147 research outputs found
Synchronization of stochastic hybrid oscillators driven by a common switching environment
Many systems in biology, physics and chemistry can be modeled through
ordinary differential equations, which are piecewise smooth, but switch between
different states according to a Markov jump process. In the fast switching
limit, the dynamics converges to a deterministic ODE. In this paper we suppose
that this limit ODE supports a stable limit cycle. We demonstrate that a set of
such oscillators can synchronize when they are uncoupled, but they share the
same switching Markov jump process. The latter is taken to represent the effect
of a common randomly switching environment. We determine the leading order of
the Lyapunov coefficient governing the rate of decay of the phase difference in
the fast switching limit. The analysis bears some similarities to the classical
analysis of synchronization of stochastic oscillators subject to common white
noise. However the discrete nature of the Markov jump process raises some
difficulties: in fact we find that the Lyapunov coefficient from the
quasi-steady-state approximation differs from the Lyapunov coefficient one
obtains from a second order perturbation expansion in the waiting time between
jumps. Finally, we demonstrate synchronization numerically in the radial
isochron clock model and show that the latter Lyapinov exponent is more
accurate
A variational method for analyzing stochastic limit cycle oscillators
We introduce a variational method for analyzing limit cycle oscillators in
driven by Gaussian noise. This allows us to derive exact
stochastic differential equations (SDEs) for the amplitude and phase of the
solution, which are accurate over times over order
, where is the amplitude of the noise
and the magnitude of decay of transverse fluctuations. Within the
variational framework, different choices of the amplitude-phase decomposition
correspond to different choices of the inner product space . For
concreteness, we take a weighted Euclidean norm, so that the minimization
scheme determines the phase by projecting the full solution on to the limit
cycle using Floquet vectors. Since there is coupling between the amplitude and
phase equations, even in the weak noise limit, there is a small but non-zero
probability of a rare event in which the stochastic trajectory makes a large
excursion away from a neighborhood of the limit cycle. We use the amplitude and
phase equations to bound the probability of it doing this: finding that the
typical time the system takes to leave a neighborhood of the oscillator scales
as
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 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 . 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 target cells ,
, with . We assume that each time the cytoneme
nucleates, it grows in a random direction so that the probability of being
oriented towards the -th target is with . 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 . The velocity state is reset to
with fixed probabilities and , where
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
of the velocity state, the resetting rate and the probability
. In particular, we show that the corresponding LDP for a Brownian
particle with resetting is recovered in the fast switching limit
. On the other hand, the behavior in the slow
switching limit depends on in the resetting protocol.Comment: 13 pages, 6 figure
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