5,839 research outputs found
Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach
We are interested in the connection between a metastable continuous state
space Markov process (satisfying e.g. the Langevin or overdamped Langevin
equation) and a jump Markov process in a discrete state space. More precisely,
we use the notion of quasi-stationary distribution within a metastable state
for the continuous state space Markov process to parametrize the exit event
from the state. This approach is useful to analyze and justify methods which
use the jump Markov process underlying a metastable dynamics as a support to
efficiently sample the state-to-state dynamics (accelerated dynamics
techniques). Moreover, it is possible by this approach to quantify the error on
the exit event when the parametrization of the jump Markov model is based on
the Eyring-Kramers formula. This therefore provides a mathematical framework to
justify the use of transition state theory and the Eyring-Kramers formula to
build kinetic Monte Carlo or Markov state models.Comment: 14 page
Fluid limit theorems for stochastic hybrid systems with application to neuron models
This paper establishes limit theorems for a class of stochastic hybrid
systems (continuous deterministic dynamic coupled with jump Markov processes)
in the fluid limit (small jumps at high frequency), thus extending known
results for jump Markov processes. We prove a functional law of large numbers
with exponential convergence speed, derive a diffusion approximation and
establish a functional central limit theorem. We apply these results to neuron
models with stochastic ion channels, as the number of channels goes to
infinity, estimating the convergence to the deterministic model. In terms of
neural coding, we apply our central limit theorems to estimate numerically
impact of channel noise both on frequency and spike timing coding.Comment: 42 pages, 4 figure
Donsker-Varadhan asymptotics for degenerate jump Markov processes
We consider a class of continuous time Markov chains on a compact metric
space that admit an invariant measure strictly positive on open sets together
with absorbing states. We prove the joint large deviation principle for the
empirical measure and flow. Due to the lack of uniform ergodicity, the zero
level set of the rate function is not a singleton. As corollaries, we obtain
the Donsker-Varadhan rate function for the empirical measure and a variational
expression of the rate function for the empirical flow
Occupation Times for Jump Processes
We consider a class of pure jump Markov processes in \rr^d whose jump
kernels are comparable to those of symmetric stable processes. We prove a
support theorem, a lower bound on the occupation times of sets, and show that
we can approximate resolvents using smooth functions
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
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