Survivability of Deterministic Dynamical Systems

The notion of a part of phase space containing desired (or allowed) states of a dynamical system is important in a wide range of complex systems research. It has been called the safe operating space, the viability kernel or the sunny region. In this paper we define the notion of survivability: Given a random initial condition, what is the likelihood that the transient behaviour of a deterministic system does not leave a region of desirable states. We demonstrate the utility of this novel stability measure by considering models from climate science, neuronal networks and power grids. We also show that a semi-analytic lower bound for the survivability of linear systems allows a numerically very efficient survivability analysis in realistic models of power grids. Our numerical and semi-analytic work underlines that the type of stability measured by survivability is not captured by common asymptotic stability measures.Comment: 21 pages, 6 figure

Dynamic Effective Connectivity of Inter-Areal Brain Circuits

Anatomic connections between brain areas affect information flow between neuronal circuits and the synchronization of neuronal activity. However, such structural connectivity does not coincide with effective connectivity (or, more precisely, causal connectivity), related to the elusive question “Which areas cause the present activity of which others?”. Effective connectivity is directed and depends flexibly on contexts and tasks. Here we show that dynamic effective connectivity can emerge from transitions in the collective organization of coherent neural activity. Integrating simulation and semi-analytic approaches, we study mesoscale network motifs of interacting cortical areas, modeled as large random networks of spiking neurons or as simple rate units. Through a causal analysis of time-series of model neural activity, we show that different dynamical states generated by a same structural connectivity motif correspond to distinct effective connectivity motifs. Such effective motifs can display a dominant directionality, due to spontaneous symmetry breaking and effective entrainment between local brain rhythms, although all connections in the considered structural motifs are reciprocal. We show then that transitions between effective connectivity configurations (like, for instance, reversal in the direction of inter-areal interactions) can be triggered reliably by brief perturbation inputs, properly timed with respect to an ongoing local oscillation, without the need for plastic synaptic changes. Finally, we analyze how the information encoded in spiking patterns of a local neuronal population is propagated across a fixed structural connectivity motif, demonstrating that changes in the active effective connectivity regulate both the efficiency and the directionality of information transfer. Previous studies stressed the role played by coherent oscillations in establishing efficient communication between distant areas. Going beyond these early proposals, we advance here that dynamic interactions between brain rhythms provide as well the basis for the self-organized control of this “communication-through-coherence”, making thus possible a fast “on-demand” reconfiguration of global information routing modalities

Synchronisation in dynamically coupled maps

The central aim of this thesis is to better understand the dynamics of a set of dynamically coupled map systems previously introduced by Ito & Kaneko in a series of papers (Phys. Rev. Lett. 88 (2002), no. 2, 028701 and Phys. Rev. E 67 (2003), no. 4, 046226). The current work extends Ito & Kaneko’s studies to clarify the changes in macrodynamics induced by the differences in microdynamics between the two systems. A third system is also introduced that has a minor change to the microdynamics from nonlinear to linear output function in the externally coupled system. The dynamics of these three dynamically-coupled maps is also compared with their simplified systems with static coupling. The previous studies of these dynamically-coupled maps showed a partitioning of the parameter space into regions of different macrodynamics. Here, an in-depth study is presented of the behaviour of the systems as they cross the boundary between one region and another. The behaviour across this boundary is shown to be much more complicated than suggested in the previous studies. These three systems of dynamically-coupled maps all differ in the form of their microscopic couplings, yet two of the systems are shown to produce similar macrodynamics, whereas the third differs dramatically by almost any measure of the macrodynamics. The time it takes for the systems to synchronise, both the dynamically-coupled and static-coupled systems, is investigated. It is shown that the introduction of dynamicalcouplings stops the systems from synchronising quasi-instantaneously. Details of potential consequences of this in the field of neuroscience are discussed. A brief study of the effect of driving the systems with external stimuli is presented. The different microscopic coupling forms cause different responses to the external stimuli. Some of the responses are similar to that observed by the visual cortex area of the brain

Spatio-temporal modelling and analysis of epileptiform EEG

In this thesis we investigate the mechanisms underlying the generation of abnormal EEG rhythms in epilepsy, which is a crucial step towards better treatment of this disorder in the future. To this end, macroscopic scale mathematical models of the interactions between neuronal populations are examined. In particular, the role of interactions between neural masses that are spatially distributed in cortical networks are explored. In addition, two other important aspects of the modelling process are addressed, namely the conversion of macroscopic model variables into EEG output and the comparison of multivariate, spatio-temporal data. For the latter, we adopt a vectorisation of the correlation matrix of windowed data and subsequent comparison of data by vector distance measures. Our modelling studies indicate that excitatory connectivity between neural masses facilitates self-organised dynamics. In particular, we report for the first time the production of complex rhythmic transients and the generation of intermittent periods of 'abnormal' rhythmic activity in two different models of epileptogenic tissue. These models therefore provide novel accounts of the spontaneous, intermittent transition between normal and pathological rhythms in primarily generalised epilepsies and the evocation of complex, self-terminating, spatio-temporal dynamics by brief stimulation in focal epilepsies. Two key properties of these models are excitability at the macroscopic level and the presence of spatial heterogeneities. The identification of neural mass excitability as an important processes in spatially extended brain networks is a step towards uncovering the multi-scale nature of the pathological mechanisms of epilepsy. A direct consequence of this work is therefore that novel experimental investigations are proposed, which in itself is a validation of our modelling approach. In addition, new considerations regarding the nature of dynamical systems as applied to problems of transitions between rhythmic states are proposed and will prompt future investigations of complex transients in spatio-temporal excitable systems.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Modelling Structure and Dynamics of Complex Systems: Applications to Neuronal Networks

Complex systems theory is a mathematical framework for studying interconnected dynamical objects. Usually these objects themselves are by construction simple, and their temporal behavior in isolation is easily predictable, but the way they are interconnected into a network allows emergence of complex, non-obvious phenomena. The emergent phenomena and their stability are dependent on both the intrinsic dynamics of the objects, the types of interactions between the objects, and the connectivity patterns between the objects. This work focuses on the third aspect, i.e., the structure of the network, although the other two aspects are inherently present in the study as well. Tools from graph theory are applied to generate and analyze the network structure, and the effect of the structure on the network dynamics is analyzed by various methods. The objects of interest are biological and physical systems, and special attention is given to spiking neuronal networks, i.e., networks of nerve cells that communicate by transmitting and receiving action potentials. In this thesis, methods for modelling spiking neuronal networks are introduced. Different point neuron models, including the integrate-and-fire model, are presented and applied to study the collective behaviour of the neurons. Special focus is placed on the emergence of network bursts, i.e., short periods of network-wide high-frequency firing. The occurrence of this behaviour is stable in certain regimes of connection strengths. This work shows that the network bursting is found to be more frequent in locally connected networks than in non-local networks, such as randomly connected networks. To gain a deeper insight, the aspects of structure that promote the bursting behaviour are analyzed by graph-theoretic means. The clustering coefficient and the maximal eigenvalue of the connectivity matrix are found the most important measures of structure in this matter, both expressing their relevance under different structural conditions. A range of different network structures are applied to confirm this result. A special class of connectivity is studied in more detail, namely, the connectivity patterns produced by simulations of growing and interconnecting neurons placed on a 2-dimensional array. Two simulators of growth are applied for this purpose. In addition, a more abstract class of dynamical systems, the Boolean networks, are considered. These systems were originally introduced as a model for genetic regulatory networks, but have thereafter been extensively used for more general studies of complex systems. In this work, measures of information diversity and complexity are applied to several types of systems that obey Boolean dynamics. The random Boolean networks are shown to possess high temporal complexity prior to reaching an attractor. Similarly, high values of complexity are found at a transition stage of another dynamical system, the lattice gas automaton, which can be formulated using the Boolean network framework as well. The temporal maximization of the complexity near the transitions between different dynamical regimes could therefore be a more general phenomenon in complex networks. The applicability of the information-theoretic framework is also confirmed in a study of bursting neuronal networks, where different types of networks are shown to be separable by the intrinsic information distance distributions they produce. The connectivities of the networks studied in this thesis are analyzed using graph-theoretic tools. The graph theory provides a mathematical framework for studying the structure of complex systems and how it affects the system dynamics. In the studies of the nervous system, detailed maps on the connections between neurons have been collected, although such data are yet scarce and laborious to obtain experimentally. This work shows which aspects of the structure are relevant for the dynamics of spontaneously bursting neuronal networks. Such information could be useful in directing the experiments to measure only the relevant aspects of the structure instead of assessing the whole connectome. In addition, the framework of generating the network structure by animating the growth of the neurons, as presented in this thesis, could serve in simulations of the nervous system as a reliable alternative to importing the experimentally obtained connectome

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

Causality and synchronisation in complex systems with applications to neuroscience

This thesis presents an investigation, of synchronisation and causality, motivated by problems in computational neuroscience. The thesis addresses both theoretical and practical signal processing issues regarding the estimation of interdependence from a set of multivariate data generated by a complex underlying dynamical system. This topic is driven by a series of problems in neuroscience, which represents the principal background motive behind the material in this work. The underlying system is the human brain and the generative process of the data is based on modern electromagnetic neuroimaging methods . In this thesis, the underlying functional of the brain mechanisms are derived from the recent mathematical formalism of dynamical systems in complex networks. This is justified principally on the grounds of the complex hierarchical and multiscale nature of the brain and it offers new methods of analysis to model its emergent phenomena. A fundamental approach to study the neural activity is to investigate the connectivity pattern developed by the brain’s complex network. Three types of connectivity are important to study: 1) anatomical connectivity refering to the physical links forming the topology of the brain network; 2) effective connectivity concerning with the way the neural elements communicate with each other using the brain’s anatomical structure, through phenomena of synchronisation and information transfer; 3) functional connectivity, presenting an epistemic concept which alludes to the interdependence between data measured from the brain network. The main contribution of this thesis is to present, apply and discuss novel algorithms of functional connectivities, which are designed to extract different specific aspects of interaction between the underlying generators of the data. Firstly, a univariate statistic is developed to allow for indirect assessment of synchronisation in the local network from a single time series. This approach is useful in inferring the coupling as in a local cortical area as observed by a single measurement electrode. Secondly, different existing methods of phase synchronisation are considered from the perspective of experimental data analysis and inference of coupling from observed data. These methods are designed to address the estimation of medium to long range connectivity and their differences are particularly relevant in the context of volume conduction, that is known to produce spurious detections of connectivity. Finally, an asymmetric temporal metric is introduced in order to detect the direction of the coupling between different regions of the brain. The method developed in this thesis is based on a machine learning extensions of the well known concept of Granger causality. The thesis discussion is developed alongside examples of synthetic and experimental real data. The synthetic data are simulations of complex dynamical systems with the intention to mimic the behaviour of simple cortical neural assemblies. They are helpful to test the techniques developed in this thesis. The real datasets are provided to illustrate the problem of brain connectivity in the case of important neurological disorders such as Epilepsy and Parkinson’s disease. The methods of functional connectivity in this thesis are applied to intracranial EEG recordings in order to extract features, which characterize underlying spatiotemporal dynamics before during and after an epileptic seizure and predict seizure location and onset prior to conventional electrographic signs. The methodology is also applied to a MEG dataset containing healthy, Parkinson’s and dementia subjects with the scope of distinguishing patterns of pathological from physiological connectivity