Network Symmetry and Binocular Rivalry Experiments
 Casey O Diekman^{1}Email author and
 Martin Golubitsky^{2}
https://doi.org/10.1186/21908567412
© C.O. Diekman, M. Golubitsky; licensee Springer 2014
Received: 10 October 2013
Accepted: 24 April 2014
Published: 7 May 2014
Abstract
Hugh Wilson has proposed a class of models that treat higherlevel 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 symmetrybreaking 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 24dot experiment performed by Kovács), and a threedot analog of the fourdot experiment. Our algorithm predicts surprising differences between the three and fourdot experiments.
Keywords
1 Introduction
In standard binocular rivalry experiments, the left and right eyes of the subject are presented dissimilar images, and the subject’s perception alternates between the two presented images [1]. Perceptual alternation has often been modeled as competition between two units, each representing a percept, with reciprocal inhibition between the units (cf. [2–6]). The point of those models is to analyze the dynamics of alternation assuming a set of percepts, rather than predicting the form of the percepts. In contrast, the goal of our analysis is to predict the form of percepts, and not the precise dynamics of alternation. To do this, we use symmetrybreaking and an extension of a class of network models proposed by Wilson [7], which we call rivalry networks. Our analysis is modelindependent; that is, we concentrate on the network architecture and not on specific equations associated with the networks.
Wilson networks generalize binocular rivalry models to multiple competing patterns [7, 8]. Originally intended as a model of deliberative decision making, these networks embody aspects of conscious brain processes that have been discussed previously by many other researchers, such as Tononi and Edelman [9], Dehaene et al. [10, 11], and Crick and Koch [12]. A Wilson network consists of a set of attributes relevant to a given decision, with reciprocal inhibition among the nodes associated with each attribute. A pattern consists of one node from each attribute, with excitatory coupling among nodes in a learned pattern. In [13], we showed that Wilson networks could be constructed to represent the binocular rivalry experiments of Kovács et al. [14], and that certain states of these Wilson networks corresponded to the unexpected percepts observed in [14]. This result suggested that Wilson networks can help classify and predict percepts for binocular rivalry experiments. In this paper, we expand and modify the ideas in [13] to demonstrate that Wilsontype models can be constructed in an algorithmic way for several binocular rivalry experiments in the literature, and that these models seem to explain the surprising percepts observed in these experiments. Using these models we also predict percepts for several binocular rivalry experiments that have yet to be performed.
Our algorithm has three parts. First, we generalize Wilson networks by introducing different types of nodes (attributes) and different types of excitatory couplings (based on features). An example of attributes of the same type is the color of discs at different geometric locations. An example of different attribute types is the color of a disc and the direction of a grating pattern in that disc. Features are properties of pairs of nodes, such as the geometric distance between two discs. We use features (which are based on Hebbian learning) to determine whether coupling between pairs of nodes are assumed to be the same or not. It is the systematic use of features, and hence of multiple types of excitatory coupling, that makes our symmetrybreaking analysis possible. Second, we use network symmetries to identify the maximal fusion states (see Definitions 1 and 2) of the network. Finally, we classify the types of fusionbreaking Hopf bifurcation from maximal fusion states; that is, Hopf bifurcations that lead to nonfusion states. The resulting periodic states lead to alternation in network patterns that we interpret as predictions of the alternations in perception a subject would be likely to experience. We claim that for Hopf bifurcation to lead to alternation between percepts, the bifurcation must take place from a fusion state. Furthermore, the expectation that fusion states exist depends on network architecture (specifically on network symmetry). This point is discussed in Sect. 2.1.
We demonstrate this procedure on the monkeytext experiments of Kovács et al. [14], the geometric rivalry stimuli experiments in Suzuki and Grabowecky [15], and the color misbinding experiments of Shevell and Hong [16, 17]. The types of periodic solutions that come from fusionbreaking bifurcations in the rivalry networks we construct are consistent with the surprising percepts reported in these experiments. Moreover, we make testable predictions about the states that we expect to be perceived in coloreddot experiments that have yet to be performed. We predict qualitatively different sets of percepts for three coloreddot and four coloreddot stimuli. In particular, for the fourdot experiments, the predictions from our theory include simple alternation between the presented stimuli, as one would intuitively expect to occur. However, for certain threedot experiments, our theory predicts that such alternation should not occur generically (see Sect. 6.3).
1.1 Wilson Networks
Wilson assumes that these networks can have multiple learned patterns and that rivalry is represented by solutions that alternate between learned patterns. In [13] we observed that network periodic solutions can exhibit patterns that were not learned, which we call derived. We proposed that the unexpected percepts in many binocular rivalry experiments exist because the dynamics of associated Wilson networks lead naturally to particular derived patterns. As noted, in this paper we generalize the definition of Wilson networks to rivalry networks, which include detail that was left out of the discussion in [13], and which is needed to systematically determine networks for binocular rivalry experiments. In doing so we use the theory of coupled cell networks developed in [19, 20].
where → indicates an excitatory learned pattern connection and ⊣ indicates an inhibitory connection. The parameters are: reciprocal learned pattern excitation between nodes $w>0$, reciprocal inhibition between nodes $\beta >0$, the external signal strength ${I}_{ij}\ge 0$ to nodes, the strength of reduction of the activity variable by the fatigue variable $g>0$, and the ratio of time scales $\epsilon <1$ on which ${\ast}^{E}$ and ${\ast}^{H}$ evolve. The gain function is usually assumed to be nonnegative and nondecreasing, and is often a sigmoid.
In this paper we assume only that each node $(i,j)$ has an activity variable ${A}_{ij}\in \mathbf{R}$. The activity variable is just the coordinate ${x}_{ij}^{E}$ in (1), but in general it might be a function of the state space variables ${A}_{ij}={A}_{j}({x}_{ij})$. As indicated, this function is assumed to be the same for each level of an attribute column. The choice of level i in an attribute column j occurs when ${A}_{ij}>{A}_{kj}$ for all $k\ne i$. A percept is assumed to be dominant when the activity variable of each node associated with the pattern is a unique maximum within its attribute column.
1.1.1 Fusion States
Definition 1 A fusion state is one where the maximum of the ${A}_{ij}$ in some column occurs simultaneously in more than one level.
is well known [20, 21] to be a flowinvariant subspace of the state space of the network admissible differential equations. So it is not surprising to find solutions (equilibria or periodic solutions) in $Fix(\Sigma )$, since those fusion solutions cannot be gotten rid of by perturbation. If one of the symmetries maps one node in an attribute column to another node in that same column, then the corresponding fixedpoint subspace can have fusion states that cannot be destroyed by perturbation of the differential equation.
Definition 2 Let Γ be the group of network symmetries for a fixed network. We call states in $Fix(\Gamma )$ maximally fused states.
Those Hopf bifurcations that can lead to a nonfused periodic solution from a maximally fused equilibrium are called fusionbreaking Hopf bifurcations. Since such bifurcations usually lead to periodic solutions with nonzero phase shifts between the previously fused nodes, these bifurcations lead to rivalrous solutions.
Note that modeling alternation between percepts (patterns) in networks of differential equations requires finding periodic solutions that alternate periods of dominance; that is, for part of the trajectory one set of nodes has maximum activity and during another part of the trajectory a different set of nodes has maximum activity. Thus, during the trajectory there must be times when the activity of multiple nodes are equal. A small amplitude periodic solution obtained via Hopf bifurcation can have this property only if the equilibrium from which the bifurcation takes place is a fusion equilibrium. We assume, as is standard in bifurcation theory in the presence of symmetry, that the states that are most likely to be observed are spawned by bifurcation from a maximally fused (or symmetric) state.
1.2 The Structure of the Paper
In this paper we discuss several binocular rivalry experiments. For each experiment we formulate a rivalry network, compute the fusionbreaking Hopf bifurcations, and discuss the derived percepts. Specifically, we discuss rivalry models for the conventional and scrambled monkeytext experiments of Kovács et al. [14] in Sect. 2; the geometric shapes of Suzuki and Grabowecky [15] in Sect. 3; and the color misbinding experiments of Shevell et al. [16, 17] in Sect. 4. The Shevell experiments require having rivalry networks with two types of nodes. In Sects. 5 and 6, we analyze experiments with three or four colored dots, which are analogs of the 24 colored dot experiments in Kovács et al. [14]. Summaries of our predictions are given in Sect. 7. This analysis requires the use of features; that is, of several types (or strengths) of excitatory coupling.
We give a formal definition of rivalry networks in Sect. 8 and summarize our approach to rivalrydriven percepts in Sect. 9.
2 MonkeyText Experiments
2.1 A Symmetry Analysis of Standard Rivalry
that is, the fixedpoint subspace of ρ consists of fusion states (those states for which $b=a$). A standard fact from dynamical systems with symmetries is that fixedpoint subspaces are flowinvariant (cf. [21]), and this fact can be checked directly for (2). It is therefore a reasonable hypothesis that a model like (2) has a fusion equilibrium.
Moreover, J acts as multiplication by $\alpha +\beta $ on ${V}^{+}$ and as multiplication by $\alpha \beta $ on ${V}^{}$. Thus, the eigenvalues of J are the eigenvalues of $\alpha +\beta $ and the eigenvalues of $\alpha \beta $. If the critical eigenvalues for J are associated to $\alpha +\beta $, then Hopf bifurcation will lead to the fusion states $(a(t),a(t))$; if the critical eigenvalues for J are associated to $\alpha \beta $, then Hopf bifurcation will lead to the nonfusion states (3).
Thus, there is only one fusionbreaking Hopf bifurcation and that bifurcation leads to alternation between the two learned images.
2.1.1 A Short Summary of Hopf Bifurcation in the Presence of Symmetry
Our discussion of Hopf bifurcation in the presence of ${\mathbf{Z}}_{2}$ symmetry is actually quite general. Observe that the symmetry ρ acts trivially on ${V}^{+}$ and as multiplication by −1 on ${V}^{}$. These two subspaces correspond to the trivial onedimensional irreducible representation of ${\mathbf{Z}}_{2}$ and the nontrivial onedimensional representation of ${\mathbf{Z}}_{2}$.
For a general group Γ there is a type of Hopf bifurcation that corresponds to each type of irreducible representation V. The general theory of Hopf bifurcation in the presence of symmetry as well as the results of these bifurcations for a number of groups Γ and their irreducible representations can be found in [21, 22]. The examples worked out in these references include all of the groups that appear in this paper. The symmetry groups that will appear in this paper are ${\mathbf{Z}}_{2}$, ${\mathbf{D}}_{n}$, and ${\mathbf{D}}_{n}\oplus {\mathbf{Z}}_{2}$, where ${\mathbf{D}}_{n}$ is the symmetry group of the 2n symmetries of a regular ngon (a rectangle when $n=2$, an equilateral triangle when $n=3$, and a square when $n=4$). Note that the symmetries of a rectangle are generated by two reflections; that is, ${\mathbf{D}}_{2}={\mathbf{Z}}_{2}\oplus {\mathbf{Z}}_{2}$. All of these groups will appear as permutation symmetries of a network. For example, ${\mathbf{D}}_{3}$ will appear as the permutation group of a ring of three identical nodes.
Finally, if a symmetry $\gamma \in \Gamma $ acts trivially on V, then all solutions $x(t)$ corresponding to the associated Hopf bifurcation will satisfy $\gamma x(t)=x(t)$. Moreover, if a symmetry $\gamma \in \Gamma $ acts as multiplication by −1 on V, then all Tperiodic solutions $x(t)$ corresponding to the associated Hopf bifurcation will satisfy $\gamma x(t)=x(t+T/2)$. If γ permutes nodes in the same attribute column, then the first result leads directly to fusion states and the corresponding Hopf bifurcation is not fusionbreaking.
2.2 Conventional MonkeyText Experiment
The conventional monkeytext experiment is a standard rivalry experiment. In this experiment there is one attribute (the image) and it has two levels (monkey and text). There is no excitatory coupling in the network and the network is the one shown in Fig. 3. As expected, the subjects in this experiment report alternation between the monkey image and the text image.
2.3 Scrambled MonkeyText Experiment
Since ${\mathbf{D}}_{2}$ acts transitively on the nodes, the maximal fusion states satisfy ${x}_{1}={x}_{2}={x}_{3}={x}_{4}$.
Our algorithm indicates that these two types of periods of alternation are possible. One type is a period of alternation between the scrambled images and the other type is a period of alternation between the reconstructed conventional images.
2.4 Comparison with the Results in [13]
In a previous paper [13], we showed that the scrambled experiment could be understood using the network in Fig. 5b. However, when one uses the rate equations (1), rivalry solutions between the derived patterns can never be stable at bifurcation. We addressed this point by adding a second form of excitatory coupling, which we called lateral coupling, to the network. We no longer believe that stability is a crucial issue, since we do not know what the equations that correspond to a rivalry network should be. Here, we focus on the existence of patterns of alternation and therefore have no need to introduce lateral coupling. These results depend only on network structure and not on specific model equations; that is, these results are modelindependent.
3 Rivalry Between Geometric Shapes
It is not clear why the brain divides the images into left halves and right halves, yet that is what appears to be happening. Note that the Wilson network for the Suzuki and Grabowecky [15] experiments is isomorphic to the Wilson network for scrambled monkeytext experiment in Fig. 5b. The network has two attributes (lefthalf image and righthalf image) and two levels (the geometric shape on the appropriate side).
There are two questions that arise when modeling a rivalry experiment: what are the states (i.e. the form of the percepts), and what is the dynamics of switching among percepts? In this paper we are focused on the former, which depends on network structure and not on specific model equations. However, Suzuki and Grabowecky [15] described an interesting property of rivalry dynamics that we would like to comment on. In their experiments, subjects exhibited a bias in the transitions among states called perceptual trapping. For example, consider the hourglass–diamond–chevrons set shown in the bottom row of Fig. 6. Whichever pair of images the subjects were shown (either the hourglass–diamond pair of Fig. 6a, or the chevron pair of Fig. 6b), the perceived shape alternated between the hourglass and the diamond and between the two chevrons more often than would be expected if the transitions among the four perceived images were random. Transitions between these pairs, i.e. from an hourglass or a diamond to a chevron or vice versa, occurred less often. Although this trapping phenomenon is not captured explicitly by our model, it is consistent with our observation that there are two different types of symmetrybreaking Hopf bifurcation that lead to rivalry in twoattribute twolevel networks such as Fig. 5b. Alternation between the rivalry associated with each of these bifurcations could occur if stochastic effects were included in the model. If the noiseinduced alternations occurred on a slower time scale than the oscillations arising from Hopf bifurcation, then overall the dynamics would resemble perceptual trapping.
4 Color Misbinding
Shevell et al. [16, 17] show that rivalry between images that have static gratings of different colors can lead to color misbinding: percepts with gratings comprised of one color from each of the learned patterns. This experiment enables us to explore the use of rivalry networks with more than one attribute type.
4.1 Two Grating Directions
We propose a rivalry network with three attributes: one grating direction and two colors (one in the top/left and one in the bottom/right). The grating direction has two levels (horizontal and vertical) and the color attributes have three levels (pink, gray, green). We assume that there is one type of excitatory coupling from grating direction to color and another from color to color.
where $u(t)$ has twice the frequency of $x(t)$ and $y(t)$ since τ implies ${u}_{1/2}={u}_{0}$. If $y>u$ (that is, bifurcation is from a fusion state where $c>b$), then the percepts will alternate between pink/gray and gray/green; that is alternation between learned images. In fact, the alternation is predicted to be somewhat more complicated; the moments in time when horizontal and vertical gratings alternate are different from the moments in time when the colors alternate. Shevell et al. [17] report the different percepts but not the order in which the percepts alternate.
On the other hand, if $y<u$ (that is, bifurcation is from a fusion state $b>c$), then the percept will be green/pink, which corresponds to color misbinding. Here, there is no alternation of colors, but there is still alternation in grating direction. Thus, the surprising color misbinding in [17] seems also to be a product of the rivalry network structure.
4.2 One Grating Direction
where ${c}_{0}={c}_{1/2}$ and ${d}_{0}={d}_{1/2}$. There are four possible percept alternations.

, ${d}_{0}>{b}_{0}$: the percept is an all gray image.${c}_{0}>{a}_{0}$

, ${d}_{0}<{b}_{0}$: the percept alternates grayorange and grayblue.${c}_{0}>{a}_{0}$

, ${d}_{0}>{b}_{0}$: the percept alternates orangegray and bluegray, which is just alternation between the original stimuli.${c}_{0}<{a}_{0}$

, ${d}_{0}<{b}_{0}$: the percept alternates orange, orangeblue, blue, and blueorange.${c}_{0}<{a}_{0}$
5 Four ColoredDot Experiments
In their review article, Tong et al. [23] suggest focusing on the four centrally located dots in these images and describe the Kovács et al. experiments in these terms. We discussed these experiments in [13], where we used a notion of lateral coupling to propose Wilsontype networks that could help explain the surprising outcomes. Here we propose a different mechanism for deriving rivalry networks for the conventional and scrambled dot experiments, one based on features that lead to different types of excitatory couplings within learned patterns. This approach creates networks that are different from the ones proposed in [13] (though they have the same symmetries). It has the advantage of being consistent with Hebbian learning and with all of the other examples discussed in this paper. We return to these points below. These modeling results can be thought of as predictions, since to our knowledge these experiments have not been performed.
5.1 FourDot Networks and Their Symmetries
Conventional pure color (${\mathbf{D}}_{4}\times {\mathbf{Z}}_{2}(\rho )$ symmetry): The two learned patterns consisting of all red and all green dots, respectively, are shown in Fig. 12a. The conventional fourdot network has two types (or strengths) of excitatory coupling, one for adjacent dots and one for diagonal dots. These coupling types are indicated in Fig. 13a by solid and dashed lines with arrows. This is an example of a feature; couplings are the same only if the dots are equidistant. See Definition 4(d).
Note that if we assume a standard rate equation model (1) for this eightnode network, then the equations corresponding to a single node j will be a twodimensional system of equations with state variables ${x}_{j}^{E}$ and ${x}_{j}^{H}$. Thus $k=2$ and the state space indicated in (7) is 16dimensional.
Scrambled diagonal (${\mathbf{D}}_{4}\times {\mathbf{Z}}_{2}(\rho )$ symmetry): Rivalry between the images in Fig. 12b leads to the network in Fig. 13b, which has four attributes, two levels, and two types of excitatory coupling, just as in the conventional fourdot experiment. The symmetry group and the maximally fused points are identical to those in the conventional fourdot network.
This symmetry group acts transitively on the network nodes; thus all node coordinates are equal in the maximal fusion states as in (8).
5.2 Hopf Bifurcation and Isotypic Components
Suppose that a group Γ acts on a vector space V with an irreducible subspace W. Then an isotypic component of that action is the sum of all representations of Γ in V that are isomorphic to W. The theory of equivariant Hopf bifurcations states that such bifurcations occur generically with a center subspace in one of the isotypic components of the symmetry group acting on phase space. Moreover, there can be a Hopf bifurcation type for each isotypic component. See [[21], Chap. XVI] or [[22], Chap. 4].
Specifically note that $dim{V}_{1}^{+}=dim{V}_{1}^{}=k$, $dim{V}_{2}^{}=2k$, and $dimW=4k$, and these dimensions add up to $dimV=8k$. Also note that each summand is invariant under the action of Γ and the first three are isotypic components. In fact, we could have decomposed W itself into three isotopic components, but we chose not to since the symmetry ρ acts trivially on W, for reasons that we now explain.
Isotypic components corresponding to representations where ρ acts as −1 in the fourdot networks
Experiment  Symmetries  V  Subspace  Kernel 

Conventional 4dot  ${\mathbf{D}}_{4}\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}^{+}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{D}}_{4}$ 
${V}_{1}^{}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  
${V}_{2}^{}$  $\left(\begin{array}{cccc}x& y& x& y\\ x& y& x& y\end{array}\right)$  1  
Scrambled diagonal  ${\mathbf{D}}_{4}\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}^{+}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{D}}_{4}$ 
${V}_{1}^{}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  
${V}_{2}^{}$  $\left(\begin{array}{cccc}x& y& x& y\\ x& y& x& y\end{array}\right)$  1  
Scrambled adjacent  ${\mathbf{D}}_{2}(\kappa ,\tau )\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{D}}_{2}$ 
${V}_{2}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{Z}}_{2}(\kappa )$  
${V}_{3}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{Z}}_{2}(\tau )$  
${V}_{4}$  $\left(\begin{array}{cccc}x& x& x& x\\ x& x& x& x\end{array}\right)$  ${\mathbf{Z}}_{2}(\kappa \tau )$  
Scrambled unbalanced  ${\mathbf{Z}}_{2}(\sigma )\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}$  $\left(\begin{array}{cccc}x& y& x& z\\ x& y& x& z\end{array}\right)$  ${\mathbf{Z}}_{2}(\sigma )$ 
${V}_{2}$  $\left(\begin{array}{cccc}x& 0& x& 0\\ x& 0& x& 0\end{array}\right)$  ${\mathbf{Z}}_{2}(\sigma \rho )$ 
Scrambled adjacent network: The isotypic components where ρ acts nontrivially are listed in Table 1. The alternations predicted by each of the associated Hopf bifurcations are as follows: ${V}_{1}$ yields Fig. 12a; ${V}_{2}$ yields Fig. 12c; ${V}_{3}$ yields Fig. 12c rotated by 90^{∘}, and ${V}_{4}$ yields Fig. 12b.
Scrambled unbalanced: There are two possible isotypic components where ρ acts nontrivially and either σ acts trivially or σ acts nontrivially. See Table 1. The solutions corresponding to ${V}_{2}$ have ${x}_{3}(t)$ and ${x}_{7}(t)$ oscillating at double frequency. However, these states are fusion states, as shown in Fig. 14b. The exact temporal ordering of percepts corresponding to ${V}_{1}$ solutions is complicated to explain. However, the percepts themselves are straightforward. The colors of dots UL and LR are always the same and alternate between red and green. The colors of the dots LL and UR also alternate between red and green, but the times of the alternations can be different. The result is that all eight fourdot percepts where the UL and LR dots have the same color can appear in a ${V}_{1}$ solution type.
5.2.1 Comparison with the FourDot Experiment Analysis in [13]
The analysis of the conventional and scrambled diagonal 4dot 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. This is a reasonable hypothesis but it leads to the conclusion that the symmetry groups of both 4dot experiments are ${\mathbf{S}}_{4}\times {\mathbf{Z}}_{2}(\rho )$; that is, permuting the attribute columns arbitrarily is permitted. Fusionbreaking Hopf bifurcation with ${\mathbf{S}}_{4}$ symmetry does not lead easily to the existence of alternation between the purecolor images in the scrambled 4dot experiment. However, in [13], when we added a second type of coupling between nodes representing the same color, which we called lateral coupling by analogy with the known architecture of the primary visual cortex, then the symmetry group in the scrambled 4dot experiment is (the desired) ${\mathbf{D}}_{4}\times {\mathbf{Z}}_{2}(\rho )$. We now see that this change in symmetry is better achieved by assuming that the excitatory couplings in our networks are the same only when certain features are preserved, which is consistent with Hebbian learning.
6 Three ColoredDot Experiments
6.1 Hopf Bifurcation and Isotypic Components
Isotypic components corresponding to representations where ρ acts as −1 in the threedot networks
Experiment  Symmetries  V  Subspace  Kernel 

Conventional 3dot  ${\mathbf{D}}_{3}\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}$  $\left(\begin{array}{ccc}x& x& x\\ x& x& x\end{array}\right)$  ${\mathbf{D}}_{3}$ 
${V}_{2}$  $\left(\begin{array}{ccc}x& y& xy\\ x& y& x+y\end{array}\right)$  1  
Scrambled unbalanced  ${\mathbf{Z}}_{2}(\kappa )\oplus {\mathbf{Z}}_{2}(\rho )$  ${V}_{1}$  $\left(\begin{array}{ccc}x& x& y\\ x& x& y\end{array}\right)$  ${\mathbf{Z}}_{2}(\kappa )$ 
${V}_{2}$  $\left(\begin{array}{ccc}x& x& 0\\ x& x& 0\end{array}\right)$  ${\mathbf{Z}}_{2}(\kappa \rho )$ 
6.2 ThreeDot Conventional Experiment

Rotating wave:$\left(\begin{array}{ccc}{x}_{0}& {x}_{2/6}& {x}_{4/6}\\ {x}_{3/6}& {x}_{5/6}& {x}_{1/6}\end{array}\right).$(11)

Two dots inphase:$\left(\begin{array}{ccc}{x}_{0}& {x}_{0}& {y}_{0}\\ {x}_{1/2}& {x}_{1/2}& {y}_{1/2}\end{array}\right).$(12)

Two dots out of phase:$\left(\begin{array}{ccc}{x}_{0}& {x}_{1/2}& {z}_{0}\\ {x}_{1/2}& {x}_{0}& {z}_{1/2}\end{array}\right).$(13)
Since z has double frequency, ${z}_{0}={z}_{1/2}$; so this final state is a partial fusion state as in Fig. 17c.
6.3 ThreeDot Scrambled Experiment
In the two Hopf bifurcations from a maximal fusion state where ρ acts nontrivially (as multiplication by −1), κ acts trivially in one and nontrivially in the other. These bifurcations lead to rivalry or partial rivalry.

κ acts trivially:$\left(\begin{array}{ccc}{x}_{0}& {x}_{0}& {y}_{0}\\ {x}_{1/2}& {x}_{1/2}& {y}_{1/2}\end{array}\right).$(14)
This solution is similar to “rivalry” between the two learned patterns, but it generically will have derived patterns in between. The times when the likecolored dots switch are different due to the phase shift between ${x}_{0}$ and ${y}_{0}$. This timeperiodic solution is the same as the one in Fig. 17b.

κ acts nontrivially:$\left(\begin{array}{ccc}{x}_{0}& {x}_{1/2}& {z}_{0}\\ {x}_{1/2}& {x}_{0}& {z}_{1/2}\end{array}\right).$(15)
In this solution, the third node is fused, and the solution is the same as the one in Fig. 17c.
In short, the only expected nonfusion percepts in the unbalanced scrambled case are the percepts in Fig. 17b. In particular, we do not expect simple alternation between the two learned patterns.
7 Summary of Colored Dot Experiments and Predictions
 (1)
Our theory suggests that the expected percepts for the first two fourdot rivalry experiments (conventional purecolor Fig. 12a and scrambled diagonal Fig. 12b) are identical.
 (2)
Both conventional and scrambled diagonal fourdot experiments admit the possibility of a discrete rotating wave (Fig. 14c), whereas the other two fourdot experiments do not.
 (3)
Suppose the single red dot in the fourdot scrambled unbalanced experiment (Fig. 12d) occurs in the UR position. Then all percepts should have the same color in the UL and LR positions. The scrambled unbalanced experiment is the only one of the fourdot experiments where simple alternation between the two learned patterns is unexpected.
 (4)
The conventional purecolor threedot experiment (Fig. 15a) leads to simple alternation between (learned pattern) purecolor percepts, but it does not lead to simple alternation between mixedcolor percepts. It can also lead to discrete rotating waves.
 (5)
The unbalanced scrambled threedot experiment (Fig. 15b) is not predicted to lead to alternation between either purecolor or mixedcolor percepts (the learned patterns). This experiment is also not expected to lead to rotating waves.
 (6)
The unbalanced scrambled threedot experiment is expected to lead to mixedcolor percept alternation occurring with purecolor percepts in between (see Fig. 17b). In this alternation the color of dots UL and LL should be the same. This experiment could also lead to percepts where UL and LL alternate with opposite colors and R has an indefinite color (fused state).
8 Definition of a Rivalry Network
The possibility that there may be several types of attributes (see the network for the Shevell et al. [17] experiment in Fig. 8) and several types of excitatory couplings (see the networks for the fourdot experiments in Fig. 12 as well as the network in Fig. 8) are the two principal differences between Wilson networks and rivalry networks. These generalizations seem to be necessary to explain the percepts found in the experiments that we have described. The need for additional coupling types was discussed in [13], where we introduced lateral couplings. Here we describe a simpler way of introducing multiple excitatory couplings, based on Hebbian learning, which we formalize through the definition of a feature below.
Following [19] a coupled cell network consists of a collection of nodes or cells , a collection of arrows or edges ℰ, equivalence classes of cells, and equivalence classes of edges.
 (a)The nodes are partitioned into a disjoint union of m attribute columns; that is,$\mathcal{C}={A}_{1}\cup \cdots \cup {A}_{m},$
 (b)
A pattern is a choice of one node from each attribute. There is a distinguished collection of learned patterns; all other patterns are called derived.
 (c)
There are two types of arrows: inhibitory and excitatory.

Each pair of nodes in the same attribute column is connected by a single inhibitory arrow.

Each pair of nodes in the same learned pattern is connected by a single excitatory arrow. (If a pair of nodes is in p patterns, then that pair will be connected by p arrows.)
Inhibitory arrows and excitatory arrows are not arrow equivalent.
Wilson network discussion: An attribute could be the color of a dot and the levels the possible colors. In binocular rivalry experiments, the network has two different learned patterns. In the language of [19], Wilson considers networks where all nodes are cell equivalent, all inhibitory arrows are arrow equivalent, and all excitatory arrows are arrow equivalent. Rivalry networks need not have these properties.
 (d)
Attribute columns partition into attribute equivalent columns such that

Nodes are cell equivalent if and only if they are in attribute equivalent columns.

Inhibitory arrows are arrow equivalent if and only if they are associated with equivalent attributes.

Attribute equivalent columns have the same number of levels.
 (e)
A feature is defined on pairs of nodes in a subset of attribute columns. Two excitatory arrows i and j are arrow equivalent if and only if the features associated to the head and tail nodes of i are identical to the features associated with the head and tail nodes of j.
 (1)
The level feature identifies pairs of nodes in two attribute equivalent columns as either having the same level or having a different level. For example, suppose ${A}_{k}$ is the attribute column defining the color of dot k. Then the level feature identifies pairs of levels in columns ${A}_{1}$ and ${A}_{2}$ as either having the same color or having different colors.
 (2)
The color dot experiments have a distance feature that assigns to any pair of nodes the geometric distance between the associated dots. Thus two excitatory arrows can be equivalent only if their head and tail nodes correspond to equidistant dots.
We have presumed that Hebbian learning strengthens excitatory connections between nodes with the same feature, so that in the end the strength of excitatory connections between pairs of nodes can be identical only if those pairs have the same set of features.
9 Conclusion
It requires some thought to define a rivalry network for any particular rivalry experiment. We have shown here that reasonable descriptions of rivalry networks for a variety of experiments lead (via the use of fusionbreaking Hopf bifurcations) to the prediction of the percepts that are perceived in these experiments. To our knowledge, the form of percepts have not previously been predicted.
Periodic states that are near Hopf bifurcation can exhibit alternation only if that Hopf bifurcation is from a fusion state. If not, the activity of one oscillating node would be bigger than the activity of other oscillating nodes for all time and alternation would not occur. It follows that alternation obtained from small amplitude periodic solutions can only occur robustly in a model network when fusion states are structurally stable in that network. For this reason, we propose that the structure inherent in rivalry networks that leads to fusion states is required to describe alternation.
Several of the rivalry experiments we have discussed exhibit interocular grouping, that is, components of the left eye and right eye images are combined to achieve a single coherent percept [24, 25]. Such groupings occur when learned images can be naturally subdivided and reassembled in other forms. The subdivision process seems to be captured by the attributes and their types (which define node types) and the features (which define excitatory coupling types), as has been shown in the experiments considered here. Together the attributes and features define the rivalry networks and their symmetries. The mathematics of symmetrybreaking Hopf bifurcation leads to an enumeration of the likely ways that the subdivisions can be reassembled, that is, to a set of interocular groupings associated with percepts.
The choice of a network for a rivalry experiment is not unique. For example, in Sect. 2 we choose to model the monkeytext experiment with a twoattribute network. A possible alternative network is one with six attributes, corresponding to the six distinct areas shown in Fig. 5a (without grouping them into blue and white regions). This leads to a 12node network as a model for this experiment, rather than the 4node network of Fig. 5b. However, there is a relationship between these two networks. The larger network has a quotient network on a flowinvariant subspace that is isomorphic to the network in Fig. 5b (see [20] and Sect. 5 of [13]). Thus, the bifurcations in the smaller model are a subset of the bifurcations in the larger model. The solution types discussed in Sect. 2 for the smaller model will also appear in the model with a more refined geometry, however, the larger model can also exhibit other more complicated dynamics not found in the simpler model. In summary, choosing an appropriate network for a given experiment involves tradeoffs and is an iterative process.
We focused here on predicting the form of percepts, which depends on network structure but not on specific model equations. A combined study of specific equations on specific networks, analogous to the work of [3, 4] on twounit models, can be contemplated.
Declarations
Acknowledgements
The authors thank Gemma Huguet, John Rinzel, Ian Stewart, and Hugh Wilson for helpful discussions. This research was supported in part by NSF Grant DMS1008412 to MG and NSF Grant DMS0931642 to the Mathematical Biosciences Institute.
Authors’ Affiliations
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