Spatiotemporal dynamics of continuum neural fields

We survey recent analytical approaches to studying the spatiotemporal dynamics of continuum neural fields. Neural fields model the large-scale dynamics of spatially structured biological neural networks in terms of nonlinear integrodifferential equations whose associated integral kernels represent the spatial distribution of neuronal synaptic connections. They provide an important example of spatially extended excitable systems with nonlocal interactions and exhibit a wide range of spatially coherent dynamics including traveling waves oscillations and Turing-like patterns

Modeling active cellular transport as a directed search process with stochastic resetting and delays

We show how certain active transport processes in living cells can be modeled in terms of a directed search process with stochastic resetting and delays. Two particular examples are the motor-driven intracellular transport of vesicles to synaptic targets in the axons and dendrites of neurons, and the cytoneme-based transport of morphogen to target cells during embryonic development. In both cases, the restart of the search process following reset has a finite duration with two components: a finite return time and a refractory period. We use a probabilistic renewal method to explicitly calculate the splitting probabilities and conditional mean first passage times (MFPTs) for capture by a finite array of contiguous targets. We consider two different search scenarios: bounded search on the interval [0, L], where L is the length of the array, with a refractory boundary at x = 0 and a reflecting boundary at x = L (model A), and partially bounded search on the half-line (model B). In the latter case there is a non-zero probability of failure to find a target in the absence of resetting. We show that both models have the same splitting probabilities, and that increasing the resetting rate r increases (reduces) the splitting probability for proximal (distal) targets. On the other hand the MFPTs for model A are monotonically increasing functions of r, whereas the MFPTs of model B are non-monotonic with a minimum at an optimal resetting rate. We also formulate multiple rounds of search-and-capture events as a G/M/∞ queue and use this to calculate the steady-state accumulation of resources in the targets

Two-dimensional droplet ripening in a concentration gradient

The discovery of various membraneless subcellular structures (biological condensates) in the cytoplasm and nucleus of cells has generated considerable interest in the effects of non-equilibrium chemical reactions on liquid–liquid phase separation and droplet ripening. Examples include the suppression of droplet ripening due to ATP-driven protein phosphorylation and the spatial segregation of droplets due to regulation by protein concentration gradients. Most studies of biological phase separation have focused on 3D droplet formation, for which mean field methods can be applied. However, mean field theory breaks down in the case of 2D systems, since the concentration around a droplet varies as ln R rather than R−1, where R is the distance from the center of a droplet. In this paper we use the asymptotic theory of diffusion in domains with small holes or exclusions (strongly localized perturbations) to study the segregation of circular droplets in gradient systems. We proceed by partitioning the region outside the droplets into a set of inner regions around each droplet together with an outer region where mean-field interactions occur. Asymptotically matching the inner and outer solutions, we derive dynamical equations for the position-dependent growth and drift of droplets. We thus show how a gradient of regulatory proteins leads to the segregation of droplets to one end of the domain, as previously found for 3D droplets

Switching diffusions and stochastic resetting

We consider a Brownian particle that switches between two different diffusion states (D0, D1) according to a two-state Markov chain. We further assume that the particle's position is reset to an initial value Xr at a Poisson rate r, and that the discrete diffusion state is simultaneously reset according to the stationary distribution ρn, n = 0, 1, of the Markov chain. We derive an explicit expression for the non-equilibrium steady state (NESS) on $\mathbb{R}$, which is given by the sum of two decaying exponentials. In the fast switching limit the NESS reduces to the exponential distribution of pure diffusion with stochastic resetting. The effective diffusivity is given by the mean $\overline{D}={\rho }_{0}{D}_{0}+{\rho }_{1}{D}_{1}$. We then determine the mean first passage time (MFPT) for the particle to be absorbed by a target at the origin, having started at the reset position Xr > 0. We proceed by calculating the survival probability in the absence of resetting and then use a last renewal equation to determine the survival probability with resetting. Similar to the NESS, we find that the MFPT depends on the sum of two exponentials, which reduces to a single exponential in the fast switching limit. Finally, we show that the MFPT has a unique minimum as a function of the resetting rate, and explore how the optimal resetting rate depends on other parameters of the system

Diffusion in a partially absorbing medium with position and occupation time resetting

In this paper we consider diffusion in a domain Ω containing a partially absorbing target $\mathcal{M}$ with position and occupation time resetting. The occupation time At is a Brownian functional that determines the amount of time that the particle spends in $\mathcal{M}$ over the time interval [0, t]. We assume that there exists some internal state ${\mathcal{U}}_{t}$ of the particle at time t which is modified whenever the particle is diffusing within $\mathcal{M}$. The state ${\mathcal{U}}_{t}$ is taken to be a monotonically increasing function of At, and absorption occurs as soon as ${\mathcal{U}}_{t}$ crosses some fixed threshold. We first show how to analyze threshold absorption in terms of the joint probability density or generalized propagator P(x, a, t|x0) for the pair (Xt, At) in the case of a non-absorbing substrate $\mathcal{M}$, where Xt is the particle position at time t and x0 is the initial position. We then introduce a generalized stochastic resetting protocol in which both the position Xt and the internal state ${\mathcal{U}}_{t}$ are reset to their initial values, Xt → x0 and ${\mathcal{U}}_{t}\to 0$, at a Poisson rate r. The latter is mathematically equivalent to resetting the occupation time, At → 0. Since resetting is governed by a renewal process, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which means that the statistics of absorption can be determined by calculating the double Laplace transform of P(x, a, t|x0) with respect to t and a. In order to develop the basic theory, we focus on one-dimensional diffusion with $\mathcal{M}$ given by a finite or semi-infinite interval, and explore how the mean first passage time with resetting depends on various model parameters. We also compare the threshold mechanism with the classical case of a constant absorption rate

Construction of stochastic hybrid path integrals using operator methods

Stochastic hybrid systems involve the coupling between discrete and continuous stochastic processes. They are finding increasing applications in cell biology, ranging from modeling promoter noise in gene networks to analyzing the effects of stochastically-gated ion channels on voltage fluctuations in single neurons and neural networks. We have previously derived a path integral representation of solutions to the associated differential Chapman–Kolmogorov equation, based on integral representations of the Dirac delta function, and used this to determine 'least action' paths in the noise-induced escape from a metastable state. In this paper we present an alternative derivation of the path integral based on operator methods, and show how this provides a more efficient and flexible framework for constructing hybrid path integrals in the weak noise limit. We also highlight the important role of principal eigenvalues, spectral gaps and the Perron–Frobenius theorem. Finally, we carry out a loop expansion of the associated moment generating functional in the weak noise limit, analogous to the semi-classical limit for quantum path integrals

Aggregation–fragmentation model of vesicular transport in neurons

We develop a mathematical model of the motor-based transport and delivery of vesicles to synaptic targets of an axon. Our model incorporates the 'stop-and-go' nature of bidirectional motor transport (which can be modeled in terms of advection–diffusion) and the reversible exchange of vesicles between motors and targets, both of which have been observed experimentally. Since motor-target interactions are reversible, it is necessary to keep track of the cluster size of vesicles bound to each motor-complex. This naturally leads to a modified version of the Becker–Doring model of aggregation–fragmentation processes. We analyze steady-state solutions of the transport model and obtain an explicit solution that supports a uniform distribution of synaptic resources along an axon. We thus establish a possible mechanism for the democratic distribution of synaptic resources along the length of an axon, based on reversible motor-target interactions. In the irreversible case, one finds that the motor-driven transport of newly synthesized proteins from the soma to presynaptic targets along the axon tends to favor the delivery of resources to more proximal synapses

Diffusion-mediated surface reactions and stochastic resetting

In this paper, we investigate the effects of stochastic resetting on diffusion in ${\mathbb{R}}^{d}{\backslash}\mathcal{U}$, where $\mathcal{U}$ is a bounded obstacle with a partially absorbing surface $\partial \mathcal{U}$. We begin by considering a Robin boundary condition with a constant reactivity κ0, and show how previous results are recovered in the limits κ0 → 0, ∞. We then generalize the Robin boundary condition to a more general probabilistic model of diffusion-mediated surface reactions using an encounter-based approach. The latter considers the joint probability density or generalized propagator P(x, ℓ, t|x0) for the pair (Xt, ℓt) in the case of a perfectly reflecting surface, where Xt and ℓt denote the particle position and local time, respectively. The local time determines the amount of time that a Brownian particle spends in a neighborhood of the boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. We construct the boundary value problem satisfied by the propagator in the presence of position resetting, and use this to derive implicit equations for the marginal density of particle position and the survival probability. We highlight the fact that these equations are difficult to solve in the case of non-constant reactivities, since resetting is not governed by a renewal process. We then consider a simpler problem in which both the position and local time are reset. In this case, the survival probability with resetting can be expressed in terms of the survival probability without resetting, which considerably simplifies the analysis. We illustrate the theory using the example of a spherically symmetric surface. In particular, we show that the effects of a partially absorbing surface on the mean first passage time (MFPT) for total absorption differs significantly if local time resetting is included. That is, the MFPT for a totally absorbing surface is increased by a multiplicative factor when the local time is reset, whereas the MFPT is increased additively when only particle position is reset

Stochastic switching in biology: from genotype to phenotype

There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1–1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a system-size expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker–Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel–Kramers–Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a self-contained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels. However, applications to other examples of biological switching are also discussed, including stochastic ion channels, diffusion in randomly switching environments, bacterial chemotaxis, and stochastic neural networks

Diffusion-mediated absorption by partially-reactive targets: Brownian functionals and generalized propagators

Many processes in cell biology involve diffusion in a domain Ω that contains a target $\mathcal{U}$ whose boundary $\partial \mathcal{U}$ is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or generalized propagator for particle position and the so-called boundary local time. The latter 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. In this paper we extend the theory of diffusion-mediated absorption to cases where the whole interior target domain $\mathcal{U}$ acts as a partial absorber rather than the target boundary $\partial \mathcal{U}$. Now the particle can freely enter and exit $\mathcal{U}$, and is only able to react (be absorbed) within $\mathcal{U}$. The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within $\mathcal{U}$) rather than the boundary local time. We show that both cases can be considered within a unified framework, which consists of a boundary value problem (BVP) for the propagator of the corresponding Brownian functional and an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target $\mathcal{U}$ located at the center of a spherical domain Ω. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing