Pixel-wise quantification of myocardial perfusion using spatial Tikhonov regularization.

Quantification of myocardial perfusion by contrast-enhanced cardiovascular magnetic resonance imaging (CMR) aims for an observer independent and reproducible risk assessment of cardiovascular disease. Currently, the data used for the pixel-wise analysis of cardiac perfusion are either filtered prior to a fitting procedure, which inherently reduces the spatial resolution of data; or all pixels are considered without any regularization or prior filtering, which yields an unstable fit in the presence of low signal-to-noise ratio. Here, we propose a new pixel-wise analysis based on spatial Tikhonov regularization which exploits the spatial smoothness of the data and ensures accurate quantification even for images with low signal-to-noise ratio. The regularization parameter is determined automatically by an L-curve criterion. We study the performance of our method on a numerical phantom and demonstrate that the method reduces significantly the root-mean square error in the perfusion estimate compared to a non-regularized fit. In patient data our method allows us to recover the myocardial perfusion and to distinguish between healthy and ischemic regions.

Synchronization patterns: from network motifs to hierarchical networks.

We investigate complex synchronization patterns such as cluster synchronization and partial amplitude death in networks of coupled Stuart-Landau oscillators with fractal connectivities. The study of fractal or self-similar topology is motivated by the network of neurons in the brain. This fractal property is well represented in hierarchical networks, for which we present three different models. In addition, we introduce an analytical eigensolution method and provide a comprehensive picture of the interplay of network topology and the corresponding network dynamics, thus allowing us to predict the dynamics of arbitrarily large hierarchical networks simply by analysing small network motifs. We also show that oscillation death can be induced in these networks, even if the coupling is symmetric, contrary to previous understanding of oscillation death. Our results show that there is a direct correlation between topology and dynamics: hierarchical networks exhibit the corresponding hierarchical dynamics. This helps bridge the gap between mesoscale motifs and macroscopic networks.This article is part of the themed issue 'Horizons of cybernetical physics'.

Adaptive time-delayed stabilization of steady states and periodic orbits.

We derive adaptive time-delayed feedback controllers that stabilize fixed points and periodic orbits. First, we develop an adaptive controller for stabilization of a steady state by applying the speed-gradient method to an appropriate goal function and prove global asymptotic stability of the resulting system. For an example we show that the advantage of the adaptive controller over the nonadaptive one is in a smaller controller gain. Second, we propose adaptive time-delayed algorithms for stabilization of periodic orbits. Their efficiency is confirmed by local stability analysis. Numerical examples demonstrate the applicability of the proposed controllers.

Controlling cluster synchronization by adapting the topology.

We suggest an adaptive control scheme for the control of in-phase and cluster synchronization in delay-coupled networks. Based on the speed-gradient method, our scheme adapts the topology of a network such that the target state is realized. It is robust towards different initial conditions as well as changes in the coupling parameters. The emerging topology is characterized by a delicate interplay of excitatory and inhibitory links leading to the stabilization of the desired cluster state. As a crucial parameter determining this interplay we identify the delay time. Furthermore, we show how to construct networks such that they exhibit not only a given cluster state but also with a given oscillation frequency. We apply our method to coupled Stuart-Landau oscillators, a paradigmatic normal form that naturally arises in an expansion of systems close to a Hopf bifurcation. The successful and robust control of this generic model opens up possible applications in a wide range of systems in physics, chemistry, technology, and life science.

Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes.

We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a Γ distribution.

Adaptation controls synchrony and cluster states of coupled threshold-model neurons.

We analyze zero-lag and cluster synchrony of delay-coupled nonsmooth dynamical systems by extending the master stability approach, and apply this to networks of adaptive threshold-model neurons. For a homogeneous population of excitatory and inhibitory neurons we find (i) that subthreshold adaptation stabilizes or destabilizes synchrony depending on whether the recurrent synaptic excitatory or inhibitory couplings dominate, and (ii) that synchrony is always unstable for networks with balanced recurrent synaptic inputs. If couplings are not too strong, synchronization properties are similar for very different coupling topologies, i.e., random connections or spatial networks with localized connectivity. We generalize our approach for two subpopulations of neurons with nonidentical local dynamics, including bursting, for which activity-based adaptation controls the stability of cluster states, independent of a specific coupling topology.

Clustering in delay-coupled smooth and relaxational chemical oscillators.

We investigate cluster synchronization in networks of nonlinear systems with time-delayed coupling. Using a generic model for a system close to the Hopf bifurcation, we predict the order of appearance of different cluster states and their corresponding common frequencies depending upon coupling delay. We may tune the delay time in order to ensure the existence and stability of a specific cluster state. We qualitatively and quantitatively confirm these results in experiments with chemical oscillators. The experiments also exhibit strongly nonlinear relaxation oscillations as we increase the voltage, i.e., go further away from the Hopf bifurcation. In this regime, we find secondary cluster states with delay-dependent phase lags. These cluster states appear in addition to primary states with delay-independent phase lags observed near the Hopf bifurcation. Extending the theory on Hopf normal-form oscillators, we are able to account for realistic interaction functions, yielding good agreement with experimental findings.

Cluster and group synchronization in delay-coupled networks.

We investigate the stability of synchronized states in delay-coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics. Using a master stability approach, we find that the master stability function shows a discrete rotational symmetry depending on the number of groups. The coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum, which helps to simplify the evaluation of the master stability function. Our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times, multiple coupling functions, but also with multiple kinds of local dynamics in the networks' nodes. We illustrate our results by calculating stability in the example of delay-coupled semiconductor lasers and in a model for neuronal spiking dynamics.

Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators.

We consider networks of delay-coupled Stuart-Landau oscillators. In these systems, the coupling phase has been found to be a crucial control parameter. By proper choice of this parameter one can switch between different synchronous oscillatory states of the network. Applying the speed-gradient method, we derive an adaptive algorithm for an automatic adjustment of the coupling phase such that a desired state can be selected from an otherwise multistable regime. We propose goal functions based on both the difference of the oscillators and a generalized order parameter and demonstrate that the speed-gradient method allows one to find appropriate coupling phases with which different states of synchronization, e.g., in-phase oscillation, splay, or various cluster states, can be selected.