Lag Synchronization in Coupled Multistable van der Pol-Duffing Oscillators

We consider the system of externally excited identical van der Pol-Duffing oscillators unidirectionally coupled in a ring. When the coupling is introduced, each of the oscillator’s trajectories is on different attractor. We study the changes in the dynamics due to the increase in the coupling coefficient. Studying the phase of the oscillators, we calculate the parameter value for which we obtain the antiphase lag synchronization of the system and also the bifurcation values for which we observe qualitative changes in the dynamics of already synchronized system. We give evidence that lag synchronization is typical for coupled multistable systems

Frequency preference and reliability of signal integration

Die Eigenschaften einzelner Nervenzellen sind von grundlegender Bedeutung für die Verarbeitung von Informationen im Nervensystem. Neuronen antworten auf Eingangsreize durch Veränderung der elektrischen Spannung über die Zellmembran. Die Spannungsantwort wird dabei durch die Dynamik der Ionenkanäle in der Zellmembran bestimmt. In dieser Arbeit untersuche ich anhand von leitfähigkeits-basierten Modellneuronen den Einfluss von Ionenkanälen auf zwei Aspekte der Signalverarbeitung: die Frequenz-Selektivität sowie die Zuverlässigkeit und zeitliche Präzision von Aktionspotentialen. Zunächst werden die zell-intrinsischen Mechanismen identifiziert, welche the Frequenz-Selektivität und die Zuverlässigkeit bestimmen. Weiterhin wird untersucht, wie Ionenkanäle diese Mechanismen modulieren können, um die Integration von Signalen zu optimieren. Im ersten Teil der Arbeit wird demonstriert, dass der Mechanismus der unterschwelligen Resonanz, so wie er bisher für periodische Signale beobachtet wurde, auch auf nicht-periodische Signale anwendbar ist und sich ebenfalls in den Feuerraten niederschlägt. Im zweiten Teil wird gezeigt, dass zeitliche Präzision und Zuverlässigkeit von Aktionspotentialen mit der Stimulusfrequenz variieren und dass, in Abhängigkeit davon, ob das Stimulusmittel über- oder unterhalb der Feuerschwelle liegt, zwei Stimulusregime unterschieden werden müssen. In beiden Regimen existiert eine bevorzugte Stimulusfrequenz, welche durch die Gesamtleitfähigkeit und die Dynamik spezifischer Ionenkanäle moduliert werden kann. Im dritten Teil wird belegt, dass Ionenkanäle die Zuverlässigkeit auch direkt über eine Veränderung der Sensitivität einer Zelle gegenüber neuronalem Rauschen bestimmen können. Die Ergebnisse der Arbeit lassen auf eine wichtige Rolle der dynamischen Regulierung der Ionenkanäle für die Frequenz-Selektivität und die zeitliche Präzision und Zuverlässigkeit der Spannungsantworten schließen.The properties of individual neurons are of fundamental importance for the processing of information in the nervous system. The generation of voltage responses to input signals, in particular, depends on the properties of ion channels in the cell membrane. Within this thesis, I employ conductance-based model neurons to investigate the effect of ionic conductances and their dynamics on two aspects of signal processing: frequency-selectivity and temporal precision and reliability of spikes. First, the cell-intrinsic mechanisms that determine frequency selectivity and spike timing reliability are identified on the basis of conductance-based model neurons. Second, it is analyzed how ionic conductances can serve to modulate these mechanisms in order to optimize signal integration. In the first part, the frequency selectivity of subthreshold response amplitudes previously observed for periodic stimuli is proven to extend to nonperiodic stimuli and to translate into firing rates. In the second part, it is demonstrated that spike timing reliability is frequency-selective and that two different stimulus regimes have to be distinguished, depending on whether the stimulus mean is below or above threshold. In both cases, resonance effects determine the most reliable stimulus frequency. It is shown that this frequency preference can be modulated by the peak conductance and dynamics of specific ion channels. In the third part, evidence is provided that ionic conductances determine spike timing reliability beyond changes in the preferred frequency. It is demonstrated that ionic conductances also exert a direct influence on the sensitivity of the timing of spikes to neuronal noise. The findings suggest an important role for dynamic neuromodulation of ion channels with regard to frequency selectivity and spike timing reliability

Evoked Patterns of Oscillatory Activity in Mean-Field Neuronal Networks

Oscillatory behaviors in populations of neurons are oberved in diverse contexts. In tasks involving working memory, a form of short-term memory, oscillations in different frequency bands have been shown to increase across varying spatial scales using recording methods such as EEG (electroencephalogram) and MEG (magnetoencephalogram). Such oscillatory activity has also been observed in the context of neural binding, where different features of objects that are perceived or recalled are associated with one another. These sets of data suggest that oscillatory dynamics may also play a key role in the maintenance and manipulation of items in working memory. Using similar recording techniques, including EEG and MEG, oscillatory neuronal activity has also been seen to occur when certain images that cause aversion and headaches in healthy human subjects or seizures in those with pattern-sensitive epilepsy are presented. The images most likely to cause such responses are those with dominant spatial frequencies near 3--5 cycles per degree, the same band of wavenumbers to which normal human vision exhibits the greatest contrast sensitivity. We model these oscillatory behaviors using mean-field, Wilson-Cowan-type neuronal networks. In the case of working memory and binding, we find that including the activity of certain long-lasting excitatory synapses in addition to the usual inhibitory and shorter-term excitatory synaptic activity allows for bistability between a low steady state and a high oscillatory state. By coupling several such populations together, both in-phase and out-of-phase oscillations arise, corresponding to distinct and bound items in working memory, respectively. We analyze the network's dynamics and dependence on biophysically relevant parameters using a combination of techniques, including numerical bifurcation analysis and weak coupling theory. In the case of spatially resonant responses to static simtuli, we employ Wilson-Cowan networks extended in one and two spatial dimensions. By placing the networks near Turing-Hopf bifurcations, we find they exhibit spatial resonances that compare well with empirical results. Using simulations, numerical bifurcation analysis, and perturbation theory, we characterize the observed dynamics and gain mathematical insight into the mechanisms that lead to these dynamics

Nonlinear synchrony dynamics of neuronal bursters

We study the appearance of a novel phenomenon for coupled identical bursters: synchronized bursts where there are changes of spike synchrony within each burst. The examples we study are for normal form elliptic bursters where there is a periodic slow passage through a Bautin (codimension two degenerate Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf bifurcation at the onset of repetitive spiking while the end of burst occurs via a fold limit cycle bifurcation. We study synchronization behavior of two Bautin-type elliptic bursters for a linear direct coupling scheme as well as demonstrating its presence in an approximation of gap-junction and synaptic coupling. We also find similar behaviour in system consisted of three and four Bautin-type elliptic bursters. We note that higher order terms in the normal form that do not affect the behavior of a single burster can be responsible for changes in synchrony pattern; more precisely, we find within-burst synchrony changes associated with a turning point in the spontaneous spiking frequency (frequency transition). We also find multiple synchrony changes in similar system by incorporating multiple frequency transitions. To explain the phenomenon we considered a burst-synchronized constrained model and a bifurcation analysis of the this reduced model shows the existence of the observed within-burst synchrony states. Within-burst synchrony change is also found in the system of mutually delaycoupled two Bautin-type elliptic bursters with a constant delay. The similar phenomenon is shown to exist in the mutually-coupled conductance-based Morris-Lecar neuronal system with an additional slow variable generating elliptic bursting. We also find within-burst synchrony change in linearly coupled FitzHugh-Rinzel 2 3 elliptic bursting system where the synchrony change occurs via a period doubling bifurcation. A bifurcation analysis of a burst-synchronized constrained system identifies the periodic doubling bifurcation in this case. We show emergence of spontaneous burst synchrony cluster in the system of three Hindmarsh-Rose square-wave bursters with nonlinear coupling. The system is found to change between the available cluster states depending on the stimulus. Lyapunov exponents of the burst synchrony states are computed from the corresponding variational system to probe the stability of the states. Numerical simulation also shows existence of burst synchrony cluster in the larger network of such system.Exeter Research Scholarship

Modeling phase synchronization of interacting neuronal populations:from phase reductions to collective behavior of oscillatory neural networks

Synchronous, coherent interaction is key for the functioning of our brain. The coordinated interplay between neurons and neural circuits allows to perceive, process and transmit information in the brain. As such, synchronization phenomena occur across all scales. The coordination of oscillatory activity between cortical regions is hypothesized to underlie the concept of phase synchronization. Accordingly, phase models have found their way into neuroscience. The concepts of neural synchrony and oscillations are introduced in Chapter 1 and linked to phase synchronization phenomena in oscillatory neural networks. Chapter 2 provides the necessary mathematical theory upon which a sound phase description builds. I outline phase reduction techniques to distill the phase dynamics from complex oscillatory networks. In Chapter 3 I apply them to networks of weakly coupled Brusselators and of Wilson-Cowan neural masses. Numerical and analytical approaches are compared against each other and their sensitivity to parameter regions and nonlinear coupling schemes is analysed. In Chapters 4 and 5 I investigate synchronization phenomena of complex phase oscillator networks. First, I study the effects of network-network interactions on the macroscopic dynamics when coupling two symmetric populations of phase oscillators. This setup is compared against a single network of oscillators whose frequencies are distributed according to a symmetric bimodal Lorentzian. Subsequently, I extend the applicability of the Ott-Antonsen ansatz to parameterdependent oscillatory systems. This allows for capturing the collective dynamics of coupled oscillators when additional parameters influence the individual dynamics. Chapter 6 draws the line to experimental data. The phase time series of resting state MEG data display large-scale brain activity at the edge of criticality. After reducing neurophysiological phase models from the underlying dynamics of Wilson-Cowan and Freeman neural masses, they are analyzed with respect to two complementary notions of critical dynamics. A general discussion and an outlook of future work are provided in the final Chapter 7

EQUADIFF 15

Equadiff 15 – Conference on Differential Equations and Their Applications – is an international conference in the world famous series Equadiff running since 70 years ago. This booklet contains conference materials related with the 15th Equadiff conference in the Czech and Slovak series, which was held in Brno in July 2022. It includes also a brief history of the East and West branches of Equadiff, abstracts of the plenary and invited talks, a detailed program of the conference, the list of participants, and portraits of four Czech and Slovak outstanding mathematicians

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”