Fold-Hopf Bursting in a Model for Calcium Signal Transduction

We study a recent model for calcium signal transduction. This model displays spiking, bursting and chaotic oscillations in accordance with experimental results. We calculate bifurcation diagrams and study the bursting behaviour in detail. This behaviour is classified according to the dynamics of separated slow and fast subsystems. It is shown to be of the Fold-Hopf type, a type which was previously only described in the context of neuronal systems, but not in the context of signal transduction in the cell.Comment: 13 pages, 5 figure

Fast-slow bursters in the unfolding of a high codimension singularity and the ultra-slow transitions of classes

Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different timescales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underly the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystems. Transitions between classes can be obtained through an ultra-slow modulation of the model's parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.Comment: 22 pages, 15 figure

Chaos at the border of criticality

The present paper points out to a novel scenario for formation of chaotic attractors in a class of models of excitable cell membranes near an Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics admits a simple and visual description in terms of the families of one-dimensional first-return maps, which are constructed using the combination of asymptotic and numerical techniques. The bifurcation structure of the continuous system (specifically, the proximity to a degenerate AHB) endows the Poincare map with distinct qualitative features such as unimodality and the presence of the boundary layer, where the map is strongly expanding. This structure of the map in turn explains the bifurcation scenarios in the continuous system including chaotic mixed-mode oscillations near the border between the regions of sub- and supercritical AHB. The proposed mechanism yields the statistical properties of the mixed-mode oscillations in this regime. The statistics predicted by the analysis of the Poincare map and those observed in the numerical experiments of the continuous system show a very good agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science (tentatively, Sept 2008

Within-burst synchrony changes for coupled elliptic bursters

We study the appearance of a novel phenomenon for linearly coupled identical bursters: synchronized bursts where there are changes of spike synchrony within each burst. The examples we study are for normal form elliptic bursters where there is a periodic slow passage through a Bautin (codimension two degenerate Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two and three Bautin-type elliptic bursters for a linear direct coupling scheme. Burst synchronization is known to be prevalent behavior among such coupled bursters, while spike synchronization is more dependent on the details of the coupling. We note that higher order terms in the normal form that do not affect the behavior of a single burster can be responsible for changes in synchrony pattern; more precisely, we find within-burst synchrony changes associated with a turning point in the spiking frequency.Comment: 17 pages, 13 figures, 2 table

Neuronal synchrony: peculiarity and generality

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their “dynamical repertoire” includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale

Toward a dynamical systems analysis of neuromodulation

This work presents some first steps toward a more thorough understanding of the control systems employed in evolutionary robotics. In order to choose an appropriate architecture or to construct an effective novel control system we need insights into what makes control systems successful, robust, evolvable, etc. Here we present analysis intended to shed light on this type of question as it applies to a novel class of artificial neural networks that include a neuromodulatory mechanism: GasNets. We begin by instantiating a particular GasNet subcircuit responsible for tuneable pattern generation and thought to underpin the attractive property of “temporal adaptivity”. Rather than work within the GasNet formalism, we develop an extension of the well-known FitzHugh-Nagumo equations. The continuous nature of our model allows us to conduct a thorough dynamical systems analysis and to draw parallels between this subcircuit and beating/bursting phenomena reported in the neuroscience literature. We then proceed to explore the effects of different types of parameter modulation on the system dynamics. We conclude that while there are key differences between the gain modulation used in the GasNet and alternative schemes (including threshold modulation of more traditional synaptic input), both approaches are able to produce tuneable pattern generation. While it appears, at least in this study, that the GasNet’s gain modulation may not be crucial to pattern generation , we go on to suggest some possible advantages it could confer