Capturing correlations in chaotic diffusion by approximation methods

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method approximates Markov partitions and transition matrices by using the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in case of dynamics where exact results for the diffusion coefficient are not available.Comment: 11 pages, 5 figure

Continuous approximations of a class of piece-wise continuous systems

In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise functions can be locally or globally approximated. The approximation results can be used to model piece-wise continuous-time dynamical systems of integer or fractional-order. In this way, by overcoming the lack of numerical methods for diffrential equations of fractional-order with discontinuous right-hand side, unattainable procedures for systems modeled by this kind of equations, such as chaos control, synchronization, anticontrol and many others, can be easily implemented. Several examples are presented and three comparative applications are studied.Comment: IJBC, accepted (examples revised

On the concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.Comment: 35 pages, LaTe

How single neuron properties shape chaotic dynamics and signal transmission in random neural networks

While most models of randomly connected networks assume nodes with simple dynamics, nodes in realistic highly connected networks, such as neurons in the brain, exhibit intrinsic dynamics over multiple timescales. We analyze how the dynamical properties of nodes (such as single neurons) and recurrent connections interact to shape the effective dynamics in large randomly connected networks. A novel dynamical mean-field theory for strongly connected networks of multi-dimensional rate units shows that the power spectrum of the network activity in the chaotic phase emerges from a nonlinear sharpening of the frequency response function of single units. For the case of two-dimensional rate units with strong adaptation, we find that the network exhibits a state of "resonant chaos", characterized by robust, narrow-band stochastic oscillations. The coherence of stochastic oscillations is maximal at the onset of chaos and their correlation time scales with the adaptation timescale of single units. Surprisingly, the resonance frequency can be predicted from the properties of isolated units, even in the presence of heterogeneity in the adaptation parameters. In the presence of these internally-generated chaotic fluctuations, the transmission of weak, low-frequency signals is strongly enhanced by adaptation, whereas signal transmission is not influenced by adaptation in the non-chaotic regime. Our theoretical framework can be applied to other mechanisms at the level of single nodes, such as synaptic filtering, refractoriness or spike synchronization. These results advance our understanding of the interaction between the dynamics of single units and recurrent connectivity, which is a fundamental step toward the description of biologically realistic network models in the brain, or, more generally, networks of other physical or man-made complex dynamical units

Mathematical approaches to differentiation and gene regulation

We consider some mathematical issues raised by the modelling of gene networks. The expression of genes is governed by a complex set of regulations, which is often described symbolically by interaction graphs. Once such a graph has been established, there remains the difficult task to decide which dynamical properties of the gene network can be inferred from it, in the absence of precise quantitative data about their regulation. In this paper we discuss a rule proposed by R.Thomas according to which the possibility for the network to have several stationary states implies the existence of a positive circuit in the corresponding interaction graph. We prove that, when properly formulated in rigorous terms, this rule becomes a theorem valid for several different types of formal models of gene networks. This result is already known for models of differential or boolean type. We show here that a stronger version of it holds in the differential setup when the decay of protein concentrations is taken into account. This allows us to verify also the validity of Thomas' rule in the context of piecewise-linear models and the corresponding discrete models. We discuss open problems as well.Comment: To appear in Notes Comptes-Rendus Acad. Sc. Paris, Biologi

Multidimensional hyperbolic billiards

The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Sim\'anyi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts