1,299 research outputs found
Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids
Numerical continuation methods for deterministic dynamical systems have been
one of the most successful tools in applied dynamical systems theory.
Continuation techniques have been employed in all branches of the natural
sciences as well as in engineering to analyze ordinary, partial and delay
differential equations. Here we show that the deterministic continuation
algorithm for equilibrium points can be extended to track information about
metastable equilibrium points of stochastic differential equations (SDEs). We
stress that we do not develop a new technical tool but that we combine results
and methods from probability theory, dynamical systems, numerical analysis,
optimization and control theory into an algorithm that augments classical
equilibrium continuation methods. In particular, we use ellipsoids defining
regions of high concentration of sample paths. It is shown that these
ellipsoids and the distances between them can be efficiently calculated using
iterative methods that take advantage of the numerical continuation framework.
We apply our method to a bistable neural competition model and a classical
predator-prey system. Furthermore, we show how global assumptions on the flow
can be incorporated - if they are available - by relating numerical
continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size
restrictions]; v2 - added Section 9 on Kramers' formula, additional
computations, corrected typos, improved explanation
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
Stochastic Representations of Ion Channel Kinetics and Exact Stochastic Simulation of Neuronal Dynamics
In this paper we provide two representations for stochastic ion channel
kinetics, and compare the performance of exact simulation with a commonly used
numerical approximation strategy. The first representation we present is a
random time change representation, popularized by Thomas Kurtz, with the second
being analogous to a "Gillespie" representation. Exact stochastic algorithms
are provided for the different representations, which are preferable to either
(a) fixed time step or (b) piecewise constant propensity algorithms, which
still appear in the literature. As examples, we provide versions of the exact
algorithms for the Morris-Lecar conductance based model, and detail the error
induced, both in a weak and a strong sense, by the use of approximate
algorithms on this model. We include ready-to-use implementations of the random
time change algorithm in both XPP and Matlab. Finally, through the
consideration of parametric sensitivity analysis, we show how the
representations presented here are useful in the development of further
computational methods. The general representations and simulation strategies
provided here are known in other parts of the sciences, but less so in the
present setting.Comment: 39 pages, 6 figures, appendix with XPP and Matlab cod
Duty Cycle Maintenance in an Artificial Neuron
Neuroprosthetics is at the intersection of neuroscience, biomedical engineering, and physics. A biocompatible neuroprosthesis contains artificial neurons exhibiting biophysically plausible dynamics. Hybrid systems analysis could be used to prototype such artificial neurons. Biohybrid systems are composed of artificial and living neurons coupled via real-time computing and dynamic clamp. Model neurons must be thoroughly tested before coupled with a living cell. We use bifurcation theory to identify hazardous regimes of activity that may compromise biocompatibility and to identify control strategies for regimes of activity desirable for functional behavior. We construct real-time artificial neurons for the analysis of hybrid systems and demonstrate a mechanism through which an artificial neuron could maintain duty cycle independent of variations in period
Tutorial of numerical continuation and bifurcation theory for systems and synthetic biology
Mathematical modelling allows us to concisely describe fundamental principles
in biology. Analysis of models can help to both explain known phenomena, and
predict the existence of new, unseen behaviours. Model analysis is often a
complex task, such that we have little choice but to approach the problem with
computational methods. Numerical continuation is a computational method for
analysing the dynamics of nonlinear models by algorithmically detecting
bifurcations. Here we aim to promote the use of numerical continuation tools by
providing an introduction to nonlinear dynamics and numerical bifurcation
analysis. Many numerical continuation packages are available, covering a wide
range of system classes; a review of these packages is provided, to help both
new and experienced practitioners in choosing the appropriate software tools
for their needs.Comment: 14 pages, 2 figures, 2 table
A continuum model for the dynamics of the phase transition from slow-wave sleep to REM sleep
Previous studies have shown that activated cortical states (awake and rapid eye-movement (REM) sleep), are associated with increased cholinergic input into the cerebral cortex. However, the mechanisms that underlie the detailed dynamics of the cortical transition from slow-wave to REM sleep have not been quantitatively modeled. How does the sequence of abrupt changes in the cortical dynamics (as detected in the electrocorticogram) result from the more gradual change in subcortical cholinergic input? We compare the output from a continuum model of cortical neuronal dynamics with experimentally-derived rat electrocorticogram data. The output from the computer model was consistent with experimental observations. In slow-wave sleep, 0.5â2-Hz oscillations arise from the cortex jumping between âupâ and âdownâ states on the stationary-state manifold. As cholinergic input increases, the upper state undergoes a bifurcation to an 8-Hz oscillation. The coexistence of both oscillations is similar to that found in the intermediate stage of sleep of the rat. Further cholinergic input moves the trajectory to a point where the lower part of the manifold in not available, and thus the slow oscillation abruptly ceases (REM sleep). The model provides a natural basis to explain neuromodulator-induced changes in cortical activity, and indicates that a cortical phase change, rather than a brainstem âflip-flopâ, may describe the transition from slow-wave sleep to REM
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