D. E. Postnov

Chaotic synchronization : applications to living systems

Coupled Nonlinear Oscillators Transverse Stability of Coupled Maps Unfolding the Riddling Bifurcation Time-Continuous Systems Coupled Pancreatic Cells Chaotic Phase Synchronization Population Dynamic Systems Clustering of Globally Maps Interacting Nephrons Coherence Resonance Oscillators.

SYNCHRONIZATION OF CHAOS

This paper is devoted to the problem of synchronization of dynamical systems in chaotic oscillations regimes. The authors attempt to use the ideas of synchronization and its mechanisms on a certain class of chaotic oscillations. These are chaotic oscillations for which one can pick out basic frequencies in their power spectra. The physical and computer experiments were carried out for the system of two coupled auto-oscillators. The experimental installation permitted one to realize both unidirectional coupling (external synchronization) and symmetrical coupling (mutual synchronization). An auto-oscillator with an inertial nonlinearity was chosen as a partial subsystem. It possesses a chaotic attractor of spiral type in its phase space. It is known that such chaotic oscillations have a distinguished peak in the power spectrum at the frequency f0 (basic frequency). In the experiments, one could make the basic frequencies of partial oscillators equal or different. The bifurcation diagrams on the plane of con...

Dynamical patterns of calcium signaling in a functional model of neuron–astrocyte networks

We propose a functional mathematical model for neuron-astrocyte networks. The model incorporates elements of the tripartite synapse and the spatial branching structure of coupled astrocytes. We consider glutamate-induced calcium signaling as a specific mode of excitability and transmission in astrocytic–neuronal networks. We reproduce local and global dynamical patterns observed experimentally.

Functional modeling of neural-glial interaction.

We propose a generalized mathematical model for a small neural-glial ensemble. The model incorporates subunits of the tripartite synapse that includes a presynaptic neuron, the synaptic terminal itself, a postsynaptic neuron, and a glial cell. The glial cell is assumed to be activated via two different pathways: (i) the fast increase of intercellular [K(+)] produced by the spiking activity of the postsynaptic neuron, and (ii) the slow production of a mediator triggered by the synaptic activity. Our model predicts the long-term potentiation of the postsynaptic neuron as well as various [Ca(2+)] transients in response to the activation of different pathways.

Role of multistability in the transition to chaotic phase synchronization

In this paper we describe the transition to phase synchronization for systems of coupled nonlinear oscillators that individually follow the Feigenbaum route to chaos. A nested structure of phase synchronized regions of different attractor families is observed. With this structure, the transition to nonsynchronous behavior is determined by the loss of stability for the most stable synchronous mode. It is shown that the appearance of hyperchaos and the transition from lag synchronization to phase synchronization are related to the merging of chaotic attractors from different families. Numerical examples using Rossler systems and model maps are given. (c) 1999 American Institute of Physics.

Noise controlled synchronization in potassium coupled neural models.

The paper applies biologically plausible models to investigate how noise input to small ensembles of neurons, coupled via the extracellular potassium concentration, can influence their firing patterns. Using the noise intensity and the volume of the extracellular space as control parameters, we show that potassium induced depolarization underlies the formation of noise-induced patterns such as delayed firing and synchronization. These phenomena are associated with the appearance of new time scales in the distribution of interspike intervals that may be significant for the spatio-temporal oscillations in neuronal ensembles.

COOPERATIVE PHASE DYNAMICS IN COUPLED NEPHRONS

The individual functional unit of the kidney (the nephron) displays oscillations in its pressure and flow regulation at two different time scales: Relatively fast oscillations associated with the myogenic dynamics of the afferent arteriole, and slower oscillations related with a delay in the tubuloglomerular feedback. Neighboring nephrons interact via vascularly propagated signals. We study the appearance of various forms of coherent behavior in a model of two such interacting nephrons. Among the observed phenomena are in-phase and anti-phase synchronization of chaotic dynamics, multistability, and partial phase synchronization in which the nephrons attain a state of chaotic phase synchronization with respect to their slow dynamics, but the fast dynamics remains desynchronized.

SYNCHRONIZATION IN DRIVEN CHAOTIC SYSTEMS : DIAGNOSTICS AND BIFURCATIONS

Abstract We investigate generic aspects of chaos synchronization in an externally forced Rossler system. By comparing different diagnostic methods, we show the existence of a well-defined cut-off of synchronization associated with the transition from weak to fully developed chaos. Chaotic synchronization is found to be lost at this cut-off only after the last band-merging bifurcation has occurred. Everywhere at the boundary of phase synchronization, one of the Lyapunov exponents becomes equal to zero. Two types of chaotic behavior, differing by the number of vanishing Lyapunov exponents, are observed outside the synchronization region.

Synchronization of tubular pressure oscillations in interacting nephrons

Abstract The pressure and flow regulation in the individual functional unit of the kidney (the nephron) tends to operate in an unstable regime. For normal rats, the regulation displays regular self-sustained oscillations, but for rats with high blood pressure the oscillations become chaotic. We explain the mechanisms responsible for this behavior and discuss the involved bifurcations. Experimental data show that neighboring nephrons adjust their pressure and flow regulation in accordance with one another. For rats with normal blood pressure, in-phase as well as anti-phase synchronization can be observed. For spontaneously hypertensive rats, indications of chaotic phase synchronization are found. Accounting for a hermodynamics as well as for a vascular coupling between nephrons that share a common interlobular artery, we present a model of the interaction of the pressure and flow regulations between adjacent nephrons. It is shown that this model, with physiologically realistic parameter values, can reproduce the different types of experimentally observed synchronization, including multistability and partial phase synchronization with respect to the slow and fast dynamics.

Chaotic bursting as chaotic itinerancy in coupled neural oscillators

We show that chaotic bursting activity observed in coupled neural oscillators is a kind of chaotic itinerancy. In neuronal systems with phase deformation along the trajectory, diffusive coupling induces a dephasing effect. Because of this effect, an antiphase synchronized solution is stable for weak coupling, while an in-phase solution is stable for very strong coupling. For intermediate coupling, a chaotic bursting activity is generated. It is a mixture of three different states: an antiphase firing state, an in-phase firing state, and a nonfiring resting state. As we construct numerically the deformed torus manifold underlying the chaotic bursting state, it is shown that the three unstable states are connected to give rise to a global chaotic itinerancy structure. Thus we claim that chaotic itinerancy provides an alternative route to chaos via torus breakdown.