Jacques Sepulchre

A comparative study of the experimental quantification of deterministic chaos

Abstract Topological and correlation dimensions of physiological chaotic attractors are evaluated. The latter are embedded in phase spaces reconstructed using the laging method, multi-channel recordings and the singular value decomposition technique. A comparative study shows that comparable results are obtained only if the correlation dimension is less than four.

A hidden feedback in signaling cascades is revealed

Cycles involving covalent modification of proteins are key components of the intracellular signaling machinery. Each cycle is comprised of two interconvertable forms of a particular protein. A classic signaling pathway is structured by a chain or cascade of basic cycle units in such a way that the activated protein in one cycle promotes the activation of the next protein in the chain, and so on. Starting from a mechanistic kinetic description and using a careful perturbation analysis, we have derived, to our knowledge for the first time, a consistent approximation of the chain with one variable per cycle. The model we derive is distinct from the one that has been in use in the literature for several years, which is a phenomenological extension of the Goldbeter-Koshland biochemical switch. Even though much has been done regarding the mathematical modeling of these systems, our contribution fills a gap between existing models and, in doing so, we have unveiled critical new properties of this type of signaling cascades. A key feature of our new model is that a negative feedback emerges naturally, exerted between each cycle and its predecessor. Due to this negative feedback, the system displays damped temporal oscillations under constant stimulation and, most important, propagates perturbations both forwards and backwards. This last attribute challenges the widespread notion of unidirectionality in signaling cascades. Concrete examples of applications to MAPK cascades are discussed. All these properties are shared by the complete mechanistic description and our simplified model, but not by previously derived phenomenological models of signaling cascades.

Multistability in networks of weakly coupled bistable units

Abstract We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a ‘propagation failure’ phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples.

Stability of discrete breathers

Abstract We approach the problem of linear stability of discrete breathers in a rigorous way for networks of not necessarily identical oscillators with general type of coupling. In the case of Hamiltonian systems, using symplectic signature theory we give general conditions on the spectrum of the monodromy map in the uncoupled limit such that the discrete breathers are l 2 -linearly stable for weak enough coupling. We consider also dissipative networks of oscillators. In this case we prove that the discrete breathers are not only stable but also attract a neighbourhood of initial data for any choice of l p -topology. Some examples are considered amongst which an instance of “roto-breather”.

Control of chaos in delay differential equations, in a network of oscillators and in model cortex

Abstract We extend the Ott-Grebogi-Yorke method to the stabilization of unstable orbits in a network of oscillators exhibiting spatiotemporal chaotic activity, wherein the perturbation is applied to the variables of the system. With the help of numerical simulations we show that a method developed by Pyragas can stabilize unstable orbits in a one variable delay differential equation and in a model cortical network with delay. We discuss the relevance of these results in the physiological processes of the brain.

Target and spiral waves in oscillatory media in the presence of obstacles

Abstract Propagation of target and spiral waves in the presence of walls and windows in a two-dimensional reaction-diffusion model is considered. The time evolution of the system is such that for a range of parameter values a supercritical Hopf bifurcation leads to bulk oscillations. It is shown that in a finite system, for sufficiently small passages, no target waves are triggered. The passage of target waves through a window induces in the next compartment spiral or target waves. In this case a new bulk frequency appears and quasi-periodic motion is observed. In presence of two windows, propagation through a large opening can inhibit the onset of waves from smaller windows.

Discrete breathers in disordered media

Abstract We describe some simple physical models where discrete breathers (nonlinear localised modes in a lattice) exist together with spatial disorder. As the models are translation invariant, both spatial localisation and spatial disorder are only due to the interplay of nonlinearity and discreteness. This result is obtained as a straightforward application of a general result of existence of discrete breathers [J.-A. Sepulchre, R.S. MacKay, Localised oscillations in conservative, or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 1–35].

Propagation into unstable states: the direct Hopf bifurcation

Abstract Front propagation into a uniform state unstable with respect to a supercritical Hopf bifurcation (kc=0) is discussed. Periodic waves are generated behind the front. The predictions are compared with solutions of one- and two-dimensional simulations of propagating front in two different reaction-diffusion models.

Retroactive Signaling in Short Signaling Pathways

In biochemical signaling pathways without explicit feedback connections, the core signal transduction is usually described as a one-way communication, going from upstream to downstream in a feedforward chain or network of covalent modification cycles. In this paper we explore the possibility of a new type of signaling called retroactive signaling, offered by the recently demonstrated property of retroactivity in signaling cascades. The possibility of retroactive signaling is analysed in the simplest case of the stationary states of a bicyclic cascade of signaling cycles. In this case, we work out the conditions for which variables of the upstream cycle are affected by a change of the total amount of protein in the downstream cycle, or by a variation of the phosphatase deactivating the same protein. Particularly, we predict the characteristic ranges of the downstream protein, or of the downstream phosphatase, for which a retroactive effect can be observed on the upstream cycle variables. Next, we extend the possibility of retroactive signaling in short but nonlinear signaling pathways involving a few covalent modification cycles.

Path Finding with Nonlinear Waves

An N*N square lattice of nonlinear oscillators may show dynamical properties such as target waves. Under appropriate conditions, the target waves have unusual properties which can be used for performing useful tasks. For example the problem of a moving robot who must choose among many openings the only door sufficiently large for it to navigate across, is adressed.