Workshop on
Coupled Cell Networks
and Dynamics
Experiments in neuroscience show that during tasks where animals have to temporarily retain some information, certain subsets of neurons keep their firing activity at high levels after the external cue is removed. This type of temporary/working memory has three important characteristics: i) it provides a short-term memory buffer which keeps relevant information while it is being used; ii) information can be encoded and preserved after very short exposures to the stimuli and; iii) information seems to be stored in a distributed manner in the neuronal network. Using mathematical modeling and computer simulations we analyze network architectures and neuron unit dynamics supporting this memory system. A key distinctive element of our approach is that we do not rely on the modulation of the connections strengths (between neuron units) to support the information storage – information is stored instead in the intrinsic dynamics of each neuron unit. Joint work with Eduardo Conde-Sousa.
This talk will look at some recent work with C Postlethwaite (Auckland) on two methods of realising heteroclinic cycles with specific graph structures in coupled cell systems. The first will allow one to embed any finite directed graph that is one-cycle and two-cycle free in a system of coupled ODEs with dimension equal to the number of vertices in the graph. The second is less restrictive in that it allows one to embed any finite graph that is one-cycle free in a system of coupled ODEs of dimension equal to the number of edges in the graph, plus one.
We consider the effects of discrete and distributed time delays on the dynamics of coupled cells. We make a distinction between the communication delays in the network and the signal processing delays within the cells. The cooperative nature of the interactions is reflected in the Laplacian matrix that describes the coupling structure and provides a connection with graph theory. We examine several types of collective behavior that arise only in the presence of delays.
A coupled cell system is a network of dynamical systems, or 'cells', each of which is given by ODEs. The network architecture is a directed graph and it represents interactions between cells. The main motivation of the study of coupled cell system, formulated by M. Golubitsky, I. Stewart et al., is to understand dynamics of the whole system from its network architecture. As the number of cells in the network grows the coupled cell system would be more and more complicated to analyse. To meet such difficulty, we develop an idea of getting at least partial information of the system from its subnetwork architecture. We show that for a certain type of network and a generic coupled cell system associated with that network, there exists a local invariant manifold in a neighborhood of equilibrium, on which the coupled cell system behaves identically the same as a coupled cell system associated with a subnetwork of the original network.
Complex systems in science and technology can often be represented as a network of interacting subsystems or subnetworks. If we follow a reductionist approach, it is natural (though not always wise!) to attempt to describe the dynamics of the network in terms of the dynamics of the subsystems of the network. Put another way, we often have a reasonable understanding of the "pieces", but how do they fit together, and what do they do collectively? In the simplest, and most studied cases, the subnetworks all run on the same clock (are updated simultaneously), and dynamics is governed by a fixed set of (usually analytic) dynamical equations: we say the network is synchronous (this is classical dynamics). In biology, especially neuroscience, and technology, for example transport networks or large distributed systems, these assumptions may not hold: components may run on different clocks, there may be switching between different dynamical equations, and most significantly, and quite unlike what happens in a classical synchronous network, component parts of the network may run independently of the rest of the network, and even stop, for periods of time. We say networks of this type are asynchronous. It is a major challenge to develop the mathematical theory of dynamics on asynchronous networks. In this talk, we describe recent work the structure and definition of a very large general class of asynchronous networks that allows for both autonomous and non-autonomous dynamics. We will present a number of examples of dynamics on asynchronous networks and point out how properties such as switching are forced by an asynchronous structure.
We represent the structure of a given general coupled cell network by a symbolic adjacency matrix, which encodes the different types of interaction. Synchrony subspaces of a given network are polydiagonals which are invariant under a linear map represented by the adjacency matrix. This leads to a computer algorithm to search all possible synchrony subspaces with matrix manipulation, and construct these into a complete lattice using the refinement relation to impose a hierarchy. We demonstrate our computational method using networks from real application problems and discuss the stability of possible synchrony subspaces which is a consequence of the details of the model systems. For regular networks, we show lattice structures have a direct link to eigenvalues and eigenvectors of their adjacency matrices, and this can be used to analyse synchrony-breaking bifurcations. Finally, I will discuss future directions of current work.
We discuss the role of different time-scales in the dynamics of two coupled FitzHugh-Nagumo equations, and the associated geometry. When two different time-scales are considered separately, the dynamics is constrained by the geometry of the slow manfold, that in this specific example is a surface in the 4-dimensional phase space.
My presentation summarizes recent advances on the role of nonlinear interactions mediated by dendrites in neural networks. Inputs arriving at the same dendrite can interact in a nonlinear manner by evoking fast dendritic spikes which strongly amplify the impact of the inputs on the neuron. The interaction requires almost simultaneous input and, in turn, generates neural output with high temporal precision. We show that this enables propagation of synchronous activity in neural networks even if they are sparse and contain only weak, biologically plausible sub-structures, or are completely random. Internal, external and induced, echo-like oscillations can further promote the propagation. This may explain prominent bursts of highly synchronous activity in the brain during sleep, as well as the associated replay of patterns from previous exploration phases. Such replay is considered crucial for memory consolidation. Further, we show that nonlinear dendrites increase the capacity and the robustness against noise in associative memory networks, already for learning rules that do not account for them. An intermediate number of nonlinear dendrites or dendritic compartments is optimal for these purposes.
We reveal the nature of the avalanche collapse of the giant viable component in multiplex networks under perturbations such as random damage. Specifically, we identify latent critical clusters associated with the avalanches of random damage. Divergence of their mean size signals the approach to the hybrid phase transition from one side, while there are no critical precursors on the other side. We find that this discontinuous transition occurs in scale-free multiplex networks whenever the mean degree of at least one of the interdependent networks does not diverge.
We consider the lifting process of regular coupled cell networks from the architectural point of view. In this approach, lifts are interpreted as networks resulting from the splitting of cells of the initial network. We present results that relate the splittings with the spectrum of the lifts. We also apply some results to the bifurcation theory of coupled cell networks. This is a joint work with A.P. Dias.
We study expanding circle maps interacting in a heterogeneous random network. Heterogeneity means that some nodes in the network are massively connected, while the remaining nodes are only poorly connected. We provide a probabilistic approach which enables us to describe the effective dynamics of the massively connected nodes when taking a weak interaction limit. More precisely, we show that for almost every random network and Lebesgue almost all initial conditions the high dimensional network governing the dynamics of the massively connected nodes can be reduced to a few macroscopic equations. Such reduction is intimately related to the ergodic properties of the expanding maps. This reduction allows one to explore the coherent properties of the network. This is a joint work with Jeroen Lamb and Sebastian van Strien.
A network structure can have a strong impact on the behaviour of a dynamical system. For example, it has been observed that networks can robustly exhibit (partial) synchronisation, multiple eigenvalues and degenerate bifurcations. In this talk I will explain how semigroups and their representations can be used to understand and predict these phenomena. As an application of our theory, I will discuss how a simple feed-forward motif can act as an amplifier. This is joint work with Jan Sanders.
Understanding the evolutionary mechanisms that promote and maintain cooperative behavior is recognized as a major theoretical problem where the intricacy increases with the complexity of the participating individuals. This is epitomized by the diverse nature of Human interactions, contexts, preferences and social structures. Here I will discuss how social diversity, in several of its flavors, catalyses cooperative behavior. From the diversity in the number of interactions an individual is involved to differences in the choice of role models and contributions, diversity is shown to significantly increase the chances of cooperation. Individual diversity leads to an overall population dynamics in which the underlying dilemma of cooperation is changed, benefiting the society as whole. In addition, I will discuss how diversity in social contexts can arise from the individual capacity for organizing their social ties. As such, Human diversity, on a grand scale, may be instrumental in shaping us as the most sophisticated cooperative entities on this planet.
Our brain is adept at interpreting noisy and ambiguous visual input. However, in certain situations multiple interpretations can be valid and are perceived in alteration. Conventional binocular rivalry occurs when two disparate images are presented to the eyes simultaneously, and, instead of seeing the two images superimposed, the observer alternately perceives one of the two stimuli in a stochastic manner. Ambiguous visual inputs may induce more than two percepts. Kovacs et al. (1996) showed that complex stimuli can lead to rivalry between two presented images and also between two reorganized images. This is an example of multistable (more than two) perceptual rivalry. Even though, there is a huge body of literature on bistable rivalry, only a few are on multistable cases. In this talk, I will first introduce a network framework, which was proposed originally by Hugh Wilson and generalized by Golubitsky et al., for multistable percepts and then I will discuss how stochaticity that was observed in experiments can be captured using mathematical models under the framework.
February
3-5
2014
Departamento de Matemática
Faculdade de Ciências
Universidade do Porto