Diffusive flux in a model of stochastically gated oxygen transport in insect respiration

Berezhkovskii, Alexander M.; Shvartsman, Stanislav Y.

doi:10.1063/1.4950769

Cited by 7 publications

(11 citation statements)

References 34 publications

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“…where g d is given by (2). We will see in section 4.3 that the mean neurotransmitter concentration in a general domain with many nerve varicosities reduces to (27).…”

Section: A Boundary Value Problem If Switching Is Markovianmentioning

confidence: 97%

“…Our local time analysis in section 4.3 gives a probabilisitic interpretation of matched asymptotic analysis of elliptic and parabolic PDEs [15,27,36] (see Remark 17). More broadly, this investigation adds to the growing body of work on diffusion in random environments [25,10,6,2,11,12] that has been driven by biological applications. Such processes combine two levels of randomness: Brownian motion at the individual particle level with a random environment.…”

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confidence: 97%

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A Probabilistic Analysis of Volume Transmission in the Brain

Lawley¹

2018

SIAM J. Appl. Math.

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Volume transmission is a fundamental neural communication mechanism in which neurons in one brain nucleus modulate the neurotransmitter concentration in the extracellular space of a second nucleus. In this paper, we formulate and analyze a mathematical model of volume transmission to calculate the neurotransmitter concentration in the extracellular space. Our model consists of the diffusion equation in a bounded two-or three-dimensional domain that contains a set of interior holes that randomly switch between being either sources or sinks. The interior holes represent nerve varicosities that are sources of neurotransmitter when firing an action potential and are sinks otherwise. To analyze this random partial differential equation, we show that each realization of its solution can be represented as a certain expected local time of a Brownian particle in a corresponding realization of a random environment. Using this representation, we prove two surprising results. First, the expected neurotransmitter concentration is approximately constant across the extracellular space. Second, by computing an explicit formula for this constant, we find that it depends on very few details in the problem. In particular, this constant does not depend on the number or arrangement of nerve varicosities, the geometry or size of the extracellular space, or firing correlations between neurons. The biological implications of these results will be explored in a forthcoming paper.

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“…where g d is given by (2). We will see in section 4.3 that the mean neurotransmitter concentration in a general domain with many nerve varicosities reduces to (27).…”

Section: A Boundary Value Problem If Switching Is Markovianmentioning

confidence: 97%

mentioning

confidence: 97%

A Probabilistic Analysis of Volume Transmission in the Brain

Lawley¹

2018

SIAM J. Appl. Math.

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“…. , c N through Poisson's equation (2). Supposing that the ion channel is always open, the boundary conditions are…”

Section: Gate Always Openmentioning

confidence: 99%

“…That is, if the potential difference is large, then the ion concentration behaves as if the channel is always open. Intuitively, one can understand this result by noting that (a) the relaxation rate of \Phi t 0 (the solution operator for an open channel) grows like V 2 4 as V grows, while (b) the relaxation rate of \Phi t 1 (the solution operator for a closed channel) vanishes as V grows. Therefore, if V \gg 1, then the ion concentration rapidly approaches equilibrium when the channel opens, but it hardly changes when the channel closes.…”

Section: Stochastic Gatingmentioning

confidence: 99%

“…The diffusion equation on an interval with randomly switching boundary conditions was first studied in [19], and additional methods of analyzing similar stochastic PDEs were then developed in [3,16]. Closely related models were later studied in the chemical physics literature [1,2]. Additional related work includes extensions of the classical Smoluchowski theory of diffusion-influenced reactions to stochastically gated reactions [4,8,26].…”

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Electrodiffusive Flux Through a Stochastically Gated Ion Channel

Lawley¹,

Keener²

2019

SIAM J. Appl. Math.

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A fundamental assumption of the Hodgkin-Huxley model and other conductancebased neuron models is that the average flux of ions through a stochastically gated ion channel is the product of (a) the flux of ions through a channel that is always open and (b) the proportion of time that the gated channel is open. In this paper, we propose and analyze a model of electrodiffusion through a stochastically gated ion channel to investigate the validity of this classical assumption. We find that this assumption is valid for typical physiological parameter regimes, and we also show that it breaks down for parameters outside of typical physiological ranges. Indeed, we show that the flux through a gated channel can be orders of magnitude larger than this classical assumption if either the gating is fast or the potential difference across the membrane is large. Mathematically, our model consists of one-dimensional advection-diffusion equations with a stochastically switching boundary condition. Employing an iterated random function approach, we prove that the solution converges in distribution at large time and find (i) the support of the solution, (ii) analytical formulas for the mean solution and mean flux, and (iii) analytical formulas for the full probability distribution of the solution in various parameter regimes. All of our analysis is accompanied by numerical simulations of the stochastic PDE.

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