# Symmetric Networks with Geometric Constraints as Models of Visual Illusions

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## Abstract

**:**

## 1. Introduction

#### 1.1. Neuroscientific and Image-Recognition Context

#### 1.2. Overview of the Paper

## 2. Wilson Networks

#### 2.1. Geometric Consistency

## 3. Rate Equations for the Dynamics

#### 3.1. Symmetry-Breaking Hopf Bifurcation

- (1)
- $W\cong V\oplus V$ where V is absolutely irreducible.
- (2)
- W is irreducible of type $\mathbb{C}$ or $\mathbb{H}$.

**Theorem**

**1.**

## 4. Examples of Illusions

#### 4.1. Necker Cube

#### 4.2. Rabbit/Duck

#### 4.3. Model-Independent Analysis

- ears + head facing right = rabbit
- beak + head facing right = transitional percept
- beak + head facing left = duck
- ears + head facing left = transitional percept

- ears + head facing right = rabbit
- ears + head facing left = transitional percept
- beak + head facing left = duck
- beak + head facing right = transitional percept

#### 4.4. Model-Dependent Analysis

## 5. 16-Node Necker Cube Network

- (1)
- Lines that are (near) vertical in the image are (near) vertical in the 3-dimensional object detected as the percept. There is much evidence that the vertical direction is special in vision; see for example Quinn [83].
- (2)
- Lines that are (near) parallel in the image are (near) parallel in the 3-dimensional object detected as the percept.
- (3)
- Lines that are not (near) parallel in the image are not (near) parallel in the 3-dimensional object detected as the percept.

#### 5.1. Symmetry-Breaking Hopf Bifurcation

## 6. Analysis of the Rate Model

**Remark**

**1.**

#### 6.1. Eigenstructure of the Adjacency Matrix

#### 6.2. Spatiotemporal Symmetries of Critical Eigenspaces

- $\rho $ Reflects the diagram left-right.
- $\tau $ Reflects the diagram top-bottom.
- $\omega $ interchanges F and B in each pair of nodes with the same number.

#### 6.3. Special Model

## 7. Analysis of the Special Model

**Definition**

**1.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**3.**

**Proof.**

#### Conditions for First Bifurcation

**Theorem**

**4.**

**Proof.**

## 8. Transitional States Are Impossible Figures

## 9. Tristable Necker-Like Figure

- A small cube (dark grey) in a ‘room’—a corner where three rectangles meet (light grey).
- A small cubical hole (dark grey) removed from a cube (light grey).
- A small cube (dark grey) in front of a large cube (light grey).

#### 9.1. Why Is No Fourth Percept Observed?

#### 9.2. Completing the Network

- If the large component is at level cube then the small component can reasonably occur at either level cube or corner, so we insert excitatory connections from node 1 to nodes 3 and 4.
- If the large component is at level corner then by the above discussion the small component can occur in a structurally stable manner only at level cube, so we insert an excitatory connection from node 2 to node 3 and an inhibitory connection from node 2 to node 4.

## 10. Further Remarks and Summary

## 11. Conclusions

- Wilson networks with natural geometric consistency conditions are capable of modelling the perception of multistable illusions.
- A relatively small number of local geometric consistency conditions can generate the observed global form of the percepts.
- A potentially important type of geometric consistency is a form of structural stability: the percept should not depend on features of the image that can be destroyed by small perturbations.
- Important features of rate models, such as the first Hopf bifurcation from a fully synchronous equilibrium, can be understood analytically, even for quite complicated networks, provided connection strengths are gain-homogeneous and the network has sufficient symmetry.
- In particular, the first bifurcation from a fusion state in the 16-node Necker cube model selects a unique spatiotemporal pattern that matches observations.
- However, in some cases (including the 16-node model) transitional states occur that do not satisfy the geometric consistency conditions used to construct the model. These percepts correspond to impossible figures, but probably occur so briefly that they would be difficult to observe.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**(

**Left**) Architecture of an untrained Wilson network. (

**Right**) Network trained on a single pattern by adding excitatory connections between the levels representing that pattern. Dashed lines are inhibitory, solid lines excitatory.

**Figure 3.**Attributes for Necker cube. At each intersection, does the horizontal line pass over or under the vertical one?

**Figure 4.**Network for 4-node Necker cube model. At each intersection, does the horizontal line pass over or under the vertical one? Lines ending in bars indicate inhibitory connections, arrows excitatory ones.

**Figure 5.**Time series for rate equations for 4-node model of Necker cube. (

**Left**) Nodes 1 (red) and 2 (blue). (

**Right**) Nodes 3 (green) and 4 (magenta). Here $\epsilon =0.67,g=1.8,I=1,\alpha =-0.6,\beta =-0.7,\gamma =0.9$. Time series for nodes 1 and 4 coincide, as do those for 2 and 3.

**Figure 6.**Network for the rabbit/duck illusion. (

**Left**) One-way connections for geometric constraints. (

**Right**) Two-way connections. Lines ending in bars indicate inhibitory connections, arrows excitatory ones.

**Figure 7.**Plot of ${x}_{1}$ (blue, initially positive), ${x}_{2}$ (blue, initially negative), ${x}_{3}$ (red, initially positive), ${x}_{4}$ (red, initially negative).

**Figure 8.**Simulation of Figure 6 (left). Here $\epsilon =0.67,g=1.8,I=1.5,\alpha =-1.5,\beta =-1.4,\gamma =0.5,$$\delta =-0.2$. Colors are: node 1 red, node 2 blue, node 3 green, node 4 magenta.

**Figure 9.**Dynamics in the 4-node network. Colors are: node 1 red, node 2 blue, node 3 green, node 4 magenta. (

**Left**) Parameter values are $\epsilon =0.67,g=1.8,I=1.5,\alpha =-0.8,\beta =-0.3,\gamma =0.4$, $\delta =-0.3$. (

**Right**) Same parameters except that $\epsilon =0.01$.

**Figure 10.**(

**Left**) Eight numbered edges of the Necker cube. (

**Right**) Network modelling the orientations of the eight numbered edges. For convenience, the ‘columns’ of this Wilson network are drawn as horizontal pairs and the network is not drawn as a single rectangular array. All connections bidirectional; arrows omitted for clarity. A subscript e indicates an excitatory connection; i indicates an inhibitory one.

**Figure 11.**The eight Hopf bifurcation patterns predicted by the equivariant Hopf theorem. White/grey shading indicates half-period phase shift. Dotted pattern in nodes 5–8 is a reminder that these nodes have a different waveform from nodes 1–4.

**Figure 12.**General model. (

**Left**) Time series for all F nodes. Parameters are ${\alpha}_{e}=0.2,{\alpha}_{i}=-0.5,{\beta}_{e}=0.4,{\beta}_{i}=-0.2,\gamma =-1.4,\epsilon =0.3,{I}_{0}=1,g=1.8$. (

**Right**) Enlarged view of the crossings, showing phase lag between the switching times. Colors correspond to nodes as follows: red = 1F, blue = 2F, green = 3F, magenta = 4F, cyan = 5F, black = 6F, brown = 7F, purple = 8F.

**Figure 13.**General model. (

**Left**) Time series nodes 1F, 2F, 3F, 4F. (

**Right**) Time series nodes 5F, 6F, 7F, 8F. Each trace shows two superposed time series of exactly synchronous nodes.

**Figure 14.**Special model. (

**Left**) Time series for all F nodes. Parameters are ${\alpha}_{e}=0.3,{\alpha}_{i}=-0.3,{\beta}_{e}=0.4,{\beta}_{i}=-0.4,\gamma =-1.4,\epsilon =0.3,{I}_{0}=1,g=1.8$. (

**Right**) Enlarged view of the crossings, showing phase lag between the switching times. Colors correspond to nodes as follows: red = 1F, blue = 2F, green = 3F, magenta = 4F, cyan = 5F, black = 6F, brown = 7F, purple = 8F.

**Figure 15.**(

**Left**) Impossible Penrose triangle. (

**Right**) Impossible rectangle formed by edges 1 and 2 of Figure 10 (left) in the transitional percept.

**Figure 16.**Tristable figure from Poston and Stewart [14].

**Figure 17.**Decomposing tristable figure from Poston and Stewart [14].

**Figure 18.**Attributes and levels for tristable figure from Poston and Stewart [14].

**Figure 19.**(

**Left**) Is the black face of the small component being viewed from above, or from beneath? If above, the image is perceived as a cube; if beneath, the image is perceived as a corner. (

**Right**) Are the black faces parallel? If they are, the image is perceived either as a small cube in a corner, or as a big cube with a corner cut off. If not, it is perceived as a small tilted cube in front of a large cube.

**Figure 21.**Dynamics in the 4-node network. (

**Left**) $\epsilon =0.67,g=1.8,I=1.5,\alpha =1.65,\beta =0.05,\gamma =0.05,\delta =0.03,\zeta =0.3,\eta =0.2$. (

**Right**) All parameters unchanged, except $\beta =0.6$. Colors are: node 1 red, node 2 blue, node 3 green, node 4 purple.

Pattern | Eigenvalue |
---|---|

1 | $\frac{1}{2}({\alpha}_{e}+{\alpha}_{i}+{\beta}_{e}+{\beta}_{i}+2\gamma \pm \sqrt{{D}_{1}})$ |

2 | $\frac{1}{2}({\alpha}_{e}+{\alpha}_{i}-{\beta}_{e}-{\beta}_{i}+2\gamma \pm \sqrt{{D}_{2}})$ |

3 | $\frac{1}{2}(-{\alpha}_{e}-{\alpha}_{i}-{\beta}_{e}-{\beta}_{i}+2\gamma \pm \sqrt{{D}_{1}})$ |

4 | $\frac{1}{2}(-{\alpha}_{e}-{\alpha}_{i}+{\beta}_{e}+{\beta}_{i}+2\gamma \pm \sqrt{{D}_{2}})$ |

5 | $\frac{1}{2}({\alpha}_{e}-{\alpha}_{i}-{\beta}_{e}+{\beta}_{i}-2\gamma \pm \sqrt{{D}_{3}})$ |

6 | $\frac{1}{2}({\alpha}_{e}-{\alpha}_{i}+{\beta}_{e}-{\beta}_{i}-2\gamma \pm \sqrt{{D}_{4}})$ |

7 | $\frac{1}{2}(-{\alpha}_{e}+{\alpha}_{i}+{\beta}_{e}-{\beta}_{i}-2\gamma \pm \sqrt{{D}_{3}})$ |

8 | $\frac{1}{2}(-{\alpha}_{e}+{\alpha}_{i}-{\beta}_{e}+{\beta}_{i}-2\gamma \pm \sqrt{{D}_{4}})$ |

**Table 2.**Spatiotemporal isotropy subgroups of the eight types of primary Hopf bifurcation. ‘No.’ refers to the position in Figure 11 reading from left to right, top to bottom. For specific choices of parameters, $u,v$ take specific values.

No. | $\mathit{\rho}$ | $\mathit{\tau}$ | $\mathit{\omega}$ | Spatial K | Eigenvector Structure (Activity Variable Only) |
---|---|---|---|---|---|

1 | + | + | + | $\mathsf{\Omega}$ | ${[u,u,v,v,v,v,u,u,u,u,v,v,v,v,u,u]}^{\mathrm{T}}$ |

2 | − | + | + | $\langle \tau ,\omega \rangle $ | ${[u,u,v,v,v,v,u,u,-u,-u,-v,-v,-v,-v,-u,-u]}^{\mathrm{T}}$ |

3 | + | − | + | $\langle \rho ,\omega \rangle $ | ${[u,u,v,v,-v,-v,-u,-u,u,u,v,v,-v,-v,-u,-u]}^{\mathrm{T}}$ |

4 | − | − | + | $\langle \omega ,\rho \tau \rangle $ | ${[u,u,v,v,-v,-v,-u,-u,-u,-u,-v,-v,v,v,u,u]}^{\mathrm{T}}$ |

5 | − | + | − | $\langle \tau ,\rho \omega \rangle $ | ${[u,-u,v,-v,v,-v,u,-u,u,-u,v,-v,v,-v,u,-u]}^{\mathrm{T}}$ |

6 | + | + | − | $\langle \rho ,\tau \rangle $ | ${[u,-u,v,-v,v,-v,u,-u,-u,u,-v,v,-v,v,-u,u]}^{\mathrm{T}}$ |

7 | − | − | − | $\langle \rho \tau ,\rho \omega \rangle $ | ${[u,-u,v,-v,-v,v,-u,u,u,-u,v,-v,-v,v,-u,u]}^{\mathrm{T}}$ |

8 | + | − | − | $\langle \rho ,\tau \omega \rangle $ | ${[u,-u,v,-v,-v,v,-u,u,-u,u,-v,v,v,-v,u,-u]}^{\mathrm{T}}$ |

Pattern | Eigenvalue | Symbol |
---|---|---|

1 | $\gamma $ [multiplicity 2] | ${\mu}_{1}^{\pm}$ |

2 | $\gamma $ [multiplicity 2] | ${\mu}_{2}^{\pm}$ |

3 | $\gamma $ [multiplicity 2] | ${\mu}_{3}^{\pm}$ |

4 | $\gamma $ [multiplicity 2] | ${\mu}_{4}^{\pm}$ |

5 | $\alpha -\beta -\gamma \pm \sqrt{5{\alpha}^{2}-2\alpha \beta +{\beta}^{2}}$ | ${\mu}_{5}^{\pm}$ |

6 | $\alpha +\beta -\gamma \pm \sqrt{5{\alpha}^{2}+2\alpha \beta +{\beta}^{2}}$ | ${\mu}_{6}^{\pm}$ |

7 | $-\alpha +\beta -\gamma \pm \sqrt{5{\alpha}^{2}-2\alpha \beta +{\beta}^{2}}$ | ${\mu}_{7}^{\pm}$ |

8 | $-\alpha -\beta -\gamma \pm \sqrt{5{\alpha}^{2}+2\alpha \beta +{\beta}^{2}}$ | ${\mu}_{8}^{\pm}$ |

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**MDPI and ACS Style**

Stewart, I.; Golubitsky, M.
Symmetric Networks with Geometric Constraints as Models of Visual Illusions. *Symmetry* **2019**, *11*, 799.
https://doi.org/10.3390/sym11060799

**AMA Style**

Stewart I, Golubitsky M.
Symmetric Networks with Geometric Constraints as Models of Visual Illusions. *Symmetry*. 2019; 11(6):799.
https://doi.org/10.3390/sym11060799

**Chicago/Turabian Style**

Stewart, Ian, and Martin Golubitsky.
2019. "Symmetric Networks with Geometric Constraints as Models of Visual Illusions" *Symmetry* 11, no. 6: 799.
https://doi.org/10.3390/sym11060799