Plato's Cave: Where in the Brain

Where in the Brain

I have described a spherical computational structure with a specific pattern of connectivity. Where in the brain might such a structure be located?

Beginning at the lower end of the phylogenetic tree, the computational structure depicted above might correspond to the spherical structure seen at the center of the insect visual system, where the base of the ommatidia meet. This model predicts therefore that neural connections in the insect visual system will be found in a pattern as shown above, designed to perform boundary and surface completion in three-dimensions. A significant feature of this representation is that it would represent not only the visible part of the visual field, but would also have a representation for the part of the world beyond the visual field, or occluded by the insect's own body, even if that sector is not covered by omatidia facing in that direction, and that collinear completion (e.g. of the horizon line) would occur across these blind sectors. This hypothesis is testable in principle.

In human vision no such spherical structure is apparent. The structure need not however be physically spherical, it need only be computationally spherical. For example, the sphere represented above might be opened up at the back and stretched out into a flat circular disk, as suggested below,

i.e. by inserting a vector from the posterior point in to the center of the sphere, and then stretching the resulting cavity outwards towards the surface of the sphere, resulting in a flattened, partially spherical surface with a finite thickness. The result of this topological transformation is shown below, where the inner surface of the perceptual sphere, representing the surface of the head, would correspond to the near face of the disk, and the outer surface of the sphere representing perceptual infinity would correspond to the far surface of the disk, while the thickness of the disk would correspond to perceptual depth. A face is sketched on the near surface of the disk in the figure to indicate that this surface represents the surface of the face, while the rest of the body appears in perspective at the bottom, beyond the cross-section of the neck, shaded with hatched lines in the figure. The outer rim of this perceptual disk represents all of space behind the observer, and the entire line of the outer perimeter represents the single point directly behind the man's head. The road which stretches to the horizon behind, stretches to the periphery of the disk to the right and left, disappearing as it reaches the perimeter of the disk on both sides.

If the neural connectivity were preserved topologically across this geometrical transformation, the computational properties of this flat disk would be identical to that of the perceptual sphere. On the other hand, a less topological transformation would result in a loss of perceptual resolution in depth relative to azimuth and elevation, and perhaps a nonlinear loss of resolution to the posterior hemisphere of space relative to the anterior half, as is seen in the visual cortex. Indeed, this structure now begins to resemble the mapping of the primary visual cortex.

The next figure shows how this transformation would affect the shape of the body map within the spatial map, inflating the face while shrinking the body.

Notice that in this model, an integral part of the representation of space is a representation of the body surface, as suggested by the [spatial robot analogy] which would be represented in the same manner as any other surface in the world, except of course that it would be updated both visually and through somatosensory signals. The phantom limb phenomenon experienced by certain amputees suggests how a somatosensory signal can generate a spatial representation of a limb in the absence of visual confirmation. This portion of the model would presumably correspond to the somatosensory cortex which represents a spatial map of the body surface.

Two issues remain to be resolved: what about the higher cortical maps in V2, V3, IT, and other areas, and how is this mapping affected by rotation of the eyes, rotation of the head, or rotation of the body? For this model suggests that a rotation of the body should result in a counter-rotation of the "image" of the external world relative to the perceptual sphere. On the other hand, if recognition of objects in the world corresponds to the activation of object recognition cells, this shifting image of the world must somehow be anchored, if those cells are to have a fixed location in the brain.

A solution to this problem was suggested earlier in the discussion of the [invariant representation] where it was shown how the presence of a pattern of activation in a matrix of cells can stimulate a global cell in a rotation, translation, and scale invariant manner. In the Bubble World model, this argument can be extended into three dimensions by way of spherical harmonics in the perceptual sphere, which can define three-dimensional forms in an invariant manner. For example an eight-way vertex in spherical harmonics would define an eight-pointed shape at a higher level, which would complete by boundary and surface filling-in at the lower levels to an eight cornered cube, as suggested below. The cube would be encoded by a resonator (i.e. tuned filter / oscillator) tuned to the frequency which would promote the emergence of this eight-way harmonic in the perceptual sphere in a rotation, translation, and scale invariant manner.

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