Parameter-dependent behaviour of periodic channels in a locus of boundary crisis

This is the final version of the article. Available from EDP Sciences via the DOI in this recordA boundary crisis occurs when a chaotic attractor outgrows its basin of attraction and suddenly disappears. As previously reported, the locus of a boundary crisis is organised by homo- or heteroclinic tangencies between the stable and unstable manifolds of saddle periodic orbits. In two parameters, such tangencies lead to curves, but the locus of boundary crisis along those curves exhibits gaps or channels, in which other non-chaotic attractors persist. These attractors are stable periodic orbits which themselves can undergo a cascade of period-doubling bifurcations culminating in multi-component chaotic attractors. The canonical diffeomorphic two-dimensional Hénon map exhibits such periodic channels, which are structured in a particular ordered way: each channel is bounded on one side by a saddle-node bifurcation and on the other by a period-doubling cascade to chaos; furthermore, all channels seem to have the same orientation, with the saddle-node bifurcation always on the same side. We investigate the locus of boundary crisis in the Ikeda map, which models the dynamics of energy levels in a laser ring cavity. We find that the Ikeda map features periodic channels with a richer and more general organisation than for the Hénon map. Using numerical continuation, we investigate how the periodic channels depend on a third parameter and characterise how they split into multiple channels with different properties

Dynamical systems analysis of spike-adding mechanisms in transient bursts

The electronic version of this article is the complete one and can be found online at: doi:10.1186/2190-8567-2-7Open Access ArticleTransient bursting behaviour of excitable cells, such as neurons, is a common feature observed experimentally, but theoretically, it is not well understood. We analyse a five-dimensional simplified model of after-depolarisation that exhibits transient bursting behaviour when perturbed with a short current injection. Using one-parameter continuation of the perturbed orbit segment formulated as a well-posed boundary value problem, we show that the spike-adding mechanism is a canard-like transition that has a different character from known mechanisms for periodic burst solutions. The biophysical basis of the model gives a natural time-scale separation, which allows us to explain the spike-adding mechanism using geometric singular perturbation theory, but it does not involve actual bifurcations as for periodic bursts. We show that unstable sheets of the critical manifold, formed by saddle equilibria of the system that only exist in a singular limit, are responsible for the spike-adding transition; the transition is organised by the slow flow on the critical manifold near folds of this manifold. Our analysis shows that the orbit segment during the spike-adding transition includes a fast transition between two unstable sheets of the slow manifold that are of saddle type. We also discuss a different parameter regime where the presence of additional saddle equilibria of the full system alters the spike-adding mechanism.Engineering and Physical Sciences Research Council (EPSRC

Continuation-based numerical detection of after-depolarization and spike-adding thresholds.

PublishedJournal ArticleResearch Support, Non-U.S. Gov'tThe changes in neuronal firing pattern are signatures of brain function, and it is of interest to understand how such changes evolve as a function of neuronal biophysical properties. We address this important problem by the analysis and numerical investigation of a class of mechanistic mathematical models. We focus on a hippocampal pyramidal neuron model and study the occurrence of bursting related to the after-depolarization (ADP) that follows a brief current injection. This type of burst is a transient phenomenon that is not amenable to the classical bifurcation analysis done, for example, for periodic bursting oscillators. In this letter, we show how to formulate such transient behavior as a two-point boundary value problem (2PBVP), which can be solved using well-known continuation methods. The 2PBVP is formulated such that the transient response is represented by a finite orbit segment for which onsets of ADP and additional spikes in a burst can be detected as bifurcations during a one-parameter continuation. This in turn provides us with a direct method to approximate the boundaries of regions in a two-parameter plane where certain model behavior of interest occurs. More precisely, we use two-parameter continuation of the detected onset points to identify the boundaries between regions with and without ADP and bursts with different numbers of spikes. Our 2PBVP formulation is a novel approach to parameter sensitivity analysis that can be applied to a wide range of problems.The research for this letter was done while J.N. was a Ph.D. student at the University of Bristol, supported by grant EP/E032249/1 from the Engineering and Physical Sciences Research Council (EPSRC). The research of K.T-A. was supported by EPSRC grant EP/I018638/1 and that of H.M.O. by grant UOA0718 of the Royal Society of NZ Marsden Fun

Prospectus, April 7, 1999

https://spark.parkland.edu/prospectus_1999/1011/thumbnail.jp