arXiv subjects

Paul C. Bressloff

Publications and source records attributed to Paul C. Bressloff.

30 records · Page 2Linked to original sources

arXiv2021

The narrow capture problem with partially absorbing targets and stochastic resetting

We consider a particle undergoing diffusion with stochastic resetting in a bounded domain $\calU\subset \R^d$ for $d=2,3$. The domain is perforated by a set of partially absorbing targets within which the particle may be absorbed at a rate $κ$. Each target is assumed to be much smaller than $|\calU|$, which allows us to use asymptotic and Green's function methods to solve the diffusion equation in Laplace space. In particular, we construct an inner solution within the interior and local exterior of each target, and match it with an outer solution in the bulk of $\calU$. This yields an asymptotic expansion of the Laplace transformed flux into each target in powers of $ν=-1/\ln ε$ ($d=2$) and $ε$ ($d=3$), respectively, where $ε$ is the non-dimensionalized target size. The fluxes determine how the mean first-passage time to absorption depends on the reaction rate $κ$ and the resetting rate $r$. For a range of parameter values, the MFPT is a unimodal function of $r$, with a minimum at an optimal resetting rate $r_{\rm opt}$ that depends on $κ$ and the target configuration.

cond-mat.stat-mech↗

arXiv2021

Search processes with stochastic resetting and partially absorbing targets

We extend the theoretical framework used to study search processes with stochastic resetting to the case of partially absorbing targets. Instead of an absorption event occurring when the search particle reaches the boundary of a target, the particle can diffuse freely in and out of the target region and is absorbed at a rate $κ$ when inside the target. In the context of cell biology, the target could represent a chemically reactive substrate within a cell or a region where a particle can be offloaded onto a nearby compartment. We apply this framework to a partially absorbing interval and to spherically symmetric targets in $\R^d$. In each case, we determine how the mean first passage time (MFPT) for absorption depends on $κ$, the resetting rate $r$, and the target geometry. For the given examples, we find that the MFPT is a monotonically decreasing function of $κ$, whereas it is a unimodal function of $r$ with a unique minimum at an optimal resetting rate $r_{\rm opt}$. The variation of $r_{\rm opt}$ with $κ$ depends on the spatial dimension $d$, decreasing in sensitivity as $d$ increases. For finite $κ$, $ r_{\rm opt}$ is a non-trivial function of the target size and distance between the target and the reset point. We also show how our results converge to those obtained previously for problems with totally absorbing targets and similar geometries when the absorption rate becomes infinite. Finally, we generalize the theory to take into account an extended chemical reaction scheme within a target.

cond-mat.stat-mech↗

arXiv2021

Spin coherent states and stochastic hybrid path integrals

Stochastic hybrid systems involve a coupling between a discrete Markov chain and a continuous stochastic process. If the latter evolves deterministically between jumps in the discrete state, then the system reduces to a piecewise deterministic Markov process (PDMP). Well known examples include stochastic gene expression, voltage fluctuations in neurons, and motor-driven intracellular transport. In this paper we use coherent spin states to construct a new path integral representation of the probability density functional for stochastic hybrid systems, which holds outside the weak noise regime. We use the path integral to derive a system of Langevin equations in the semi-classical limit, which extends previous diffusion approximations based on a quasi-steady-state reduction. We then show how in the weak noise limit the path integral is equivalent to an alternative representation that was previously derived using Doi-Peliti operators. The action functional of the latter is related to a large deviation principle for stochastic hybrid systems.

cond-mat.stat-mech↗

arXiv2020

Construction of stochastic hybrid path integrals using "quantum-mechanical'' operators

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 the use of bra-kets and ``quantum-mechanical'' operators. We show how the operator method provides a more efficient and flexible framework for constructing hybrid path integrals, which eliminates certain ad hoc steps from the previous derivation and provides more context with regards the general theory of stochastic path integrals. We also highlight the important role of principal eigenvalues, spectral gaps and the Perron-Frobenius theorem. We then use perturbation methods to develop various approximation schemes for hybrid path integrals and the associated moment generating functionals. First, we consider Gaussian approximations and loop expansions in the weak noise limit, analogous to the semi-classical limit for quantum path integrals. Second, we identify the analog of a weak-coupling limit by treating the stochastic hybrid system as the nonlinear perturbation of an Ornstein-Uhlenbeck process. This leads to an expansion of the moments in terms of products of free propagators.

cond-mat.stat-mech↗

arXiv2020

Target competition for resources under diffusive search-and-capture

In this paper we use asymptotic analysis to determine the steady-state mean number of resources in each of $N$ small interior targets within a three-dimensional bounded domain. The accumulation of resources is based on multiple rounds of search-and-capture events; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position (resetting after capture). The searcher is then resupplied with cargo and a new search process is initiated after a random delay. Assuming that the accumulation of resources is counterbalanced by degradation, one can derive general expressions for the moments of the resource distribution. We use this to show that the mean number of resources in a target is proportional to its effective "shape capacitance." We then extend the analysis to the case of diffusive search with stochastic resetting before capture, where the position of the searcher is reset to its initial position at a random sequence of times that is statistically independent of the ongoing search process, in contrast to the sequence of resetting times after capture.

cond-mat.stat-mech↗

arXiv2020

Stochastic Turing pattern formation in a model with active and passive transport

We investigate Turing pattern formation in a stochastic and spatially discretized version of a reaction diffusion advection (RDA) equation, which was previously introduced to model synaptogenesis in \textit{C. elegans}. The model describes the interactions between a passively diffusing molecular species and an advecting species that switches between anterograde and retrograde motor-driven transport (bidirectional transport). Within the context of synaptogenesis, the diffusing molecules can be identified with the protein kinase CaMKII and the advecting molecules as glutamate receptors. The stochastic dynamics evolves according to an RDA master equation, in which advection and diffusion are both modeled as hopping reactions along a one-dimensional array of chemical compartments. Carrying out a linear noise approximation of the RDA master equation leads to an effective Langevin equation, whose power spectrum provides a means of extending the definition of a Turing instability to stochastic systems, namely, in terms of the existence of a peak in the power spectrum at a non-zero spatial frequency. We thus show how noise can significantly extend the range over which spontaneous patterns occur, which is consistent with previous studies of RD systems.

cond-mat.stat-mech↗

arXiv2018

Phase Reduction of Stochastic Biochemical Oscillators

A common method for analyzing the effects of molecular noise in chemical reaction networks is to approximate the underlying chemical master equation by a Fokker-Planck equation, and to study the statistics of the associated chemical Langevin equation. This so-called diffusion approximation involves performing a perturbation expansion with respect to a small dimensionless parameter $ε=Ω^{-1}$, where $Ω$ characterizes the system size. For example, $Ω$ could be the mean number of proteins produced by a gene regulatory network. In the deterministic limit $Ω\rightarrow \infty$, the chemical reaction network evolves according to a system of ordinary differential equations based on classical mass action kinetics. In this paper we develop a phase reduction method for chemical reaction networks that support a stable limit cycle in the deterministic limit. We present a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the system size $Ω$. This contrasts with the phase equation obtained under the diffusion approximation, which is only accurate up to times $O(Ω)$. In particular, we show that for a constant $C$, the probability that the system leaves an $O(ζ)$ neighborhood of the limit cycle before time $T$ scales as $T\exp\big(-CΩbζ^2 \big)$, where $b$ is the rate of attraction to the limit cycle. We illustrate our analysis using the example of a chemical Brusselator.

math.PR↗

arXiv2016

Coupling sample paths to the partial thermodynamic limit in stochastic chemical reaction networks

Many biochemical systems appearing in applications have a multiscale structure so that they converge to piecewise deterministic Markov processes in a thermodynamic limit. The statistics of the piecewise deterministic process can be obtained much more efficiently than those of the exact process. We explore the possibility of coupling sample paths of the exact model to the piecewise deterministic process in order to reduce the variance of their difference. We then apply this coupling to reduce the computational complexity of a Monte Carlo estimator. In addition to rigorous results concerning the asymptotic computational complexity of the Monte Carlo estimator, numerical simulations are performed on some simple biological models confirming that computational gains are made.

physics.comp-ph↗

arXiv2015

Nonlinear Langevin equations for wandering patterns in stochastic neural fields

We analyze the effects of additive, spatially extended noise on spatiotemporal patterns in continuum neural fields. Our main focus is how fluctuations impact patterns when they are weakly coupled to an external stimulus or another equivalent pattern. Showing the generality of our approach, we study both propagating fronts and stationary bumps. Using a separation of time scales, we represent the effects of noise in terms of a phase-shift of a pattern from its uniformly translating position at long time scales, and fluctuations in the pattern profile around its instantaneous position at short time scales. In the case of a stimulus-locked front, we show that the phase-shift satisfies a nonlinear Langevin equation (SDE) whose deterministic part has a unique stable fixed point. Using a linear-noise approximation, we thus establish that wandering of the front about the stimulus-locked state is given by an Ornstein-Uhlenbeck (OU) process. Analogous results hold for the relative phase-shift between a pair of mutually coupled fronts, provided that the coupling is excitatory. On the other hand, if the mutual coupling is given by a Mexican hat function (difference of exponentials), then the linear-noise approximation breaks down due to the co-existence of stable and unstable phase-locked states in the deterministic limit. Similarly, the stochastic motion of mutually coupled bumps can be described by a system of nonlinearly coupled SDEs, which can be linearized to yield a multivariate OU process. As in the case of fronts, large deviations can cause bumps to temporarily decouple, leading to a phase-slip in the bump positions.

q-bio.NC↗

arXiv2013

Breakdown of fast-slow analysis in an excitable system with channel noise

We consider a stochastic version of an excitable system based on the Morris-Lecar model of a neuron, in which the noise originates from stochastic Sodium and Potassium ion channels opening and closing. One can analyze neural excitability in the deterministic model by using a separation of time scales involving a fast voltage variable and a slow recovery variable, which represents the fraction of open Potassium channels. In the stochastic setting, spontaneous excitation is initiated by ion channel noise. If the recovery variable is constant during initiation, the spontaneous activity rate can be calculated using Kramer's rate theory. The validity of this assumption in the stochastic model is examined using a systematic perturbation analysis. We find that in most physically relevant cases, this assumption breaks down, requiring an alternative to Kramers theory for excitable systems with one deterministic fixed point. We also show that an exit time problem can be formulated in an excitable system by considering maximum likelihood trajectories of the stochastic process.

q-bio.NC↗

arXiv2013

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.

cond-mat.dis-nn↗

arXiv2011

Hebbian learning of recurrent connections: a geometrical perspective

We show how a Hopfield network with modifiable recurrent connections undergoing slow Hebbian learning can extract the underlying geometry of an input space. First, we use a slow/fast analysis to derive an averaged system whose dynamics derives from an energy function and therefore always converges to equilibrium points. The equilibria reflect the correlation structure of the inputs, a global object extracted through local recurrent interactions only. Second, we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs. Indeed, multidimensional scaling methods make it possible to project the final connectivity matrix on to a distance matrix in a high-dimensional space, with the neurons labelled by spatial position within this space. The resulting network structure turns out to be roughly convolutional. The residual of the projection defines the non-convolutional part of the connectivity which is minimized in the process. Finally, we show how restricting the dimension of the space where the neurons live gives rise to patterns similar to cortical maps. We motivate this using an energy efficiency argument based on wire length minimization. Finally, we show how this approach leads to the emergence of ocular dominance or orientation columns in primary visual cortex. In addition, we establish that the non-convolutional (or long-range) connectivity is patchy, and is co-aligned in the case of orientation learning.

q-bio.NC↗