Martin Hasler

Connectivity in ad-hoc and hybrid networks

We consider a large-scale wireless network, but with a low density of nodes per unit area. Interferences are then less critical, contrary to connectivity. This paper studies the latter property for both a purely ad-hoc network and a hybrid network, where fixed base stations can be reached in multiple hops. We assume here that power constraints are modeled by a maximal distance above which two nodes are not (directly) connected. We find that the introduction of a sparse network of base stations does significantly help in increasing the connectivity, but only when the node density is much larger in one dimension than in the other. We explain the results by percolation theory. We obtain analytical expressions of the probability of connectivity in the 1D case. We also show that at a low spatial density of nodes, bottlenecks are unavoidable. Results obtained on actual population data confirm our findings.

Autonomous cellular neural networks: a unified paradigm for pattern formation and active wave propagation

This tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ecology, engineering, and physics. Each CNN is defined mathematically by its cell dynamics (e.g., state equations) and synaptic law, which specifies each cells interaction with its neighbors. We focus on reaction-diffusion CNNs having a linear synaptic law that approximates a spatial Laplacian operator. Such a synaptic law can be realized by one or more layers of linear resistor couplings. An autonomous CNN made of third-order universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves, target (concentric) waves, spiral waves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell to its neighboring cells, the resulting CNN can be used to generate various turing patterns. >

Communication by chaotic signals : the inverse system approach

The inverse of a non-linear dynamical system is introduced and its synchronization with the original system is discussed. If the original system is chaotic and if despite this the inverse system synchronizes, the two systems can be used respectively as a modulator and a demodulator for a chaotic carrier signal. The transmitted signal is then hidden in chaos. We show that the inverse system can have a different order from the original system. The difference corresponds to the relative degree of the original system. In an inverse system with reduced order it is easier to achieve synchronization; on the other hand, such a system may distort a noisy input signal considerably. Examples from the literature are discussed from the point of view of order reduction. New circuit examples are given where the input and the output are not variables at the same port, as is the case for all published examples in the context of communication with chaotic signals. Finally, a quite general system structure together with its inverse that can be designed for synchronization is given.

Synchronization of chaotic systems and transmission of information

An overview over the methods currently under investigation for the transmission of information hidden in a chaotic signal is given. First, the notion of synchronization is discussed, then the coupling of two systems to achieve synchronization is presented and then four methods to mix the information signal with a chaotic carrier are described. Finally, the problem to send several information bearing chaotic signals through the same communication channel is posed as a challenge.

SYNCHRONIZATION AND GRAPH TOPOLOGY

This paper clarifies the relation between synchronization and graph topology. Applying the Connection Graph Stability method developed by Belykh et al. [2004a] to the study of synchronization in networks of coupled oscillators, we show which graph properties matter for synchronization. In particular, while we explicitly link the stability of synchronization with the average path length for a wide class of coupling graphs, we prove by a simple argument that the average path length is not always the crucial quantity for synchronization. We also show that synchronization in scale-free networks can be described by means of regular networks with a star-like coupling structure. Finally, by considering an example of coupled Hindmarsh–Rose neuron models, we demonstrate how global stability of synchronization depends on the parameters of the individual oscillator.

Pattern formation properties of autonomous Cellular Neural Networks

We use the Cellular Neural Network (CNN) to study the pattern formation properties of large scale spatially distributed systems. We have found that the Cellular Neural Network can produce patterns similar to those found in Ising spin glass systems, discrete bistable systems, and the reaction-diffusion system. A thorough analysis of a 1-D CNN whose cells are coupled to immediate neighbors allows us to completely characterize the patterns that can exist as stable equilibria, and to measure their complexity thanks to an entropy function. In the 2-D case, we do not restrict the symmetric coupling between cells to be with immediate neighbors only or to have a special diffusive form. When larger neighborhoods and generalized diffusion coupling are allowed, it is found that some new and unique patterns can be formed that do not fit the standard ferro-antiferromagnetic paradigms. We have begun to develop a theoretical generalization of these paradigms which can be used to predict the pattern formation properties of given templates. We give many examples. It is our opinion that the Cellular Neural Network model provides a method to control the critical instabilities needed for pattern formation without obfuscating parameterizations, complex nonlinearities, or high-order cell states, and which will allow a general and convenient investigation of the essence of the pattern formation properties of these systems. >

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.

Topography of EEG multivariate phase synchronization in early Alzheimer's disease

Alzheimers disease (AD) is likely to disrupt the synchronization of the bioelectrical processes in the distributed cortical networks underlying cognition. We analyze the surface topography of the multivariate phase synchronization (MPS) of multichannel EEG in 17 patients (Clinical Dementia Rating (CDR) Scale: 0.5-1; Functional Assessment Staging (FAST): 3-4) compared to 17 controls by applying a combination of global and regional MPS measures to the resting EEG. In early AD, whole-head mapping reveals a specific landscape of synchronization characterized by a decrease in MPS over the fronto-temporal region and an increase over the temporo-parieto-occipital region predominantly of the left hemisphere. These features manifest themselves through the EEG delta-beta bands and discriminate patients from controls with an accuracy of up to 94%. Moreover, the abnormal MPS in both anterior and posterior clusters correlates with the Mini Mental State Examination score, binding regional EEG synchronization to cognitive decline in AD patients. The MPS technique reveals that the EEG phenotype of early AD is relevant to the clinical picture and may ultimately become its sensitive and specific biomarker.

Chaos communication over noisy channels

The problem of transmitting digital information using chaotic signals over a channel with Gaussian white noise perturbation is introduced rigorously. It is shown that discrete time base-band chaotic communication systems with discrete time Gaussian white noise in the channel are sufficiently general in this context. The optimal receiver is given explicitly in terms of conditional probabilities. For the example of chaos shift keying using iterations of the tent map, the optimal classifier is constructed explicitly. Finally, it is shown how previously published methods, in particular those based on chaos synchronization, fit into this framework.

ENGINEERING CHAOS FOR ENCRYPTION AND BROADBAND COMMUNICATION

We present the different methods that have been proposed in the literature for sending information by means of a chaotic signal. They are based on three different ways to synchronize a receiver system with a chaotic transmitter system. Three different methods to modulate the transmitter system with an information carrying signal are also presented and some of their advantages and drawbacks are discussed.