Making a shape net

Introduction

We have talked about layer objects and their defining shape nets at references 1, 2 and 3. Shape nets which we suppose to be neither very big nor very complicated; maybe less than a hundred nodes and links, taken together, altogether, at any one time.

We return here to their activation, to their projection into consciousness from the layer of cortex which is home to LWS-N.

Figure 1

A simple shape net, for a layer object in four parts, is illustrated in blue above, adapted from reference 3, with the supporting texture nets in green. Note the slight awkwardness of the central node, an awkwardness to which we shall return below.

Figure 2

An example of an image from the outside world (left) with something like a shape net superimposed on it (right). Taken from the neural network challenge at reference 5, a picture taken by an electron microscope of a section of a nerve cord of a fruit fly larva (Drosophila).

For the present we suppose our shape nets to be static, at least for the duration of a frame of consciousness, a second or so, a supposition which we expect to be able to relax in due course.

Electrical aspects and the field of consciousness

We start with the notion of a one dimensional travelling wave, the sort of thing described at reference 4, and we suppose that our shape nets are animated by such waves, for which purpose we suppose the links of our shape nets to be directed. The amplitude of the wave at any point in two dimensional space and time is the firing frequency of the neurons at that point, a frequency derived or defined by some kind of limiting or averaging process. The existence of such waves implies some synchronisation of the firing of the neurons involved and results in the oscillating electrical field which is hypothesised to amount to consciousness.

At reference 1, we suggested that both nodes and links were made up of numbers of neurons, with nodes being a strongly connected clump of closely packed neurons and with links being a more linear, directed structure; the whole being an idealisation of the rather more messy, real world of the human brain. We suppose that these neurons are firing in bursts, synchronised enough for that firing to aggregate to something like a sinusoidal wave travelling along that linear structure, not necessarily a straight line, but unlikely to be anything complicated. With the wave possibly pausing briefly at a node, before moving out along the outbound links, in the case that there were any.

The core proposition of LWS-N is that the subjective experience of consciousness is the product of the electrical field generated by the activation of the neurons in the small bit of cortical sheet that is its host. Small that is, but macroscopic rather than microscopic, say around 5 square centimetres and containing of the order of 100 million neurons  .

Activation which rises above what must be the enormous amount of electrical noise in something like the brain. Which rises above by virtue of the coherence of the activation activity, resulting in something which one might call active stability, perhaps the stability of some sort of limit cycle in some sort of two dimensional phase space in some sort of dynamical system. There is a lot going on, there is a lot of content, but that going on is expressed in travelling waves (and stationary waves at the limit), so there is a lot of redundancy too.

Put another way, in order that these travelling waves survive into the ambient electric field, we require that all the other activity in the vicinity, all the noise, adds up to something close to a firing distribution which is uniform in both time and space, against which background our travelling waves remain visible, can be experienced.

Consciousness exists in the region between too much redundancy and too little, between too much activity and too little. There is an upper limit on what can be held in consciousness and a lower limit on what becomes conscious at all. It is hard to be conscious of the void, at least unless there is a self of some sort perched over the abyss, in which case we no longer have a void.

Geometrical aspects

The overlaying of direction on to the links of a shape net gives rise to sinks and sources, possibly implemented by column objects. A mandatory sink is a node with only inbound links. A mandatory source is a node with only outbound links. While an optional sink is a node with at least one inbound link and an optional source is a node with at least one outbound link.

Figure 3

By way of example we offer above a three part shape net (in blue), adapted from Figure 1,with stalks, the ends of which are necessarily either sinks or sources. In which the two minimal polygons define two parts and the bit of texture net below (in green) defines the third part, a part with uncertain – not to say absent – south eastern perimeter. An uncertainty which might arise because this object is partially occluded by some other object. Note that there are both nodes and links which are outside the maximum polygon and which are inside a minimal polygon. Also that a shape net has at most one maximum polygon, but zero, one or more minimal polygons – and if it has one it necessarily has the other. Also that, Figure 3 above notwithstanding, we expect texture nets to have a lot more nodes and links than their supporting shape nets. Shape nets are coarse grained while the texture nets are fine grained.

We have one mandatory and three desirable requirements on our directed shape nets.

(M1) There should be sufficient sources – at least one –  in the shape net for the activation from them to traverse the entire net. There is a volume control for this activation.

(D1) We would like waves arriving at a node to be in phase and meeting this requirements may mean a little tweaking at the edges, either of the geometry or of the waves.

(D2) We would like it would be possible to traverse the whole of the perimeter of any minimal polygon of a shape net in one direction or the other, clockwise or anticlockwise. It is this traversing which we suppose to give rise to the subjective sense of shape.

(D3) We would like it be possible to traverse the whole of the perimeter of the maximum polygon of a shape net in one direction or the other, clockwise or anticlockwise.

The subjective experience of the shape net is strong and clear to the extent that these requirements are met, requirements which place some limits on the geometry of the shape nets.

Figure 4

So, at the top, we have the activation of a very simple shape net, with the nodes denoted by the blue spots and the direction of travel of the wave along the links denoted by the arrows. The minimum and maximum polygon coincide and it can be traversed. Both desiderata 2 and 3 are met.

The shape net we get by putting two such rectangles side by side is not going to work, as we have the link in the middle having two directions. With one answer being to reverse the direction of flow of the second, as shown at the bottom – with the catch that we meet desideratum 2 but not desideratum 3. However, a device which can be used to build the two tilings which follow.

Figure 5

Two tilings, with the repeat indicated in red. With the right-hand tiling buying its simpler repeat with the additional diagonal links. Bearing in mind that shape nets are not usually about tilings, more the province of texture nets.

Figure 6

Devices which are well enough as far as they go, and we can traverse the perimeters of all the minimal polygons in the figure above – but the five quadrilaterals on the five sides of the central pentagon have to be kept separate. We cannot, for example, collapse the two red nodes onto the green node. And we are a long way from meeting the third desideratum.

Figure 7

A problem which can be analysed in terms of the T-junction circled at A above – a formation of which there are plenty of examples in Figure 2 above.

One solution is to split the lower rectangle into two parts, as in B, two parts which do not really exist in whatever it is that is being imaged. Another is to introduce a narrow triangle, breaking the deadlock by allowing anti-clockwise circulation on both upper left hand and upper right hand parts of the original formation. A third possibility is to allow two way traffic, implicitly dividing the neurons underlying the link into two groups, one for each direction, but with the resultant field degenerating to a standing wave, rather than the travelling waves we desire.

A slightly different presentation of the same problem follows.

Figure 8

This simple shape net, with its associated texture nets, taken from reference 3, does not present a problem as regards the second desideratum. The activity can circulate the right hand part perimeter clockwise, the left hand part perimeter anticlockwise, with the synchronised waves adding along the shared link.

Figure 9

But when we add a third part underneath the first two (lower left) we have our T-junction problem, solved here by the addition of a narrow intermediary part, separating the two clockwise parts. An arrangement which solves the topological problem, but giving us two waves travelling in opposite directions, rather close together. And the closer they are, the closer the resultant electrical field is going to be that arising from the stationary wave which is the simple sum of two such travelling waves.

Another price that we pay is that we do not meet desideratum 3, we cannot traverse the perimeter of the shape net as a whole in one direction, weakening the sense of shape of that whole.

Figure 10

So is it better just to go the whole hog and make the red stretch bidirectional?
Nevertheless, there are some configurations in which it is possible both to traverse the parts and the whole, without needing any links to be bidirectional.

Figure 11

Which amounts to a slightly specialised version of the two colour mapping theorem, a junior relation of the well known four colour mapping problem, with one colour denoting clockwise and the other colour denoting anticlockwise. And there are theorems in graph theory about this sort of thing.

Generally speaking though, our shape nets are only going to approximate to our desiderata and we are not going to be able to traverse both the parts and the whole.

Perhaps we like shape nets to the extent that they meet all our requirements, rather in the way that the ear likes octaves.

Estimation

Let us suppose we have a directed shape net (S) with one or more sources and zero, one or more sinks. We suppose that our shape net can be spanned from its sources, which is to say that one can get to any node on the shape net by following links, in the right direction, from a source. We suppose travelling waves of neuronal excitation to be coming out of the sources and flowing across the net.

Remembering that both nodes and links are assemblies of neurons on our patch of cortical sheet, we define a conduit as a directed, non-repeating chain of individual neurons. The geometric length (G-length) of the conduit is the sum of the lengths of the links, usually more than distance between the start and end nodes, and only equal when the conduit traces something close to a straight line. The synaptic length (S-length) of the conduit is the number of links, one less than the number of neurons. We suppose that the average distance between successive neurons in a conduit is of the order of 0.1mm.

A conduit for a directed link starts at the initial node and ends at the terminal node. The time it takes a travelling wave to span a conduit can be estimated as C1*<S-length> + C2*<G-length>, that is to say a weighted sum of the synaptic latencies and the travel time. We might, for an initial stab, suppose the latency to be 2ms (C1) and the speed of travel 2mm/ms (C2). Stabs which have been informed by references 7 and 8.

A link can be described by the maximum number of disjoint conduits that there are through it, by the minimum, mean and maximum lengths of those conduits. We can talk about the G-length of a link and the S-length of a link, although we do need to come up with proper definitions. The time it takes a travelling wave to span a link (T(L)) can then be estimated as C1*<S-length> + C2*<G-length>.

In diagrams we have portrayed links as straight lines, and while this need not necessarily be the case, we suppose them to be not far removed from straight lines, perhaps arcs of large circles, so the G-length of a link will be quite close to the simple distance between the two nodes which define it.

We want to estimate the time it takes for our net (T(S)) to be spanned by the travelling waves being pumped out of its sources. Some links in the net will be stronger than others, both in the sense that they have more conduits and in that they will be spanned more than once in the time it takes to span the entire net. In spanning the entire net there will be a maximal chain and we denote this chain by C(S).

The G-length of this chain is the sum of the constituent G-lengths, rather than the distance between start and end, which may well be zero. The S-length of this chain is also the sum of the constituent S-lengths. The G-length of the shape net is the G-length of this chain. The S-length of the shape net is the S-length of this chain. Thus giving us T(C(S)).

Then T(S) is defined to be T(C(S)).

The strength of the shape net (S) is inversely proportional to its time (T(S)). A shape net which is being spanned many times in the course of our frame of consciousness, in duration something of the order of one second, makes a stronger impression than one that is not. Giving us a trade-off between complexity of shape net and strength of impression.

Our patch of cortex is supposed to be 5 square centimetres in area. So we can suppose that our shape net has a maximum diameter of 3cm and that the G-length of C(S) is at most 3cm and the S-length of the order of 300. Which gives us a T(S) of 660ms, which is rather too much. So there is work to do here.

We might build more detailed estimates by saying that if there is at least one inbound wave at a node, then all outbound links will acquire outbound waves. We might say something about the amplitude of those waves. Alternatively, we might say that if there is at least one inbound wave at a node, then outbound links acquire outbound waves on a probabilistic basis, depending, inter alia, on the number of inbound waves.

The strength of the shape net is also dependant on the extent to which it meets desiderata D1, D2 and D3.

Some other matters

Dimensions

Part of the LWS-N proposition is that the content of consciousness is built out of two dimensions. Our patch of cortex may well include a lot of topical organisation, but that patch is a two dimensional structure and layer objects are two dimensional structures, although we have thought about allowing two dimensional surfaces of three dimensional objects, squashed down into two dimensions. Notwithstanding, any sense of a third dimension is a clever illusion, perhaps comparable to the illusion given by the large flats of an old fashioned stage set or the small flats of a perspective box.

Differentiation and integration

We say that two nodes in LWS-N are connected if they may be joined by some sequence of links and column objects.

Consciousness is sometimes described as being both differentiated and integrated. See, for example, reference 6. We dispute both claims. A minimal experience, not described above, might arise from a uniform texture net, the sort of thing suggested at Figure 5 above. Not much differentiation here, perhaps not much more than the existence of options. Perhaps we only experience a uniform world of red, something that conceptual artists sometimes try to create in installations in the South Bank Centre, because there is the alternative of a uniform world of blue, perhaps lurking in the background on some other layer. And regarding integration, there might well be two patches of activity on our cortical sheet, both giving rise to subjective experience, but quite unconnected. So not much integration either, at least not necessarily.

Duration

In order for their to be a subjective experience which the subject can report on, we expect the field to have to be sustained, with the waves of activation spanning our cortical sheets many times during a frame of consciousness.

We note as an aside that one might argue that, in a healthy human adult, there is only a subjective experience when it can be reported on, in some way or another, in some fashion or another. When it leaves a trace which is persistent enough to be reported on, a trace which takes a little time to generate.

Transformations

We think here about various manipulations of the subjective experience yielding field generated by LWS-N.

We expect that rigid rotations and translations of the field will not change the subjective experience. There is no interaction with the earth’s magnetic field or anything like that.

We wonder about affine transformations of the field. While it seems likely that some of the experience would be preserved, we wonder, for example, about the topical organisation of the layers carrying the visual field.

We expect that uniform inflation of the field in space would destroy the experience, which depends, in some way yet to be determined, on a lot of action in a small space.

Speculation

We have been thinking about the cerebellum  and about whether it might be a home for LWS-N. An upside would be that it is a fairly homogenous organ packed with a lot more neurons than the cerebral cortex – by maybe a factor of ten.

A downside would be that there are a small number of adults in the world (say less than 20 reported) who seem to be able to get along without one. A difficulty which might be answered by saying that these people either never had a cerebellum or lost it in early in their development, either way giving the developing brain time to compensate, perhaps with some structure in the midbrain, perhaps using material which would have otherwise provided communications with the cerebellum.

Conclusions

We have sketched the way in which activation might span a shape net and given a preliminary estimate for how long this might take. Work in progress.

References

Reference 1: http://psmv3.blogspot.com/2017/09/geometry-and-activation-in-world-of.html.

Reference 2: http://psmv3.blogspot.com/2018/05/a-modest-change-to-layer-objects-of-lws.html.

Reference 3: https://psmv3.blogspot.com/2018/07/from-neurons-to-layer-objects.html.

Reference 4: http://resonanceswavesandfields.blogspot.com/2016/03/three-types-of-waves-traveling-waves.html.

Reference 5: https://imagej.net/Segmentation_of_neuronal_structures_in_EM_stacks_challenge_-_ISBI_2012.

Reference 6: Consciousness: here, there and everywhere - Tononi & Koch – 2015.

Reference 7: https://biology.stackexchange.com/questions/36651/what-is-the-latency-between-paired-neuronal-responses-in-the-brain.

Reference 8: Neural firing latency. Using 'Horizontal Synaptic Connections in Monkey Prefrontal Cortex: An In Vitro Electrophysiological Study - Guillermo González-Burgos, German Barrionuevo, David A. Lewis – 2000.

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