MBI Publications

MBI Publications for Marty Golubitsky (13)

  • M. Golubitsky and R. Lauterbach
    Bifurcations from synchrony in homogeneous networks: linear theory
    SIAM J. Appl.Vol. 8 No. 1 (2009) pp. 40-75

    Abstract

  • M. Aguiar, M. Aguiar, A. Dias, A. Dias, M. Golubitsky and M. Leite
    Bifurcations from regular quotient networks: A first insight
    Physica D.Vol. 238 No. 2 (2009) pp. 137-155

    Abstract

  • M. Golubitsky, C. Postlethwaite, L. Shiau and Y. Zhang
    The feed-forward chain as a filter amplifier motif. In: Coherent Behavior in Neuronal Networks
    Springer (2009) pp. 95-120

    Abstract

  • N. Filipski and M. Golubitsky
    The abelian Hopf H mod K theorem
    SIAM J. Appl. Dynam. Sys.Vol. 9 No. 2 (2010) pp. 283-291

    Abstract

  • M. Golubitsky, D. Romano and Y. Wang
    Network periodic solutions: full oscillation and rigid synchrony
    NonlinearityVol. 23 (2010) pp. 3227-3243

    Abstract

  • Y. Zhang and M. Golubitsky
    Periodically forced Hopf bifurcation
    SIAM J. Appl. Dynam. Sys.Vol. 10 (2011) pp. 1272-1306

    Abstract

  • I. Stewart and M. Golubitsky
    Synchrony-breaking bifurcation at a simple real eigenvalue for regular networks 1: 1-dimensional cells
    SIAM J. Appl. Dynam. Sys.Vol. 10 No. 4 (2011) pp. 1404-1442

    Abstract

  • M. Golubitsky and C. Postlethwaite
    Feed-forward networks, center manifolds, and forcing
    Discrete and Continuous Dynamical Systems - Series AVol. 32 (2012) pp. 2913-2935

    Abstract

  • M. Golubitsky, D. Romano and Y. Wang
    Network periodic solutions: patterns of phase-shift synchrony
    NonlinearityVol. 25 (2012) pp. 1045-1074

    Abstract

  • M. Golubitsky and A. Comanici
    Patterns on growing square domains via mode interactions
    Dynamical SystemsVol. 23 No. 2 (2012) pp. 167-206

    Abstract

  • Y. Wang, T. McMillen, M. Golubitsky and C. Diekman
    Reduction and dynamics of a generalized rivalry network with two learned patterns
    SIAM Journal of Applied Dynamical SystemsVol. 11 (2012) pp. 1270-1309

    Abstract

    We use the theory of coupled cell systems to analyze a neuronal network model for generalized rivalry posed by H. Wilson. We focus on the case of rivalry between two patterns and identify conditions under which large networks of n attributes and m intensity levels can reduce to a model consisting of two or three cells depending on whether or not the patterns have any attribute levels in common. (The two-cell reduction is equivalent to certain recent models of binocular rivalry.) Notably, these reductions can lead to large recurrent excitation in the reduced network even though the individual cells in the original network may have none. We also show that symmetry-breaking Takensâ??Bogdanov (TB) bifurcations occur in the reduced networks, and this allows us to further reduce much of the dynamics to a planar system. We analyze the dynamics of the quotient systems near the TB singularity, discussing how variation of the input parameter I organizes the dynamics. This variation leads to a degenerate path through the unfolding of the TB point. We also discuss how the network structure affects recurrent excitation in the reduced networks, and the consequences for the dynamics.
  • C. Diekman, M. Golubitsky and Y. Wang
    Derived patterns in binocular rivalry networks.
    Journal of mathematical neuroscienceVol. 3 No. 1 (2013) pp. 6

    Abstract

    Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Wilson proposed a class of neuronal network models that generalize rivalry to multiple competing patterns. The networks are assumed to have learned several patterns, and rivalry is identified with time periodic states that have periods of dominance of different patterns. Here, we show that these networks can also support patterns that were not learned, which we call derived. This is important because there is evidence for perception of derived patterns in the binocular rivalry experiments of Kovács, Papathomas, Yang, and Fehér. We construct modified Wilson networks for these experiments and use symmetry breaking to make predictions regarding states that a subject might perceive. Specifically, we modify the networks to include lateral coupling, which is inspired by the known structure of the primary visual cortex. The modified network models make expected the surprising outcomes observed in these experiments.
  • C. Diekman and M. Golubitsky
    Network symmetry and binocular rivalry experiments.
    Journal of mathematical neuroscienceVol. 4 (2014) pp. 12

    Abstract

    Hugh Wilson has proposed a class of models that treat higher-level decision making as a competition between patterns coded as levels of a set of attributes in an appropriately defined network (Cortical Mechanisms of Vision, pp. 399-417, 2009; The Constitution of Visual Consciousness: Lessons from Binocular Rivalry, pp. 281-304, 2013). In this paper, we propose that symmetry-breaking Hopf bifurcation from fusion states in suitably modified Wilson networks, which we call rivalry networks, can be used in an algorithmic way to explain the surprising percepts that have been observed in a number of binocular rivalry experiments. These rivalry networks modify and extend Wilson networks by permitting different kinds of attributes and different types of coupling. We apply this algorithm to psychophysics experiments discussed by Kovács et al. (Proc. Natl. Acad. Sci. USA 93:15508-15511, 1996), Shevell and Hong (Vis. Neurosci. 23:561-566, 2006; Vis. Neurosci. 25:355-360, 2008), and Suzuki and Grabowecky (Neuron 36:143-157, 2002). We also analyze an experiment with four colored dots (a simplified version of a 24-dot experiment performed by Kovács), and a three-dot analog of the four-dot experiment. Our algorithm predicts surprising differences between the three- and four-dot experiments.

View Publications By