On the effect of circadian oscillations on biochemical cell signaling by NF-κB.

We report the results of a numerical investigation of a mathematical model for NF-κB oscillations, described by a set of ordinary nonlinear differential equations, when perturbed by a circadian oscillation. The main result is that a circadian rhythm, even when it represents a weak perturbation, enhances the signaling capabilities of NF-κB oscillations. This is done by turning rest states into periodic oscillations, and periodic oscillations into quasiperiodic oscillations. Strong perturbations result in complex periodic oscillations and even in chaos. Circadian rhythms would then result in a NF-κB dynamics that is more complex than the simple oscillations and rest states, initially reported for this model. This renders it more amenable for information coding.

Nonlinear dynamics of the membrane potential of a bursting pacemaker cell.

This article presents the results of an exploration of one two-parameter space of the Chay model of a cell excitable membrane. There are two main regions: a peripheral one, where the system dynamics will relax to an equilibrium point, and a central one where the expected dynamics is oscillatory. In the second region, we observe a variety of self-sustained oscillations including periodic oscillation, as well as bursting dynamics of different types. These oscillatory dynamics can be observed as periodic oscillations with different periodicities, and in some cases, as chaotic dynamics. These results, when displayed in bifurcation diagrams, result in complex bifurcation structures, which have been suggested as relevant to understand biological cell signaling.

Block structured dynamics and neuronal coding.

When certain control parameters of nervous cell models are varied, complex bifurcation structures develop in which the dynamical behaviors available appear classified in blocks, according to criteria of dynamical likelihood. This block structured dynamics may be a clue to understand how activated neurons encode information by firing spike trains of their action potentials.

Generalized synchronization in directionally coupled systems with identical individual dynamics.

A simple chaotic flow is presented, which when driven by an identical copy of itself, for certain initial conditions, is able to display generalized synchronization instead of identical synchronization. Being that the drive and the response are observed in exactly the same coordinate system, generalized synchronization is demonstrated by means of the auxiliary system approach and by the conditional Lyapunov spectrum. This is interpreted in terms of changes in the structure of the system stationary points, caused by the coupling, which modify its global behavior.

Amplitude envelope synchronization in coupled chaotic oscillators.

A peculiar type of synchronization has been found when two Van der Pol-Duffing oscillators, evolving in different chaotic attractors, are coupled. As the coupling increases, the frequencies of the two oscillators remain different, while a synchronized modulation of the amplitudes of a signal of each system develops, and a null Lyapunov exponent of the uncoupled systems becomes negative and gradually larger in absolute value. This phenomenon is characterized by an appropriate correlation function between the returns of the signals, and interpreted in terms of the mutual excitation of new frequencies in the oscillators power spectra. This form of synchronization also occurs in other systems, but it shows up mixed with or screened by other forms of synchronization, as illustrated in this paper by means of the examples of the dynamic behavior observed for three other different models of chaotic oscillators.

Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model.

Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.