Oscillations and temporal signalling in cells

The development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration. Here we review the different mechanisms which can produce oscillations in gene expression or protein concentration, using a framework of simple mathematical models. We focus on three eukaryotic genetic regulatory networks which show "ultradian" oscillations, with time period of the order of hours, and involve, respectively, proteins important for development (Hes1), apoptosis (p53) and immune response (NFkB). We argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour

A delay differential model of ENSO variability: Parametric instability and the distribution of extremes

We consider a delay differential equation (DDE) model for El-Nino Southern Oscillation (ENSO) variability. The model combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform stability analyses of the model in the three-dimensional space of its physically relevant parameters. Our results illustrate the role of these three parameters: strength of seasonal forcing $b$, atmosphere-ocean coupling $\kappa$, and propagation period $\tau$ of oceanic waves across the Tropical Pacific. Two regimes of variability, stable and unstable, are separated by a sharp neutral curve in the $(b,\tau)$ plane at constant $\kappa$. The detailed structure of the neutral curve becomes very irregular and possibly fractal, while individual trajectories within the unstable region become highly complex and possibly chaotic, as the atmosphere-ocean coupling $\kappa$ increases. In the unstable regime, spontaneous transitions occur in the mean ``temperature'' ({\it i.e.}, thermocline depth), period, and extreme annual values, for purely periodic, seasonal forcing. The model reproduces the Devil's bleachers characterizing other ENSO models, such as nonlinear, coupled systems of partial differential equations; some of the features of this behavior have been documented in general circulation models, as well as in observations. We expect, therefore, similar behavior in much more detailed and realistic models, where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure

Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems

We present three examples of delayed bifurcations for spike solutions of reaction-diffusion systems. The delay effect results as the system passes slowly from a stable to an unstable regime, and was previously analysed in the context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that the instability would not be fully realized until the system had entered well into the unstable regime. The bifurcation is said to have been "delayed" relative to the threshold value computed directly from a linear stability analysis. In contrast, we analyze the delay effect in systems of PDE's. In particular, for spike solutions of singularly perturbed generalized Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of delay resulting from slow passage into regimes of oscillatory and competition instability. In the first example, for the GM model on the infinite real line, we analyze the delay resulting from slowly tuning a control parameter through a Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a finite one-dimensional domain. In this scenario, as opposed to the extrinsic tuning of a system parameter through a bifurcation value, we analyze the delay of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the third example, we consider competition instabilities of the GS model triggered by the extrinsic tuning of a feed rate parameter. In all cases, we find that the system must pass well into the unstable regime before the onset of instability is fully observed, indicating delay. We also find that delay has an important effect on the eventual dynamics of the system in the unstable regime. We give analytic predictions for the magnitude of the delays as obtained through analysis of certain explicitly solvable nonlocal eigenvalue problems. The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen

Asymptotic methods for delay equations.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006

Bifurcation analysis of a semiconductor laser with filtered optical feedback

We study the dynamics and bifurcations of a semiconductor laser with delayed filtered optical feedback, where a part of the output of the laser reenters after spectral filtering. This type of coherent optical feedback is more challenging than the case of conventional optical feedback from a simple mirror, but it provides additional control over the output of the semiconductor laser by means of choosing the filter detuning and the filter width. This laser system can be modeled by a system of delay differential equations with a single fixed delay, which is due to the travel time of the light outside the laser. In this paper we present a bifurcation analysis of the filtered feedback laser. We first consider the basic continuous wave states, known as the external filtered modes (EFMs), and determine their stability regions in the parameter plane of feedback strength versus feedback phase. The EFMs are born in saddle-node bifurcations and become unstable in Hopf bifurcations. We show that for small filter detuning there is a single region of stable EFMs, which splits up into two separate regions when the filter is detuned. We then concentrate on the periodic orbits that emanate from Hopf bifurcations. Depending on the feedback strength and the feedback phase, two types of oscillations can be found. First, there are undamped relaxation oscillations, which are typical for semiconductor laser systems. Second, there are oscillations with a period related to the delay time, which have the remarkable property that the laser frequency oscillates while the laser intensity is almost constant. These frequency oscillations are only possible due to the interaction of the laser with the filter. We determine the stability regions in the parameter plane of feedback strength versus feedback phase of the different types of oscillations. In particular, we find that stable frequency oscillations are dominant for nonzero values of the filter detuning. © 2007 Society for Industrial and Applied Mathematics

Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization

Diffusion-induced turbulence in spatially extended oscillatory media near a supercritical Hopf bifurcation can be controlled by applying global time-delay autosynchronization. We consider the complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime and analytically investigate the stability of uniform oscillations depending on the feedback parameters. We show that a noninvasive stabilization of uniform oscillations is not possible in this type of systems. The synchronization diagram in the plane spanned by the feedback parameters is derived. Numerical simulations confirm the analytical results and give additional information on the spatiotemporal dynamics of the system close to complete synchronization.Comment: 19 pages, 10 figures submitted to Physica

Spike frequency adaptation affects the synchronization properties of networks of cortical oscillators

Oscillations in many regions of the cortex have common temporal characteristics with dominant frequencies centered around the 40 Hz (gamma) frequency range and the 5–10 Hz (theta) frequency range. Experimental results also reveal spatially synchronous oscillations, which are stimulus dependent (Gray&Singer, 1987;Gray, König, Engel, & Singer, 1989; Engel, König, Kreiter, Schillen, & Singer, 1992). This rhythmic activity suggests that the coherence of neural populations is a crucial feature of cortical dynamics (Gray, 1994). Using both simulations and a theoretical coupled oscillator approach, we demonstrate that the spike frequency adaptation seen in many pyramidal cells plays a subtle but important role in the dynamics of cortical networks. Without adaptation, excitatory connections among model pyramidal cells are desynchronizing. However, the slow processes associated with adaptation encourage stable synchronous behavior