Self-Emerging and Turbulent Chimeras in Oscillator Chains

We report on a self-emerging chimera state in a homogeneous chain of nonlocally and nonlinearly coupled oscillators. This chimera, i.e. a state with coexisting regions of complete and partial synchrony, emerges via a supercritical bifurcation from a homogeneous state and thus does not require preparation of special initial conditions. We develop a theory of chimera basing on the equations for the local complex order parameter in the Ott-Antonsen approximation. Applying a numerical linear stability analysis, we also describe the instability of the chimera and transition to a phase turbulence with persistent patches of synchrony

Moran’s I quantifies spatio-temporal pattern formation in neural imaging data

Motivation: Neural activities of the brain occur through the formation of spatio-temporal patterns. In recent years, macroscopic neural imaging techniques have produced a large body of data on these patterned activities, yet a numerical measure of spatio-temporal coherence has often been reduced to the global order parameter, which does not uncover the degree of spatial correlation. Here, we propose to use the spatial autocorrelation measure Moran’s I, which can be applied to capture dynamic signatures of spatial organization. We demonstrate the application of this technique to collective cellular circadian clock activities measured in the small network of the suprachiasmatic nucleus (SCN) in the hypothalamus.Results: We found that Moran’s I is a practical quantitative measure of the degree of spatial coherence in neural imaging data. Initially developed with a geographical context in mind, Moran’s I accounts for the spatial organization of any interacting units. Moran’s I can be modified in accordance with the characteristic length scale of a neural activity pattern. It allows a quantification of statistical significance levels for the observed patterns. We describe the technique applied to synthetic datasets and various experimental imaging time-series from cultured SCN explants. It is demonstrated that major characteristics of the collective state can be described by Moran’s I and the traditional Kuramoto order parameter R in a complementary fashion

Dynamik und Stabilität von Pulsen und Pulsfolgen in Erregbaren Medien

Die vorliegende Dissertation handelt von der Dynamik und Stabilität von Pulsen und Pulsfolgen in erregbaren Medien. Diese Pulse stellen nichtlineare Wellen in aktiven Systemen fern vom thermodynamischen Gleichgewicht dar und sind wichtig für das Verständnis solcher Phänomene wie die Erregungsleitung in neuronalen Systemen, intrazelluläre Kalziumwellen, elektrische Erregungswellen im Herzmuskel, Musterbildung in der Belousov-Zhabotinsky Reaktion und vieler anderer. Zwei qualitativ verschiedene Pulstypen werden untersucht. Bei dem ersten hat der Einzelpuls einen monotonen Ausläufer, während der zweite Typ kleinamplitudige Oszillationen im Ausläufer besitzt. Dieser Unterschied im Ausläufer des Einzelpulses beeinflusst wesentlich die Wechselwirkung zwischen Pulsen innerhalb einer räumlich periodischen Pulsfolge oder eines Pulspaares. Die Oszillationen im Ausläufer verursachen die bislang unbekannte Koexistenz von Pulsfolgen mit gleicher Wellenlänge aber unterschiedlichen Ausbreitungsgeschwindigkeiten. Diese Koexistenz findet ihren Ausdruck in bistabilen Bereichen in der Dispersionskurve von räumlich periodischen Pulsfolgen. Ein grosser Teil der Dissertation befasst sich mit der Untersuchung der Stabilität der Pulsfolgen in den bistabilen Bereichen der Dispersionskurve. Wir zeigen, dass Pulse mit oszillatorischen Ausläufern typischerweise in der Nähe des Übergangsbereiches zwischen Trigger- und Phasenwellen existieren und beschreiben mehrere Stufen dieses Überganges. Mit zunehmender Erregbarkeit entstehen zunächst gedämpfte Oszillationen im Ausläufer des Einzelpulses, die mit weiterer Zunahme der Erregbarkeit ungedämpft werden. In Übereinstimmung damit zeigen die räumlich periodischen Pulsfolgen gedämpfte bzw. ungedämpfte Oszillationen in der Dispersionskurve. Der Übergang wird durch eine Kollision der Dispersionkurven von Phasen- und Triggerwellen abgeschlossen. Die Beschreibung des Überganges erforderte Stabilitätsuntersuchungen von Trigger- und Phasenwellen im Übergangsbereich. Pulse mit monotonen Ausläufern unter dem Einfluss von nicht-lokaler Kopplung werden auch untersucht. In zahlreichen Systemen wie in neuronalen Geweben oder bei elektrochemischen Reaktionen sind neben kurzreichweitigen auch langreichweitige räumliche Kopplungen zu berücksichtigen. Es wurde gezeigt, dass eine beliebig schwache, exponentiell abklingende nichtlokale Kopplung zur Ausbildung von Pulspaaren führt. Die Entstehung von Pulspaaren ist unabhängig vom verwendeten Modell, vorausgesetzt, dieses besitzt ohne nichtlokale Kopplung stabile Einzelpulse.The present Thesis deals with the dynamics and stability of pulses and spatially periodic pulse trains in excitable media. Such pulses represent nonlinear waves in active systems far from the thermodynamic equilibrium. They are very important for the understanding of such phenomena as the conduction of excitation in neuronal systems, intracellular calcium dynamics, electrical activity in the heart muscle, structure formation in the Belousov-Zhabotinsky reaction and many others. We analyze two qualitatively different types of solitary pulses. The first one has a monotonous decay behind the high-amplitude pulse head and the second one decays in an oscillatory manner. This difference in the decay essentially influences the interaction between pulses in a spatially periodic pulse train or in a pulse pair. The presence of the tail oscillations leads to the so far unknown coexistence of pulse trains of the same wavelength and different velocities. This coexistence is reflected in the bistable domains in the dispersion relation of spatially periodic pulse trains. A large part of the dissertation addresses the stability of the pulse trains in such bistable domains. We show that oscillatory decay is typical for excitation pulses close to the transition between trigger and phase waves and describe several stages of the transition. First, we observe the emergence of small-amplitude damped oscillations in the wake of the solitary pulse under increase of the excitability of the system. Further increase of the excitability results in the emergence of undamped tail oscillations. Dispersion curve of spatially periodic pulse trains displays damped and undamped oscillations as well in accordance to the decay type behind the solitary pulse. The depiction of the transition between trigger and phase waves demanded a detailed analysis of the stability of the waves in the transition region. We also studied pulses with monotonous tails under influence of non-local coupling. In numerous systems nonlocal coupling must be considered besides local diffusive coupling. Examples for such systems include brain tissue and electro-chemical reactions. We show that an arbitrary small, exponentially decaying nonlocal coupling leads to the emergence of bound states. The emergence of bound states is model-independent given that the model supports propagation of stable solitary pulses

Human chronotypes from a theoretical perspective.

The endogenous circadian timing system has evolved to synchronize an organism to periodically recurring environmental conditions. Those external time cues are called Zeitgebers. When entrained by a Zeitgeber, the intrinsic oscillator adopts a fixed phase relation ψ to the Zeitgeber. Here, we systematically study how the phase of entrainment depends on clock and Zeitgeber properties. We combine numerical simulations of amplitude-phase models with predictions from analytically tractable models. In this way we derive relations between the phase of entrainment ψ to the mismatch between the endogenous and Zeitgeber period, the Zeitgeber strength, and the range of entrainment. A core result is the "180° rule" asserting that the phase ψ varies over a range of about 180° within the entrainment range. The 180° rule implies that clocks with a narrow entrainment range ("strong oscillators") exhibit quite flexible entrainment phases. We argue that this high sensitivity of the entrainment phase contributes to the wide range of human chronotypes

Mathematical modeling in chronobiology

International audienceCircadian clocks are autonomous oscillators entrained by external Zeitgebers such as light-dark and temperature cycles. On the cellular level, rhythms are generated by negative transcriptional feedback loops. In mammals, the suprachiasmatic necleus (SCN) in the anterior part of the hypothalamus plays the role of the central circadian pacemaker. Coupling between individual neurons in the SCN leads to precise self-sustained oscillations even in the absence of external signals. These neuronal rhythms orchestrate the phasing of circadian oscillations in peripheral organs. Altogether, the mammalian circadian system can be regarded as a network of coupled oscillators. In order to understand the dynamic complexity of these rhythms, mathematical models successfully complement experimental investigations. Here we discuss basic ideas of modeling on three di erent levels: (i) rhythm generation in single cells by delayed negative feedbacks, (ii) synchronization of cells via external stimuli or cell-cell coupling, and (iii) optimization of chronotherapy

Flexibility of entrainment phases due to small variations of the endogenous period.

<p>The inserts show normally distributed periods with a standard deviation of 0.2 h. Simulations of amplitude-phase models illustrate the flexibility of entrainment phases for strong oscillators (left) compared to weak oscillators (right).</p

Sinusoidal phase response curve (PRC) and associated phase transition curves (PTCs) according to Eq. (5).

<p>Applying -periodic pulses, stationary entrainment phases are given by the intersections of the PTC with the diagonal <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0059464#pone.0059464-Glass1" target="_blank">[15]</a>. Upper graphs: Vanishing frequency mismatch leads to a stable entrainment phase h. Lower graphs: Period mismatches correspond to the borderlines of the entrainment range. The corresponding entrainment phases of 18 h and 6 h are associated to the extrema of the sine-function and are 12 h (or 180° ) apart.</p

Phases of entrainment within the entrainment regions.

<p>Left: Numerical simulations of amplitude-phase model. Right: Entrainment phases from Eq. (2) derived from the Kuramoto phase equation. In both cases the entrainment phase varies from −6 h to 6 h between the borderlines of entrainment. Note that the lines with h are very close to those with h and are not marked separately for the sake of clarity of the graphical representation.</p

Phase response curves of strong (left) and weak (right) oscillators.

<p>The extrema marked by A and B are related to the borders of the entrainment range (compare <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0059464#pone-0059464-g001" target="_blank">Figure 1</a>) as explained in the text.</p