A story which is very much a reprise of that at reference 5.
Fields in general
In mathematics, a field is a set on which we have defined addition, subtraction, multiplication and division. With a zero for addition & subtraction and with a one for multiplication & division. And generally behaving, in many ways at least, like the (field of) real numbers. Or the rational numbers, or the complex numbers. Such a field is a construct of algebra.
In physics, a field is something quite different. A field is about some physical phenomenon – perhaps the gravitational field of some massive object – and the fields in which we are interested here all have extent in some perfectly ordinary three dimensional space and can be expressed as an often time varying map from that space to some value space – perhaps the set of positive real numbers. Values which can always be postulated, which can sometimes be computed and which can sometimes be measured – with it sometimes being relevant, sometimes being important, that the business of measurement can change that which is being measured.
More formally, one of these fields of physics is a map φ from some space-like domain of interest D to a range of values R. By which we mean that each point in D has a position in some three dimensional Euclidean space – that is to say the position of each such point can be described by three real numbers – often called the x, y and z coordinates – and has a value in R.
Note that, while in what follows we will talk of continuity, that is to say talk of the map φ preserving nearness, at least most of the time, we will not talk of φ preserving addition. More or less normal vector addition is available in both D and R, but φ does not preserve that addition. It is not the case that φ(a+b) = φ(a) + φ(b), where the first ‘+’ is addition in D and the second ‘+’ is addition in R. D is about space, about geometry, and is not about doing sums.
So if our phenomenon is gravity – the Newton rather than the Einstein sort – we might put a massive object at the centre of our three dimensional space and the value of the field at some other point in that space is a vector in that same space, with the length of that vector being the magnitude of the gravitational pull of our massive object at that point and with its direction being the direction of that pull, in this simple case, always towards the centre. Our field is the gravitational field generated by our massive object. Or we might want the gravitational field generated by a number of massive objects, with this field determining the probably complex behaviour of those objects in time. Or we might want the electrostatic field generated by a number of charged objects – the link here being that electric charge involves an inverse square rule near identical to that of gravity. With one of the differences being that, despite there being at base just the negatively charged electron, we allow charge to be positive or negative and we allow oppositely charged objects to be very close together – to make electric dipoles – like the molecules of water – which does not happen in the case of gravity, where all charge, that is to say mass, is positive and all forces are attractive. Another difference being that the electrostatic attraction between a proton and an electron is getting on for 1040 times bigger than the corresponding gravitational attraction – bearing in mind that electrons are a rather concentrated form of charge, which one does not find at scale.
Both the fields of gravity and of electrostatics allow addition of a different sort to that mentioned above. So if we have a body at A, we can talk of the value of the field at B. If we have bodies at A1 and A2, we can talk of the fields generated by A1 and A2 at B – or we can add them together to make a unified, a combined field. The two vectors of force arising from the two fields (φ and χ) can be added together, as vectors, to produce a combined vector of force, which can be considered to arise from a combined field (ψ). So we do not have ‘φ(a+b) = φ(a) + φ(b)’ but we do have ‘φ(a) + χ(a) = (φ+χ)(a) = ψ(a)’.
Or if our phenomenon is the temperature at ground level of the earth, our domain is the surface of the globe – which we might approximate as the surface of a sphere, quite a small subset of three dimensional space as a whole – and our values are simply real numbers, temperature in some units or other, often degrees centigrade. Or it might be the wind at ground level, in which case our values are vectors which are tangent to the surface of our sphere, vectors which give us both force and direction.
Or if we are astrophysicists, our domain might be the whole of space time.
Or if we are neurologists, our domain might be the interior of someone’s brain, what we can get at with an fMRI machine. Or it might be the surface of someone’s brain, what we can get at with an EEG machine. Very small fractions both of the whole of space time.
So our values might be scalars, say real numbers. They might also be vectors, that is to say pairs or triples of real numbers. There are plenty of other possibilities, but they do not need concern us here.
There will be a natural map from R to some one, two or three dimensional Euclidean space. R might map onto the unit line, onto the unit square or it might map onto the unit circle – that is to say the circle rather than the disc, the circle excluding the interior. Values which might be the phase angle of something or other, probably expressed modulo 2π. Winfree has written, for example at reference 2, of some interesting fields which take values of this sort. Or to take another example, the amphidromic points of tides, the points where tides vanish, a modest number of which are scattered about the oceans.
In any event, we are able to define distances, in the ordinary way, on both D and R. We then require our fields to be continuous, continuous in the sense that they preserve nearness, as defined by those distances, almost all the time. So if two points are close in D we require their values to be close in R.
Noting that the electrostatic field is not continuous at a point charge and that the gravitational field is not continuous at a point mass.
We expect the derivatives of these values in both space and time to be continuous more or less everywhere.
In LWS-N, we are interested in an electrical field φ of some sort which takes scalar values. Not to be confused with the Φ (aka phi) of reference 3, which might be about consciousness, but which is something else altogether.
Moreover, we are interested in the values taken by our field inside the space spanned by the (charged) ions which are generating that field by moving around the neurons of our unit square. We are not so interested in the values that it might take at a distance, in the way of the measurements made by an EEG headset, very much outside rather than inside the brain. Put another way, we are interested in what is going on inside the head, not in what can be seen from outside – it seeming quite likely that the latter is going to fall well short of the former for a good many years yet.
We do not concern ourselves here with problems of measurement, with the need for those purposes to divide space, time and values into discrete bins.
We do not expect to have a mathematical function which describes our field. It is enough that the field should exist and that we should know something about it – but that knowledge does not need to extend to definition of the field function – as a Taylor series, a Fourier series or anything else – defined on space and time.
But we do expect the LWS-N field to oscillate in both space and time, with higher frequencies in time, lower frequencies in space and with waves travelling in space. We expect the amplitude of high frequency components of the these oscillations, say those of more than 1,000Hz, to be small. Stuff which can be regarded as noise rather than as signal.
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| Figure 1 |
Note that on the LWS-N hypothesis proper, our unit square is of the order of a few square centimetres in area and contains maybe 100 million neurons. Against that, a good deal of neural infrastructure will be required to support the organisation and activation (to be described in due course) of our 1,000,000 neurons, infrastructure which will eat into the other 99 million.
Note that the dots of Figure 3 below, while suggestive of the neurons of our unit square, are also misleading in that the spaces spanned by real neurons overlap, with their dendrites and axons all mixed up in space, an arrangement which is necessary if they are to talk to each other.
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| Figure 2 |
From time to time a neuron will fire, with the resultant peak in potential lasting about a millisecond (a thousandth of a second). This duration does not vary much. And while, in theory, a neuron can fire (at least for a while) at rates approaching 1,000Hz, that is to say every millisecond, across the brain as a whole the average is probably less than 10Hz, sleeping, waking or anything else.
Note that the amplitude does not vary much either. If you want amplitude in your field, you need a lot of neurons more or less in one place, firing in phase.
Without pretending to understand the processes involved, we suppose that the firing of all our neurons generates what we call an electrical field (φ), to distinguish it from the electric or electrostatic field and the magnetic field, two proper fields. Our electrical field is a scalar field taking real values, positive or negative, in space and time, where by space we mean the unit square defined above, not one of the three dimensional domains which physicists usually work with. This despite that fact that the bit of cortical sheet implementing LWS-N is supposed to have an area of a few square centimetres and a thickness of a couple of millimetres or so: more like a roof tile or a domino than the sheet of paper suggested by the term ‘cortical sheet’ – although we do have it that the cortical sheet is itself organised in a fairly uniform way into 6 layers – with the magic number 7 having failed to make this particular cut.
A simplification that reflects our hypothesis that LWS-N is organised topically, with a lot of information being held in the positions of our neurons, in much the same way that a lot of information is held on our retinas and a lot of information is held on the somatosensory and motor cortices on the top of the brain. Information which we believe to be essentially two dimensional, or, with the layers of LWS-N, at best two and a bit dimensions, certainly not three dimensional. We allow that a lot of information in the brain is not held topically, it is broken down and scattered – but we also claim that the LWS-N compiler puts it all back together again, for delivery to our field, to consciousness.
At any point in space and time, this field may be active or inactive. In the case that it is inactive, it just takes some mean value, which we might arrange to be zero, otherwise the resting value. In the case that the field is active, it oscillates around that mean, and in consequence has the properties of frequency, amplitude and phase.
The electrical field generated by all these action potentials is additive in the sense that if all our neurons are firing at random, the field across the unit square would be more or less uniform and unchanging, disturbed only by a bit of random noise, perhaps white noise. Inactive in the sense of the previous paragraph. No subjective experience.
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| Figure 3 |
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| Figure 4 |
Or we night have two such regions, together taking up the whole of the unit square and amounting to a rather different subjective experience. This is suggested by Figure 4. In both cases we have something more interesting than a field only offering a bit of white noise, rather something which varies in a chunky and interesting way in both space and time. With the hypothesis being that it is this variation in confined space and time which generates the subjective experience. We find the analogy of the confinement needed to attain either fission or fusion suggestive.
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| Figure 5 |
We do not yet know what might happen at the boundary between two regions, what the subjective experience of that boundary might be - beyond the idea that there is a fairly abrupt change in the pattern of activation across the boundary line. Or when the compiler might think it appropriate to extend the boundary line into a boundary region, a region in its own right. Perhaps analogous to a problem that painters have to think about; that is to say whether or not to highlight a boundary or an edge with a thin stripe of some third colour.
All quite different from the Φ mentioned above, in which our understanding is that you only get consciousness when there is a great deal of information flying about. The present hypothesis opts for a slower start, with Figures 3 and 4 above starting near the beginning, with just two such regions, just two primary frequencies. In the next part of the story we will go on to suggest how we might go on to implement the layer and column objects of LWS-N.
Moving things about
In which we are not talking about the movement of an object out in the world, for example a pedestrian tramping across our field of vision or a plate being pushed across a table, rather moving the active region of our field from one position on the unit square to another. Where we use the phrase ‘active region’ to mean all the active regions of our field, taken together, perhaps only amounting to a small fraction of the unit square as a whole.
We might start by simple rigid translations and rotations in the plane. As in, as it happens, moving a plate around on a table. We might then add expansions and contractions. And then reflections across a line. Noting that in these expansions, contractions and reflections we are introducing special lines and special points. Where are these special lines and points to come from? Do we only allow some lines and points?
A related question is what happens if we move the plane itself about, if we move the head, given that we suppose our bounded square to occupy a fixed position, somewhere in the middle of the brain? Is the subjective experience of the taste of the ice cream the same if we stand on our head, with our unit plane the wrong way up? The obvious answer to this being yes, at least if we put aside changes arising from any discomfort or any drips of ice cream, changes which result in changes within the active region of the unit plane, not the same thing at all. We suggest that the subjective experience is invariant under changes of position or orientation of the unit plane as a whole.
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| Figure 6 |
There is no relevant activity in the white space, so that is contributing nothing to the experience.
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| Figure 7 |
We might add that the size of the active part of our unit square reflects the size and importance, in the mind of the subject, of whatever it is that is being imaged, in whatever mode. Seeing, hearing, tasting or whatever. That is to say considerably less in Figure 7 than in Figure 4.
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| Figure 8 |
When we are seeing, the image depends on the orientation of the eyes, with that orientation being the combination of position of the eyes in their sockets, the position of the head on the shoulders and the orientation of the shoulders. Noting that all three things are lined up when we are what would be called pointing in dog, attending to something straight head of us; lined up and ready for action. When we are hearing, the image depends to a much lesser extent on the orientation of the head, that is to say of the ears. While the other senses are only weakly, if at all, dependent on the orientation of the head.
The present suggestion is that the complier does know about the orientation of the unit square and the directions suggested at Figure 5 above will tend to add up to the vertical, so the active region as a whole is orientated with regard to the vertical, with a north, as it were. This might, for example, simply be the result of the neuronal wiring, with the energy needed for all this, the activation which is driving the whole show, coming in at the bottom of the unit square and going out at the top. There might be all kinds of wriggling around on the way, but the general drift has to be upwards.
So the various orientations of the circle suggested in Figure 8 would result in different experiences, and not just differences of size and importance. This vertical would be well defined in the case of pointing, described above, less well defined otherwise. And possibly given by the orientation of the eyes, the head or the shoulders. Or something in between. Different people may well have different abilities in this department.
Miscellaneous matters
We return to the primary hypothesis, the hypothesis that it is the mere existence of this electrical field, probably somewhere deep inside the brain, which results in the subjective experience. The fact that this field is probably, at the present state of the art, invisible from outside the brain, is not material – although we grant that it does make testing the hypothesis more difficult. The sort of thing that reference 4 talks of in the context of modelling the signals recorded by an EEG headset from outside the brain does not arise here: ‘… Thus, the current sources in the brain that generate EEG can be modelled in terms of dipole moment per unit volume ... For convenience of this discussion, the brain volume may be parcelled into N small tissue masses of volume ΔV (e.g., 3 mm x 3mm x 3mm), each producing its vector dipole moment…’. And goes on to talk of N being of the order of hundreds of thousands for the whole brain – while we in LWS-N are talking about a couple of dozen of such tissue masses at most. And in which such vector dipole moments would hide all the detail of present interest.
At the limit, the granularity of this experience is limited by both by the finite numbers involved, that is to say of the neurons involved and their finite rate of firing, and by the noise in the system which masks the higher frequencies; there is a lower limit to the distinctions that are or can be experienced – but remembering here that human vision is capable of distinguishing very fine gradations of colour, when those colours are placed side by side. All that said, our present guess is that the average frame of consciousness does not contain more information than the average snap taken by a mobile phone, that is to say around 10Mb.
It may seem rather odd that consciousness should be hypothesised to be the result of the oscillations of a not particularly complicated electrical field. Against that, consciousness and neurons clearly both exist and the electrical activation of the latter seems a reasonable place to look for the former.
A hypothesis which might appear to be grounded in vision, which indeed it is. But we remain confident that the model offered by vision can be extended to the other senses – and with the importance of vision among most of the animals commonly thought to be conscious being suggestive.
Conclusions
We have said something about the field which we suppose to be generated by the neurons of LWS-N. A field which we have supposed can be considered, at least for present purposes, as having extent in the plane rather than in space, with this second alternative being more usual with both electricity and magnetism.
The next part of the story will be saying something about how that field might carry the information which is the subject matter of conscious experience. Or, more precisely, to build on the elementary Figure 4 above to get to something which can carry the layer and column objects of previous posts.
References
Reference 1: http://psmv3.blogspot.com/2018/01/an-introduction-to-lws-n.html.
Reference 2: The geometry of biological time – Winfree – 1980.
Reference 3: Consciousness: here, there and everywhere – Tononi & Koch – 2015.
Reference 4: Electroencephalography (EEG): neurophysics, experimental methods, and signal processing - Nunez, Michael; Nunez, Paul; Srinivasan, Ramesh – 2016. Chapter 7 of a book edited by others.
Reference 5: https://psmv4.blogspot.com/2019/01/making-shape-net.html.
Reference 6: http://psmv4.blogspot.com/2019/10/old-and-new.html. Notice of some other background reading. As usual, rather cursory background reading.
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