Leonhard Lücken

Reduction of interaction delays in networks

Delayed interactions are a common property of coupled natural systems and therefore arise in a variety of different applications. For instance, signals in neural or laser networks propagate at finite speed giving rise to delayed connections. Such systems are often modeled by delay differential equations with discrete delays. In realistic situations, these delays are not identical on different connections. We show that by a componentwise timeshift transformation it is often possible to reduce the number of different delays and simplify the models without loss of information. We identify dynamic invariants of this transformation, determine its capabilities to reduce the number of delays and interpret these findings in terms of the topology of the underlying graph. In particular, we show that networks with identical sums of delay times along the fundamental semicycles are dynamically equivalent and we provide a normal form for these systems. We illustrate the theory using a network motif of coupled Mackey-Glass systems with 8 different time delays, which can be reduced to an equivalent motif with three delays.

Two-cluster bifurcations in systems of globally pulse-coupled oscillators

Abstract For a system of globally pulse-coupled phase-oscillators, we derive conditions for stability of the completely synchronous state and all stationary two-cluster states and explain how the different states are naturally connected via bifurcations. The coupling is modeled using the phase-response-curve (PRC), which measures the sensitivity of each oscillator’s phase to perturbations. For large systems with a PRC, which is zero at the spiking threshold, we are able to find the parameter regions where multiple stable two-cluster states coexist and illustrate this by an example. In addition, we explain how a locally unstable one-cluster state may form an attractor together with its homoclinic connections. This leads to the phenomenon of intermittent, asymptotic synchronization with abating beats away from the perfect synchrony.

Desynchronization boost by non-uniform coordinated reset stimulation in ensembles of pulse-coupled neurons

Several brain diseases are characterized by abnormal neuronal synchronization. Desynchronization of abnormal neural synchrony is theoretically compelling because of the complex dynamical mechanisms involved. We here present a novel type of coordinated reset (CR) stimulation. CR means to deliver phase resetting stimuli at different neuronal sub-populations sequentially, i.e., at times equidistantly distributed in a stimulation cycle. This uniform timing pattern seems to be intuitive and actually applies to the neural network models used for the study of CR so far. CR resets the population to an unstable cluster state from where it passes through a desynchronized transient, eventually resynchronizing if left unperturbed. In contrast, we show that the optimal stimulation times are non-uniform. Using the model of weakly pulse-coupled neurons with phase response curves, we provide an approach that enables to determine optimal stimulation timing patterns that substantially maximize the desynchronized transient time following the application of CR stimulation. This approach includes an optimization search for clusters in a low-dimensional pulse coupled map. As a consequence, model-specific non-uniformly spaced cluster states cause considerably longer desynchronization transients. Intriguingly, such a desynchronization boost with non-uniform CR stimulation can already be achieved by only slight modifications of the uniform CR timing pattern. Our results suggest that the non-uniformness of the stimulation times can be a medically valuable parameter in the calibration procedure for CR stimulation, where the latter has successfully been used in clinical and pre-clinical studies for the treatment of Parkinsons disease and tinnitus.

Noise-enhanced coupling between two oscillators with long-term plasticity.

Spike timing-dependent plasticity is a fundamental adaptation mechanism of the nervous system. It induces structural changes of synaptic connectivity by regulation of coupling strengths between individual cells depending on their spiking behavior. As a biophysical process its functioning is constantly subjected to natural fluctuations. We study theoretically the influence of noise on a microscopic level by considering only two coupled neurons. Adopting a phase description for the neurons we derive a two-dimensional system which describes the averaged dynamics of the coupling strengths. We show that a multistability of several coupling configurations is possible, where some configurations are not found in systems without noise. Intriguingly, it is possible that a strong bidirectional coupling, which is not present in the noise-free situation, can be stabilized by the noise. This means that increased noise, which is normally expected to desynchronize the neurons, can be the reason for an antagonistic response of the system, which organizes itself into a state of stronger coupling and counteracts the impact of noise. This mechanism, as well as a high potential for multistability, is also demonstrated numerically for a coupled pair of Hodgkin-Huxley neurons.

Pattern reverberation in networks of excitable systems with connection delays

We consider the recurrent pulse-coupled networks of excitable elements with delayed connections, which are inspired by the biological neural networks. If the delays are tuned appropriately, the network can either stay in the steady resting state, or alternatively, exhibit a desired spiking pattern. It is shown that such a network can be used as a pattern-recognition system. More specifically, the application of the correct pattern as an external input to the network leads to a self-sustained reverberation of the encoded pattern. In terms of the coupling structure, the tolerance and the refractory time of the individual systems, we determine the conditions for the uniqueness of the sustained activity, i.e., for the functionality of the network as an unambiguous pattern detector. We point out the relation of the considered systems with cyclic polychronous groups and show how the assumed delay configurations may arise in a self-organized manner when a spike-time dependent plasticity of the connection delays is assumed. As excitable elements, we employ the simplistic coincidence detector models as well as the Hodgkin-Huxley neuron models. Moreover, the system is implemented experimentally on a Field-Programmable Gate Array.

The Dynamical Impact of a Shortcut in Unidirectionally Coupled Rings of Oscillators

We study the destabilization mechanism in a unidirectional ring of identical oscillators, perturbed by the introduction of a long-range connection. It is known that for a homogeneous, unidirectional ring of identical Stuart-Landau oscillators the trivial equilibrium undergoes a sequence of Hopf bifurcations eventually leading to the coexistence of multiple stable periodic states resembling the Eckhaus scenario. We show that this destabilization scenario persists under small non-local perturbations. In this case, the Eckhaus line is modulated according to certain resonance conditions. In the case when the shortcut is strong, we show that the coexisting periodic solutions split up into two groups. The first group consists of orbits which are unstable for all parameter values, while the other one shows the classical Eckhaus behavior.

Multi-jittering Instability in Oscillatory Systems with Pulse Coupling

In oscillatory systems with pulse coupling regular spiking regimes may destabilize via a peculiar scenario called “multi-jitter instability”. At the bifurcation point numerous so-called “jittering” regimes with distinct inter-spike intervals emerge simultaneously. Such regimes were first discovered in a single oscillator with delayed pulse feedback and later were found in networks of coupled oscillators. The present chapter reviews recent results on multi-jitter instability and discussed its features.

Experimental study of jittering chimeras in a ring of excitable units

A new type of chimera-like regime is reported that we call “jittering chimera”. The regime is observed in a ring of excitable units in which the excitation is invoked by an oscillator included into the ring. The jittering chimera is characterized by the presence of two domains, one with regular spiking and the other with irregular. A method to set and control desired chimera states in a physically implemented electronic circuit is developed.