E Erik Steur

Synchronization in Networks of Diffusively Time-Delay Coupled (Semi-)Passive Systems

We consider networks with a general topology which consist of nonlinear systems that interact via diffusive coupling with constant time-delays. Using the notion of (semi-)passivity we prove under some mild assumptions that passive systems will synchronize and that the solutions of interconnected semi-passive systems will be bounded. Furthermore we prove that identical strictly semi-passive systems, whose internal dynamics are stable, always will synchronize given that the coupling between the systems is sufficiently strong and a possible constant time delay is sufficiently small. We demonstrate our results using numerical simulations of a network consisting of linear systems and a network consisting Hindmarsh-Rose neurons.

Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters

We consider the problem of asymptotic reconstruction of the state and parameter values in systems of ordinary differential equations. A solution to this problem is proposed for a class of systems of which the unknowns are allowed to be nonlinearly parameterized functions of state and time. Reconstruction of state and parameter values is based on the concepts of weakly attracting sets and non-uniform convergence and is subjected to persistency of excitation conditions. In the absence of nonlinear parametrization the resulting observers reduce to standard estimation schemes. In this respect, the proposed method constitutes a generalization of the conventional canonical adaptive observer design.

Network complexity and synchronous behavior an experimental approach

We discuss synchronization in networks of Hindmarsh-Rose neurons that are interconnected via gap junctions, also known as electrical synapses. We present theoretical results for interactions without time-delay. These results are supported by experiments with a setup consisting of sixteen electronic equivalents of the Hindmarsh-Rose neuron. We show experimental results of networks where time-delay on the interaction is taken into account. We discuss in particular the influence of the network topology on the synchronization.

Partial synchronization in diffusively time-delay coupled oscillator networks.

We study networks of diffusively time-delay coupled oscillatory units and we show that networks with certain symmetries can exhibit a form of incomplete synchronization called partial synchronization. We present conditions for the existence and stability of partial synchronization modes in networks of oscillatory units that satisfy a semipassivity property and have convergent internal dynamics.

Nonuniform Small-Gain Theorems for Systems with Unstable Invariant Sets

We consider the problem of asymptotic convergence to invariant sets in interconnected nonlinear dynamical systems. Standard approaches often require that the invariant sets be uniformly attracting, e.g., stable in the Lyapunov sense. This, however, is neither a necessary requirement nor is always useful. Systems may, for instance, be inherently unstable (e.g., intermittent, itinerant, meta-stable) or the problem statement may include requirements that cannot be satisfied with stable solutions. This is often the case in general optimization problems and in nonlinear parameter identification or adaptation. Conventional techniques for these cases either rely on detailed knowledge of the systems vector-fields or require boundedness of its states. The presently proposed method relies only on estimates of the input-output maps and steady-state characteristics. The method requires the possibility of representing the system as an interconnection of a stable and contracting part with an unstable and exploratory part. We illustrate with examples how the method can be applied to problems of analyzing the asymptotic behavior of locally unstable systems as well as to problems of parameter identification and adaptation in the presence of nonlinear parametrizations. The relation of our results to conventional small-gain theorems is discussed.

Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology

Abstract We consider networks of time-delayed diffusively coupled systems and relate conditions for synchronization of the systems in the network to the topology of the network. First we present sufficient conditions for the solutions of the time-delayed coupled systems to be bounded. Next we give conditions for local synchronization and we show that the values of the coupling strength and time-delay for which there is local synchronization in any network can be determined from these conditions. In addition we present results on global synchronization in relation to the network topology for networks of a class of nonlinear systems. We illustrate our results with examples of synchronization in networks with FitzHugh–Nagumo model neurons and Hindmarsh–Rose neurons.

Spatially constrained adaptive rewiring in cortical networks creates spatially modular small world architectures

A modular small-world topology in functional and anatomical networks of the cortex is eminently suitable as an information processing architecture. This structure was shown in model studies to arise adaptively; it emerges through rewiring of network connections according to patterns of synchrony in ongoing oscillatory neural activity. However, in order to improve the applicability of such models to the cortex, spatial characteristics of cortical connectivity need to be respected, which were previously neglected. For this purpose we consider networks endowed with a metric by embedding them into a physical space. We provide an adaptive rewiring model with a spatial distance function and a corresponding spatially local rewiring bias. The spatially constrained adaptive rewiring principle is able to steer the evolving network topology to small world status, even more consistently so than without spatial constraints. Locally biased adaptive rewiring results in a spatial layout of the connectivity structure, in which topologically segregated modules correspond to spatially segregated regions, and these regions are linked by long-range connections. The principle of locally biased adaptive rewiring, thus, may explain both the topological connectivity structure and spatial distribution of connections between neuronal units in a large-scale cortical architecture.

Brief paper: Synchronization and activation in a model of a network of β-cells

Islets of pancreatic @b-cells are of utmost importance in the understanding of diabetes mellitus. We consider here a model of a network of such pancreatic @b-cells which are globally coupled via gap junctions. Some of the cells in the islet produce bursting oscillations while other cells are inactive. We prove that the cells in the islet synchronize if the coupling is sufficiently large and all cells are active (or inactive). If the islet consists of both active and inactive cells and the coupling is sufficiently large, an active cluster and an inactive cluster emerge. We show that activity of the islet depends on the coupling strength and the number of active cells compared to the number of inactive cells. If too few cells are active the islet becomes inactive.

Leaders do not look back, or do they?

We study the effect of adding to a directed chain of interconnected systems a directed feedback from the last element in the chain to the first. The problem is closely related to the fundamental question of how a change in network topology may influence the behavior of coupled systems. We begin the analysis by investigating a simple linear system. The matrix that specifies the system dynamics is the transpose of the network Laplacian matrix, which codes the connectivity of the network. Our analysis shows that for any nonzero complex eigenvalue � of this matrix, the following inequality holds: |ℑ�| |ℜ�| � cot � n . This bound is sharp, as it becomes an equality for an eigenvalue of a simple directed cycle with uniform interaction weights. The latter has the slowest decay of oscillations among all other network configurations with the same number of states. The result is generalized to directed rings and chains of identical nonlinear oscillators. For directed rings, a lower boundc for the connection strengths that guarantees asymptotic synchronization is found to follow a similar pattern: �c = 1 1−cos(2�/n) . Numerical

Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems

We provide Lyapunov-like characterizations of boundedness and convergence of nontrivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation, and adaptive control. In addition to providing boundedness and convergence criteria, the results allow us to derive domains of initial conditions corresponding to solutions leaving a given neighborhood of the origin at least once. In contrast to other works addressing convergence issues in unstable systems, our results require neither input-output characterizations for the stable part nor estimates of convergence rates. The results are illustrated with examples, including the analysis of phase synchronization of neural oscillators with heterogeneous coupling.