512 research outputs found
Wave nucleation rate in excitable systems in the low noise limit
Motivated by recent experiments on intracellular calcium dynamics, we study
the general issue of fluctuation-induced nucleation of waves in excitable
media. We utilize a stochastic Fitzhugh-Nagumo model for this study, a
spatially-extended non-potential pair of equations driven by thermal (i.e.
white) noise. The nucleation rate is determined by finding the most probable
escape path via minimization of an action related to the deviation of the
fields from their deterministic trajectories. Our results pave the way both for
studies of more realistic models of calcium dynamics as well as of nucleation
phenomena in other non-equilibrium pattern-forming processes
Transient localized wave patterns and their application to migraine
Transient dynamics is pervasive in the human brain and poses challenging
problems both in mathematical tractability and clinical observability. We
investigate statistical properties of transient cortical wave patterns with
characteristic forms (shape, size, duration) in a canonical reaction-diffusion
model with mean field inhibition. The patterns are formed by a ghost near a
saddle-node bifurcation in which a stable traveling wave (node) collides with
its critical nucleation mass (saddle). Similar patterns have been observed with
fMRI in migraine. Our results support the controversial idea that waves of
cortical spreading depression (SD) have a causal relationship with the headache
phase in migraine and therefore occur not only in migraine with aura (MA) but
also in migraine without aura (MO), i.e., in the two major migraine subforms.
We suggest a congruence between the prevalence of MO and MA with the
statistical properties of the traveling waves' forms, according to which (i)
activation of nociceptive mechanisms relevant for headache is dependent upon a
sufficiently large instantaneous affected cortical area anti-correlated to both
SD duration and total affected cortical area such that headache would be less
severe in MA than in MO (ii) the incidence of MA is reflected in the distance
to the saddle-node bifurcation, and (iii) the contested notion of MO attacks
with silent aura is resolved. We briefly discuss model-based control and means
by which neuromodulation techniques may affect pathways of pain formation.Comment: 14 pages, 11 figure
Resonantly Forced Inhomogeneous Reaction-Diffusion Systems
The dynamics of spatiotemporal patterns in oscillatory reaction-diffusion
systems subject to periodic forcing with a spatially random forcing amplitude
field are investigated. Quenched disorder is studied using the resonantly
forced complex Ginzburg-Landau equation in the 3:1 resonance regime. Front
roughening and spontaneous nucleation of target patterns are observed and
characterized. Time dependent spatially varying forcing fields are studied in
the 3:1 forced FitzHugh-Nagumo system. The periodic variation of the spatially
random forcing amplitude breaks the symmetry among the three quasi-homogeneous
states of the system, making the three types of fronts separating phases
inequivalent. The resulting inequality in the front velocities leads to the
formation of ``compound fronts'' with velocities lying between those of the
individual component fronts, and ``pulses'' which are analogous structures
arising from the combination of three fronts. Spiral wave dynamics is studied
in systems with compound fronts.Comment: 14 pages, 19 figures, to be published in CHAOS. This replacement has
some minor changes in layout for purposes of neatnes
Nucleation of reaction-diffusion waves on curved surfaces
We study reaction-diffusion waves on curved two-dimensional surfaces, and
determine the influence of curvature upon the nucleation and propagation of
spatially localized waves in an excitable medium modelled by the generic
FitzHugh-Nagumo model. We show that the stability of propagating wave segments
depends crucially on the curvature of the surface. As they propagate, they may
shrink to the uniform steady state, or expand, depending on whether they are
smaller or larger, respectively, than a critical nucleus. This critical nucleus
for wave propagation is modified by the curvature acting like an effective
space-dependent local spatial coupling, similar to diffusion, thus extending
the regime of propagating excitation waves beyond the excitation threshold of
flat surfaces. In particular, a negative gradient of Gaussian curvature
, as on the outside of a torus surface (positive ), when the
wave segment symmetrically extends into the inside (negative ), allows
for stable propagation of localized wave segments remaining unchanged in size
and shape, or oscillating periodically in size
Nonlocal control of pulse propagation in excitable media
We study the effects of nonlocal control of pulse propagation in excitable
media. As a generic example for an excitable medium the FitzHugh-Nagumo model
with diffusion in the activator variable is considered. Nonlocal coupling in
form of an integral term with a spatial kernel is added. We find that the
nonlocal coupling modifies the propagating pulses of the reaction-diffusion
system such that a variety of spatio-temporal patterns are generated including
acceleration, deceleration, suppression, or generation of pulses, multiple
pulses, and blinking pulse trains. It is shown that one can observe these
effects for various choices of the integral kernel and the coupling scheme,
provided that the control strength and spatial extension of the integral kernel
is appropriate. In addition, an analytical procedure is developed to describe
the stability borders of the spatially homogeneous steady state in control
parameter space in dependence on the parameters of the nonlocal coupling
Convective Instability and Boundary Driven Oscillations in a Reaction-Diffusion-Advection Model
In a reaction-diffusion-advection system, with a convectively unstable
regime, a perturbation creates a wave train that is advected downstream and
eventually leaves the system. We show that the convective instability coexists
with a local absolute instability when a fixed boundary condition upstream is
imposed. This boundary induced instability acts as a continuous wave source,
creating a local periodic excitation near the boundary, which initiates waves
traveling both up and downstream. To confirm this, we performed analytical
analysis and numerical simulations of a modified Martiel-Goldbeter
reaction-diffusion model with the addition of an advection term. We provide a
quantitative description of the wave packet appearing in the convectively
unstable regime, which we found to be in excellent agreement with the numerical
simulations. We characterize this new instability and show that in the limit of
high advection speed, it is suppressed. This type of instability can be
expected for reaction-diffusion systems that present both a convective
instability and an excitable regime. In particular, it can be relevant to
understand the signaling mechanism of the social amoeba Dictyostelium
discoideum that may experience fluid flows in its natural habitat.Comment: 10 pages, 13 figures, published in Chaos: An Interdisciplinary
Journal of Nonlinear Scienc
On polymorphic logical gates in sub-excitable chemical medium
In a sub-excitable light-sensitive Belousov-Zhabotinsky chemical medium an
asymmetric disturbance causes the formation of localized traveling
wave-fragments. Under the right conditions these wave-fragment can conserve
their shape and velocity vectors for extended time periods. The size and life
span of a fragment depend on the illumination level of the medium. When two or
more wave-fragments collide they annihilate or merge into a new wave-fragment.
In computer simulations based on the Oregonator model we demonstrate that the
outcomes of inter-fragment collisions can be controlled by varying the
illumination level applied to the medium. We interpret these wave-fragments as
values of Boolean variables and design collision-based polymorphic logical
gates. The gate implements operation XNOR for low illumination, and it acts as
NOR gate for high illumination. As a NOR gate is a universal gate then we are
able to demonstrate that a simulated light sensitive BZ medium exhibits
computational universality
- …