4,792 research outputs found
Relaxation oscillations in a class of delay-differential equations.
We study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics. Under certain circumstances the model exhibits self-sustained oscillations, with periods which can be significantly longer than the basic cell cycle time. We show that the long periods in the oscillations occur when the cell generation rate is small, and we provide an asymptotic analysis of the model in this case. This analysis bears a close resemblance to the analysis of relaxation oscillators (such as the Van der Pol oscillator), except that in our case the slow manifold is infinite dimensional. Despite this, a fairly complete analysis of the problem is possible
Dynamic behavior of stochastic gene expression models in the presence of bursting
This paper considers the behavior of discrete and continuous mathematical
models for gene expression in the presence of transcriptional/translational
bursting. We treat this problem in generality with respect to the distribution
of the burst size as well as the frequency of bursting, and our results are
applicable to both inducible and repressible expression patterns in prokaryotes
and eukaryotes. We have given numerous examples of the applicability of our
results, especially in the experimentally observed situation that burst size is
geometrically or exponentially distributed.Comment: 22 page
What measurable zero point fluctuations can(not) tell us about dark energy
We show that laboratory experiments cannot measure the absolute value of dark
energy. All known experiments rely on electromagnetic interactions. They are
thus insensitive to particles and fields that interact only weakly with
ordinary matter. In addition, Josephson junction experiments only measure
differences in vacuum energy similar to Casimir force measurements. Gravity,
however, couples to the absolute value. Finally we note that Casimir force
measurements have tested zero point fluctuations up to energies of ~10 eV, well
above the dark energy scale of ~0.01 eV. Hence, the proposed cut-off in the
fluctuation spectrum is ruled out experimentally.Comment: 4 page
Oscillations in a maturation model of blood cell production.
We present a mathematical model of blood cell production which describes both the development of cells through the cell cycle, and the maturation of these cells as they differentiate to form the various mature blood cell types. The model differs from earlier similar ones by considering primitive stem cells as a separate population from the differentiating cells, and this formulation removes an apparent inconsistency in these earlier models. Three different controls are included in the model: proliferative control of stem cells, proliferative control of differentiating cells, and peripheral control of stem cell committal rate. It is shown that an increase in sensitivity of these controls can cause oscillations to occur through their interaction with time delays associated with proliferation and differentiation, respectively. We show that the characters of these oscillations are quite distinct and suggest that the model may explain an apparent superposition of fast and slow oscillations which can occur in cyclical neutropenia. © 2006 Society for Industrial and Applied Mathematics
An augmented moment method for stochastic ensembles with delayed couplings: I. Langevin model
By employing a semi-analytical dynamical mean-field approximation theory
previously proposed by the author [H. Hasegawa, Phys. Rev. E {\bf 67}, 041903
(2003)], we have developed an augmented moment method (AMM) in order to discuss
dynamics of an -unit ensemble described by linear and nonlinear Langevin
equations with delays. In AMM, original -dimensional {\it stochastic} delay
differential equations (SDDEs) are transformed to infinite-dimensional {\it
deterministic} DEs for means and correlations of local as well as global
variables. Infinite-order DEs arising from the non-Markovian property of SDDE,
are terminated at the finite level in the level- AMM (AMM), which
yields -dimensional deterministic DEs. Model calculations have been made
for linear and nonlinear Langevin models. The stationary solution of AMM for
the linear Langevin model with N=1 is nicely compared to the exact result. The
synchronization induced by an applied single spike is shown to be enhanced in
the nonlinear Langevin ensemble with model parameters locating at the
transition between oscillating and non-oscillating states. Results calculated
by AMM6 are in good agreement with those obtained by direct simulations.Comment: 18 pages, 3 figures, changed the title with re-arranged figures,
accepted in Phys. Rev. E with some change
Irreversible Thermodynamics in Multiscale Stochastic Dynamical Systems
This work extends the results of the recently developed theory of a rather
complete thermodynamic formalism for discrete-state, continuous-time Markov
processes with and without detailed balance. We aim at investigating the
question that whether and how the thermodynamic structure is invariant in a
multiscale stochastic system. That is, whether the relations between
thermodynamic functions of state and process variables remain unchanged when
the system is viewed at different time scales and resolutions. Our results show
that the dynamics on a fast time scale contribute an entropic term to the
"internal energy function", , for the slow dynamics. Based on the
conditional free energy , one can then treat the slow dynamics as if
the fast dynamics is nonexistent. Furthermore, we show that the free energy,
which characterizes the spontaneous organization in a system without detailed
balance, is invariant with or without the fast dynamics: The fast dynamics is
assumed to reach stationarity instantaneously on the slow time scale; they have
no effect on the system's free energy. The same can not be said for the entropy
and the internal energy, both of which contain the same contribution from the
fast dynamics. We also investigate the consequences of time-scale separation in
connection to the concepts of quasi-stationaryty and steady-adiabaticity
introduced in the phenomenological steady-state thermodynamics
On the Exponentials of Some Structured Matrices
In this note explicit algorithms for calculating the exponentials of
important structured 4 x 4 matrices are provided. These lead to closed form
formulae for these exponentials. The techniques rely on one particular Clifford
Algebra isomorphism and basic Lie theory. When used in conjunction with
structure preserving similarities, such as Givens rotations, these techniques
extend to dimensions bigger than four.Comment: 19 page
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