Metastability in a stochastic neural network modeled as a velocity jump Markov process

One of the major challenges in neuroscience is to determine how noise that is present at the molecular and cellular levels affects dynamics and information processing at the macroscopic level of synaptically coupled neuronal populations. Often noise is incorprated into deterministic network models using extrinsic noise sources. An alternative approach is to assume that noise arises intrinsically as a collective population effect, which has led to a master equation formulation of stochastic neural networks. In this paper we extend the master equation formulation by introducing a stochastic model of neural population dynamics in the form of a velocity jump Markov process. The latter has the advantage of keeping track of synaptic processing as well as spiking activity, and reduces to the neural master equation in a particular limit. The population synaptic variables evolve according to piecewise deterministic dynamics, which depends on population spiking activity. The latter is characterised by a set of discrete stochastic variables evolving according to a jump Markov process, with transition rates that depend on the synaptic variables. We consider the particular problem of rare transitions between metastable states of a network operating in a bistable regime in the deterministic limit. Assuming that the synaptic dynamics is much slower than the transitions between discrete spiking states, we use a WKB approximation and singular perturbation theory to determine the mean first passage time to cross the separatrix between the two metastable states. Such an analysis can also be applied to other velocity jump Markov processes, including stochastic voltage-gated ion channels and stochastic gene networks

Renewal equations for single-particle diffusion through a semipermeable interface

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 $\kappa_0$ can be modeled in terms of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of partially reflecting 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 $\kappa_0$. 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. 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.Comment: 18 pages, 6 figure

Directional search-and-capture model of cytoneme-based morphogenesis

In this paper we develop a directional search-and-capture model of cytoneme-based morphogenesis. We consider a single cytoneme nucleating from a source cell and searching for a set of $N$ target cells $\Omega_k\subset \R^d$, $k=1,\ldots,N$, with $d\geq 2$. We assume that each time the cytoneme nucleates, it grows in a random direction so that the probability of being oriented towards the $k$-th target is $p_k$ with $\sum_{k=1}^Np_k<1$. Hence, there is a non-zero probability of failure to find a target unless there is some mechanism for returning to the nucleation site and subsequently nucleating in a new direction. We model the latter as a one-dimensional search process with stochastic resetting, finite returns times and refractory periods. We use a renewal method to calculate the splitting probabilities and conditional mean first passage times (MFPTs) for the cytoneme to be captured by a given target cell. We then determine the steady-state accumulation of morphogen over the set of target cells following multiple rounds of search-and-capture events and morphogen degradation. This then yields the corresponding morphogen gradient across the set of target cells, whose steepness depends on the resetting rate. We illustrate the theory by considering a single layer of target cells, and discuss the extension to multiple cytonemes.Comment: 20 pages, 11 figure

Truncated stochastically switching processes

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 to exceed a given number of resets, respectively.Comment: 14 pages, 6 figure

Occupation time of a run-and-tumble particle with resetting

We study the positive occupation time of a run-and-tumble particle (RTP) subject to stochastic resetting. Under the resetting protocol, the position of the particle is reset to the origin at a random sequence of times that is generated by a Poisson process with rate $r$. The velocity state is reset to $\pm v$ with fixed probabilities $\rho_1$ and $\rho_{-1}=1-\rho_1$, where $v$ is the speed. We exploit the fact that the moment generating functions with and without resetting are related by a renewal equation, and the latter generating function can be calculated by solving a corresponding Feynman-Kac equation. This allows us to numerically locate in Laplace space the largest real pole of the moment generating function with resetting, and thus derive a large deviation principle (LDP) for the occupation time probability density using the Gartner-Ellis theorem. We explore how the LDP depends on the switching rate $\alpha$ of the velocity state, the resetting rate $r$ and the probability $\rho_1$. In particular, we show that the corresponding LDP for a Brownian particle with resetting is recovered in the fast switching limit $\alpha\rightarrow \infty$. On the other hand, the behavior in the slow switching limit depends on $\rho_1$ in the resetting protocol.Comment: 13 pages, 6 figure

Drift-diffusion on a Cayley tree with stochastic resetting: the localization-delocalization transition

In this paper we develop the theory of drift-diffusion on a semi-infinite Cayley tree with stochastic resetting. In the case of a homogeneous tree with a closed terminal node and no resetting, it is known that the system undergoes a classical localization-delocalization (LD) transition at a critical mean velocity $v_c= -(D/L)\ln (z-1)$ where $D$ is the diffusivity, $L$ is the branch length and $z$ is the coordination number of the tree. If $v <v_c$ then the steady state concentration at the terminal node is non-zero (drift-dominated localized state), whereas it is zero when $v >v_c$ (diffusion-dominated delocalized state). This is equivalent to the transition between recurrent and transient transport on the tree, with the mean first passage time (MFPT) to be absorbed by an open terminal node switching from a finite value to infinity. Here we show how the LD transition provides a basic framework for understanding analogous phase transitions in optimal resetting rates. First, we establish the existence of an optimal resetting rate $r^{**}(z)$ that maximizes the steady-state solution at a closed terminal node. In addition, we show that there is a phase transition at a critical velocity $v_c^{**}(z)$ such that $r^{**}>0$ for $v>v_c^{**}$ and $r^{**}=0$ for $v<v_c^{**}$. We then identify a critical velocity $v^*(z)$ for a phase transition in a second optimal resetting rate $r^*$ that minimizes the MFPT to be absorbed by an open terminal node. Previous results for the semi-infinite line are recovered on setting $z=2$. The critical velocity of the LD transition provides an upper bound for the other critical velocities such that $v_c^*(z)<v_c^{**}(z)<v_c(z)$ for all finite $z$. Only $v_c(z)$ has a simple universal dependence on the coordination number $z$. We end by considering the combined effects of quenched disorder and stochastic resetting.Comment: 25 pages, 11 figure