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Article ; Online: Global density equations for interacting particle systems with stochastic resetting: From overdamped Brownian motion to phase synchronization.

Bressloff, Paul C

2024 Volume 34, Issue 4

Abstract: A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic "agents." Kinetic methods for reducing the ... ...

Abstract	A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic "agents." Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years, there has been considerable interest in the mean field limit of interacting particle systems with long-range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper, we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case, we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson noise process, reflecting the fact that resetting is common to all of the particles and, thus, induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold.
Language	English
Publishing date	2024-04-01
Publishing country	United States
Document type	Journal Article
ZDB-ID	1472677-4
ISSN	1089-7682 ; 1054-1500
ISSN (online)	1089-7682
ISSN	1054-1500
DOI	10.1063/5.0196626
Database	MEDical Literature Analysis and Retrieval System OnLINE

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Article ; Online: Truncated stochastically switching processes.

Bressloff, Paul C

Physical review. E

2024 Volume 109, Issue 2-1, Page(s) 24103

Abstract: There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov ... ...

Abstract	There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.
Language	English
Publishing date	2024-03-15
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.109.024103
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Book: Waves in neural media

Bressloff, Paul C.

from single neurons to neural fields

(Lectures notes on mathematical modelling in the life sciences)

2014

Author's details	Paul C. Bressloff
Series title	Lectures notes on mathematical modelling in the life sciences
Language	English
Size	XIX, 436 S. : Ill., graph. Darst.
Publisher	Springer
Publishing place	New York u.a.
Publishing country	United States
Document type	Book
HBZ-ID	HT017901732
ISBN	978-1-4614-8865-1 ; 9781461488668 ; 1-4614-8865-6 ; 1461488664
Database	Catalogue ZB MED Medicine, Health

In stock of ZB MED Cologne/Königswinter

Shelf mark	Loan status	Location
2013 A 3604	available for loan (reading room)	Lesesaal EG

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Article ; Online: Close encounters of the sticky kind: Brownian motion at absorbing boundaries.

Bressloff, Paul C

Physical review. E

2023 Volume 107, Issue 6-1, Page(s) 64121

Abstract: Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we ... ...

Abstract	Encounter-based models of diffusion provide a probabilistic framework for analyzing the effects of a partially absorbing reactive surface, in which the probability of absorption depends upon the amount of surface-particle contact time. In this paper we develop a class of encounter-based models that deal with absorption at sticky boundaries. Sticky boundaries occur in a diverse range of applications, including cell biology, colloidal physics, finance, and human crowd dynamics. They also naturally arise in active matter, where confined active particles tend to spontaneously accumulate at boundaries even in the absence of any particle-particle interactions. We begin by constructing a one-dimensional encounter-based model of sticky Brownian motion (BM), which is based on the zero-range limit of nonsticky BM with a short-range attractive potential well near the origin. In this limit, the boundary-contact time is given by the amount of time (occupation time) that the particle spends at the origin. We calculate the joint probability density or propagator for the particle position and the occupation time, and then identify an absorption event as the first time that the occupation time crosses a randomly generated threshold. We illustrate the theory by considering diffusion in a finite interval with a partially absorbing sticky boundary at one end. We show how various quantities, such as the mean first passage time (MFPT) for single-particle absorption and the relaxation to steady state at the multiparticle level, depend on moments of the random threshold distribution. Finally, we determine how sticky BM can be obtained by taking a particular diffusion limit of a sticky run-and-tumble particle (RTP).
Language	English
Publishing date	2023-07-19
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.107.064121
Database	MEDical Literature Analysis and Retrieval System OnLINE

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Article ; Online: Renewal equations for single-particle diffusion through a semipermeable interface.

Bressloff, Paul C

Physical review. E

2023 Volume 107, Issue 1-1, Page(s) 14110

Abstract: Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional ...

Abstract	Diffusion through semipermeable interfaces has a wide range of applications, ranging from molecular transport through biological membranes to reverse osmosis for water purification using artificial membranes. At the single-particle level, one-dimensional diffusion through a barrier with constant permeability κ_{0} can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflected BMs that are restricted to either the left or right of the barrier. Each round is killed (absorbed) at the barrier when its Brownian local time exceeds an exponential random variable parameterized by κ_{0}. A new round is then immediately started in either direction with equal probability. It has recently been shown that the probability density for snapping out BM satisfies a renewal equation that relates the full density to the probability densities of partially reflected BM on either side of the barrier. Moreover, generalized versions of the renewal equation can be constructed that incorporate non-Markovian, encounter-based models of absorption. In this paper we extend the renewal theory of snapping out BM to single-particle diffusion in bounded domains and higher spatial dimensions. In each case we show how the solution of the renewal equation satisfies the classical diffusion equation with a permeable boundary condition at the interface. That is, the probability flux across the interface is continuous and proportional to the difference in densities on either side of the interface. We also consider an example of an asymmetric interface in which the directional switching after each absorption event is biased. Finally, we show how to incorporate an encounter-based model of absorption for single-particle diffusion through a spherically symmetric interface. We find that, even when the same non-Markovian model of absorption applies on either side of the interface, the resulting permeability is an asymmetric time-dependent function with memory. Moreover, the permeability functions tend to be heavy tailed.
Language	English
Publishing date	2023-02-16
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.107.014110
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Book ; Online: Global density equations for interacting particle systems with stochastic resetting

Bressloff, Paul C

from overdamped Brownian motion to phase synchronization

2024

Abstract	A wide range of phenomena in the natural and social sciences involve large systems of interacting particles, including plasmas, collections of galaxies, coupled oscillators, cell aggregations, and economic ``agents'. Kinetic methods for reducing the complexity of such systems typically involve the derivation of nonlinear partial differential equations for the corresponding global densities. In recent years there has been considerable interest in the mean field limit of interacting particle systems with long range interactions. Two major examples are interacting Brownian particles in the overdamped regime and the Kuramoto model of coupled phase oscillators. In this paper we analyze these systems in the presence of local or global stochastic resetting, where the position or phase of each particle independently or simultaneously resets to its original value at a random sequence of times generated by a Poisson process. In each case we derive the Dean-Kawasaki (DK) equation describing hydrodynamic fluctuations of the global density, and then use a mean field ansatz to obtain the corresponding nonlinear McKean-Vlasov (MV) equation in the thermodynamic limit. In particular, we show how the MV equation for global resetting is driven by a Poisson shot noise process, reflecting the fact that resetting is common to all of the particles and thus induces correlations that cannot be eliminated by taking a mean field limit. We then investigate the effects of local and global resetting on nonequilibrium stationary solutions of the macroscopic dynamics and, in the case of the Kuramoto model, the reduced dynamics on the Ott-Antonsen manifold. Comment: 53 pages, 13 figures
Keywords	Condensed Matter - Statistical Mechanics ; Mathematics - Dynamical Systems ; Mathematics - Probability
Subject code	612
Publishing date	2024-01-07
Publishing country	us
Document type	Book ; Online
Database	BASE - Bielefeld Academic Search Engine (life sciences selection)

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Article ; Online: Morphogen gradient formation in partially absorbing media.

Bressloff, Paul C

Physical biology

2022 Volume 19, Issue 6

Abstract: Morphogen gradients play an essential role in the spatial regulation of cell patterning during early development. The classical mechanism of morphogen gradient formation involves the diffusion of morphogens away from a localized source combined with some ...

Abstract	Morphogen gradients play an essential role in the spatial regulation of cell patterning during early development. The classical mechanism of morphogen gradient formation involves the diffusion of morphogens away from a localized source combined with some form of bulk absorption. Morphogen gradient formation plays a crucial role during early development, whereby a spatially varying concentration of morphogen protein drives a corresponding spatial variation in gene expression during embryogenesis. In most models, the absorption rate is taken to be a constant multiple of the local concentration. In this paper, we explore a more general class of diffusion-based model in which absorption is formulated probabilistically in terms of a stopping time condition. Absorption of each particle occurs when its time spent within the bulk domain (occupation time) exceeds a randomly distributed threshold
MeSH term(s)	Morphogenesis ; Models, Biological ; Diffusion ; Embryonic Development
Language	English
Publishing date	2022-10-25
Publishing country	England
Document type	Journal Article
ZDB-ID	2133216-2
ISSN	1478-3975 ; 1478-3967
ISSN (online)	1478-3975
ISSN	1478-3967
DOI	10.1088/1478-3975/ac95ea
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Article ; Online: Local accumulation time for diffusion in cells with gap junction coupling.

Bressloff, Paul C

Physical review. E

2022 Volume 105, Issue 3-1, Page(s) 34404

Abstract: In this paper we analyze the relaxation to steady state of intracellular diffusion in a pair of cells with gap-junction coupling. Gap junctions are prevalent in most animal organs and tissues, providing a direct diffusion pathway for both electrical and ... ...

Abstract	In this paper we analyze the relaxation to steady state of intracellular diffusion in a pair of cells with gap-junction coupling. Gap junctions are prevalent in most animal organs and tissues, providing a direct diffusion pathway for both electrical and chemical communication between cells. Most analytical models of gap junctions focus on the steady-state diffusive flux and the associated effective diffusivity. Here we investigate the relaxation to steady state in terms of the so-called local accumulation time. The latter is commonly used to estimate the time to form a protein concentration gradient during morphogenesis. The basic idea is to treat the fractional deviation from the steady-state concentration as a cumulative distribution for the local accumulation time. One of the useful features of the local accumulation time is that it takes into account the fact that different spatial regions can relax at different rates. We consider both static and dynamic gap junction models. The former treats the gap junction as a resistive channel with effective permeability μ, whereas the latter represents the gap junction as a stochastic gate that randomly switches between an open and closed state. The local accumulation time is calculated by solving the diffusion equation in Laplace space and then taking the small-s limit. We show that the accumulation time is a monotonically increasing function of spatial position, with a jump discontinuity at the gap junction. This discontinuity vanishes in the limit μ→∞ for a static junction and β→0 for a stochastically gated junction, where β is the rate at which the gate closes. Finally, our results are generalized to the case of a linear array of cells with nearest-neighbor gap junction coupling.
Language	English
Publishing date	2022-04-10
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.105.034404
Database	MEDical Literature Analysis and Retrieval System OnLINE

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Article ; Online: Narrow capture problem: An encounter-based approach to partially reactive targets.

Bressloff, Paul C

Physical review. E

2022 Volume 105, Issue 3-1, Page(s) 34141

Abstract: A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically ... ...

Abstract	A general topic of current interest is the analysis of diffusion problems in singularly perturbed domains with small interior targets or traps (the narrow capture problem). One major application is to intracellular diffusion, where the targets typically represent some form of reactive biochemical substrate. Most studies of the narrow capture problem treat the target boundaries as totally absorbing (Dirichlet), that is, the chemical reaction occurs immediately on first encounter between particle and target surface. In this paper, we analyze the three-dimensional narrow capture problem in the more realistic case of partially reactive target boundaries. We begin by considering classical Robin boundary conditions. Matching inner and outer solutions of the single-particle probability density, we derive an asymptotic expansion of the Laplace transformed flux into each reactive surface in powers of ε, where ερ is a given target size. In turn, the fluxes determine the splitting probabilities for target absorption. We then extend our analysis to more general types of reactive targets by combining matched asymptotic analysis with an encounter-based formulation of diffusion-mediated surface reactions. That is, we derive an asymptotic expansion of the joint probability density for particle position and the so-called boundary local time, which characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. Robin boundary conditions are recovered in the special case of an exponential law for the stopping local times. Finally, we illustrate the theory by exploring how the leading-order contributions to the splitting probabilities depend on the choice of surface reactions. In particular, we show that there is an effective renormalization of the target radius of the form ρ→ρ-Ψ[over ̃](1/ρ), where Ψ[over ̃] is the Laplace transform of the stopping local time distribution.
Language	English
Publishing date	2022-04-10
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.105.034141
Database	MEDical Literature Analysis and Retrieval System OnLINE

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Article ; Online: Stochastically switching diffusion with partially reactive surfaces.

Bressloff, Paul C

Physical review. E

2022 Volume 106, Issue 3-1, Page(s) 34108

Abstract: In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a ... ...

Abstract	In this paper we develop a hybrid version of the encounter-based approach to diffusion-mediated absorption at a reactive surface, which takes into account stochastic switching of a diffusing particle's conformational state. For simplicity, we consider a two-state model in which the probability of surface absorption depends on the current particle state and the amount of time the particle has spent in a neighborhood of the surface in each state. The latter is determined by a pair of local times ℓ_{n,t}, n=0,1, which are Brownian functionals that keep track of particle-surface encounters over the time interval [0,t]. We proceed by constructing a differential Chapman-Kolmogorov equation for a pair of generalized propagators P_{n}(x,ℓ_{0},ℓ_{1},t), where P_{n} is the joint probability density for the set (X_{t},ℓ_{0,t},ℓ_{1,t}) when N_{t}=n, where X_{t} denotes the particle position and N_{t} is the corresponding conformational state. Performing a double Laplace transform with respect to ℓ_{0},ℓ_{1} yields an effective system of equations describing diffusion in a bounded domain Ω, in which there is switching between two Robin boundary conditions on ∂Ω. The corresponding constant reactivities are κ_{j}=Dz_{j} and j=0,1, where z_{j} is the Laplace variable corresponding to ℓ_{j} and D is the diffusivity. Given the solution for the propagators in Laplace space, we construct a corresponding probabilistic model for partial absorption, which requires finding the inverse Laplace transform with respect to z_{0},z_{1}. We illustrate the theory by considering diffusion of a particle on the half-line with the boundary at x=0 effectively switching between a totally reflecting and a partially absorbing state. We calculate the flux due to absorption and use this to compute the resulting MFPT in the presence of a renewal-based stochastic resetting protocol. The latter resets the position and conformational state of the particle as well as the corresponding local times. Finally, we indicate how to extend the analysis to higher spatial dimensions using the spectral theory of Dirichlet-to-Neumann operators.
Language	English
Publishing date	2022-10-20
Publishing country	United States
Document type	Journal Article
ZDB-ID	2844562-4
ISSN	2470-0053 ; 2470-0045
ISSN (online)	2470-0053
ISSN	2470-0045
DOI	10.1103/PhysRevE.106.034108
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