Homogeneous three-cell networks

Leite, Maria C. A.; Golubitsky, Martin

doi:10.1088/0951-7715/19/10/004

Cited by 53 publications

(65 citation statements)

References 18 publications

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“…That is, for every input arrow i ∈ I(c) A groupoid (Brandt [11], Brown [12], Higgins [47]) is an algebraic structure rather like a group, except that the product of two elements is not always defined. Note that the union in (5.5) There are four connected 3-cell networks with identical coupling and valence 1 (see Figure 21) and 34 such networks with valence 2 (see Leite [55,56] and Figure 22). It is possible for two networks to generate the same systems of differential equations [18,19,42].…”

Section: An Element Of I(c) Is Called An Input Edge or Input Arrow Ofmentioning

confidence: 99%

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Nonlinear dynamics of networks: the groupoid formalism

Golubitsky

Stewart

2006

Bull. Amer. Math. Soc.

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370

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Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos.Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the 'input sets'. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of 'robust' patterns of synchrony in terms of the combinatorial structure of the network.Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables-the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology-which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.

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Section: An Element Of I(c) Is Called An Input Edge or Input Arrow Ofmentioning

confidence: 99%

“…One might be tempted to attribute the non-trivial Jordan block to the skew-product structure forced by the feed forward architecture, but the truth must be somewhat subtler [55,56]. Consider network 32 in Figure 22 …”

Section: Synchrony-breaking Bifurcationsmentioning

confidence: 99%

Nonlinear dynamics of networks: the groupoid formalism

Golubitsky

Stewart

2006

Bull. Amer. Math. Soc.

Self Cite

347

370

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“…Bifurcation theory has been applied to the theory of coupled cell networks in various ways (e.g. [6][7][8][9]). …”

Section: (C) Motivationmentioning

confidence: 99%

“…The results in Leite & Golubitsky [7] and Golubitsky & Lauterbach [8] relate the eigenvalues of the Jacobian J to the eigenvalues of the adjacency matrix of the network. More specifically, if μ 1 , .…”

Section: (C) Motivationmentioning

confidence: 99%

On bifurcations in lifts of regular uniform coupled cell networks

Moreira

2014

Proc. R. Soc. A.

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A lift of a given network is a network that admits the first network as quotient. Assuming that a bifurcation occurs for a coupled cell system consistent with the structure of a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type), it is well known that some lifts exhibit new bifurcating branches of solutions. In this work, we approach this problem restricting attention to uniform networks, that is, networks that have no loops and no multiple arrows. We show that, from the bifurcation point of view, rings and their lifts are special networks. We also prove that generically there are lifts that just exhibit the bifurcating branches determined by the quotient network and, moreover, we identify all generic situations where lifts exist that may exhibit bifurcating branches that do not appear in the quotient itself.

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“…As in previous examples, the reduced AMEE model provides a model for such dynamics. A detailed study of its structure, symmetries and bifurcations, should reveal the conditions for balance/unbalance of such a network [45,46,64].…”

Section: Gap Junctionsmentioning

confidence: 99%

Neural Activity Measures and Their Dynamics

Shlizerman¹,

Schröder²,

Kutz³

2012

SIAM J. Appl. Math.

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Abstract. We provide an asymptotically justified derivation of activity measure evolution equations (AMEE) for a finite size neural network. The approach takes into account the dynamics for each isolated neuron in the network being modeled by a biophysical model, i.e. Hodgkin-Huxley equations or their reductions. By representing the interacting network as self and pairwise interactions, we propose a general definition of spatial projections of the network, called activity measures, that quantify the activity of a network. We show that the evolution equations that govern the dynamics of the activity measure shadow the activity measure of the network (i.e. the two quantities stay close to each other for all times) for general interactions and various asymptotic dynamics. The AMEE effectively serve as a dimensionality reduction technique for the complex network when spatial synchrony and coherence are present and allow to a priori predict network dynamics that would not be guessed from individual neuron behavior. To demonstrate an explicit derivation of such a reduction, we consider the mean measure for a network of interacting FitzHugh-Nagumo neurons.Computational results comparing the full network dynamics with the mean AMEE model of identical and nonidentical FitzHugh-Nagumo and FitzHugh-Rinzel neurons validate the shadowing theorems and expose the various resulting AMEE models that allow to describe the mean of the network.

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Homogeneous three-cell networks

Cited by 53 publications

References 18 publications

Nonlinear dynamics of networks: the groupoid formalism

Nonlinear dynamics of networks: the groupoid formalism

On bifurcations in lifts of regular uniform coupled cell networks

Neural Activity Measures and Their Dynamics

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