Derived Patterns in Binocular Rivalry Networks

Diekman, Casey O.; Golubitsky, Martin; Wang, Yunjiao

doi:10.1186/2190-8567-3-6

Cited by 21 publications

(44 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The analysis of the conventional and scrambled diagonal 4-dot experiments presented here is related to, but definitely different from, the one that we gave in [13]. In [13], we followed Wilson [7] by assuming that all excitatory couplings between attribute nodes in the learned patterns were the same.…”

Section: Four Colored-dot Experimentsmentioning

confidence: 99%

Network Symmetry and Binocular Rivalry Experiments

Diekman

Golubitsky

2014

The Journal of Mathematical Neuroscience

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Four Colored-dot Experimentsmentioning

confidence: 99%

Network Symmetry and Binocular Rivalry Experiments

Diekman

Golubitsky

2014

The Journal of Mathematical Neuroscience

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here, conflicting images are presented to the two eyes, and the visual system interprets this combination in sometimes surprising ways. Diekman et al 36 observed that Wilson networks are useful for understanding rivalry itself. A Wilson network is trained on a set of signals or patterns, and its architecture is designed to detect these patterns and distinguish them from others.…”

Section: Application: Rivalrymentioning

confidence: 99%

“…However, some also reported an alternation between two "whole" images, neither of which occurred in the training set; see Figure 11 (right). Diekman et al 34,36 considered a natural model using the Wilson network in Figure 12. This has two attributes, each with two levels, corresponding to which parts of the images appear in the "jigsaw" decomposition shown.…”

Section: Application: Rivalrymentioning

confidence: 99%

See 1 more Smart Citation

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts Rene Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network. Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems-systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have ...

show abstract

“…Rate models in a network with the same symmetries are analysed in Diekman et al [25] in a model of binocular rivalry using Wilson networks (the "monkey/text" experiment). They assume a more general form of the gain function, but take β = 0, γ < 0 in Figure 1 because some connections in Wilson networks are always inhibitory, and the others are determined by a learning process.…”

mentioning

confidence: 99%

Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator

Stewart

2014

Symmetry

View full text Add to dashboard Cite

The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Z 2 × Z 2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.

show abstract

Derived Patterns in Binocular Rivalry Networks

Cited by 21 publications

References 30 publications

Network Symmetry and Binocular Rivalry Experiments

Network Symmetry and Binocular Rivalry Experiments

Recent advances in symmetric and network dynamics

Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator

Contact Info

Product

Resources

About