Saccadic eye movements: oculomotor control in the superior colliculus

• Introduction
• Method
• Implementation
• Discussion
• References

Introduction

Method

Figure 1: Schematic of the architecture of the model. White nodes represent the stimulus, dark grey nodes represent the direction of the saccade. In the centre, the light grey nodes represent the lattice units. Not all lattice units are shown.

“...we describe the correspondence between the visual stimulus and the resultant saccade simply by a pair of values $\left({\mathbf{w}}_r,{\mathbf{w}}_r^{\left(\text{out}\right)}\right)$ of the centrally localized neuron. In reality, the resultant saccade is determined by a group of excited neurons localized at $r$ .”

Ritter et al [1992]: 146

“Every neural unit has two radial and two circumferential neighbours.”

Ritter et al [1992]: 150

1. Present a vector ${\mathbf{v}}^L$ in accordance with the probability distribution $\mathrm{P}\left({\mathbf{v}}^L\right)$ above.
2. Find the centre of excitation $c^L$ in the lattice according to the “winner-takes-all” condition
   $\parallel {\mathbf{v}}^L-{\mathbf{w}}_{c^L}^L\parallel \le \parallel {\mathbf{v}}^L-{\mathbf{w}}_i^L\parallel \mathrm{\forall i}$
3. Perform the learning step for the lattice weights to form a Kohonen-like topology-conserving map onto the lattice
   $\Delta {\mathbf{w}}_i^L={\eta }^L{h}_{{i,c}^L}^L({\mathbf{v}}^L-{\mathbf{w}}_i^L)\mathrm{\forall i}$
4. Trigger the saccade centred on $c^L$ so that the new position ${\mathbf{v}}^S$ of the image occurs at
   ${\mathbf{v}}^S={\mathbf{v}}^L+\frac{\sum _{\mathrm{i}}{h}_{{i,c}^L}^S{\mathbf{w}}_i^S}{\sum _{\mathrm{i}}{h}_{{i,c}^L}^S}$
5. If the image is now in the fovea, i.e. $\parallel {\mathbf{v}}^S\parallel <r_{\text{fovea}}$ , go to step (1)
6. Find the new centre of excitation $c^S$ in the lattice for the stimulus ${\mathbf{v}}^S$ according to
   $\parallel {\mathbf{v}}^S-{\mathbf{w}}_{c^S}^L\parallel \le \parallel {\mathbf{v}}^S-{\mathbf{w}}_i^L\parallel \mathrm{\forall i}$
7. Trigger the saccade centred on $c^S$ so that the new position ${\mathbf{v}}^{\mathrm{S\prime }}$ of the image occurs at
   ${\mathbf{v}}^{\mathrm{S\prime }}={\mathbf{v}}^S+\frac{\sum _{\mathrm{i}}{h}_{{i,c}^L}^S{\mathbf{w}}_i^S}{\sum _{\mathrm{i}}{h}_{{i,c}^S}^S}$
8. If this corrective saccade is an improvement, i.e. $\parallel {\mathbf{v}}^{\mathrm{S\prime }}\parallel <\parallel {\mathbf{v}}^S\parallel$ , perform the learning step for the saccade weights according to
   $\Delta {\mathbf{w}}_i^S={\eta }^S{h}_{{i,c}^L}^S\left(\right({\mathbf{w}}_{c^L}^S+{\mathbf{w}}_{c^L}^S)-{\mathbf{w}}_i^S)\mathrm{\forall i}$
9. Go to step (1)

• ${\eta }^L\left(\mathrm{t}\right)=0.3\cdot \exp \left(-0.0002t\right)$
• ${\sigma }^L\left(\mathrm{t}\right)=10\cdot \exp \left(-0.0003t\right)$
• ${\sigma }^L\left(\mathrm{t}\right)=10\cdot \exp \left(-0.0003t\right)$
• ${\sigma }^S\left(\mathrm{t}\right)=3\cdot \exp \left(-0.0003t\right)$

“The precise form of the distance measure between two neural units in the lattice is inconsequential for the organizational process to converge; however, sometimes a certain metric may fit a problem better than other distance measures. ...we used the ‘Manhattan’ rather than the Euclidean metric. ... it is the distances in the lattice which determine the spatial interaction between the neurons themselves and, thereby, determine the distance-dependent adaptation steps in the model. ... (I)n the vicinity of the fovea ... receptors that are directly opposite to each other lie close together but have to learn saccades that differ as much as saccades that belong to receptive fields directly opposite ... Therefore, it makes sense to use the ‘Manhattan’ metric which yields, for the foveal and peripheral pair, the same lattice distance between the diametrically opposite neural units”.

Ritter et al [1992]: 155

Implementation

Figure 2: Lattice and saccade weight space before any stimuli

Figure 3: Lattice and saccade weight space after 4,000 stimulus presentations

Figure 4: Lattice and saccade weight space after 8,000 stimulus presentations

Figure 5: Lattice and saccade weight space after 16,000 stimulus presentations

Discussion

“Compared to these observations, our model learns its saccadic eye motions much too ‘well’ because at the end of the learning process all of our saccades precisely lead in the fovea.”

Ritter et al [1992]: 160

References