Reliability and robustness of oscillations in some slow-fast chaotic systems.

A variety of nonlinear models of biological systems generate complex chaotic behaviors that contrast with biological homeostasis, the observation that many biological systems prove remarkably robust in the face of changing external or internal conditions. Motivated by the subtle dynamics of cell activity in a crustacean central pattern generator (CPG), this paper proposes a refinement of the notion of chaos that reconciles homeostasis and chaos in systems with multiple timescales. We show that systems displaying relaxation cycles while going through chaotic attractors generate chaotic dynamics that are regular at macroscopic timescales and are, thus, consistent with physiological function. We further show that this relative regularity may break down through global bifurcations of chaotic attractors such as crises, beyond which the system may also generate erratic activity at slow timescales. We analyze these phenomena in detail in the chaotic Rulkov map, a classical neuron model known to exhibit a variety of chaotic spike patterns. This leads us to propose that the passage of slow relaxation cycles through a chaotic attractor crisis is a robust, general mechanism for the transition between such dynamics. We validate this numerically in three other models: a simple model of the crustacean CPG neural network, a discrete cubic map, and a continuous flow.

Effects of polarization induced by non-weak electric fields on the excitability of elongated neurons with active dendrites.

An externally-applied electric field can polarize a neuron, especially a neuron with elongated dendrites, and thus modify its excitability. Here we use a computational model to examine, predict, and explain these effects. We use a two-compartment Pinsky-Rinzel model neuron polarized by an electric potential difference imposed between its compartments, and we apply an injected ramp current. We vary three model parameters: the magnitude of the applied potential difference, the extracellular potassium concentration, and the rate of current injection. A study of the Time-To-First-Spike (TTFS) as a function of polarization leads to the identification of three regions of polarization strength that have different effects. In the weak region, the TTFS increases linearly with polarization. In the intermediate region, the TTFS increases either sub- or super-linearly, depending on the current injection rate and the extracellular potassium concentration. In the strong region, the TTFS decreases. Our results in the weak and strong region are consistent with experimental observations, and in the intermediate region, we predict novel effects that depend on experimentally-accessible parameters. We find that active channels in the dendrite play a key role in these effects. Our qualitative results were found to be robust over a wide range of inter-compartment conductances and the ratio of somatic to dendritic membrane areas. In addition, we discuss preliminary results where synaptic inputs replace the ramp injection protocol. The insights and conclusions were found to extend from our polarized PR model to a polarized PR model with I h dendritic currents. Finally, we discuss the degree to which our results may be generalized.

A period-doubling cascade precedes chaos for planar maps.

A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic. Our method of proof in showing there is one cascade is to show there must be infinitely many cascades. We discuss in detail two-dimensional families like those which arise as a time-2π maps for the Duffing equation and the forced damped pendulum equation.

The structure of synchronization sets for noninvertible systems.

Unidirectionally coupled systems (x,y) --> (f(x),g(x,y)) occur naturally, and are used as tractable models of networks with complex interactions. We analyze the structure and bifurcations of attractors in the case the driving system is not invertible, and the response system is dissipative. We discuss both cases in which the driving system is a map, and a strongly dissipative flow. Although this problem was originally motivated by examples of nonlinear synchrony, we show that the ideas presented can be used more generally to study the structure of attractors, and examine interactions between coupled systems.

The geometry of chaos synchronization.

Chaos synchronization in coupled systems is often characterized by a map phi between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set--by which we mean graph(phi)--can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. In brief, synchronization sets can in general become nondifferentiable, and in the more severe case of noninvertible dynamics, they might even be multivalued. We suggest two different ways to quantify these features, and we discuss possible failures in detecting chaos synchrony using standard continuity-based methods when these features are present.

Limits to the experimental detection of nonlinear synchrony.

Chaos synchronization is often characterized by the existence of a continuous function between the states of the components. However, in coupled systems without inherent symmetries, the synchronization set can be extremely complicated. We describe and illustrate three typical complications that can arise, and we discuss how existing methods for detecting synchronization will be hampered by the presence of these features.