Yi Ming Lai. Rachel Nicks. Pedro Soares. Dynamical Systems with Noise from Renewal Processes. Synchrony Branching Lemma for Regular Networks

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1 Yi Ming Lai Dynamical Systems with Noise from Renewal Processes One important area in the area of dynamics on networks is the study of temporal patterns of edges, for example, when edges appear or disappear due to a time-dependent or stochastic process. While there has been much progress using the assumption that such stochasticity is Poisson in nature, many systems deviate from this by being non-stationary or even non-markovian. There has been recent work studying such networks using a generalization of the Montroll-Weiss equation (which was originally used in the context of anomalous diffusion). By adapting this formalism, we have been able to find a new way to study stochastic dynamical systems subject to noise from an arbitrary renewal process. We draw examples from neuronal population modelling and queuing theory. Rachel Nicks Clusters in nonsmooth oscillator networks For coupled oscillator networks with Laplacian coupling the Master Stability Function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory this approach has recently been extended to treat more general cluster states. However, the MSF and its generalisations require the determination of a set of Floquet multipliers from variational equations obtained by linearisation around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by the framework is often explored using numerical techniques. Here we show that further insight into network dynamics can be obtained by focusing on piece-wise linear (pwl) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of saltation matrices. By augmenting the variational approach for studying smooth dynamical system with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration we analyse an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled networks built from planar pwl nodes. We use these examples to emphasise that the stability of network cluster states depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Pedro Soares Synchrony Branching Lemma for Regular Networks Coupled cell systems are dynamical systems associated to a network and synchrony subspaces, given by balanced colorings of the network, are invariant subspaces for every coupled cell systems associated to that network. Golubitsky and Lauterbach (SIAM J. Applied Dynamical Systems, 8 (1) 2009, 40-75) prove an analogue of the Equivariant Branching Lemma in the context of regular

2 networks. We generalize this result proving the generic existence of steady-state bifurcation branches for regular networks with maximal synchrony. We also give necessary and sufficient conditions for the existence of steady-state bifurcation branches with some submaximal synchrony. Those conditions only depend on the network structure, but the lattice structure of the balanced colorings is not sufficient to decide which synchrony subspaces support a steady-state bifurcation branch. Manuela Aguiar The lattice of synchrony subspaces for networks with asymmetric inputs One of the main aims in the study of coupled cell systems is to understand the dynamics forced by the associated network structure. A relevant example are the synchrony subspaces for a network - subspaces defined by equalities of cell coordinates and that are flow-invariant by all the coupled cell systems that are compatible with the network structure. The existence of synchrony subspaces can have a strong impact on the dynamics, for example, in terms of bifurcation and robust heteroclinic phenomena. As proved by Ian Stewart, the set of synchrony subspaces for a coupled cell network is a complete lattice. We show how to obtain that lattice for the particular case of coupled cell networks with asymmetric inputs - each cell receives at most one input edge of each type. This involves, in particular, results about the lattice of synchrony subspaces for the disjoint union of networks and for 1-input networks. Peter Ashwin Noisy network attractors: design and computational properties Networks in physical space can give rise to networks (of different structures) in phase space in a robust and attracting manner, although it is not always easy to see how which networks in phase space may arise from a given physical network. This talk will look at an approach to "reverse engineer" or design physical networks that have explicit network attractors, and to explore their computation properties. In doing so we are able to embed finite state models for computational systems (Turing machines) into the dynamics of networks with noise and non-autonomous inputs. Joint work with Claire Postlethwaite (Auckland). Stephen Coombes Networks of Nonsmooth Neural Oscillators "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 [1]. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling. To gain insight into the behaviour of neural networks when phase-oscillator descriptions are not appropriate we turn instead to the study

3 of tractable piece-wise linear (pwl) systems. There has been an appreciation for some time in the applied sciences, and particularly in electrical engineering, of the benefits of studying caricatures of complex systems built from pwl and possibly discontinuous dynamical systems. Although a beautifully simplistic modelling perspective the necessary loss of smoothness precludes the use of many results from the standard toolkit of smooth dynamical systems, and one must be careful to correctly determine conditions for existence, uniqueness and stability of solutions. In this talk I will describe a variety of pwl neural oscillators, and show how to analyse periodic orbits. Building on this approach I will show how to analyse network states, with a focus on synchrony. I will make use of an extension of the master stability function approach (popular in the Physics community), and show how this framework is very amenable to explicit calculations when considering networks of pwl oscillators [2]. To highlight the usefulness of such an approach (to determine possible bifurcations of network states) I will discuss an inverse period-doubling route to synchrony, under variation in coupling strength, in gap-junction (linearly) coupled networks for which the node dynamics is poised near a homoclinic bifurcation. I will contrast this with node dynamics poised near a non-smooth Andronov-Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs. I will also show how the framework can treat integrate-and-fire neuron models with balanced synaptic coupling, as well as networks of Wilson-Cowan population models. [1] P Ashwin, S Coombes and R Nicks 2016 Mathematical frameworks for oscillatory network dynamics in neuroscience, Journal of Mathematical Neuroscience, 6:2 [2] S Coombes and R Thul 2016 Synchrony in networks of coupled nonsmooth dynamical systems: Extending the master stability function, European Journal of Applied Mathematics, Vol 27(6), Ana Dias Networks: weighted or not? Coupled cell systems are networks of dynamical systems (the cells), where the links between the cells are described through the network structure, the coupled cell network. Synchrony subspaces are spaces defined in terms of equalities of certain cell coordinates that are flow-invariant for all coupled cell systems associated with a given network structure. The enumeration method for networks with a given quotient network and the lifting bifurcation problem, the lattice of synchrony subspaces of a network and the possible synchrony- breaking bifurcations are some of the problems that have been addressed in the last years for networks where the weights are zero or positive integers. In this talk we plan to review some of the results that have been achieved in this setup and to question those issues in the setup of weighted networks networks where the connections have an associated weight which does not have to be zero neither a positive integer.

4 Michael Field Functional Asynchronous Networks of feed-forward type After a discussion of some of the issues involved in modelling dynamical phenomena using networks of coupled dynamical systems, we present (1) a result (joint with Chris Bick, Oxford and Exeter) giving a natural feed-forward structure on a class of functional asynchronous networks, and (2) a model of unsupervised adaptive learning that involves local times, weight dynamics and a feed-forward functional asynchronous network that can be viewed as a dynamical version of a neural net (related to current work with Manuel Aguiar and Ana Dias, Porto). Marty Golubitsky Binocular Rivalry and Patterns of Rigid Phase Shifts This talk reviews work on models for generalized rivalry proposed by Hugh Wilson. Wilson's proposal includes the specification of a class of coupled cell networks that we call "Wilson networks." We use these networks to model experiments in binocular rivalry. The specific application shows how rigid phase-shifts in periodic solutions of admissible maps for Wilson networks can predict observed percepts in many binocular rivalry experiments. Ernest Montbrió Synchronization patterns in firing rate models with synaptic delay Population models of neuronal activity have become an standard tool of analysis in computational neuroscience. Rather than focus on the microscopic dynamics of neurons, these models describe the collective properties of large numbers of neurons, typically in terms of the mean firing rate of a neuronal ensemble. In general, such population models, often called firing rate equations (FRE) are obtained using heuristic mean-field arguments. Despite their heuristic nature, Heuristic-FRE often show remarkable qualitative agreement with the dynamics in equivalent networks of spiking neurons. However, this agreement breaks down once a significant fraction of the neurons in the population fires spikes synchronously. In my presentation, I will introduce a recent theory to derive the exact FRE for a large population of heterogeneous Quadratic Integrate and Fire (QIF) neurons. In contrast with traditional H-FRE, these QIF-FRE adequately capture collective synchronization. To illustrate this, I will use the QIF-FRE to investigate the emergence of synchronous oscillations in a population of identical inhibitory neurons with synaptic delays. The analysis reveals the presence of an intriguing class of partially synchronized states, which display periodic and even chaotic collective dynamics. Finally, the relationship of the model s dynamics with fast neuronal oscillations will be discussed.

5 David Spivak Sheaf-theoretic analysis of networked dynamical systems "We discuss a ""behavioral approach"" to dynamical systems using the category-theoretic notion of sheaves. Each sort of dynamical system including discrete, continuous, and hybrid has an associated behavior type, by which we mean a set of possible behaviors over any duration of time. Such behavior types are formalized as sheaves, a notion which includes far more variety than the three sorts of dynamical systems mentioned above. For instance, the set of rational numbers forms a sheaf, as does our internal model of Time itself. We consider a system to be a behavior type together with an interface, which gives the ports by which the internal behavior is made known to the environment. As various interfaces are linked into a network, the behaviors of the inhabiting systems interact and constrain each other, and together these behaviors combine to form a global behavior of the system of systems. For example, in the National Airspace System, airplanes, pilots, traffic collision avoidance systems, and radars interact to ensure that the whole airspace enjoys the property called ""safe separation"". Such a guarantee of safe separation can be made as a formal statement in a new sort of higher-order temporal logic for behaviors, based on topos theory. This language is fully compositional, even when the subsystems are dynamical systems of very different natures, as in the example mentioned above. We assume no category-theoretic background for this talk." Erik Steur Emergence of oscillations in networks of time-delay coupled inert systems We consider networks of single-input-single-output systems that interact via linear, time-delay coupling functions. The systems itself are inert, that is, their solutions converge to a globally stable equilibrium. However, in the presence of coupling, the network of systems exhibits ongoing oscillatory activity. We study the emergence of these oscillations by finding conditions for 1. the solutions of the time-delay coupled systems to be bounded, 2. the network equilibrium to be unique, and 3. the network equilibrium to be unstable. The network of time-delay coupled inert systems is oscillatory if the aforementioned three conditions are satisfied. In addition, we identify the patterns of synchronized oscillation that may emerge. This is joint work with Alexander Pogromsky. Ian Stewart Networks and the Perception of Visual Illusions "Marty Golubitsky plans to talk on binocular rivalry, where the two eyes are presented with conflicting information. This is a companion talk on a similar phenomenon, illusions, where one or both eyes are presented with ambiguous information. The classic example is the Necker cube, a line drawing of a skeletal cube that is perceived in two alternating orientations. In both cases, the visual system has to decide which image it perceives, and percepts switch between different alternatives. The decision process has been modelled using Wilson networks and their generalisations, in which

6 attributes of the image (colour, orientation, and so on) are represented by mutually inhibitory nodes corresponding to levels or values of those attributes (red, blue, green, and so on). Learned or perceived patterns are represented by adding excitatory connections between nodes associated in the patterns. The dynamics of these networks determines the outcome of the decision process. The talk will introduce a number of standard illusions, construct associated networks, and analyse their dynamics." Sören Schwenker Generic steady state bifurcations in monoid equivariant dynamical systems Rink, Sanders and Nijholt have proved a number of striking results on dynamics in homogeneous coupled cell networks in the last couple of years. Using their methods, one can present network ODEs as (sub-systems of) systems that are symmetric with respect to a monoid representation. In the past most studies on equivariant dynamics were based on group representation theory. The work of Rink et al. draws attention to very exciting generalizations of representation theory and equivariant dynamics. The type of generic steady state bifurcations has so far been clearified only in parts: either in the direct context of the networks or in a special case of the monoid representation. It was anticipated however, that generic steady state bifurcations are connected to subrepresentations of the monoid not only in that special case but for all suitable monoid representations. We employ techniques based on the work by Rink, Sanders and Nijholt to prove this generalization. Kyle Wedgwood Coherent behaviour in spiking neural networks "Spatially coherent behaviour, in various guises, has been shown to be important in a variety of aspects of neural processing. For example, persistent localised activity, colloquially known as bumps, are associated with working memory. Propagating, but localised, is associated with switching visual perceptions, but are also associated with seizure generation in epilepsy. There is thus a desire to understand the origin of such patterned neural behaviour. Neural field models have provided great insight into the mechanisms giving rise to this behaviour, and in particular, in showing what conditions are necessary for them. However, these represent averaged activity, giving by the firing rate of a population of neurons, and as such, they ignore the important spiking activity of the neutrons, which may contribute significantly to the overall information processing in neural tissue. Here, I will present work which extends the use of interface dynamics, commonly used in neural field models to quantify localised activity, to spiking neural networks, and demonstrate how tools from non-smooth analysis and equation-free modelling can be used to capture patterned behaviour in these networks." Matthias Wolfrum Chimera states in systems of coupled phase oscillators

7 Chimera states are self-organized patterns of coherence and incoherence that can be observed in systems of coupled oscillators. We give an overview over different types of such patterns in one and two dimensional arrays of oscillators with non-local coupling. In the continuum limit, they appear as relative equilibria of a continuity equation and their stability can be studied in the framework of the Ott-Antonsen reduction. In finite oscillator systems they have the properties of spatially extensive chaos and show additional finite size effects. In very small systems, classical chimera states are unstable, but can be stabilized by adding a control term. Their dynamics can be described as selflocalized excitability, inducing a variety of stationary, propagating regular and chaotic excitation patterns.

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