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.

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.

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.

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.

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.

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.