117 research outputs found
Stripe to spot transition in a plant root hair initiation model
A generalised Schnakenberg reaction-diffusion system with source and loss
terms and a spatially dependent coefficient of the nonlinear term is studied
both numerically and analytically in two spatial dimensions. The system has
been proposed as a model of hair initiation in the epidermal cells of plant
roots. Specifically the model captures the kinetics of a small G-protein ROP,
which can occur in active and inactive forms, and whose activation is believed
to be mediated by a gradient of the plant hormone auxin. Here the model is made
more realistic with the inclusion of a transverse co-ordinate. Localised
stripe-like solutions of active ROP occur for high enough total auxin
concentration and lie on a complex bifurcation diagram of single and
multi-pulse solutions. Transverse stability computations, confirmed by
numerical simulation show that, apart from a boundary stripe, these 1D
solutions typically undergo a transverse instability into spots. The spots so
formed typically drift and undergo secondary instabilities such as spot
replication. A novel 2D numerical continuation analysis is performed that shows
the various stable hybrid spot-like states can coexist. The parameter values
studied lead to a natural singularly perturbed, so-called semi-strong
interaction regime. This scaling enables an analytical explanation of the
initial instability, by describing the dispersion relation of a certain
non-local eigenvalue problem. The analytical results are found to agree
favourably with the numerics. Possible biological implications of the results
are discussed.Comment: 28 pages, 44 figure
Predictions from a stochastic polymer model for the MinDE dynamics in E.coli
The spatiotemporal oscillations of the Min proteins in the bacterium
Escherichia coli play an important role in cell division. A number of different
models have been proposed to explain the dynamics from the underlying
biochemistry. Here, we extend a previously described discrete polymer model
from a deterministic to a stochastic formulation. We express the stochastic
evolution of the oscillatory system as a map from the probability distribution
of maximum polymer length in one period of the oscillation to the probability
distribution of maximum polymer length half a period later and solve for the
fixed point of the map with a combined analytical and numerical technique. This
solution gives a theoretical prediction of the distributions of both lengths of
the polar MinD zones and periods of oscillations -- both of which are
experimentally measurable. The model provides an interesting example of a
stochastic hybrid system that is, in some limits, analytically tractable.Comment: 16 page
Discrete embedded solitons
We address the existence and properties of discrete embedded solitons (ESs),
i.e., localized waves existing inside the phonon band in a nonlinear
dynamical-lattice model. The model describes a one-dimensional array of optical
waveguides with both the quadratic (second-harmonic generation) and cubic
nonlinearities. A rich family of ESs was previously known in the continuum
limit of the model. First, a simple motivating problem is considered, in which
the cubic nonlinearity acts in a single waveguide. An explicit solution is
constructed asymptotically in the large-wavenumber limit. The general problem
is then shown to be equivalent to the existence of a homoclinic orbit in a
four-dimensional reversible map. From properties of such maps, it is shown that
(unlike ordinary gap solitons), discrete ESs have the same codimension as their
continuum counterparts. A specific numerical method is developed to compute
homoclinic solutions of the map, that are symmetric under a specific reversing
transformation. Existence is then studied in the full parameter space of the
problem. Numerical results agree with the asymptotic results in the appropriate
limit and suggest that the discrete ESs may be semi-stable as in the continuous
case.Comment: A revtex4 text file and 51 eps figure files. To appear in
Nonlinearit
Modelling the vascular response to sympathetic postganglionic nerve activity
AbstractThis paper explores the influence of burst properties of the sympathetic nervous system on arterial contractility. Specifically, a mathematical model is constructed of the pathway from action potential generation in a sympathetic postganglionic neurone to contraction of an arterial smooth muscle cell. The differential equation model is a synthesis of models of the individual physiological processes, and is shown to be consistent with physiological data.The model is found to be unresponsive to tonic (regular) stimulation at typical frequencies recorded in sympathetic efferents. However, when stimulated at the same average frequency, but with repetitive respiratory-modulated burst patterns, it produces marked contractions. Moreover, the contractile force produced is found to be highly dependent on the number of spikes in each burst. In particular, when the model is driven by preganglionic spike trains recorded from wild-type and spontaneously hypertensive rats (which have increased spiking during each burst) the contractile force was found to be 10-fold greater in the hypertensive case. An explanation is provided in terms of the summative increased release of noradrenaline. Furthermore, the results suggest the marked effect that hypertensive spike trains had on smooth muscle cell tone can provide a significant contribution to the pathology of hypertension
Conveyance of texture signals along a rat whisker
Neuronal activities underlying a percept are constrained by the physics of sensory signals. In the tactile sense such constraints are frictional stickāslip events, occurring, amongst other vibrotactile features, when tactile sensors are in contact with objects. We reveal new biomechanical phenomena about the transmission of these microNewton forces at the tip of a ratās whisker, where they occur, to the base where they engage primary afferents. Using high resolution videography and accurate measurement of axial and normal forces at the follicle, we show that the conical and curved rat whisker acts as a sign-converting amplification filter for moment to robustly engage primary afferents. Furthermore, we present a model based on geometrically nonlinear Cosserat rod theory and a friction model that recreates the observed whole-beam whisker dynamics. The model quantifies the relation between kinematics (positions and velocities) and dynamic variables (forces and moments). Thus, only videographic assessment of acceleration is required to estimate forces and moments measured by the primary afferents. Our study highlights how sensory systems deal with complex physical constraints of perceptual targets and sensors
Standard and Embedded Solitons in Nematic Optical Fibers
A model for a non-Kerr cylindrical nematic fiber is presented. We use the
multiple scales method to show the possibility of constructing different kinds
of wavepackets of transverse magnetic (TM) modes propagating through the fiber.
This procedure allows us to generate different hierarchies of nonlinear partial
differential equations (PDEs) which describe the propagation of optical pulses
along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium
and derive an extended Nonlinear Schrodinger equation (eNLS) with a third order
derivative nonlinearity, governing the dynamics for the amplitude of the
wavepacket. In this derivation the dispersion, self-focussing and diffraction
in the nematic are taken into account. Although the resulting nonlinear
may be reduced to the modified Korteweg de Vries equation (mKdV), it also has
additional complex solutions which include two-parameter families of bright and
dark complex solitons. We show analytically that under certain conditions, the
bright solitons are actually double embedded solitons. We explain why these
solitons do not radiate at all, even though their wavenumbers are contained in
the linear spectrum of the system. Finally, we close the paper by making
comments on the advantages as well as the limitations of our approach, and on
further generalizations of the model and method presented.Comment: "Physical Review E, in press
Nonlinear parametric instability in double-well lattices
A possibility of a nonlinear resonant instability of uniform oscillations in
dynamical lattices with harmonic intersite coupling and onsite nonlinearity is
predicted. Numerical simulations of a lattice with a double-well onsite
anharmonic potential confirm the existence of the nonlinear instability with an
anomalous value of the corresponding power index, 1.57, which is intermediate
between the values 1 and 2 characterizing the linear and nonlinear (quadratic)
instabilities. The anomalous power index may be a result of competition between
the resonant quadratic instability and nonresonant linear instabilities. The
observed instability triggers transition of the lattice into a chaotic
dynamical state.Comment: A latex text file and three pdf files with figures. Physical Review
E, in pres
Coarse-grained dynamics of an activity bump in a neural field model
We study a stochastic nonlocal PDE, arising in the context of modelling
spatially distributed neural activity, which is capable of sustaining
stationary and moving spatially-localized ``activity bumps''. This system is
known to undergo a pitchfork bifurcation in bump speed as a parameter (the
strength of adaptation) is changed; yet increasing the noise intensity
effectively slowed the motion of the bump. Here we revisit the system from the
point of view of describing the high-dimensional stochastic dynamics in terms
of the effective dynamics of a single scalar "coarse" variable. We show that
such a reduced description in the form of an effective Langevin equation
characterized by a double-well potential is quantitatively successful. The
effective potential can be extracted using short, appropriately-initialized
bursts of direct simulation. We demonstrate this approach in terms of (a) an
experience-based "intelligent" choice of the coarse observable and (b) an
observable obtained through data-mining direct simulation results, using a
diffusion map approach.Comment: Corrected aknowledgement
Increased intrinsic excitability of muscle vasoconstrictor preganglionic neurons may contribute to the elevated sympathetic activity in hypertensive rats
Hypertension is associated with pathologically increased sympathetic drive to the vasculature. This has been attributed to increased excitatory drive to sympathetic preganglionic neurons (SPN) from brainstem cardiovascular control centers. However, there is also evidence supporting increased intrinsic excitability of SPN. To test this hypothesis, we made whole cell recordings of muscle vasoconstrictor-like (MVC(like)) SPN in the working-heart brainstem preparation of spontaneously hypertensive (SH) and normotensive Wistar-Kyoto (WKY) rats. The MVC(like) SPN have a higher spontaneous firing frequency in the SH rat (3.85 Ā± 0.4 vs. 2.44 Ā± 0.4 Hz in WKY; P = 0.011) with greater respiratory modulation of their activity. The action potentials of SH SPN had smaller, shorter afterhyperpolarizations (AHPs) and showed diminished transient rectification indicating suppression of an A-type potassium conductance (I(A)). We developed mathematical models of the SPN to establish if changes in their intrinsic properties in SH rats could account for their altered firing. Reduction of the maximal conductance density of I(A) by 15ā30% changed the excitability and output of the model from the WKY to a SH profile, with increased firing frequency, amplified respiratory modulation, and smaller AHPs. This change in output is predominantly a consequence of altered synaptic integration. Consistent with these in silico predictions, we found that intrathecal 4-aminopyridine (4-AP) increased sympathetic nerve activity, elevated perfusion pressure, and augmented Traube-Hering waves. Our findings indicate that I(A) acts as a powerful filter on incoming synaptic drive to SPN and that its diminution in the SH rat is potentially sufficient to account for the increased sympathetic output underlying hypertension
Creation of gap solitons in Bose-Einstein condensates
We discuss a method to launch gap soliton-like structures in atomic
Bose-Einstein condensates confined in optical traps. Bright vector solitons
consisting of a superposition of two hyperfine Zeeman sublevels can be created
for both attractive and repulsive interactions between the atoms. Their
formation relies on the dynamics of the atomic internal ground states in two
far-off resonant counterpropagating sigma^+ sigma^- polarized laser beams which
form the optical trap. Numerical simulations show that these solitons can be
prepared from a one-component state provided with an initial velocity.Comment: 6 pages, 3 figure
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