Attraction Rates, Robustness, and Discretization of Attractors

We investigate necessary and sufficient conditions for the convergence of attractors of discrete time dynamical systems induced by numerical one{step approximations of ordinary differential equations (ODEs) to an attractor for the approximated ODE. We show that both the existence of uniform attraction rates (i.e., uniform speed of convergence towards the attractors) and uniform robustness with respect to perturbations of the numerical attractors are necessary and sufficient for this convergence property. In addition, we can conclude estimates for the rate of convergence in the Hausdorff metric

On Dynamics of Integrate-and-Fire Neural Networks with Conductance Based Synapses

We present a mathematical analysis of a networks with Integrate-and-Fire neurons and adaptive conductances. Taking into account the realistic fact that the spike time is only known within some \textit{finite} precision, we propose a model where spikes are effective at times multiple of a characteristic time scale $\delta$, where $\delta$ can be \textit{arbitrary} small (in particular, well beyond the numerical precision). We make a complete mathematical characterization of the model-dynamics and obtain the following results. The asymptotic dynamics is composed by finitely many stable periodic orbits, whose number and period can be arbitrary large and can diverge in a region of the synaptic weights space, traditionally called the "edge of chaos", a notion mathematically well defined in the present paper. Furthermore, except at the edge of chaos, there is a one-to-one correspondence between the membrane potential trajectories and the raster plot. This shows that the neural code is entirely "in the spikes" in this case. As a key tool, we introduce an order parameter, easy to compute numerically, and closely related to a natural notion of entropy, providing a relevant characterization of the computational capabilities of the network. This allows us to compare the computational capabilities of leaky and Integrate-and-Fire models and conductance based models. The present study considers networks with constant input, and without time-dependent plasticity, but the framework has been designed for both extensions.Comment: 36 pages, 9 figure

Resonances in a chaotic attractor crisis of the Lorenz Flow

Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle--Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

Aspects of stochastic resonance in reaction-diffusion systems: The nonequilibrium-potential approach

We analyze several aspects of the phenomenon of stochastic resonance in reaction-diffusion systems, exploiting the nonequilibrium potential's framework. The generalization of this formalism (sketched in the appendix) to extended systems is first carried out in the context of a simplified scalar model, for which stationary patterns can be found analytically. We first show how system-size stochastic resonance arises naturally in this framework, and then how the phenomenon of array-enhanced stochastic resonance can be further enhanced by letting the diffusion coefficient depend on the field. A yet less trivial generalization is exemplified by a stylized version of the FitzHugh-Nagumo system, a paradigm of the activator-inhibitor class. After discussing for this system the second aspect enumerated above, we derive from it -through an adiabatic-like elimination of the inhibitor field- an effective scalar model that includes a nonlocal contribution. Studying the role played by the range of the nonlocal kernel and its effect on stochastic resonance, we find an optimal range that maximizes the system's response.Comment: 16 pages, 15 figures, uses svjour.cls and svepj-spec.clo. Minireview to appear in The European Physical Journal Special Topics (issue in memory of Carlos P\'erez-Garc\'{\i}a, edited by H. Mancini

Synchronization in multicell systems exhibiting dynamic plasticity

Collective behaviour in multicell systems arises from exchange of chemicals/ signals between cells and may be different from their intrinsic behaviour. These chemicals are products of regulated networks of biochemical pathways that underlie cellular functions, and can exhibit a variety of dynamics arising from the non-linearity of the reaction processes. We have addressed the emergent synchronization properties of a ring of cells, diffusively coupled by the end product of an intracellular model biochemical pathway exhibiting non-robust birhythmic behaviour. The aim is to examine the role of intercellular interaction in stabilizing the non-robust dynamics in the emergent collective behaviour in the ring of cells. We show that, irrespective of the inherent frequencies of individual cells, depending on the coupling strength, the collective behaviour does synchronize to only one type of oscillations above a threshold number of cells. Using two perturbation analyses, we also show that this emergent synchronized dynamical state is fairly robust under external perturbations. Thus, the inherent plasticity in the oscillatory phenotypes in these model cells may get suppressed to exhibit collective dynamics of a single type in a multicell system, but environmental influences can sometimes expose this underlying plasticity in its collective dynamics

Directed transport in a ratchet with internal and chemical freedoms

We consider mechanisms of directed transport in a ratchet model comprising, besides the external freedom where transport occurs, a chemical freedom that replaces the familiar external driving by an autonomous dynamics providing energy input, and an internal freedom representing a functional mode of a motor molecule. The dependence of the current on various parameters is studied in numerical simulations of our model. In particular, we point out the role of the internal freedom as a buffer between energy input and output of mechanical work that allows a temporary storage of injected energy and can contribute to the efficiency of current generation.Comment: 7 pages, 9 figure