We move up from LWS-N, the neuron-centric version of our LWS system, our local and layered workspace system which is contrasted with the global workspace of others, to LWS-R, region-centric version. This was heralded at references 2 and 3. The thought here being that the building brick of one of our layer objects would be the region, each defined and characterised by some sort of distinctive travelling wave rolling across it. With the present post informed by an excursion into the world of waves, starting at reference 1 and moving on into Wikipedia and elsewhere. No shortage of suitable material out there, some of it half a century old.
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Figure 1: a layer object
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The figure above suggests a layer object made of seven regions, distinguished one from another, and from everything else by the details, by the parameters of their waves.
We say that a region is maximal with respect to its wave. No adjacent region may use the same wave. Note that the blue is just there for contrast, and while suggestive of the shape nets of LWS-N, these last have no place here in LWS-R, where the only building brick is the region. While the red patterns are suggestive of the texture nets of LWS-N. We associate to the challenge of colouring a map of countries or counties with as few colours as possible – and the huge amount of effort that went into proving that four was enough.
But there is a residue. Our patch of cortex cannot manage an instantaneous transition (either in space or time) between one wave and another, there are going to be boundaries where there is no wave, at least no information bearing signal. And the geometry of these boundaries does amount, in some sense at least, to the more positively defined shape nets that we had before. And it may well be that a sharp, clean transition between regions is expensive in terms of the number of neurons needed to support it – so not something that the brain can afford too much of.
We note the possibility that two regions with much the same wave might be adjacent but distinguished mostly by the presence of such a boundary region. Supplemented by a change of phase across that boundary?
Waves in general
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Figure 2a: the sine function
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Figure 2b: alternating current
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Waves of one sort or another are pervasive. They crop up all over the place. Many of us will associate to the sine wave snapped above, pretty much the same as the alternating electrical current of domestic supply. But despite their being common, physics textbooks mostly define waves in a rather vague way as a mechanism for moving energy about without moving matter. So very roughly speaking, a wave on the surface of the sea might move energy from A to B, but individual drops of water move in relatively small circles and certainly do not move from A to B. Moving water about is a matter for currents not waves.
While the standing waves in the stretched (one-dimensional) strings of stringed instruments or in the stretched (two-dimensional) membranes of drums are propagated into the surrounding air as sound waves, in which last the air is vibrating rather than the string or the membrane.
An important property of many kinds of waves – but not all waves – is that they can be added up, sometimes in just the same way as sines and cosines can be added up in trigonometry. A property known as the principle of superposition and waves which have this property are called linear. From whence we get the Fourier analysis which runs through a great deal of science.
Another property of present interest is periodicity, and waves which have this property repeat themselves at some fixed interval of time, in the way of the sine wave above.
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Figure 3: the tan function
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We leave aside the wider ramifications of periodicity – say of the circadian rhythms of many plants and animals – addressed in reference 7 – in which connection there is much interest in the sort of stimulations which reset the phases of those rhythms. Reference 7 also addresses the interesting complications arising from the necessary discontinuity of many maps from two-dimensional space to a circle, rather different topologically from an interval, many of which crop up in the lives of said plants and animals. Discontinuities which are apt to be expressed as points of singularity, not so unlike those of the humble tan function in trigonometry, snapped above.
We note the slightly disturbing possibility of generating complicated geometry which is difficult to visualise from simple beginnings in rhythms and waves in two dimensions. Plenty of this in reference 7. Other books, for example reference 8, address interesting questions of propagation, the way that pulses of waves propagate in space, through a medium – be it a solid, a liquid or a gas – or through a vacuum. Neither complications nor questions which arise in what follows, where the interest is in travelling waves established for short periods of time on the small patches of two-dimensional cortex here called regions. We will worry about their establishment on another occasion.
Our waves
Light is a transverse travelling wave, but human eyes convert light to neural signals to do with grey, red, green and blue, and any simple link between wave properties of light and wave goings on in the brain seems improbable.
We also have machinery to both receive and generate sound, which comes and goes in the form of longitudinal travelling waves. Human ears convert sound to neural signals to do with frequency. But again, despite there being some overlap between the frequency range of audible sound (say 20Hz to 20KHz) and the frequency range of neural activity (say 0.05Hz to 500Hz), any simple link between wave properties of sound and wave goings on in the brain seems improbable.
And there is plenty of physiological rhythm to be found in reference 7, a lot of it with periods of days, (lunar) months or years. But it is a stretch to call these rhythms waves.
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Figure 4: heart and related rhythms
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Perhaps more relevant, the brain generates the signals needed to make the heart beat in a reasonably regular way, at a rate between 1Hz and 2Hz. So in the composite above, on the left we have the spiky, electrical waveforms recorded by a ECG machine. In the middle, the corresponding changes in blood pressure. And on the right, a longer term take on blood pressure.
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Figure 5: fibrillation
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For those curious about hearts, we think the snap above is of a 5Hz ventricular fibrillation – undesirable, but much more like a sine wave than the series of spikes you get from a healthy heart.
Even more relevant, the electrical signals at the surface of the brain recorded by an EEG machine, the signals which are the subject of the book at reference 4. There is clearly plenty of wave activity in the brain – but waves which are rarely as simple as the fibrillation above. There is too much going on for the waves recorded at the surface of the brain to be either sinusoidal or simple.
The present waves
Two-dimensional travelling waves are an important and well analysed class of wave, with the waves on water already mentioned being a well known example. Sadly difficult to illustrate without moving pictures, but reference 5 is an example of the sort of thing that can be done with them. Beyond the reach of our mid range Microsoft Office skills.
Such a travelling wave might be defined as the sum of a small number of travelling waves of the form:
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Figure 6
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Where A (for amplitude), ky, ky and ω are real constants. Minus for waves moving from left to right, plus for waves moving from right to left. And x and y are the coordinates in two dimensional space and t is the coordinate in one dimension of time, chosen so that V is zero at the origin. While V stands for the scalar value of the wave function at some point in space and time. Maybe something like pressure, height or electrical potential. We omit complications arising from choice of coordinate systems. We put aside consideration of whether the subjective experience of a wave moving from left to right is the same as that of an otherwise identical wave moving from right to left.
In many physical systems where there are lots of these basic waves, there is an equation linking the kx, ky and ω. Often called the dispersion relation, this because the form of this equation determines whether and how the waves disperse over time.
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Figure 7
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This may not be appropriate or necessary here. In any event, if we are allowed up to two or three of these waves for any one region, we have quite a lot of degrees of freedom; the resultant wave form is carrying quite a lot of information.
Let our population of such wave forms be called P. One can define all kinds of binary relations on P and one could, no doubt, define some kind of a metric so that one could say how close one (composite) wave form was to another. And one would need to think about how close the wave forms of adjacent regions could be, before the subjective experience became that of one region.
The regions
The patch of cortex supporting LWS-R is guessed at 5 square centimetres and 100 million neurons. Perhaps 2% and 5% of the respective totals, excluding the neuron rich cerebellum.
A region might have non-exclusive occupation of a proportion of that patch, say a proportion varying between (say) 1% and 100%.
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| Figure 8: some regions |
A region has a reasonably uncomplicated shape, lying somewhere being convex and being locally convex. For present purposes it will usually be enough to suggest a region with a polygon with a modest number of edges, say less than 10. We suppose topologically closed, although we presently doubt whether the distinction between open and closed, important in topology, is important here.
So in the snap above the blue shapes are clearly in. The two blue shapes top left are convex in a conventional way and the round blue shape is nearly so. The red shapes are also in, but they are pushing things a bit. Second left by having a hole in the middle. Bottom middle by being rather extravagantly shaped. The long thin line by being too line like: our waves need at least some extent in two dimensions to be viable – and the extent to which the boundary between two regions is perceived as a line, without the need for some more explicit structure is for consideration. Then top right and bottom right by not being connected, with bottom right being a bit extravagant: the defining wave form will need to be strong and distinctive for the subjective experience to be that of a single region.
We suppose that there is a limit to how much of this pushing one could be conscious of at the same time. One complicated region, perhaps the centre of attention, perhaps with waves with the biggest (A) amplitude or the biggest (ω) frequency, is one thing. Lots of complicated regions is another.
We suppose also that one could define this pushing in some mathematical way and propose a limit, perhaps a limit which varies, within bounds, from person to person, but we do not attempt that here.
From regions to layer objects
Then a region is a maximal subset of our bit of cortical sheet with respect to some wave form in P. And the various different wave forms of the seven regions in the object of Figure 1 each generate their own subjective experience, perhaps of colour, texture or both.
A layer object is then a maximal collection of regions with respect to some property of those waveforms, perhaps amplitude, perhaps ω, this last being directly related to the firing frequency of the underlying neurons. Whatever property we choose, we will have a small number of bands of value, suitably distanced one from another, so that one object is clearly distinguished from another. We associate to the parcelling out of radio frequencies into bands, each allocated to some broadcast service or function. Such an object is suggested in Figure 1 above – bearing in mind that the blue is only there to heighten the contrast. Not part of an LWS-R layer object at all.
We suppose that most regions will be like those of Figure 1: they will be not be complicated, they will be connected and will not have holes. Similarly, we suppose that most layer objects will be like that of Figure 1: they will be not be complicated, they will be connected and will not have holes.
The blue is also suggestive of the shape nets we had before. While the red textures of the seven parts is suggestive of the texture nets we had before.
So a large part of the information payload of a region is its position on the patch of cortical sheet which underpins LWS-R. Position which is clearly central to vision and not absent from hearing, touch and smell. Another part is the parameters of the one or more travelling waves which have been put on that region.
And we have not excluded the possibility of regions from different objects occupying the same space. Thus allowing, inter alia, for seeing the fish in the fish pond.
Speculations
Physics texts go to a lot of trouble to explain how materials like water come to exhibit behaviour which can be approximated by equations – functions of sines – of the form given above. Explanations which are rooted in the physical properties of water and the world in which it lives.
While computer scientists, perhaps the chap who wrote the code underneath the visualisation at reference 5, are able to write code which will model those equations without needing physical properties at all, without getting wet at all.
We are supposing that the brain is somewhere in between and the LWS-R compiler is able to exploit the electrical properties of neurons so as to get them to produce the travelling waves suggested above, to order. The compiler is able to use these neurons as an output device, a device under its own control, not the blind servant, the blind product of the laws of physics.
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Figure 9a: Dürer's Melancholia of 1514
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Figure 9b: detail of same
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We hope to go on to consider how the LWS-R compiler would cope with an image like the famous Dürer engraving (on copper) snapped above. What sort of spatial resolution can it manage and what happens at the transition from hatching lines to grey scale?
Hopefully a more focussed, an easier challenge than that at reference 9.
Conclusions
We have rounded out the story of the waves on the regions of LWS-R started at references 2 and 3.
PS: one of the old texts we consulted was reference 8, where some of the introductory material was helpful. Curiously, this book, near a half century old, still appears to be something of a standard text, commanding serious prices on both Amazon and ebay. On the other hand someone has thought to upload a scanned image onto the Internet and make it freely available – more than good enough to meet our very modest needs. And if we got really keen, we could get the 650 or so pages printed off at the print shop in Epsom High Street (reference 10) and at, say, 5p a page it would still be of the order of half what the actual book would cost.
References
Reference 1: University Physics – Harris Benson – 1995.
Reference 2: http://psmv4.blogspot.com/2019/10/the-field-of-lws-n.html.
Reference 3: http://psmv4.blogspot.com/2019/11/more-on-making-regions-into-objects-and.html.
Reference 4: Rhythms of the brain - György Buzsáki – 2011.
Reference 5: https://youtu.be/AETMRNbeVkI. A visualisation.
Reference 6: https://youtu.be/y53z2zVipAs. Waves in water from the Open University.
Reference 7: The geometry of biological time – Winfree – 1980 .
Reference 8: Linear and Nonlinear Waves – G.B. Whitham – 1974.
Reference 9: https://psmv4.blogspot.com/2019/01/texture-nets.html.
Reference 10: https://aiprinters.co.uk/.
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