Interplay between network topology and synchrony-breaking bifurcation: homogeneous four-cell coupled networks

Complex networks are studied across many fields of science. Much progress has been made on static and statistical features of networks, such as small world and scale-free networks. However, general studies of network dynamics are comparatively rare. Synchrony is one commonly observed dynamical behaviour in complex networks. Synchrony breaking is where a fully synchronised network loses coherence, and breaks up into multiple clusters of self-synchronised sub-networks. Mathematically this can be described as a bifurcation from a fully synchronous state, and in this thesis we investigate the effect of network topology on synchrony-breaking bifurcations. Coupled cell networks represent a collection of individual dynamical systems (termed cells) that interact with each other. Each cell is described by an ordinary differential equation (ODE) or a system of ODEs. Schematically, the architecture of a coupled cell network can be represented by a directed graph with a node for each cell, and edges indicating cell couplings. Regular homogeneous networks are a special case where all the nodes/cells and edges are of the same type, and every node has the same number of input edges, which we call the valency of the network. Classes of homogeneous regular networks can be counted using an existing group theoretic enumeration formula, and this formula is extended here to enumerate networks with more generalised structures. However, this does not generate the networks themselves. We therefore develop a computer algorithm to display all connected regular homogeneous networks with less than six cells and analysed synchrony-breaking bifurcations for four-cell regular homogeneous networks. Robust patterns of synchrony (invariant synchronised subspaces under all admissible vector fields) describe how cells are divided into multiple synchronised clusters, and their existence is solely determined by the network topology. These robust patterns of synchrony have a hierarchical relationship, and can be treated as a partially ordered set, and expressed as a lattice. For each robust pattern of synchrony (or lattice point) we can reduce the original network to a smaller network, called a quotient network, by representing each cluster as a single combined node. Therefore, the lattice for a given regular homogeneous network provides robust patterns of synchrony and corresponding quotient networks. Some lattice structures allow a synchrony breaking bifurcation analysis based solely on the dynamics of the quotient networks, which are lifted to the original network using the robust patterns of synchrony. However, in other cases the lattice structure also tells us of the existence and location of additional synchrony-breaking bifurcating branches not seen in the quotient networks. In conclusion the work undertaken here shows that the invariant synchronised subspaces that arise from a network topology facilitate the classification of synchrony-breaking bifurcations of networks

Synchrony and Elementary Operations on Coupled Cell Networks

Given a finite graph (network), let every node (cell) represent an individual dynamics given by a system of ordinary differential equations, and every arrow (edge) encode the dynamical influence of the tail node on the head node. We have then defined a coupled cell system that is associated with the given network structure. Subspaces that are defined by equalities of cell coordinates and left invariant under every coupled cell system respecting the network structure are called synchrony subspaces. They are completely determined by the network structure and form a complete lattice under set inclusions. We analyze the transition of the lattice of synchrony subspaces of a network that is caused by structural changes in the network topology, such as deletion and addition of cells or edges, and rewirings of edges. We give sufficient, and in some cases both sufficient and necessary, conditions under which lattice elements persist or disappear

States and sequences of paired subspace ideals and their relationship to patterned brain function

It is found here that the state of a network of coupled ordinary differential equations is partially localizable through a pair of contractive ideal subspaces, chosen from dual complete lattices related to the synchrony and synchronization of cells within the network. The first lattice is comprised of polydiagonal subspaces, corresponding to synchronous activity patterns that arise from functional equivalences of cell receptive fields. This lattice is dual to a transdiagonal subspace lattice ordering subspaces transverse to these network-compatible synchronies. Combinatorial consideration of contracting polydiagonal and transdiagonal subspace pairs yields a rich array of dynamical possibilities for structured networks. After proving that contraction commutes with the lattice ordering, it is shown that subpopulations of cells are left at fixed potentials when pairs of contracting subspaces span the cells' local coordinates - a phenomenon named glyph formation here. Treatment of mappings between paired states then leads to a theory of network-compatible sequence generation. The theory's utility is illustrated with examples ranging from the construction of a minimal circuit for encoding a simple phoneme to a model of the primary visual cortex including high-dimensional environmental inputs, laminar speficicity, spiking discontinuities, and time delays. In this model, glyph formation and dissolution provide one account for an unexplained anomaly in electroencephalographic recordings under periodic flicker, where stimulus frequencies differing by as little as 1 Hz generate responses varying by an order of magnitude in alpha-band spectral power. Further links between coupled-cell systems and neural dynamics are drawn through a review of synchronization in the brain and its relationship to aggregate observables, focusing again on electroencephalography. Given previous theoretical work relating the geometry of visual hallucinations to symmetries in visual cortex, periodic perturbation of the visual system along a putative symmetry axis is hypothesized to lead to a greater concentration of harmonic spectral energy than asymmetric perturbations; preliminary experimental evidence affirms this hypothesis. To conclude, connections drawn between dynamics, sensation, and behavior are distilled to seven hypotheses, and the potential medical uses of the theory are illustrated with a lattice depiction of ketamine xylazine anaesthesia and a reinterpretation of hemifield neglect

Interplay between network topology and synchrony-breaking bifurcation : homogeneous four-cell coupled networks

Complex networks are studied across many fields of science. Much progress has been made on static and statistical features of networks, such as small world and scale-free networks. However, general studies of network dynamics are comparatively rare. Synchrony is one commonly observed dynamical behaviour in complex networks. Synchrony breaking is where a fully synchronised network loses coherence, and breaks up into multiple clusters of self-synchronised sub-networks. Mathematically this can be described as a bifurcation from a fully synchronous state, and in this thesis we investigate the effect of network topology on synchrony-breaking bifurcations. Coupled cell networks represent a collection of individual dynamical systems (termed cells) that interact with each other. Each cell is described by an ordinary differential equation (ODE) or a system of ODEs. Schematically, the architecture of a coupled cell network can be represented by a directed graph with a node for each cell, and edges indicating cell couplings. Regular homogeneous networks are a special case where all the nodes/cells and edges are of the same type, and every node has the same number of input edges, which we call the valency of the network. Classes of homogeneous regular networks can be counted using an existing group theoretic enumeration formula, and this formula is extended here to enumerate networks with more generalised structures. However, this does not generate the networks themselves. We therefore develop a computer algorithm to display all connected regular homogeneous networks with less than six cells and analysed synchrony-breaking bifurcations for four-cell regular homogeneous networks. Robust patterns of synchrony (invariant synchronised subspaces under all admissible vector fields) describe how cells are divided into multiple synchronised clusters, and their existence is solely determined by the network topology. These robust patterns of synchrony have a hierarchical relationship, and can be treated as a partially ordered set, and expressed as a lattice. For each robust pattern of synchrony (or lattice point) we can reduce the original network to a smaller network, called a quotient network, by representing each cluster as a single combined node. Therefore, the lattice for a given regular homogeneous network provides robust patterns of synchrony and corresponding quotient networks. Some lattice structures allow a synchrony breaking bifurcation analysis based solely on the dynamics of the quotient networks, which are lifted to the original network using the robust patterns of synchrony. However, in other cases the lattice structure also tells us of the existence and location of additional synchrony-breaking bifurcating branches not seen in the quotient networks. In conclusion the work undertaken here shows that the invariant synchronised subspaces that arise from a network topology facilitate the classification of synchrony-breaking bifurcations of networks.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Two-particle bosonic-fermionic quantum walk via 3D integrated photonics

Quantum walk represents one of the most promising resources for the simulation of physical quantum systems, and has also emerged as an alternative to the standard circuit model for quantum computing. Up to now the experimental implementations have been restricted to single particle quantum walk, while very recently the quantum walks of two identical photons have been reported. Here, for the first time, we investigate how the particle statistics, either bosonic or fermionic, influences a two-particle discrete quantum walk. Such experiment has been realized by adopting two-photon entangled states and integrated photonic circuits. The polarization entanglement was exploited to simulate the bunching-antibunching feature of non interacting bosons and fermions. To this scope a novel three-dimensional geometry for the waveguide circuit is introduced, which allows accurate polarization independent behaviour, maintaining a remarkable control on both phase and balancement.Comment: 4 pages, 5 figures + supplementary informatio

Synchrony patterns in Laplacian networks

A network of coupled dynamical systems is represented by a graph whose vertices represent individual cells and whose edges represent couplings between cells. Motivated by the impact of synchronization results of the Kuramoto networks, we introduce the generalized class of Laplacian networks, governed by mappings whose Jacobian at any point is a symmetric matrix with row entries summing to zero. By recognizing this matrix with a weighted Laplacian of the associated graph, we derive the optimal estimates of its positive, null and negative eigenvalues directly from the graph topology. Furthermore, we provide a characterization of the mappings that define Laplacian networks. Lastly, we discuss stability of equilibria inside synchrony subspaces for two types of Laplacian network on a ring with some extra couplings

Rigid patterns of synchrony for equilibria and periodic cycles in network dynamics

We survey general results relating patterns of synchrony to network topology, applying the formalism of coupled cell systems. We also discuss patterns of phase-locking for periodic states, where cells have identical waveforms but regularly spaced phases. We focus on rigid patterns, which are not changed by small perturbations of the differential equation. Symmetry is one mechanism that creates patterns of synchrony and phase-locking. In general networks, there is another: balanced colorings of the cells. A symmetric network may have anomalous patterns of synchrony and phase-locking that are not consequences of symmetry. We introduce basic notions on coupled cell networks and their associated systems of admissible differential equations. Periodic states also possess spatio-temporal symmetries, leading to phase relations; these are classified by the H/K theorem and its analog for general networks. Systematic general methods for computing the stability of synchronous states exist for symmetric networks, but stability in general networks requires methods adapted to special classes of model equations