Fatihcan M. Atay

Delays, connection topology, and synchronization of coupled chaotic maps.

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor connections and their small-world variants generally exhibit poor synchronization. On the other hand, connection delays can actually be conducive to synchronization, so that it is possible for the delayed system to synchronize where the undelayed system does not. Furthermore, the delays determine the synchronized dynamics, leading to the emergence of a wide range of new collective behavior which the individual units are incapable of producing in isolation.

Distributed delays facilitate amplitude death of coupled oscillators

Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold, the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.

STABILITY AND BIFURCATIONS IN NEURAL FIELDS WITH FINITE PROPAGATION SPEED AND GENERAL CONNECTIVITY

A stability analysis is presented for neural field equations in the presence of finite propagation speed along axons and for a general class of connectivity kernels and synaptic properties. Sufficient conditions are given for the stability of equilibrium solutions. It is shown that the propagation delays play a significant role in nonstationary bifurcations of equilibria, whereas the stationary bifurcations depend only on the connectivity kernel. In the case of nonstationary bifurcations, bounds are determined on the frequencies of the resulting oscillatory solutions. A perturbative scheme is used to calculate the types of bifurcations leading to spatial patterns, oscillations, and traveling waves. For high propagation speeds a simple method is derived that allows the determination of the bifurcation type by visual inspection of the Fourier transforms of the kernel and its first moment. Results are numerically illustrated on a class of neurologically plausible systems with combinations of Gaussian excitat...

Synchronization of discrete-time dynamical networks with time-varying couplings

We study the local complete synchronization of discrete-time dynamical networks with time-varying couplings. Our conditions for the temporal variation of the couplings are rather general and include variations in both the network structure and the reaction dynamics; the reactions could, for example, be driven by a random dynamical system. A basic tool is the concept of the Hajnal diameter, which we extend to infinite Jacobian matrix sequences. The Hajnal diameter can be used to verify synchronization, and we show that it is equivalent to other quantities which have been extended to time-varying cases, such as the projection radius, projection Lyapunov exponents, and transverse Lyapunov exponents. Furthermore, these results are used to investigate the synchronization problem in coupled map networks with time-varying topologies and possibly directed and weighted edges. In this case, the Hajnal diameter of the infinite coupling matrices can be used to measure the synchronizability of the network process. As ...

Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers

Stability analysis of a single-lane microscopic car-following model is studied analytically from the perspective of delayed reactions of human drivers. In the literature, the delayed reactions of the drivers are modeled with discrete delays, which assume that drivers make their control decisions based on the stimuli they receive from a point of time in the history. We improve this model by introducing a distribution of delays, which assumes that the control actions are based on information distributed over an interval of time in history. Such an assumption is more realistic, as it takes into consideration the memory capabilities of the drivers and the inevitable heterogeneity of their delay times. We calculate exact stability regions in the parameter space of some realistic delay distributions. Case studies are provided demonstrating the application of the results.The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean. MSC: 34K35, 93C23, 93D15, 34K20

Bifurcation analysis of neural mass models: Impact of extrinsic inputs and dendritic time constants

Neural mass models (NMMs) explain dynamics of neuronal populations and were designed to strike a balance between mathematical simplicity and biological plausibility. They are currently widely used as generative models for noninvasive electrophysiological brain measurements; that is, magneto- and electroencephalography (M/EEG). Here, we systematically describe the oscillatory regimes which a NMM of a single cortical source with extrinsic input from other cortical and subcortical areas to each subpopulation can explain. For this purpose, we used bifurcation analysis to describe qualitative changes in system behavior in response to quantitative input changes. This approach allowed us to describe sequences of oscillatory regimes, given some specific input trajectory. We systematically classified these sequential phenomena and mapped them into parameter space. Our analysis suggests a principled scheme of how complex M/EEG phenomena can be modeled parsimoniously on two time scales: While the system displays fast oscillations, it slowly traverses phase space to another qualitatively different oscillatory regime, depending on the input dynamics. The resulting scheme is useful for applications where one needs to model an ordered sequence of switching between qualitatively different oscillatory regimes, for example, in pharmacological interventions, epilepsy, sleep, or context-induced state changes.

Synchronization of networks with prescribed degree distributions

We show that the degree distributions of graphs do not suffice to characterize the synchronization of systems evolving on them. We prove that, for any given degree sequence satisfying certain conditions, there exists a connected graph having that degree sequence for which the first nontrivial eigenvalue of the graph Laplacian is arbitrarily close to zero. Consequently, complex dynamical systems defined on such graphs have poor synchronization properties. The result holds under quite mild assumptions, and shows that there exists classes of random, scale-free, regular, small-world, and other common network architectures which impede synchronization. The proof is based on a construction that also serves as an algorithm for building nonsynchronizing networks having a prescribed degree distribution.

Network synchronization: Spectral versus statistical properties

We consider synchronization of weighted networks, possibly with asymmetrical connections. Focusing on causal relations rather than the observed correlations, we show that the synchronizability of networks cannot be directly inferred from their statistical properties. Small local changes in the network structure can sensitively affect the eigenvalues relevant for synchronization, while the gross statistical network properties remain essentially unchanged. Consequently, commonly used statistical properties, including the degree distribution, degree homogeneity, average degree, average distance, degree correlation and clustering coefficient, can fail to characterize the synchronizability of networks in terms of causal relations, despite the observed correlations. (c) 2006 Elsevier B.V. All rights reserved.

Total and partial amplitude death in networks of diffusively coupled oscillators

Networks of weakly nonlinear oscillators are considered with diffusive and time-delayed coupling. Averaging theory is used to determine parameter ranges for which the network experiences amplitude death, whereby oscillations are quenched and the equilibrium solution has a large domain of attraction. The amplitude death is shown to be a common phenomenon, which can be observed regardless of the precise nature of the nonlinearities and under very general coupling conditions. In addition, when the network consists of dissimilar oscillators, there exist parameter values for which only parts of the network are suppressed. Sufficient conditions are derived for total and partial amplitude death in arbitrary network topologies with general nonlinearities, coupling coefficients, and connection delays.

Modeling brain resonance phenomena using a neural mass model

Stimulation with rhythmic light flicker (photic driving) plays an important role in the diagnosis of schizophrenia, mood disorder, migraine, and epilepsy. In particular, the adjustment of spontaneous brain rhythms to the stimulus frequency (entrainment) is used to assess the functional flexibility of the brain. We aim to gain deeper understanding of the mechanisms underlying this technique and to predict the effects of stimulus frequency and intensity. For this purpose, a modified Jansen and Rit neural mass model (NMM) of a cortical circuit is used. This mean field model has been designed to strike a balance between mathematical simplicity and biological plausibility. We reproduced the entrainment phenomenon observed in EEG during a photic driving experiment. More generally, we demonstrate that such a single area model can already yield very complex dynamics, including chaos, for biologically plausible parameter ranges. We chart the entire parameter space by means of characteristic Lyapunov spectra and Kaplan-Yorke dimension as well as time series and power spectra. Rhythmic and chaotic brain states were found virtually next to each other, such that small parameter changes can give rise to switching from one to another. Strikingly, this characteristic pattern of unpredictability generated by the model was matched to the experimental data with reasonable accuracy. These findings confirm that the NMM is a useful model of brain dynamics during photic driving. In this context, it can be used to study the mechanisms of, for example, perception and epileptic seizure generation. In particular, it enabled us to make predictions regarding the stimulus amplitude in further experiments for improving the entrainment effect.