Bifurcation Analysis and Spatiotemporal Patterns of Nonlinear Oscillations in a Ring Lattice of Identical Neurons with Delayed Coupling

We investigate the dynamics of a delayed neural network model consisting of n identical neurons. We first analyze stability of the zero solution and then study the effect of time delay on the dynamics of the system. We also investigate the steady state bifurcations and their stability. The direction and stability of the Hopf bifurcation and the pitchfork bifurcation are analyzed by using the derived normal forms on center manifolds. Then, the spatiotemporal patterns of bifurcating periodic solutions are investigated by using the symmetric bifurcation theory, Lie group theory and S1-equivariant degree theory. Finally, two neural network models with four or seven neurons are used to verify our theoretical results

Clustering behaviour in networks with time delayed all-to-all coupling

Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches. We first use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specfi c examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies

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

Dynamics of neural systems with time delays

Complex networks are ubiquitous in nature. Numerous neurological diseases, such as Alzheimer's, Parkinson's, epilepsy are caused by the abnormal collective behaviour of neurons in the brain. In particular, there is a strong evidence that Parkinson's disease is caused by the synchronisation of neurons, and understanding how and why such synchronisation occurs will bring scientists closer to the design and implementation of appropriate control to support desynchronisation required for the normal functioning of the brain. In order to study the emergence of (de)synchronisation, it is necessary first to understand how the dynamical behaviour of the system under consideration depends on the changes in systems parameters. This can be done using a powerful mathematical method, called bifurcation analysis, which allows one to identify and classify different dynamical regimes, such as, for example, stable/unstable steady states, Hopf and fold bifurcations, and find periodic solutions by varying parameters of the nonlinear system. In real-world systems, interactions between elements do not happen instantaneously due to a finite time of signal propagation, reaction times of individual elements, etc. Moreover, time delays are normally non-constant and may vary with time. This means that it is vital to introduce time delays in any realistic model of neural networks. In this thesis, I consider four different models. First, in order to analyse the fundamental properties of neural networks with time-delayed connections, I consider a system of four coupled nonlinear delay differential equations. This model represents a neural network, where one subsystem receives a delayed input from another subsystem. The exciting feature of this model is the combination of both discrete and distributed time delays, where distributed time delays represent the neural feedback between the two sub-systems, and the discrete delays describe neural interactions within each of the two subsystems. Stability properties are investigated for different commonly used distribution kernels, and the results are compared to the corresponding stability results for networks with no distributed delays. It is shown how approximations to the boundary of stability region of an equilibrium point can be obtained analytically for the cases of delta, uniform, and gamma delay distributions. Numerical techniques are used to investigate stability properties of the fully nonlinear system and confirm our analytical findings. In the second part of this thesis, I consider a globally coupled network composed of active (oscillatory) and inactive (non-oscillatory) oscillators with distributed time delayed coupling. Analytical conditions for the amplitude death, where the oscillations are quenched, are obtained in terms of the coupling strength, the ratio of inactive oscillators, the width of the uniformly distributed delay and the mean time delay for gamma distribution. The results show that for uniform distribution, by increasing both the width of the delay distribution and the ratio of inactive oscillators, the amplitude death region increases in the mean time delay and the coupling strength parameter space. In the case of the gamma distribution kernel, we find the amplitude death region in the space of the ratio of inactive oscillators, the mean time delay for gamma distribution, and the coupling strength for both weak and strong gamma distribution kernels. Furthermore, I analyse a model of the subthalamic nucleus (STN)-globus palidus (GP) network with three different transmission delays. A time-shift transformation reduces the model to a system with two time delays, for which the existence of a unique steady state is established. Conditions for stability of the steady state are derived in terms of system parameters and the time delays. Numerical stability analysis is performed using traceDDE and DDE-BIFTOOL in Matlab to investigate different dynamical regimes in the STN-GP model, and to obtain critical stability boundaries separating stable (healthy) and oscillatory (Parkinsonian-like) neural ring. Direct numerical simulations of the fully nonlinear system are performed to confirm analytical findings, and to illustrate different dynamical behaviours of the system. Finally, I consider a ring of n neurons coupled through the discrete and distributed time delays. I show that the amplitude death occurs in the symmetric (asymmetric) region depending on the even (odd) number of neurons in the ring neural system. Analytical conditions for linear stability of the trivial steady state are represented in a parameter space of the synaptic weight of the self-feedback and the coupling strength between the connected neurons, as well as in the space of the delayed self-feedback and the coupling strength between the neurons. It is shown that both Hopf and steady-state bifurcations may occur when the steady state loses its stability. Stability properties are also investigated for different commonly used distribution kernels, such as delta function and weak gamma distributions. Moreover, the obtained analytical results are confirmed by the numerical simulations of the fully nonlinear system

Role of coupling delay in oscillatory activity in autonomous networks of excitable neurons with dissipation

We study numerically the effects of time delay in networks of delay-coupled excitable FitzHugh Nagumo systems with dissipation. The generation of periodic self-sustained oscillations and its threshold are analyzed depending on the dissipation of a single neuron, the delay time, and random initial conditions. The peculiarities of spatiotemporal dynamics of time-delayed bidirectional ring-structured FitzHugh-Nagumo neuronal systems are investigated in cases of local and nonlocal coupling topology between the nodes, and a first-order nonequilibrium phase transition to synchrony is established. It is shown that the emergence of oscillatory activity in delay-coupled FitzHugh-Nagumo neurons is observed for smaller values of the coupling strength as the dissipation parameter decreases. This can provide the possibility of controlling the spatiotemporal behavior of the considered neuronal networks. The observed effects are quantified by plotting distributions of the maximal Lyapunov exponent and the global order parameter in terms of delay and coupling strength.Comment: 14 pages, 17 figure

Information processing in biological complex systems: a view to bacterial and neural complexity

This thesis is a study of information processing of biological complex systems seen from the perspective of dynamical complexity (the degree of statistical independence of a system as a whole with respect to its components due to its causal structure). In particular, we investigate the influence of signaling functions in cell-to-cell communication in bacterial and neural systems. For each case, we determine the spatial and causal dependencies in the system dynamics from an information-theoretic point of view and we relate it with their physiological capabilities. The main research content is presented into three main chapters. First, we study a previous theoretical work on synchronization, multi-stability, and clustering of a population of coupled synthetic genetic oscillators via quorum sensing. We provide an extensive numerical analysis of the spatio-temporal interactions, and determine conditions in which the causal structure of the system leads to high dynamical complexity in terms of associated metrics. Our results indicate that this complexity is maximally receptive at transitions between dynamical regimes, and maximized for transient multi-cluster oscillations associated with chaotic behaviour. Next, we introduce a model of a neuron-astrocyte network with bidirectional coupling using glutamate-induced calcium signaling. This study is focused on the impact of the astrocyte-mediated potentiation on synaptic transmission. Our findings suggest that the information generated by the joint activity of the population of neurons is irreducible to its independent contribution due to the role of astrocytes. We relate these results with the shared information modulated by the spike synchronization imposed by the bidirectional feedback between neurons and astrocytes. It is shown that the dynamical complexity is maximized when there is a balance between the spike correlation and spontaneous spiking activity. Finally, the previous observations on neuron-glial signaling are extended to a large-scale system with community structure. Here we use a multi-scale approach to account for spatiotemporal features of astrocytic signaling coupled with clusters of neurons. We investigate the interplay of astrocytes and spiking-time-dependent-plasticity at local and global scales in the emergence of complexity and neuronal synchronization. We demonstrate the utility of astrocytes and learning in improving the encoding of external stimuli as well as its ability to favour the integration of information at synaptic timescales to exhibit a high intrinsic causal structure at the system level. Our proposed approach and observations point to potential effects of the astrocytes for sustaining more complex information processing in the neural circuitry

Stochastic neural network dynamics: synchronisation and control

Biological brains exhibit many interesting and complex behaviours. Understanding of the mechanisms behind brain behaviours is critical for continuing advancement in fields of research such as artificial intelligence and medicine. In particular, synchronisation of neuronal firing is associated with both improvements to and degeneration of the brain’s performance; increased synchronisation can lead to enhanced information-processing or neurological disorders such as epilepsy and Parkinson’s disease. As a result, it is desirable to research under which conditions synchronisation arises in neural networks and the possibility of controlling its prevalence. Stochastic ensembles of FitzHugh-Nagumo elements are used to model neural networks for numerical simulations and bifurcation analysis. The FitzHugh-Nagumo model is employed because of its realistic representation of the flow of sodium and potassium ions in addition to its advantageous property of allowing phase plane dynamics to be observed. Network characteristics such as connectivity, configuration and size are explored to determine their influences on global synchronisation generation in their respective systems. Oscillations in the mean-field are used to detect the presence of synchronisation over a range of coupling strength values. To ensure simulation efficiency, coupling strengths between neurons that are identical and fixed with time are investigated initially. Such networks where the interaction strengths are fixed are referred to as homogeneously coupled. The capacity of controlling and altering behaviours produced by homogeneously coupled networks is assessed through the application of weak and strong delayed feedback independently with various time delays. To imitate learning, the coupling strengths later deviate from one another and evolve with time in networks that are referred to as heterogeneously coupled. The intensity of coupling strength fluctuations and the rate at which coupling strengths converge to a desired mean value are studied to determine their impact upon synchronisation performance. The stochastic delay differential equations governing the numerically simulated networks are then converted into a finite set of deterministic cumulant equations by virtue of the Gaussian approximation method. Cumulant equations for maximal and sub-maximal connectivity are used to generate two-parameter bifurcation diagrams on the noise intensity and coupling strength plane, which provides qualitative agreement with numerical simulations. Analysis of artificial brain networks, in respect to biological brain networks, are discussed in light of recent research in sleep theor