Column objects

 At reference 1, we mentioned column objects, along with layer objects, but deferred their new definition in the world of LWS-R regions. Here we give some options and come to a proposal. We end with a supplement to the material offered at references 1 and 2.

Recall that we have a world of layers, each layer occupying the whole of our patch of cortex, and layer objects, usually divided into regions on those layers. The primary function of column objects is to provide a direct link between objects and regions on different layers. A secondary function may be to provide an indirect link between objects and regions on the same layer.

Figure 1

At reference 1 we introduction the wave element reproduced above which was used to define layers, layer objects and regions. With a quantity called ω (omega) fixed across a layer. We start here with the idea of frequency modulation.

Option 1

At reference 1, we talked of regions, that is to say the constituents of layer objects, which were defined by adding up a small number of wave elements of the form above, with a layer being defined by its value of ω. These waves and the regions they are defined on are usually fixed for the duration of a frame.

Figure 2

Suppose now we wish to link region K of object L on layer M defined by ω=N – sketched in the figure above, adapted from reference 1 – to region Q of object R on layer S defined by ω=T.

We take out a small area, Z, in the interior of both region K and region Q, which are thus required to overlap substantially in space, where instead of some combination of waves all with ω=N, we have a rather simpler wave – but one where ω oscillates regularly in time between N and T, thus serving as a link between region K of object L and region Q of object R.

Figure 3

Z is distinguished from other, layer-bound structures in LWS-R by ω not being fixed, being modulated by φ. The figure above suggests just one way of doing this, a sine within a sine, delivering an ω which oscillates between N and T.

Z is also a defect in the expression of both object L and object R – the price we have to pay for linking the two together. No electrical link without some disturbance of both L and R. We associate to the optic nerve of the eye.

We are also looking for some kind of transfer of energy here, for pulses of energy to be travelling between region K & object L and region Q & object R. For there to be a kind of oscillation of attention between them. But this idea is not here developed.

Option 2

Figure 4

Under this option the column object is defined as a special region, rather than as a hole in a region. The column object is not related to a particular region, more to the layer object as a whole, although it will be adjacent to, nearer to some regions than others.

Otherwise, the wave modulation would be as in option 1.

An advantage here being the re-use of region, rather than introducing a new device.

Option 3

Having got to re-use of region, here we use coincidence in space of two regions to define a column object, rather than introducing a modulated wave. A region on one layer is linked to a region on another layer if they share the same space. One might allow the strength of that link to vary with the degree of coincidence.

Note that two layer objects are deemed to be linked if they have a region in common. There is no need for the objects as a whole to coincide – although they may do that as well – a point to which we return below.

This geometric definition may be enough, but it may prove convenient to add restrictions to the wave elements in the two object. One might, for example, require there to be just one element defining each of the two regions, only differing by their value of ω. A disadvantage of such restrictions would be the flaws in both objects, flaws which we did not, for some reason, notice.

For the present, this is the preferred option. We will see how it works out.

Option 4

Figure 5

As things stand, column objects can only be used to link regions at the same place on different layers. We could generalise, by using an intermediate layer object, the main purpose of which was to act as a holder for column objects. In this way one could, in two steps rather than one, link more or less any pair of regions.

So in the figure above, the blue object left is linked to the green object right by the brown object in the middle. Blue and brown must be on different layers, as must brown and green, but blue and green can be on the same layer. For simplicity of presentation, we omit the regional breakdown of these objects.

A complication is that on the LWS-R hypothesis, the brown object is apt to be in consciousness, as well as the blue and green objects. A distracting meta-object. But there may be some device which lets it do its linking work in the background, rather than in the foreground.

An option which we do not further pursue here.

Supplement

Uniformity

The patch of cortex on which LWS-R resides is supposed, for the present anyway, to be uniform. That the mix of neurons – of which there is a regular menagerie in the brain at large  – and everything else is the same all over the patch. Where here we are talking about the matrix of neurons which are generating our travelling waves, participating in regions and layer objects. There will be plenty of other neurons in supporting roles, not least those to do with the compiler. Uniformity as defined here does not need to extend to them.

More or less flat

We also suppose the patch to be more or less flat, not intricately folded in the way of, for example, the cerebellum. This because we suppose that on an intricately folded surface the interaction between the fields generated by the different parts of the patch would destroy much of the signal.

We do not know where in the brain one might find such a patch.

We note that the retina does not exhibit this sort of uniformity with systematic variation of the mix of cells from centre, at and around the fovea, to the periphery.

Refinements and versions

Figure 6

We may have dropped the shape nets of LWS-N, in the sense of having large numbers of neurons defining the nodes and edges of such nets, but something equivalent to shape nets is implicit in the connected set of regions making up a layer object. On the assumption that the regions of a layer object partition the space occupied by a layer object, the boundaries of the regions, taken together, amount to the shape net, this being suggested by the figure above, derived from the right hand object in Figure 9 below.

Note that the boundaries between regions, while quite possibly constrained to being reasonably simple, need not be the straight lines shown above.

Such shape nets can be related in a graph theoretic way, just paying attention to nodes and edges. Noting, that such graphs, for the moment anyway, are all planar; that is to say that they can be expressed in the plane without any crossing edges. A restriction we have in the past suggested relaxing, to allow a degree of three dimensional vision.

Figure 7

To which one might add geometric congruence: a region does not just have the same nodes and edges as another, it occupies the same, or a congruent place, in space. So in the figure above, the left hand pair of objects are congruent, differences in texture notwithstanding. The middle and right hand pairs are not.

We note in passing that one difference between upper and lower middle objects, apart from their different textures, is that the smaller size comes with lower resolution. The packing of the neurons on our patch is fixed and finite, so resolution comes down with size. Not like a technical drawing package on a computer where the resolution everywhere is more or less infinite.

Which ideas on congruence we use to define copy, version and refinement.

One layer object is a copy of another if they and their constituent regions are congruent. That is to say, congruent as shape nets: we say nothing about whether the constituent regions have the same textures, the same waves.

One layer object is a version of another if the two objects occupy the same space on different layers. We have yet to explore whether requiring version to have, in addition, congruent regional structure is helpful. 

We may expand option 3 above so that if one object is a version of the other, even if there is no pair of coincident regions, the objects are deemed to be linked, are experienced as being linked. An expansion qualified by the idea that the more regions that are coincident, the stronger the link.

Figure 8

In any event, versions is a way of allowing, for example, consciousness to hold two visual images of the same thing at the same time, which we believe will be useful when considering the subjective experience generated by pictures and diagrams, for example that above, taken from reference 3.

Figure 9

Lastly, in the figure above, give or take the vagaries of Powerpoint, the right hand object is a refinement of the left hand object. That is to say, the two objects taken as wholes are congruent to each other and all the parts on the left are either congruent to the corresponding part of the object on the right or to the union of a small number of parts of the object on the right. So the region marked A left is broken down into two regions right, and the region marked B left is broken down into three regions right. The right hand object is a refinement of the left hand object.

Note that this refinement is a transitive relation but not a commutative relation.

Conclusions

We have suggested some ways of doing column objects, a matter left over from references 1 and 2. We have come up with a preferred option.

We have added some other supplementary material.

References

Reference 1: http://psmv4.blogspot.com/2020/09/an-updated-introduction-to-lws-r.html.

Reference 2: http://psmv4.blogspot.com/2020/08/waved-up-regions.html. 

Reference 3: http://psmv4.blogspot.com/2019/11/more-on-making-regions-into-objects-and.html. 

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