Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback

International audienceMotivated by the recent development of Deep Brain Stimulation (DBS) for neurological diseases, we study a network of interconnected oscillators under the influence of mean-field feedback and analyze the robustness of its phase-locking with respect to general inputs. Under standard assumptions, this system can be reduced to a modified version of the Kuramoto model of coupled nonlinear oscillators. In the first part of the paper we present an analytical study on the existence of phase-locked solutions under generic interconnection and feedback configurations. In particular we show that, in general, no oscillating phaselocked solutions can co-exist with any non-zero proportional mean-field feedback. In the second part we prove some robustness properties of phase-locked solutions (namely total stability). Thisgeneral result allows in particular to justify the persistence of practically phase-locked states if sufficiently small feedback gains are applied, and to give explicit necessary conditions on the intensity of a desynchronizing mean-field feedback. Furthermore, the Lyapunov function used in the analysis provides a new characterization of the robust phase-locked configurations in the Kuramoto system with symmetric interconnections

In Synch but Not in Step: Circadian Clock Circuits Regulating Plasticity in Daily Rhythms

The suprachiasmatic nucleus (SCN) is a network of neural oscillators that program daily rhythms in mammalian behavior and physiology. Over the last decade much has been learned about how SCN clock neurons coordinate together in time and space to form a cohesive population. Despite this insight, much remains unknown about how SCN neurons communicate with one another to produce emergent properties of the network. Here we review the current understanding of communication among SCN clock cells and highlight a collection of formal assays where changes in SCN interactions provide for plasticity in the waveform of circadian rhythms in behavior. Future studies that pair analytical behavioral assays with modern neuroscience techniques have the potential to provide deeper insight into SCN circuit mechanisms

Adaptive Parameter Selection for Deep Brain Stimulation in Parkinson’s Disease

Each year, around 60,000 people are diagnosed with Parkinson’s disease (PD) and the economic burden of PD is at least $14.4 billion a year in the United States. Pharmaceutical costs for a Parkinson’s patient can be reduced from$12,000 to $6,000 per year with the addition of neuromodulation therapies such as Deep Brain Stimulation (DBS), transcranial Direct Current Stimulation (tDCS), Transcranial Magnetic Stimulation (TMS), etc. In neurodegenerative disorders such as PD, deep brain stimulation (DBS) is a desirable approach when the medication is less effective for treating the symptoms. DBS incorporates transferring electrical pulses to a specific tissue of the central nervous system and obtaining therapeutic results by modulating the neuronal activity of that region. The hyperkinetic symptoms of PD are associated with the ensembles of interacting oscillators that cause excess or abnormal synchronous behavior within the Basal Ganglia (BG) circuitry. Delayed feedback stimulation is a closed loop technique shown to suppress this synchronous oscillatory activity. Deep Brain Stimulation via delayed feedback is known to destabilize the complex intermittent synchronous states. Computational models of the BG network are often introduced to investigate the effect of delayed feedback high frequency stimulation on partially synchronized dynamics. In this work, we developed several computational models of four interacting nuclei of the BG as well as considering the Thalamo-Cortical local effects on the oscillatory dynamics. These models are able to capture the emergence of 34 Hz beta band oscillations seen in the Local Field Potential (LFP) recordings of the PD state. Traditional High Frequency Stimulations (HFS) has shown deficiencies such as strengthening the synchronization in case of highly fluctuating neuronal activities, increasing the energy consumed as well as the incapability of activating all neurons in a large-scale network. To overcome these drawbacks, we investigated the effects of the stimulation waveform and interphase delays on the overall efficiency and efficacy of DBS. We also propose a new feedback control variable based on the filtered and linearly delayed LFP recordings. The proposed control variable is then used to modulate the frequency of the stimulation signal rather than its amplitude. In strongly coupled networks, oscillations reoccur as soon as the amplitude of the stimulus signal declines. Therefore, we show that maintaining a fixed amplitude and modulating the frequency might ameliorate the desynchronization process, increase the battery lifespan and activate substantial regions of the administered DBS electrode. The charge balanced stimulus pulse itself is embedded with a delay period between its charges to grant robust desynchronization with lower amplitude needed. The efficiency and efficacy of the proposed Frequency Adjustment Stimulation (FAS) protocol in a delayed feedback method might contribute to further investigation of DBS modulations aspired to address a wide range of abnormal oscillatory behaviors observed in neurological disorders. Adaptive stimulation can open doors towards simultaneous stimulation with MRI recordings. We additionally propose a new pipeline to investigate the effect of Transcranial Magnetic Stimulation (TMS) on patient specific models. The pipeline allows us to generate a full head segmentation based on each individual MRI data. In the next step, the neurosurgeon can adaptively choose the proper location of stimulation and transmit accurate magnetic field with this pipeline

Adaptive dynamical networks

It is a fundamental challenge to understand how the function of a network is related to its structural organization. Adaptive dynamical networks represent a broad class of systems that can change their connectivity over time depending on their dynamical state. The most important feature of such systems is that their function depends on their structure and vice versa. While the properties of static networks have been extensively investigated in the past, the study of adaptive networks is much more challenging. Moreover, adaptive dynamical networks are of tremendous importance for various application fields, in particular, for the models for neuronal synaptic plasticity, adaptive networks in chemical, epidemic, biological, transport, and social systems, to name a few. In this review, we provide a detailed description of adaptive dynamical networks, show their applications in various areas of research, highlight their dynamical features and describe the arising dynamical phenomena, and give an overview of the available mathematical methods developed for understanding adaptive dynamical networks

Synchrony and bifurcations in coupled dynamical systems and effects of time delay

Dynamik auf Netzwerken ist ein mathematisches Feld, das in den letzten Jahrzehnten schnell gewachsen ist und Anwendungen in zahlreichen Disziplinen wie z.B. Physik, Biologie und Soziologie findet. Die Funktion vieler Netzwerke hängt von der Fähigkeit ab, die Elemente des Netzwerkes zu synchronisieren. Mit anderen Worten, die Existenz und die transversale Stabilität der synchronen Mannigfaltigkeit sind zentrale Eigenschaften. Erst seit einigen Jahren wird versucht, den verwickelten Zusammenhang zwischen der Kopplungsstruktur und den Stabilitätseigenschaften synchroner Zustände zu verstehen. Genau das ist das zentrale Thema dieser Arbeit. Zunächst präsentiere ich erste Ergebnisse zur Klassifizierung der Kanten eines gerichteten Netzwerks bezüglich ihrer Bedeutung für die Stabilität des synchronen Zustands. Folgend untersuche ich ein komplexes Verzweigungsszenario in einem gerichteten Ring von Stuart-Landau Oszillatoren und zeige, dass das Szenario persistent ist, wenn dem Netzwerk eine schwach gewichtete Kante hinzugefügt wird. Daraufhin untersuche ich synchrone Zustände in Ringen von Phasenoszillatoren die mit Zeitverzögerung gekoppelt sind. Ich bespreche die Koexistenz synchroner Lösungen und analysiere deren Stabilität und Verzweigungen. Weiter zeige ich, dass eine Zeitverschiebung genutzt werden kann, um Muster im Ring zu speichern und wiederzuerkennen. Diese Zeitverschiebung untersuche ich daraufhin für beliebige Kopplungsstrukturen. Ich zeige, dass invariante Mannigfaltigkeiten des Flusses sowie ihre Stabilität unter der Zeitverschiebung erhalten bleiben. Darüber hinaus bestimme ich die minimale Anzahl von Zeitverzögerungen, die gebraucht werden, um das System äquivalent zu beschreiben. Schließlich untersuche ich das auffällige Phänomen eines nichtstetigen Übergangs zu Synchronizität in Klassen großer Zufallsnetzwerke indem ich einen kürzlich eingeführten Zugang zur Beschreibung großer Zufallsnetzwerke auf den Fall zeitverzögerter Kopplungen verallgemeinere.Since a couple of decades, dynamics on networks is a rapidly growing branch of mathematics with applications in various disciplines such as physics, biology or sociology. The functioning of many networks heavily relies on the ability to synchronize the network’s nodes. More precisely, the existence and the transverse stability of the synchronous manifold are essential properties. It was only in the last few years that people tried to understand the entangled relation between the coupling structure of a network, given by a (di-)graph, and the stability properties of synchronous states. This is the central theme of this dissertation. I first present results towards a classification of the links in a directed, diffusive network according to their impact on the stability of synchronization. Then I investigate a complex bifurcation scenario observed in a directed ring of Stuart-Landau oscillators. I show that under the addition of a single weak link, this scenario is persistent. Subsequently, I investigate synchronous patterns in a directed ring of phase oscillators coupled with time delay. I discuss the coexistence of multiple of synchronous solutions and investigate their stability and bifurcations. I apply these results by showing that a certain time-shift transformation can be used in order to employ the ring as a pattern recognition device. Next, I investigate the same time-shift transformation for arbitrary coupling structures in a very general setting. I show that invariant manifolds of the flow together with their stability properties are conserved under the time-shift transformation. Furthermore, I determine the minimal number of delays needed to equivalently describe the system’s dynamics. Finally, I investigate a peculiar phenomenon of non-continuous transition to synchrony observed in certain classes of large random networks, generalizing a recently introduced approach for the description of large random networks to the case of delayed couplings

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.Comment: Final version published in Physics Reports. More information available at http://synchronets.googlepages.com

Modelos matemáticos para lesões em redes neurais com padrões complexos de conectividade

Orientador: Prof. Dr. Ricardo Luiz VianaTese (Doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Física. Defesa: Curitiba, 18/09/2015Inclui referências : f. 139-146Resumo: O cérebro contem cerca de cem bilhões de neurônios que se conectam através de um padrão complexo de conectividade, que opera no sentido de otimizar o processo de transmissão de informação. Os neurônios, ao estabelecerem uma conexão entre si podem muitas vezes exibir sincronização. A presença de traumas e doenças degenerativas em regiões específicas do cérebro podem, através de efeitos locais, danificar o funcionamento cerebral como um todo. O propósito deste trabalho é tentar responder a questões do tipo: em uma rede neural, quais são as formas de lesões que causam maior impacto na dinâmica da rede? É possível identificar um tipo de lesão a partir do seu efeito? Existem topologias de rede que são mais robustas à lesões? Neste sentido, analisamos a sincronização de fase para uma rede de neurônios com diferentes topologias de rede, comparamos os resultados com um modelo de osciladores e analisamos diferentes tipos de lesões. Nossos resultados apontam que para o estudo de sincronização de fase, os neurônios podem ser considerados como osciladores, porém, o comportamento das frequências no estado sincronizado em redes neurais, em geral, não é similar ao comportamento de osciladores. No estudo das lesões, do ponto de vista dinâmico, para cada tipo de rede existe um comportamento distinto aos diferentes tipos de lesões. Entre neurônios globalmente acoplados, é possível distinguir a partir da dinâmica global se a lesão destrói apenas as conexões ou destrói os neurônios. Em redes complexas, o efeito das lesões é maior quando a lesão afeta os neurônios mais conectados ou com maior centralidade de intermediação. Em redes de pequeno mundo, a diferença entre os tipos de lesão é perceptível, porém, mais sutil do que para redes aleatórias e sem escala.Abstract: The brain is composed of around one hundread billion of neurons connected through synapses forming a complex pattern of connectivity. This complex connectivity is responsible to optimize the information process. When neurons are connected among themselves they can exhibit synchronization. The presence of traumas and neurodegenerative diseases in some brain areas causes not only local effects, but in the whole brain. The purpose of this work is to answer questions like: which are the type of lesions with bigger dynamical effects in the neural network? Is it possible to identify a type of lesion just looking at its dynamical effects in the network? Are there topologies against lesions which are more robust than others? In this sense, we analyse phase synchronization in a neural network with different network topologies. We compare the obtained results with a model of phase oscillators and we analysed different types of lesions. Our results show that neuronal phase synchronization is similar to phase synchronization in oscillators, however, frequency synchronization usually is different in both models. Related to lesions, from the dynamical point of view, for each type of network there is a distinct behavior for each type of lesion. Among globally coupled neurons, it is possible to dynamically distinguish when the lesion either disrupt or destroy the neurons. For complex networks, the most effective lesions are those that affects the most connected neurons or those with the largest betweenness. For small-world networks, the difference among types of lesions are distinguishable, though, they are subtle in comparison with random and scale-free networks

Perspectives on adaptive dynamical systems

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems like the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems. We reflect on the notion and terminology of adaptivity in different disciplines and discuss which role adaptivity plays for various fields. We highlight common open challenges, and give perspectives on future research directions, looking to inspire interdisciplinary approaches.Comment: 46 pages, 9 figure