MacLaurin (2018) review stochastic hybrid methods, which allow for the detailed analysis of partially deterministic Markov processes (PDMPs) that emerge from models in cellular neuroscience [3]. Considering variability and uncertainty is also important when fitting parameterized models to data.…”

“…Ion channels open and close and can be modeled by a continuous time Markov process, where the finite number of channels causes spontaneous spiking in conductance-based models due to channel fluctuations [12, 15]. Bressloff and MacLaurin (2018) review the mathematical framework needed to analyze such stochastic systems, leveraging their piecewise deterministic nature [3]. Stochastic transitions occur at discrete time, but in between the dynamics evolves deterministically.…”

“…Such metastability can emerge in stochastic hybrid systems which in the mean field limit become conductance-based neuron models. The stochastic evolution of action potentials in such models can be characterized by analyzing the corresponding CK equations describing the population evolution of ion channel states [3]. The authors also discuss models with stochastic gap junctions and diffusion in randomly switching environments [16], as well as models of intracellular transport in axons and dendrites [17].…”

“…The tutorials thus provide complementary analyses and techniques for addressing stochasticity in neural systems. Bressloff and MacLaurin (2018) analyze mechanistic models accounting for stochasticity at the microscopic level and discuss ways of scaling these models up to more macroscopic quantities [3]. Alternatively, Moye and Diekmann (2018) suggest means of dealing with such stochasticity as a statistical problem [4], so noisy data can be filtered properly to reliably fit deterministic models—these can be the mean field counterparts of stochastic neuronal models.…”

“…1), including balanced networks, information theory and geometry, efficient coding in spiking networks, plasticity, and stochastic hybrid systems. In line with these efforts, we present two tutorial reviews, one on stochastic hybrid methods [3] and the other on data assimilation methods in neuron models [4].

Figure 1Selected tutorial, plenary, and parallel presentations from ICMNS 2017.

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