Uncoiling The Spiral Of Being: Math, Escher And Hallucinations
September 8, 2010
Last night I had a dream… A spiralling coil of color unfolded its serpentine and slithering body before my mind’s inner eye. Patterns danced in such an orchestrated synchronicity, so as to complect into a delicate and balanced interplay of form and function, all actions and reactions – though it was hard to tell which were which anymore, because of the temporal passage that had clouded all the previous causes to things – feeding back into the source, creating new snaking forms of colorful displays… Never ending, never repeating exactly, so interdependent on everything else around them, looking for nourishment and inspiration in themselves and those around them, every part of it guided by a wild and beating leviathan heart, a heart that was run by the only certainty I could ever find… That of uncertainty… That of chaos… An open ended function that was the only pure motivation for all universal being and which itself, alone, could only describe and create such a miraculous and highly dynamical order… “I” was a part of it… And in reflecting all of what “I” saw unraveling around me, this rhythm of chaotic movement began to shine through my very Being, allowing me to try to define myself in self-similar patterns… Patterns similar to those that “I” witnessed going on around me, allowing me a vain hope to understand what “I” is… While providing me with all I needed to partake in this dance of joyous wonderment before me; understandings were nothing more that rippling imaginations that carelessly skittered over and through the patterns of consciousness… Shape that had been fluxing within my brain’s complex and structured form… And still, I could only try to understand why, like almost everyone else, “I” tried to find similar reasons for Being in the ocean of delusion that swelled and sank around me… It was the only way “I” had known how to be throughout my entire life… And this was how the wonder twisted through my living, convoluting flow… A pattern that embraced every aspect of our Being, clutching “my” particle-like body into the blossom of its infinite totality….
Perhaps this was what many people before me had decided to call “God…” Mainly because they hadn’t properly understood its essence and nature… After, the wise mystics of the East followed the way of this unspeakable, indescribable beast. The Tao, they called it. “The Way.” And still it remains the only way to be, to dream and to live in harmony with all under heaven. Riddled with self-similarity, it writhed and pulsed to various rhythms running through its Being, all running inside and outside of each other, layering into and out of itself, fluxing with such precision that it might have been a silken fabric so finely woven, that the very threads we but atomic braids of molecular chains, of which any movement could upset the natural order and cause a mighty ripple to undulate throughout itself.
In all honesty I can’t remember how long this phantasm of interconnected geometry lasted… All I know is that I woke with a sudden jolt to find myself in bed with the covers strewn half on me and half on the floor. In someways I was relieved to find myself back home… But also I had a distinct sense of underlying melancholy that seemed to underpin my sleepy head… Sort of like when one departs the company of good friends. Slowly as my mind came back into focus, I found myself thinking of M. C. Escher‘s work. The seeming parallels that ran through my mind joined my dream up with Escher’s precise visions of nature’s “natural” symmetry. These in turn linked up with my own personal first hand experiences with mescaline, psilocybin, DMT and LSD… I haven’t tripped in a long, long time now. And I doubt I will need to ever again. What I had to learn from these powerful allies of the plant world, I did. They have kindly shown me all that I need to see. Within their own tapestries of mind, from the altered states of consciousness that they seem to so gracefully and naturally induce, I found myself faced with patterns as complex as those that I had seen on the Alhambra.
Yes… That’s it. That’s what all this reminds me of… The Moorish architecture of the Alhambra… There is so much of divine Moorish masonry to be found in Granada… And funnily enough it’s almost a year ago to the day that I arrived back from there… Perhaps, this is where my dream came from… Parallels in our orbit around our star, echoing through the structure of my brain. Perhaps I should provide a brief setting for this slight tangent… Between 710 and 713 A.D., Spain had been overrun by the Moors (populations of Berber, Black African and Arab peoples from Northern Africa), and these Islamic conquerors naturally introduced the ornate Moorish, or Moresque or Hispano-Moresque, style of design to the Iberian Peninsula, and is especially noted in the architecture of Southern Spain, which is centred and personified in the Alhambra, located in the city of Granada.
The Moors were not entirely driven out of the Southern provinces until 1610, but in the nine hundred years intervening, the Moresque style flourished sporadically throughout many portions of Spain. And one can see why… The splendour of this mode of design brought nearly everyone who saw it closer to a true sense of wonder regarding the creation of all things than anything else at the time. During the Romanesque period a large part of the country was still under Moorish rule… Here the balanced European form mingled with Islamic sensibilities, producing wondrous Romanesque structures laced with Moresque imagery and pattern. This marriage of form inspired the late M. C. Escher so much during his first visit in 1922, that he is reportedly to have said, “I have never before seen such concentrated inspiration in all the world!” After this his works of art began to take a very different turn. From the Italian country side sketches and etchings, he slowly incorporated this Moorish symmetry into his designs. While the Moors we forbidden to use any human or animal forms in their art – mainly because humans and animals were considered to be the divine and perfect work of Allah, and any human representation could only ever be an imperfect representation of the creator’s master work, and thus a blaspheme – Escher began to break this mould and used images of animals and plants in tessellations of wondrous cunning. These tessellations began to feature predominantly throughout most of the work of his later life. And rather than limiting them to just the snug, tightly fitting geometries of mathematical sensibilities… He opened them up with his imagination into metamorphosing consternations. It was almost as if Escher had seen the key to the universe, and had unlocked the door, through which it began to speak through him.
I know… I know… Sounds like a sort of far fetched fantasy derived from a dream I had… However, I’m going to present an idea in the form of an article that I found on the Twitter vine not too long ago. It is entitled “Uncoiling The Spiral: Math And Hallucinations” and was written by Marianne Freiberger.
Uncoiling The Spiral: Math And Hallucinations
Think drug-induced hallucinations, and the whirly, spirally, tunnel-vision-like patterns of psychedelic imagery immediately spring to mind. But it’s not just hallucinogenic drugs like LSD, cannabis or mescaline that conjure up these geometric structures. People have reported seeing them in near-death experiences, as a result of disorders like epilepsy and schizophrenia, following sensory deprivation, or even just after applying pressure to the eyeballs. So common are these geometric hallucinations, that in the last century scientists began asking themselves if they couldn’t tell us something fundamental about how our brains are wired up. And it seems that they can.
Geometric hallucinations were first studied systematically in the 1920s by the German-American psychologist Heinrich Klüver. Klüver’s interest in visual perception had led him to experiment with peyote, that cactus made famous by Carlos Castaneda, whose psychoactive ingredient mescaline played an important role in the shamanistic rituals of many central American tribes. Mescaline was well-known for inducing striking visual hallucinations. Popping peyote buttons with his assistant in the laboratory, Klüver noticed the repeating geometric shapes in mescaline-induced hallucinations and classified them into four types, which he called form constants: tunnels and funnels, spirals, lattices including honeycombs and triangles, and cobwebs.
In the 1970s the mathematicians Jack D. Cowan and G. Bard Ermentrout used Klüver’s classification to build a theory describing what is going on in our brain when it tricks us into believing that we are seeing geometric patterns. Their theory has since been elaborated by other scientists, including Paul Bressloff, Professor of Mathematical and Computational Neuroscience at the newly established Oxford Centre for Collaborative Applied Mathematics.
How The Cortex Got Its Stripes…
In humans and mammals the first area of the visual cortex to process visual information is known as V1. Experimental evidence, for example from fMRI scans, suggests that Klüver’s patterns, too, originate largely in V1, rather than later on in the visual system. Like the rest of the brain, V1 has a complex, crinkly, folded-up structure, but there’s a surprisingly straight-forward way of translating what we see in our visual field to neural activity in V1. “If you imagine unfolding [V1],” says Bressloff, “You can think of it as neural tissue a few millimetres thick with various layers of neurons. To a first approximation, the neurons through the depth of the cortex behave in a similar way, so if you compress those neurons together, you can think of V1 as a two-dimensional sheet.”
An object or scene in the visual world is projected as a two-dimensional image on the retina of each eye, so what we see can also be treated as flat sheet: the visual field. Every point on this sheet can be pin-pointed by two coordinates, just like a point on a map, or a point on the flat model of V1. The alternating regions of light and dark that make up a geometric hallucination are caused by alternating regions of high and low neural activity in V1 — regions where the neurons are firing very rapidly and regions where they are not firing rapidly. To translate visual patterns to neural activity, what is needed is a coordinate map, a rule which links each point in the visual field to a point on the flat model of V1. In the 1970s scientists including Cowan came up with just such a map, based on anatomical knowledge of how neurons in the retina communicate with neurons in V1 (see the box on the right for more detail). For each light or dark region in the visual field, the map identifies a region of high or low neural activity in V1.
So how does this retino-cortical map transform Klüver’s geometric patterns? It turns out that hallucinations comprising spirals, circles, and rays that emanate from the centre correspond to stripes of neural activity in V1 that are inclined at given angles. Lattices like honeycombs or chequer-boards correspond to hexagonal activity patterns in V1. This in itself might not have appeared particularly exciting, but there was a precedent: stripes and hexagons are exactly what scientists had seen when modelling other instances of pattern formation, for example convection in fluids, or, more strikingly, the emergence of spots and stripes in animal coats. The mathematics that drives this pattern formation was well known, and it now suggested a mechanism for modelling the workings of the visual cortex too.
…And How The Leopard Got Its Spots
The first model of pattern formation in animal coats goes back to Alan Turing, better known as the father of modern computer science and Bletchley Park code breaker. Turing was interested in how a spatially homogeneous system, such as a uniform ball of cells making up an animal embryo, can generate a spatially inhomogeneous but static pattern, such as the stripes of a zebra.
Turing hypothesised that these animal patterns are a result of a reaction-diffusion process. Imagine an animal embryo which has two chemicals living in its skin. One of the two chemicals is an inhibitor, which suppresses the production of both itself and the other chemical. The other, an activator, promotes the production of both.
Initially, at time zero in Turing’s model, the two chemicals exactly balance each other — they are in equilibrium, and their concentrations at the various points on the embryo do not change over time. But now imagine that, for some reason or other, the concentration of activator increases slightly at one point. This small perturbation sets the system into action. The higher local concentration of activator means that more activator and inhibitor are produced there — this is a reaction. But both chemicals also diffuse through the embryo skin, inhibiting or activating production elsewhere.
For example, if the inhibitor diffuses faster than the activator, then it quickly spreads around the point of perturbation and decreases the concentration of activator there. So you end up with a region of high activator concentration bordered by high inhibitor concentration — in other words, you have a spot of activator on a background of inhibitor. Depending on the rates at which the two chemicals diffuse, it is possible that such a spotty pattern arises all over the skin of the embryo, and eventually stabilises. If the activator also promotes the generation of a pigment in the skin of the animal, then this pattern can be made visible. (See the Plus article How the leopard got its spots for more detail.)
Turing wrote down a set of differential equations which describe the competition between the two chemicals (see the box on the right), and which you can let evolve over time, to see if any patterns emerge. The equations depend on parameters capturing the rate at which the two chemicals diffuse: if you choose them correctly, the system will eventually stabilise on a particular pattern, and you can vary the pattern by varying the parameters. Here is an applet (kindly provided by Chris Jennings) which allows you to play with different parameters and see the patterns form.
Patterns In The Brain
Neural activity in the brain isn’t a reaction-diffusion process, but there are analogies to Turing’s model. “Neurons send signals to each other via their output lines called axons,” says Bressloff. Neurons respond to each other’s signals, so we have a reaction. “[The signals] propagate so quickly relative to the process of pattern formation, that you can think of them as instantaneous interactions.” So rather than diffusion, which is a local process, we have instantaneous interaction at a distance in this case. The roles of activator and inhibitor are played by two different classes of neurons. “There are neurons which are excitatory — they make other neurons more likely to become active — and there are inhibitory neurons, which make other neurons less likely to become active,” says Bressloff. “The competition between the two classes of neurons is the analogue of the activator-inhibitor mechanism in Turing’s model.”
Inspired by the analogies to Turing’s process, Cowan and Ermentrout constructed a model of neural activity in V1, using a set of equations that had been formulated by Cowan and Hugh Wilson. Although the equations are more complicated than Turing’s, you can still play the same game, letting the system evolve over time and see if patterns in neural activity evolve. “You find that, under certain circumstances, if you turn up a parameter which represents, for example, the effect of a drug on the cortex, then this leads to a growth of periodic patterns,” says Bressloff.
Cowan and Ermentrout’s model suggests that geometric hallucinations are a result of an instability in V1: something, for example the presence of a drug, throws the neural network off its equilibrium, kicking into action a snowballing interaction between excitatory and inhibitory neurons, which then stabilises in a stripy or hexagonal pattern of neural activity in V1. In the visual field we then “see” this pattern in the shape of the geometric structures described by Klüver.
Symmetries In The Brain
In reality, things aren’t quite as simple as in Cowan and Ermentrout’s model, because neurons don’t only respond to light and dark images. Through the thickness of V1, neurons are arranged in collections of columns, known as hypercolumns, with each hypercolumn roughly responding to a small region of the visual field. But the neurons in a hypercolumn aren’t all the same: apart from detecting light and dark regions, each neuron specialises in detecting local edges — the separation lines between light and dark regions in a part of an image — of a particular orientation. Some detect horizontal edges, others detect vertical edges, others edges that are inclined at a 45° angle, and so on. Each hypercolumn contains columns of neurons of all orientation preferences, so that a hypercolumn can respond to edges of all orientations from a particular region of the visual field. It is the lay-out of hypercolumns and orientation preferences that enables us to detect contours, surfaces and textures in the visual world.
Over recent years, much anatomical evidence has accumulated showing just how neurons with various orientation preferences interact. Within their own hypercolumn, neurons tend to interact with most other neurons, regardless of their orientation preference. But when it comes to neurons in other hypercolumns they are more selective, only interacting with those of similar orientations and in a way which ensures that we can detect continuous contours in the visual world.
Bressloff, in collaboration with Cowan, the mathematician Martin Golubitsky and others, has generalised Cowan and Ermentrout’s original model to take account of this new anatomical evidence. They again used the plane as the basis for a model of V1: each hypercolumn is represented by a point (x, y) on the plane, while each point (x, y) in turn corresponds to a hypercolumn. Neurons with a given orientation preference Θ (where Θ is an angle between 0 and π) are represented by the location (x, y) of the hypercolumn they’re in, together with the angle Θ, that is, they are represented by three bits of information, (x, y, Θ). So in this model V1 is not just a plane, but a plane together with a full set of orientations for each point.
In keeping with anatomical evidence, Bressloff and his colleagues assumed that a neuron represented by (x0, y0, Θ0) interacts with all other neurons in the same hypercolumn (x0, y0). But it only interacts with neurons in other hypercolumns, if these hypercolumns lie in their preferred direction Θ0: on the plane, draw a line through (x0, y0) of inclination Θ0. Then the neurons represented by (x0, y0, Θ0) interact only with neurons in hypercolumns that lie on this line, and which have the same preferred orientation Θ0.
This interaction pattern is highly symmetric. For example, the pattern doesn’t appear any different if you shift the plane along in a given direction by a given distance: if two elements (x0, y0, Θ0) and (s0, t0, ϕ0) interact, then the elements you get to by shifting along, that is (x0 + a, y0 + b, Θ0) and (s0 + t, y0 + b, ϕ0) for some and , interact in the same way. In a similar way, the pattern is also invariant under rotations and reflections of the plane.
Bressloff and his colleagues used a generalised version of the equations from the original model to let the system evolve. The result was a model that is not only more accurate in terms of the anatomy of V1, but can also generate geometric patterns in the visual field that the original model was unable to produce. These include lattice tunnels, honeycombs and cobwebs that are better characterised in terms of the orientation of contours within them, than in terms of contrasting regions of light and dark.
What’s more, the model is sensitive to the symmetries in the interaction patterns between neurons: the mathematics shows that it is these symmetries that drive the formation of periodic patterns of neural activity. So the model suggests that it is the lay-out of hypercolumns and orientation preferences, in other words the mechanisms that enable us to detect edges, contours, surfaces and textures in the visual world, that generate the hallucinations. It is when these mechanism become unstable, for example due to the influence of a drug, that patterns of neural activity arise, which in turn translate to the visual hallucinations.
Bressloff’s model does not only provide insight into the mechanisms that drive visual hallucinations, but also gives clues about brain architecture in a wider sense. In collaboration with his wife, an experimental neuroscientist, Bressloff has looked at the connection circuits between hypercolumns in normal vision, to see how visual images are processed. “People used to think that neurons in V1 just detect local edges, and that you have to go to higher levels in the brain to put these edges together to detect more complicated features like contours and surfaces. But the work I have done with my wife shows that these structures in V1 actually allow the earlier visual cortex to detect contours and do more global processing. It used to be thought that you process more and more complex aspects of an image as you go higher up in the brain. But now it’s realised that there is a huge amount of feedback between higher and lower cortical areas. It’s not a simple hierarchical process, but an incredibly complicated and active system it will take many years to understand.”
Practical applications of this work include computer vision — computer scientists are already building the inter-connectivity structures that Bressloff and his colleagues played around with into their models, with the aim of teaching computers to detect contours and textures. On a more speculative note, Bressloff’s research may also one day help to restore vision to visually impaired people. “The question here is if you can somehow stimulate part of the visual cortex, [bypassing the eye], and use that to guide a blind person,” says Bressloff. “If one can understand how the cortex is wired up and responds to stimulation, perhaps one would then have a better way of stimulating it in the right way.”
There are even applications that have nothing at all to do with the brain. Bressloff applied the insights from this work to other situations in which objects are located in space and also have an orientation, for example fibroblast cells found in human and animal tissue. He showed that under certain circumstances these interacting cells and molecules can line up and form patterns analogous to those that arise in V1.
People have reported seeing visual hallucinations since the dawn of time and in almost all human cultures — hallucinatory images have even been found in petroglyphs and cave paintings. In shamanistic traditions around the world they have been regarded as messages from the spirit world. Few neuroscientists today would agree that spirits have anything to do with it, but as messengers from a hidden world — this time the hidden world of our brain — these hallucinations seem to have lost none of their potency.
by Marianne Freiberger
For me that article just magically linked up some seemingly random dots that had been lingering in my mind… Ones that were loosely drifting around on a plane of understanding that seemed to – only at the best of times – be based on flights of fancy and mathematical musings of divine symmetry… Could the reason why I, and others, are so drawn to these tapestries of geometrical wonder be because this pattern is naturally residing in the brain’s architecture? Could the key to our modes of perception regarding the surrounding universe be found – amazingly enough – in the roots of our minds? Is the mysticism lying behind the Alhambra’s amazing architecture linked to the patterns locked deep within the brains structure? Is that where our notions of God and the divine come from i.e. the imagery of divine knowing and interrelatedness that came to haunt my dream last night?
For me there is no doubt that there is a strong link between the spiritual ecstasy that I have experienced in altered states of consciousness and while viewing Escher’s works of art… Perhaps those followers of Allah, who invaded the Iberian Peninsula and left their indelible mark on the Spanish people’s cities and towns, saw a similar connection too. Certainly it is mentioned that the prophet Muhammad experienced visions while meditating within a cave for several weeks every year. It is here in this cave on Mount Hira, near Mecca, that he apparently experienced a direct countenance with the angel Gabriel who revealed many things to him. Certainly adherents and prophets of other religions also recount similar marvels and revelatory experiences (see Aldous Huxley‘s “The Perennial Philosophy”).
While I am not religious… I am aware of a pattern of mind that links these spiritual experiences into a similar and all encompassing perennial philosophy. Perhaps the key to this insight lies within ourselves through direct experience, rather than in notions and metaphors of an omniscient and omnipotent god/group of gods. Perhaps it’s time we forgot our differences and looked for the key to understanding our experiences through consciousness itself… Where we relate to one another through our patterns of mind and body… A view that would be free from delusion and ‘self’ impossed egocentric understandings… ? Perhaps psychedelics are a type of direct key to seeing this pattern of the divine… ? And perhaps our notions of an eternal creator is nothing more than the same patterns we see springing forth in the mind in altered states of consciousness… Perhaps this direct experience of the divine is so powerful that it leaves us reeling with a deep feeling of connect… Mainly because it is what we really are at base… And thus we dedicate such intricate, beautiful and inspiring architecture – a testament to the divine nature of our being – to those ideals of God that many of us hold so high. Perhaps this is why some many of us find the Mandelbrot set so mesmerising… Perhaps Escher knew this deep down… ?
If you would like to see where I sourced the article, entitled “Uncoiling The Spiral: Maths And Hallucinations,” from, please click here.
If you’d like to learn more about Marianne Freiberger, then please click here.
Or if you’d like to learn more about M. C. Escher and his life’s work, please click here.