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Novel Symmetry, Self-Similarity(Fractal) & Recursivity decomposition of the structures and processes of Brain and Mind leads to several key unifying insights.

Abstract

Brain and Mind science is characterized by dizzying complexity and a myriad diversity of facts and findings. We demonstrate that through the systematic application of the inter-related and fundamental concepts of symmetry, self-similarity and recursivity; we can show that the complexity of the brain reduces to a small number invariant organizational patterns and processes. These hidden symmetries span not just the physical substrate of brain but also the emergent structures of mind and allow us to conceptualize neuroscience and psychology as one continuum.

We formalize our approach using the non-Euclidean geometry of binary N-Space and the hierarchical language of binary trees and binary combinatorial spaces. We show that this formalism corresponds to a wealth of empirical evidence and gives us a powerful and very general way of describing the structures and processes of brain and mind. One which is binary, bifurcating, doubling, digital, grid-like and discrete. This unifying language then allows us to conceptualize all the various aspects of brain and mind as a single all encompassing hierarchical structure. So that the motivational, homeostatic, symbolic, sub-symbolic and neural substrate are brought together in a single conception and unity.

With this unifying organizational principle, concept and language, we are then able to show that all the seemingly separate processes of brain and mind, are really the expression of a single underlying recursive process. Furthermore this symmetry of process extends not just to how brains and minds work, but also to how brains come into being, i.e. neurogenesis. So that the entire brain theory is able to be succinctly expressed as a single recursive process deriving from a single atom of recursion. In the same way that our brains and minds derive from the originator recursive atom of the fertilized egg.

The preceding, allows us to explain and implement recursive self modification and recursive self reference, which are some essential features of intelligence and consciousness. This is through the introduction of a feedback loop whereby, the underlying recursive process creates and augments structure through instantiating mappings in combinatorial space. In turn the structures so created, themselves encode mappings which go on to further express the original process but in augmented form. So in the strangest of strange loops, the original recursive atomic seed, extends itself through mapping, then feeds back on itself recursively to better extend it self, and so on and so forth. Finally we use this to explain the full significance of the Polar Frontal Cortex, a brain region involved in recursive thought processes and show why this facility is key to understanding consciousness and explaining human intelligence over that of animals.

Introduction :

The complex and difficult problem of understanding the brain and mind

The puzzle of how the brain works and the nature of the mind, remains one of the biggest unsolved problems in science and exists as one of its last great frontiers. It is the holy grail of modern times and its final solving would have massive implications, commercial, educational, medical, philosophical, social and political. Not least of the implications would be the creation of true artificial intelligence and the impact this would have on the world.

After earlier optimism in the 1950s and 1960s when cognitive scientists thought the problem of mind would be solved in a decade; instead the problem has turned out to be quite intractable and fiendishly difficult. In a recent interview Noam Chomsky said, 'The work in [AI] of about 60 years has not really given any insight, to speak of, into the nature of thought and organization of action and so on.' and that, 'We're eons away from [a theory of being smart]'. In a similar comment the Physicist and popular science writer David Deutsche said, "No brain on Earth is yet close to knowing what brains do. The enterprise of achieving it artificially — the field of 'artificial intelligence' has made no progress whatever during the entire six decades of its existence." In 1979 Francis Crick wrote that ‘[The Study of the Brain] is conspicuously lacking a broad framework of ideas’, yet 35 years later it is still looking for this 'broad framework'. This sort of sentiment is also echoed by several other prominent researchers and thinkers. But what is the impasse? Why has this, yes complex, but finite object frustrated all attempts to understand its workings? Why have the answers been so elusive? This is one of the things we seek to answer.

Today when a lot of specialists are asked when will we finally come to understand the brain and create true AI, a typical answer is sometime 50 to 100 years in the future. Sometimes a more optimistic estimate is given, say 10 years or so, but this is never accompanied by any sort of clear picture of how this might come about. Perhaps the lowest bound estimate was given by John McCarthy one of the pioneers of AI and the person credited with coining the expression 'artificial intelligence'. He said words to the effect that there is the possibility that someone has already figured it all out, but 'he hasn't told us about it yet'. Along similar lines David Deutsche, while being quite negative about the state of AI and theoretical brain science, he has also expressed the hopeful view that, ‘[ I ] can agree with the AGI[Artificial General Intelligence] is imminent camp', and that it is, ‘plausible just a single idea stands between us and the breakthrough, but it will have to be one of the best ideas ever.'

I believe we already know quite a lot about this single 'break-through' idea and also we already know that it is one of the best ideas ever, because this idea is really the key idea behind all of science. This idea is called Symmetry. And Symmetry is the key idea behind all science because when we ask what is science, then a fair and immediate answer would be that science is the process of discovering the patterns in nature and physical reality, and then describing those patterns in the form of mathematical constructs. But science is more that that, because it involves not just capturing the patterns in nature but also the patterns behind the patterns, the meta-patterns and unifying patterns. That is finding the overarching patterns which show that all the seemingly separate and unrelated patterns are really just different manifestations of the same underlying pattern. The word which best describes this process is Symmetry and it is the property which enables Science.

The concept of symmetry in science allows us to conceptualize and understand seemingly separate phenomena as really being the same thing. So for instance electricity and magnetism were considered as separate things but then came the idea of electromagnetism and that behind these different manifestations are the same underlying. Later on the concept of electromagnetic spectrum brought together the phenomena of electricity, magnetism and light. It means there is an underlying symmetry which is behind all them. And they transform into each other, so for instance light from the sun via a solar panel can put electric charge into the battery of an electric car which then uses magnetic force to translate that electricity into motion. That motion may be translated back into electricity when the car brakes through the action of the magnets in the braking dynamos and stored back into the battery of the car. Later on that electric charge may be converted back into light through the cars headlights. So here we have many instances of transformation but all the while something stays the same or invariant in the laws of physics which describes in one conception all these separate manifestations.

This unifying process using the property of symmetry proceeds in science. Physicist have already unified the electromagnetic force with some of the other forces of nature, i.e. the strong and weak nuclear forces, and a unification together with the force of gravity still pending. This is still a work in progress and is driven by a belief that there exist unifying symmetries in nature.. So that what is initially seen and diverse and different, will turn out to be really various facets of the same simplifying underlying theory. We have a directly analogous situation in the brain and mind sciences, with a vast and myriad array of data, facts and findings waiting for some sort of simplifying and unifying understanding to clarify matters. When we understand that it is symmetry that is behind the process of science then there is no reason to suppose that it should be otherwise with respect to brain and mind science.

Symmetry, Self Similarity and Recursivity

We are proposing is that it is this property of Symmetry together with the related properties of Self Similarity and Recursivity which when used together will enable us to properly understand the structure and workings of the brain and mind. We’ll introduce the idea of Self-Similarity here but explain it in more detail a little later. When some ‘thing’ or some object has the property of Self Similarity then it is said to be 'Fractal', an expression coined by the Mathematician Benoit Mandelbrot in the 1970s who is credited with inventing Fractal Geometry. Implicit in notion of Self Similarity is the idea of nested symmetry and also recursivity; this we’ll also explain soon. So a convenient shorthand for expressing the systematic application of the ideas of symmetry, self similarity and recursivity for the purpose of understanding the brain and mind is to call what comes out of this process a 'Fractal Brain Theory’. Though its full title would be the Fully Symmetrical, Self Similar and Recursive Theory of the Brain and Mind. All of these terms may be unfamiliar to a lot of readers, so in we'll be elaborating more fully on what is meant by Symmetry, Self-Similarity and Recursivity in the context of understanding the brain and mind.

Symmetry, Self Similarity and Recursivity are really such fundamental concepts and as we will show, they are also extremely power tools for understanding what is the nature of mind and how the brain works. It often doesn’t seem obvious at all how these concepts can be applied to understanding the brain. With all the asymmetry and dis-symmetry that exists in the organization of the brain it may seem inappropriate to even consider the principle of symmetry in relation to thinking about brain and mind. However often it is the case that superficial and complex asymmetries masks the underlying and simple symmetry behind it all. Also the self-similar or fractal nature of brain organization is not immediately apparent, and sometimes there can be objection to the very notion of a ‘fractal brain’ theory. Once again it is not through superficial inspection that the self-similar nature of the brain is revealed. We are proposing that self-similarity, symmetry and the related concept of recursivity; are really the fundamental organizing principles of the brain and the key to understanding the process of mind. If the symmetry and self-similarity inherent in the structures and processes of brain and mind were obvious then we would have worked out how the brain works many years ago and the advent of true artificial intelligence would be a historical detail. Therefore we first need to make clear what we mean by symmetry, self similarity and recursivity in order that we then be able to systematically apply these principles to understanding the brain and mind. We will show that there exists a unifying conception of brain and mind which is fully symmetrical, self similar and recursive. And furthermore that these principles apply all the way from the scale of the entire brain right down to the most basic representational building blocks of brain and mind. When it is fully understood how fundamental these principles are then likewise an all encompassing understanding of brain and mind derived from them will have some very special, fundamental and foundational properties.

What is Symmetry?

Two key concepts behind the idea of Symmetry are that of Transformation and Invariance. So what is a transformation? An object, shape, representation, mathematical equation or some form of abstract entity may undergo a variety of different forms of change. For instance an object can be twisted, stretched or otherwise distorted and rules of derivation and logic can be applied to mathematical equations to generate new ones. Also genes can undergo mutations and also other forms of more pre-programmed modification. These are all instances of transformation. While these transformations are able to generate potentially infinite variation and difference, i.e. a myriad number of different shapes, mathematical equations or gene sequences. At the same time underlying all the differences are qualities which are preserved or conserved, and which are said to stay invariant. So in the case of a simple shape undergoing the basic transformations of translation, scaling and rotation. Even though from the original shape a variety of different patterns are formed, behind all this is still the same invariant original shape and if we reverse the transformations then we arrive back at exactly the same original form. See diagram below. And likewise the same sort of reasoning can be applied to the domain of mathematical equations, genes and abstract forms. These transformations are known as symmetries, and if we fully understood the nature of these transformations then we would likewise see fully the invariance or commonality of pattern, underlying all the seeming variety. The application of the principle of symmetry is discovering what these transformations are in order to simplify and understand complex and varied phenomena in terms of a more manageable and preferably minimal set of invariant factors. i.e. basic geometric shapes, simple mathematical equations describing the laws of physics, or the letters of the genetic code from which all DNA is constructed.

This way of looking at Symmetry may seem a bit different from what is taught in high school maths lessons, where we learn about a shape undergoing certain transformations and which as a result appears exactly the same as the original instance. So for instance we learn that a square being rotated at right angle increments will appear as exactly the same square and that it has rotation symmetry with itself. And the same square by reflecting along the vertical y axis going down the middle of the square will produce a transformed image that looks virtually identical to the original and so is said to have mirror symmetry. Here we have the idea of transformation and invariance in its purest and simplest form. Where the transformations are the most basic and the invariance is most apparent. In these most basic instances of symmetry, a shape undergoes some basic transformations, i.e. rotations and reflections, and end up appearing to the eye exactly the same.

When we talk about symmetry being applied to understanding the brain and being the key concept behind science, we are a talking about this basic high school conception of symmetry but in a much extended, augmented and embellished form. But the same basic idea remains, of things being transformed and at the same time something staying the same or invariant. And we can even extend this idea to different members of the same animal species, where of course no two animals are exactly the same, and there will be an infinity of differences between them. And then we can further extend the idea to looking at the structures of the brain to find the underlying commonality behind the diversity. So when we are applying the idea of symmetry to understanding the brain then we are no longer dealing with simple perfect geometric forms such as perfect squares or perfect equilateral triangles but the more messy world of neurons and brain circuits. But still the same principles of transformations producing an array of variation while preserving an underlying invariance, can still be applied. Later we shall learn that with a few levels of abstracting away some of the more superficial and random seeming features of the brain, then we will arrive at a very compact description of the entire brain, which does have perfect mathematical symmetry. This along with perfect self-similarity and perfect recursivity. We need to introduce a lot of experimental evidence and also some bridging concepts in order to do this. This we will do in due course. But for now we’ll move on to explaining self-similarity.

What is Self-Similarity?

Self-Similarity is a property of an object or thing when it is made up of smaller copies of itself at various scales and potentially to infinitesimally tiny scales or resolutions. So for instance you can take a basic shape and make smaller copies of that shape and place these smaller copies within itself. And then in turn you can take each of these smaller copies and likewise make even smaller copies of these copies and similarly place them within their parent copies from which they were derived. And you could imagine continuing this process without limit. This would produce patterns or shapes which would have the property of being self similar at all scales or can be said to be a fractal. So fractals are conceptual objects or abstract shapes which have this property of being self similar at all scales right down the infinitesimally small. In physical reality things can never be self-similar to infinite levels of consideration but still the application of the idea of self-similarity is very useful when trying to understand natural phenomena. An interesting perspective on what is the nature of self-similarity is to think of it as nested symmetry, which would then relate to the earlier discussion. See diagram below where we had the origin brain shape from earlier, but this time we’ve made smaller copies of itself and placed these copies within itself. We’ve then repeated this process a few more times, i.e. recursively. We'll be discussing recursivity more.

Another way to visualize self-similarity is to imagine russian dolls but with many smaller russian dolls contained in the next size down. The diagram below illustrates this concept with what we normally think of as russian dolls at the top of the diagram but with our augmented version of the concept below it. The two augmented russian dolls more towards the right of the diagram represent progressively ‘zoomed’ in versions of the original biggest and enclosing doll, drawn to the left most side.

And below are a few geometric forms to further illustrate the concept of self-similarity or fractalicality. The object to the right most side below is a vegetable called romanesque broccoli or cauliflower and is an example of a natural fractal. Note here again we have structures which are made up from smaller 'self-similar' copies of the overall form.

What is Recursivity?

Recursivity or recursion refers to the property that certain processes have when they are repeated applied upon themselves. So that the result of the process is repeatedly fed back into itself to produce another result, and so on and so forth. This can be in the form of mathematical equations, where we feed in a number into an equation to get another number, and in turn put this new number back into the equation to get a third etc. Also natural processes such as mitosis or cell division can be described as recursive. This is the process of the cells in our body repeatedly and recursively dividing into two. It is what enables a single cell that is the fertilized egg, to become all the 35 trillion or so cells in our body. So from the single cell, through the recursive process of mitosis we get two cells. The process is then repeated on these two cells to get 4, then repeated again to get 8 and so on. We may also call mitosis a divergent recursive process because of the way it starts from a single point and produces a spreading or diverging set of myriad new points. An example of a process working in the opposite converging way would be the idea of idea in a sports tournament where 8 competitors paired off to play each other in a quarter final round, the 4 winners of which are paired off in the semi-finals, to produce 2 finalists made to compete in the finals, which produces the eventual winner. See diagram below. We could extend this idea to encompass arbitrary numbers of contestants. And this process would be convergent because we start from a whole load of contestants and end up with a single winner.

So these three examples of an iterative mathematical function, biological cell division and a sporting arrangement; gives us an idea of what is meant by a recursive process. When this recursivity is combined with the ideas of self-similarity and symmetry, then we have the most important fundamental concepts for understanding the brain and mind in place.

We will now talk about the brain itself, and start our analysis by trying to find some common organizational patterns or ‘fuzzy’ symmetries, to gain an initial sense of order and overarching pattern. Later we translate our fuzzy symmetries into something more crystalline and definite.

Some underlying and recurring organizational patterns behind the brains complexity.

The brain has been called the most complex object known to humankind and at times it seems bewilderingly so. With its around 10s of billion neurons, trillions of synapses and 100 000+ kilometres of wiring all contained in the space of our skulls. To some brain researchers it seems like a total mess, a ‘kluge’ or the result of a long series of haphazard accidents. But underneath this superficial disorganization and mind numbing complexing lies a stunning simplicity. Through the systematic application of the principles of symmetry, self similarity and recursivity we can show that there exist organizational principles and common underlying patterns, which allow us to fully comprehend brain and mind.

So we start with some obvious repeating patterns in the brain, then move to demonstrating ones which are increasingly less obvious. Some of these patterns have only been able to be discerned as a result of quite recent research in the neurosciences. And then we move to underlying common patterns or symmetries which are quite abstract.

First of all, we consider the cerebral cortex or neo-cortex (two completely interchangeable terms) and the neo-cortical columns from which it is entirely constructed. This part of the brain comprises most of the brain mass in the skull ~80% and is what we normally understand as brain. It consists of all the crinkly folds and ridges we see when looking at a brain. But its real structure is that of a flat sheet a few millimetres in thickness and covering an area of approximately 2 and a half feet squared. By scraping out all the white matter which are the nerve fibre tracts connecting up all the different regions and columns of the neo-cortex then it is possible to literally flatten it out. The diagrams below left show a depiction of a flattened out human cerebral cortex and the size of this ‘flat sheet’ relative to a whole brain. The diagram below left does the same for one hemisphere of a macaque monkey cerebral cortex.

This flat sheet is made up of a few million cortical columns which are like cylinders, with a diameter of ~0.3 millimetres and a length corresponding to that of the cortex itself i.e. 2-3 millimetres. This makes up the ‘grey matter’ of the cerebral cortex, as the cell bodies give a greyish appearance to naked eye. All the white matter or fibre tracts connecting up the cerebral cortex, arises from neurons located within the cortical columns. The really amazing thing is that all of these cortical columns share a same basic design and contain roughly the same number of neurons arranged in a stereotypical manner within all the columns. With certain neuron types and certain patterns of connectivity arranged in a very ordered and stereotypical manner, throughout all the cortical columns making up the neo-cortex. Of course no two cortical columns will be exactly the same, and there does exist a lot of variation between columns existing in different regions of the neo-cortex. Nonetheless underlying all the variation and difference is commonality and invariance in basic design. What’s more, this basic blueprint seems to be preserved through the evolutionary process, so that the cerebral cortex of a mouse brain is likewise made up of the same columnar building blocks, of roughly the same diameter, containing roughly the same number of neurons, the same neuron types, and following the same stereotypical blueprint also found in the human brains. So that even though the neocortical is stupendously complex, at the same time there exists a lot of repeating pattern, symmetry and organization.

Below left is a diagrammatic presentation of some cortical columns. Below middle is a diagram of the distribution and size of neuron cell bodies from various regions of cerebral cortex showing some variation. And below right is a diagram showing the main neuron classes found within the cell columns.

The same line of reasoning and analysis we’ve just applied to the neo-cortex can also be applied to the cerebellar cortex and also the striatum, which together with the auxiliary structure of the thalamus, makes up most of the rest of the brain. The cerebellar cortex can likewise be flattened into a sheet like structure, which is also made up of a stereo-typical modular architecture. These are referred to cerebellar ‘complexes’ and ‘micro-complexes’. With these basic modular building blocks which are around 0.5 millimetres in length, the entire flat sheet of the cerebellar cortex is constructed. The structure of a cerebellar complex is simpler than that of a cortical column of the neo-cortex. The design is also evolutionarily more ancient, so that the stereotypical design of the basic cerebellar circuit, has been preserved from shark brains right up to human brain. Going even further back, even the brain of a lamprey, a primitive form of fish and very ancient animal, contains rudimentary cerebellar circuitry.

The diagrams below left, show the the relative position of the cerebellum to the rest of the brain and a cross section of it shows if extremely folded nature. The diagram below centre shows a drawing of an animal cerebellum. Here some body maps of the animal are drawn in to show which parts of the cerebellum handle which parts of the animals body. The cerebellum displays topographic mapping in its representation of the body. The diagram below right depicts a flattened human cerebellum. We can see from the scale indicator that the extent of this flattened cerebellum from top to bottom is about 60 cm or 2 feet. We can see why it has to be so intricately folded in order to cram this considerable surface area into such a small space.

The diagram left below show the the striatum in human brain, depicted in green and shows where it is located within the brain.The same sort of reasoning that we applied to the neocortex and cerebellum likewise applies to the striatum, So again we find stereotypical organizational patterns throughout all the striatal areas in the brain. The stereotypical pattern of striatal organization and neuronal makeup as found in a human brain is also the same basic design found in the striatum of birds and reptiles. And so like the neocortex and cerebellum we are seeing brain structures preserved through the course of evolution. The diagram below right depicts a part of the striatum called the putamen which both humans and many other mammals possess. A body map is drawn onto the diagram of an animal putamen, showing which zones within it, handle the control which parts of the animals body. Here again we see topographical mapping, with relative spatial relationships between different parts of the body preserved.

The analysis so far has mainly dealt with quite obvious basic symmetries or invariant patterns. We can proceed now with exposing commonalities of pattern which are less obvious. A very important one ties together the three main structures of the brain we have so far considered and gives an important clue as to their functioning as an integrated unity. It involves the way both the cerebellum and striatum interact with the neo-cortex. Both these structures separately form what are known as ‘closed loops’ between themselves and the neo-cortex and in both cases the thalamus is used as an intermediate relaying structure. There are many such closed loops working in parallel and separate ones for each different sub-region of the neo-cortex. Such that specific subregions of the neo-cortex will work in conjunction with specific sub-regions of cerebellar cortex and striatum etc. Later on we be explaining how this loops may represent time but for now the main thing is this symmetry of closed loop structuring that links the cerebral cortex to the striatum and cerebellum.

We bring now into our discussion of the main structures of the brain, what can be described as the emotion centres. These are the centres of the brain involved in motivation and homeostasis and include such structures as the hypothalamus, amygdaloid nuclei and parts of the cerebral cortex such as the orbital frontal cortex and subgenual anterior cingulate cortex. We have already described how the striatum forms closed loops with regions of the cerebral cortex. What was discovered and confirmed quite recently is the idea that this striatal loop structure also extends to the emotion centres. So that the hypothalamus and the amygdala structures also have their corresponding striatal closed loops with which they interact. So in the case of the hypothalamus, its various nuclei interact with a once mysterious structure called the lateral septum. We now know this structure has the same stereotypical architecture as the rest of the striatum and forms separate closed loops with each of the hypothalamic nuclei; each separate region of the hypothalamus working in conjunction with its own area of lateral septum. And in relation to the amygdaloid nuclei, we now know that a sub-region known as the central nucleus of the amygdala also can be understood as ‘striatal’ and forming the same closed loops with the rest of the amygdala. In fact all the emotion centres we mentioned share the same underlying striatal-like organization. We’ll be discussing the emotional centres a lot more later in this paper.

What we have then is a common striatal organizational pattern which is found through-out large parts of the brain and covers all the anterior and prefrontal neocortex together with the emotion centres. Also a similar extension of our understanding of the cerebellar cortex has also occurred in recent years so that a part of the brain once thought to be involved mainly in the coordination of movement (like the striatum) has likewise been extended to cover the emotion centres and also the cognitive areas. What this means is that we have common organizational principles that cover most of the brain and encompass, neo-cortex, striatal structures, cerebellar cortex and the emotion centres. We will demonstrate later on and in much more detail, important symmetries which span all the emotion centres together with neo-cortex, cerebellum and striatal structures and show how they can be conceptualized as sharing the same underlying design and mode of operation.

We’ll complete our initial quite superficial overview of brain anatomy with a description of another recurring pattern found throughout the brain which brings together everything we’ve talked about so far. It is a basic recurring ‘fuzzy’ symmetry which we’ll elaborate upon much more later and show that it is key to understanding what it is that brains and mind are doing. And we’ll also be formalizing these fuzzy and vague symmetries into very definite and precise mathematical ones.

A Basic Overarching Pattern in Brain Organization

The basic pattern which seems to be recurring all over the brain is that of a tree, like the sort we find in our gardens and in parks but with a special addition which we’ll describe in a moment. So if we imagine a tree then we have its trunk, which is its body, and then we have all the main branches coming off the trunk, and the progressing branching into the twigs and leaves of the tree. But there are also the roots which we don’t normally see because they are underground. And like the branches above ground, the roots also branch out progressively, further and further into the depths of the earth. Apart from superficial appearance, there is a very important difference between the roots of a tree, versus the branches and leaves. It has to do with the main flow of nutrients going through the tree. That is, water and nutrients from the soil, flow through the roots in a converging recursive process towards the trunk or body of the tree. In turn, the flow continues through the branches and leaves of a tree but in a diverging recursive process away from the trunk. This is analogous to the earlier sports tournament and cell division examples used earlier to illustrate convergent and divergent recursive processes respectively. As we shall see, this inflow and outflow, tree like converging and diverging branching pattern is something that is found everywhere in the organization of the brain. And whereas in the case of botanical trees, it is water and plant nutrient that is being moved around, in the trees of brain and mind it is information that is flowing in and flowing out. If we needed an simple overall concept, or analogy for understanding brain and mind, then this would be it. But the trees of the brain would have an additional feature that is not found in our everyday trees. This is the idea that the tree structures and processes found all over the brain, loop back upon themselves so that some of the outflow of the divergent branches are feed back into the ‘roots’ of the corresponding convergent inflow going back to the original starting place. This is best illustrated by consider examples of this in the brain. So we’ll go through a load of brain structures existing at various scales to support these ideas, ending with a look at the entire brain.

So in the diagram below, diagram 1./ depicts a typical pyramidal type neuron which makes up around 80% of all the neurons in the cerebral cortex and is really the ‘work horse’ neuron of the brain. Most of the high level information processing in the brain critically involves these neurons. And all the major nerve fibre bundles, that is the white matter found underneath the surface of the neocortex, consists of the axons emanating from pyramidal neurons. So the diagram depicts a small pyramid shaped cell body, hence their name and the dendrites surround it. The arrow heading straight down from the cell body is the axon of the pyramidal cell which will traverse some distance to another part of the neocortex and then branch profusely all around its area of termination. This is not depicted. What is shown is a recurrent connection that splits off from the outward bound axon and which loops back to make connections in the vicinity of the neuron from which it came. However this recurrent part of the axon generally won’t make direct contact with the cell body of the originating neuron. Diagram 2./ depicts a cortical column which would be made up of around 1000 neurons and we have in composite, the same pattern that we described for a single pyramidal neuron. We could have an aggregates of connections flowing in and also flowing out, from and to other cortical columns. And we would likewise have an aggregate feedback loop made up of hundreds of individual nerve fibres splitting off from the output projections to other columns and flowing back into the same column. In Diagram 3./ we depict entire cortical patches each made up of many thousands of cortical columns. At this scale we can also make out a converging flowing in and diverging flowing out pattern, but this time involving entire patches of neocortex. At this scale there is an aggregate looping back of the recurrent loops deriving from individual columns but what is more important are the closed loops that many patches of neocortex, particularly in the frontal cortex, will form with the striatum and cerebellum.

The diagram below depicts a more complex arrangement of neocortical patches, which would correspond to a whole processing nexus centred in one of the prefrontal cortical regions involved some specific task such as speech or spatial processing. Here the central patch would have its own striatal loop and also a cerebellar loop(not drawn). It would receive various inflows of information from cortical patches around the brain each with their own little nexus of inflow, outflow and looping back activity. Our central prefrontal patch will in turn feed its output to various other centres that are themselves little processing centres with a similar set of inputs and outputs.

When we come to the entire brain then we can also conceptualize it is a tree structure with an overall recurrent loop. See diagram below. The converging roots would correspond to all the sensory hierarchies starting from the primary sensory cortices of sight, touch, sound etc. All these modalities will form their respective hierarchies with vision being the most complex and elaborated. Also the modalities will converge in some places to form cross modal or multimodal aggregated hierarchies. Eventually all these hierarchies flow inwards and converge upon the hippocampus, with each individual or multimodal sensory input, representing the distilled information coming from the various senses, making up the thickest root segments closest to the zone of ultimate convergence. This is actually the emotion centres and their full integration into the sensory and motor hierarchies will be explained in detail later. From here all the sensory flow of information is feed forwards to the cingulate cortex, which is the start of the diverging out motor and cognitive hierarchies. From here the flow of information spreads outwards. So at this end the initial thick branches of the cingulate cortex will correspond to modalities of action and mental processing. Different regions of the cingulate will drive different activities. So for instance the retrosplenial cortex will be involved in memory access and the posterior cingulate will be involved in internal spatial processing and first person perspective visualization. The anterior & dorsal-medial cingulate cortex will drive reasoning, cognition and language. The regions posterior to these regions are involved movement and motor control. So each of these modes will be like the initial thick branches of a tree, progressively radiating out to greater levels of detail ending up as specific thoughts, language output, movement and behaviours. And even at this whole brain scale we discover fibres tracts emanating out from cingulate areas and looping back to the inflow input zone, i.e. the hippocampus. This is depicted by the blue arrow in the diagram below. In this way the brain forms a massive recurrent loop between its diverging action and converging sensing hierarchies.

So we see this quite basic and simple recurring tree like patterning from the level of neurons right up to the level of the entire brain and various scales in between too. And together with this tree structuring we also find our recurrent loops all over the brain and at various scales, again from the level of neurons to the entire brain.

We’re starting to get an idea of some overarching patterns and modular design principles existing in the brain. Where before we seemed to have a mass of unrelated and hard to fathom physiological and anatomical data. We started off by describing the stereo-typical modular design of the cerebral cortex, cerebellar cortex and extended striatum. Then we described the stereotypical way in which these parts of the brain are wired together, i.e. with the striatum and cerebellum forming closed loops of circular connectivity with the cerebral cortex. We also briefly talked about the emotion centres and described how this striatal and cerebellar closed loop architecture also applied to these important brain areas as well. Thereby highlighting some more recurring organizational patterns. After that we introduced our basic overall organizing pattern found throughout the brain and at various scales i.e. the tree like inflow, outflow branching; and looping back arrangement. All our earlier stereotypically patterned modules which make up the cerebral cortex, cerebellum and striatum, and the cerebellar and striatal closed loops would totally fit into and be accommodated by our overall tree like organizing principle.

So we are getting a sense of least some order amid the bewildering complexity of the brain. However in our analysis of the brain and attempt to demonstrate underlying patterns we are constrained by the vagaries of language and verbal descriptions. Even with our visual and diagrammatic representations of brain we are limited. Towards our goal of fully understanding the brain and mind, we can only go so far with this sort of analogous explanation and graphical means of reasoning. This ‘mathematics of the eye’ as Benoit Mandelbrot described it. We need a more formal mathematical language and means of capturing our ideas in order to take things further. We need to geometrize the problem in order to more clearly see how all the different aspects of the puzzle relate to one another in order that they may be solved. We need to convert our fuzzy symmetries and rough analogies into something totally explicit and definite. So towards this end we need to introduce some abstract ideas that will allow us to completely formalize and precisely describe all the neural organizing patterns we have described so far. Later on we will completely revisit all the neuroscience and brain anatomy that we have been talking about so far. We will be elaborating on it much and then translate all of it into a unified theory of how the brain is organized and how it works.

The importance of abstracting, formalizing and geometrizing

We find that in the history of science it is often a process of geometrization which has enabled us to see things more clearly and explicitly. So it was through the analytical geometry of Rene Descartes which allowed for a more systematic study of the motion of the planets and which paved the way for the invention of the differential calculus by Isaac Newton and Gottfried Liebnitz. And likewise Einstein’s theory of relativity was only made possible by the application of a peculiar sort of non-Euclidean geometry where parallel lines may intersect. And so the same sort of idea may be useful for the task of understanding the brain and mind. What we need is some sort of formal language and abstract geometry that will allow us to fit together all the separate pieces of this vast jigsaw puzzle and see patterns which would otherwise be hidden from us.

In the late 1990s Rodney Brooks, who was then the director of the MIT (Massachusetts Institute of Technology) AI lab, predicted the coming of an, ‘Organizational principle, concept or language, that may revitalize the Mind Sciences in [the 21st Century]’. And separately in a 1996 publication titled, ‘Fractals of Brain, Fractals of Mind: In Search of a Symmetry Bond’, one of the contributing editors talked about the possible existence of what was described as a ‘secret symmetry’, secret in the sense of being undiscovered. Its uncovering would provide the ‘symmetry bond’ referred to in the subtitle of the book, which would allow us to understand brain and mind as existing on the same continuum where as they put it, ‘mind/brain performs as an indistinguishable one from a formal, neurological and psychological point of view’. The formal abstract geometry we are about to describe will be both an ‘organizational principle, concept or language’, and also one which will allow us to not just bring together all the details of the brain, but also unify brain and mind into a single conception. So what is this formalism? It is the language of binary trees and binary combinatorial codes. These are really the simplest and most basic concepts in computer science and may on first consideration seem unimpressive and limited. Also it will not be immediate clear why these concepts should provide the key to understanding the brain and mind. But science is really the process of explaining the most with the least and as we shall see, there is a very powerful way of looking at the brain which is completely binary, bifurcating, doubling, octaving, griddy and discretized. Also later on we shall demonstrate that these properties are pretty much ubiquitous throughout the brain and also in the description of the data structures of mind. But we first need to explain more clearly what are binary trees and binary combinatorial spaces.

Binary Trees and Binary Combinatorial Spaces

The binary tree is one of the most elementary ideas in computer science along with binary combinatorial spaces. And binary codes are the most basic unit of consideration in Claude Shannon’s information theory. Quite simply a binary tree is like the sort of tree we see in our gardens and parks but they have the property that every branching may only fork into two, never 3, 5 or even 4 directly. To the nodes of binary trees we can assign the numbers 0 or 1 and this allows us to represent combinatorial codes. Each of these codes is basically a binary base 2 number where the only digits allowed are 0 or 1, as opposed to the base 10 numbers we normally understand i.e. consisting of digits from 0 and 1 up to 9 and all the numbers in between. With our base 10 numbers we are specifying decimal codes which can theoretical represent any number we could possibly imagine. We could also equivalently represent these quantities in binary base 2 and this is the way digital computers work. So this is what is meant by binary coding. But what is a binary combinatorial space? Here is where we diverge a little from the world of base 10 and decimal numbers.

A binary combinatorial space is where first we consider every combinatorial possibility that a sequence of binary digits may represent, i.e. every number that may be represented by that sequence of binary bits by allocating to it, either 0 or 1. We then treat each combinatorial possibility as a point in abstract space. We then define a distance measure between all these points in abstract space corresponding to every different combinatorial code that may be represented by our sequence of binary digits. This is simply what’s known as the ‘Hamming distance’ between any two different binary sequences. And this is the difference not in the number that the binary sequences represent, but rather the difference in bit pattern when we take two binary sequences, put them side by side, and make a comparison at each corresponding point of the two sequences to see whether they contain the same binary digit or not, i.e. 0 or 1. If both the sequences contain the same digit, i.e. both 0 or both 1, then there is no difference and no additional distance. If however they contain non matching bits, then this point along the two sequences contributes 1 unit of distance to the overall distance measure. And so we do this for every point along the two binary sequences we’re comparing, and we end up with a total tally of the number of bits where the two sequences differ. This is the hamming distance. If both binary sequences were identical then the hamming distance would be zero. If they were not identical then the hamming distance would be greater than zero and this would give an indicator of how different the two binary sequences were.

Because these sequences are being thought of as points in an abstract space, then we now have a way of completely specifying the distance between every single point in this abstract space. This way of looking at things satisfies certain conditions so that mathematicians who study this sort of thing would call this a non-Euclidean metric space. Sometimes this binary combinatorial space is referred to as ‘binary N-space’ with the ‘N’ corresponding to an arbitrary number of binary digits or length of the sequences that we might take into consideration. In this abstract binary N-space we may define points, lines, circles, regions, attractor basins etc. However they will have very different properties from the lines and circles we might define in our normal ‘x, y, z’, Euclidean or Cartesian space, which we normally perceive and think of as space. Some of these special properties of binary N-space make it very suitable for representing things and tolerating errors and also for understanding the brain.

Reconciling Binary Combinatorial N-space with ‘normal’ Euclidean space

We have seen how binary trees give rise to an abstract non-euclidean metric space based on binary codes corresponding with binary digits allocated to binary tree nodes. But our brains and minds need to function in euclidean space, i.e. space as we normally perceive and understand it. We need to be able to use binary trees to represent these normal spatial relationships and the spatially defined objects existing in the space of the reality we inhabit. Fortunately there is a very straight forward way to do this through the recursive sub-dividing of lengths in normal space into two and then allocating these two separated compartments to the two branchings of a node of a binary tree.

The simple diagram below would explain this most effectively. Here we see a series of squares subdivided into a 2-dimensional grid. Drawn together with the blue 2d squares and grids representing normal euclidean space is a binary tree drawn in red. It is drawn in such a way that it is easy to see the correspondence between the nodes of the binary tree and the blue squares representing normal space. It is easy to imagine how the grids can be specified to any resolution and details, i.e. with more branchings of the binary tree. It is also possible to use the same principle to include a third spatial dimension, though this is harder to depict in a static 2 dimensional diagram. In computer science this way of representing space is called Quad Trees. Something similar also occurs in the implementation of Wavelet Transforms which are used in data compression algorithms.

The next diagram below also depicts a binary tree except it is drawn flattened out in the 2d plane. It shows in a different way how we may maps the nodes of a binary tree into normal euclidean space in order to represent it. The binary trees in the diagram below and above are actually topologically equivalent.

The next diagram shows 2-dimensional grids of increasing resolution, represented by binary trees of increasing branching depth.

Furthermore we may also represent time using binary trees. So that instead of recursively partitioning space into binary subdivisions we instead carry out the same process along the time dimension. So in the next diagram below, instead of using binary trees to represent space, each of the branches subdivide the time dimension. At the bottom end nodes of the temporal tree are depicted eight 2d grids representing spatial information. So taken together this effectively hybrid temporal-spatial binary tree would be able to represent the succession and evolution of spatial information over time.

Now that we’ve described how we can specify euclidean space and time using binary trees, we can then allocate these spaces or contexts, binary values, i.e. 0 or 1, so that these spatial temporal descriptions may also contain a combinatorial code. Or put another way, they become spatial temporal combinatorial codes. We have then a merging of the earlier description of binary combinatorial space, with our binary tree representation of euclidean space. It means that we may take projections from euclidean space and map them onto our binary tree and binary combinatorial space language. This sounds a little complex but a simple example will greatly clarify what is meant by this. When we take a digital photograph, then patterns of light from objects existing in reality are projected by the camera lens onto a 2d surface which converts the light into digital information. This is an example of a projection from euclidean space onto combinatorial space, whereby a photographic scene of converted into a computer file in the camera. All we’re adding on top of this simple commonly used everyday process is a binary tree hierarchical structure on top of the 2d photo plane.

There is a slight problem with mappings from euclidean space to binary combinatorial space. This is that slight changes or transformations in euclidean space can lead to massive hamming distance changes in binary combinatorial space. The solution is that we include all the simple affine transformations in euclidean space, i.e. translation, rotation, scaling and inversion, into our binary trees, and also represent these transformations as binary codes. So if our binary trees represent squares in euclidean space, as depicted in the diagrams above, then these squares are allowed to transform themselves i.e. shift, enlarge/contract and rotate. If a picture is sometimes worth a thousand words, then a short video animation even more so. A relatively simple computer program shows this process in action. See… https://www.youtube.com/watch?v=1CNjZGhWqC0&feature=youtu.be

The Binary Brain and Neuropsychology

In this next section we will demonstrate how it is that the binary tree and binary coded way of looking at things is extremely relevant for understanding the brain and mind. Going through quite a diverse array of empirical findings, we show that there is a very powerful way of looking at the brain and structures of mind, which is completely binary, bifurcating, octaving, doubling, griddy and quantized. Furthermore this perspective on things is pretty ubiquitous and generalizing throughout the brain and mind.

Binary Space

We start with considering how we form mental maps of space and the outside world in our heads. A clue as to the workings of this essential facility is provided by case studies of patients suffering from a neurological deficit called hemi-neglect. Sometimes we come to learn better about how something works when it breaks down. This is a case in point. People suffering from hemi-neglect usually have some form of damage to the parietal lobes, which are an area of neocortex which our coordinates movements in space and is known to be involved in forming our mental maps of the worlds. The classic symptom of hemi-neglect syndrome is that those afflicted with it literally lose half their world; both either to perceive it or to imagine it. So the pictures below show what happens when someone with hemineglect is asked to copy some shape or picture. They literally can't place or register details on the neglected side. Asked to draw a clock, they can only place the numbers on one side. Asked to read a word they only register the letters on one side.

When asked to eat a plate of food, they will eat only the food on one side of the plate and stop when that’s all gone seemingly oblivious to the food on the other side of the plate. But when the plate is rotated 180 degrees, so that the remaining food comes into view of the non-neglected side then the test subject starts eating again, only to stop half way again leaving all the food on the non-neglected side untouched. It seems as if there is an underlying binary sub-divisioning in our ability to perceive objects including plates of food and the food on the plate. See diagrams below.

In an experiment on hemineglect sufferers, patients were asked to imagine walking down some street well known. They were asked to name the various buildings located on either side of the street. As might be expected they were unable to recall any of the buildings located on the neglected side. However when they were asked to imagine walking down to the end of the street, turning completely around and walking back the other way, now they were able to name all the previously unrecallable buildings which couldn’t be imagined. But also this time round, the patients had extreme difficulty recollecting the building on the opposite side of the street, which were easily described when imagining walking in the original direction This relates to what psychologists refer to as egocentric spatial mapping. It is our view of the world from the first person perspective and so here again we see a binary partitioning of our egocentric maps. Not just in our ability to perceive but also in our ability to imagine.

When we consider the micro-features of our visual perception which are spatial frequency detectors (which also act as bar detectors) then we discover an interesting doubling in frequency arrangement. The diagram below left shows what is meant by spatial frequency. From what are known as 'Just noticeable difference’ (JND) experiments it seems that the spatial frequencies which are detected by our brains are spaced one octave apart. It is possible to manipulate each sub-band separately from the adjacent sub-bands above and below. So that when a spatial frequency grating with a specific wavelength was repeatedly presented as a target stimulus, it became progressively harder to detect with each presentation. It is as if a specific frequency processing sub-band was getting steadily fatigued and so altering its ability to respond. However when spatial frequency gratings either double in frequency and half the wavelength, or else half the frequency and double the wavelength, i.e. one octave above or one octave below the original spatial frequency, then these new patterns were able to be detected more easily, in the same way the original stimulus was initially easier to detect. We may map frequencies all an octave apart to correspond with the nodes of a binary tree which thereby encode this doubling and halving pattern. See diagram below right.

Another brain system also displaying a similar frequency doubling pattern to the spatial frequency detectors of the mammalian visual system, is the centres of the brain involved in what is referred to as ‘allocentric’ spatial representation. These involve the recently discovered grid cells found in the hippocampus and auxiliary structures in particular the dentate gyrus. Allocentric spatial mapping refers to where we are in a physical coordinate system and frame of reference. So for instance our GPS (global positioning system) location is an example of an allocentric point of reference. This is kind of spatial processing is also involved in procedure known as ‘dead reckoning’, where direction, velocity and times of travel can be used to work out where we are in physical reality. It is our x, y location or point on the map. This is different from our egocentric spatial map or first person perspective as discussed earlier. We may stay in the same allocentric location, i.e. standing on the same spot, but we might turn our heads and scan our eyes all over the place to give lots of different egocentric frames of reference, but tied to one unchanging allocentric reference frame. What was discovered in recent years is that there exist special neurons which seem to map out lattice like reference grids over our allocentric x, y floor space. These neurons will only fire in response to the animal being in specific allocentric locations as the test subject moves around an open area of floor space.

The amazing thing is that these special locations where a particular 'grid' neuron fires the most, closely correspond to the vertices of equilateral triangles arranged in a perfectly tessellating grid projected onto the floor space. See diagram below. Alternatively we can think of these special firing zones as being in the centres of perfect hexagons similarly arranged in a perfectly tessellating grid. Furthermore these tessallating grids only come in certain sizes. What’s really interesting for our current purposes is that the scaling factor from one grid size to the next one above or below in size is roughly 1.42. This is a very recent finding and is significant because 1.42 is roughly the square root of 2. This means that if we multiply by 1.42 twice then we come to double the scale of the equilateral triangles or hexagons.

So we see this frequency doubling phenomenon with respect to the micro-features of visual perception, i.e. spatial frequency and also in the context of allocentric spatial mapping where in a sense a spatial frequency mesh is projected onto our perception of physical space.

When we turn our attention to the auditory system, the part of the brain most associated with processing frequency and oscillating phenomena, then in the arrangement of the cochlear-topic or tonotopic maps of the auditory cortex then again we discover this frequency doubling arrangement. The frequency processing sub-bands of the auditory cortex are spaced 1 octave apart. See diagram below.

More generally, it is possible to take any sort of cortical topographic map or any region of the cortex and represent them using binary trees; thus also making them conform to our binary bifurcating and doubling view of things. This would include visual retinotopic maps, auditory tonotopic maps, the somatotopic maps representing our bodies, the musculature-topic map representing our muscle groups and all the other areas of neocortex involved in more abstract functions. In general all these sorts of topographic brain maps which preserve spatial relationships are involved in a very simple form of spatial mapping. The 2d-laminae which these topographic maps can be thought of as, represent in a sort of pictographic form our senses, musculature, memories and thoughts. Even if this is not intuitive or obviously apparent as in the case of more abstract representations. And so too is it the case that we can use binary trees to reduce and represent any 2d topographic map or any sub-area of cerebral cortex.

For as we showed earlier there is a very straight forward way to map binary trees onto any bounded two dimensional plane. All cortical maps are effectively laminar or sheet-like 2d planes, and therefore we can represent them easily using our binary tree spatial mapping scheme described earlier. This gives us a rather boxy and square spatial structuring and some readers may object that the components of real cortical maps are better thought of as being arranged in a more hexagonal pattern. After all, if the brain is made of roughly cylindrical columns, then wouldn’t this pack into a more hexagonal grid? Also from what we find in empirical studies of neocortex and also in relation to our allocentric grids, it seems hexagons or equilateral triangles seem more the norm. Wouldn't this be incompatible with our square view of things? The simple answer is that we can make these alternative tessellating patterns conform to our binary and square way of looking at things. We do this because it allows us to develop the more unified perspective which we seek, while also keeping our grounding in empirical evidence. A diagram or two can save a lot of verbal description so below is depicted a way in which a hexagonal mapping scheme can be perfectly represented with our binary tree and boxy, griddy way of doing things. See diagrams below. The centres of each hexagon may be mapped to a superimposed square grid, drawn in blue, and which may in turn be specified using binary trees.

What we have shown so far is that there is an underlying binary and doubling pattern to a host of brain functions and facilities of mind, that may seem very unrelated and disparate. From egocentric spatial maps, to allocentric spatial maps to all the varieties of topographic cortical map. And from the micro-features of visual perception to the or brain’s processing of auditory sound information. We see throughout all these different ways of thinking about space, this underlying binary, doubling, octaving pattern. When we turn our attention to time then likewise we see pervading underneath and in all manner of ways, this sort of binary structuring. We have discussed the different sorts of spatial representation and we will now discuss temporal representation.

Binary Time

At the start of our discussion on the binary nature of time processing and representation in the brain, first we need to show it discrete nature. In the same way that space is demarcated by the peaks and troughs of spatial frequency patterns or the binary sub-divisioning of egocentric maps and compartmentalized around these reference points; likewise time is also partitioned and compartmentalized. Though subjectively we experience a continuous flow of time, we know from physiological data and evidence from neuropsychological experiments, that time is actually chopped up into separated segments and is non-continuously processed by our brains.

Brain waves are pretty much all pervasive in the brains of just about any living creature imaginable. The peaks and troughs of these waves synchronize brain activity but also enable and dis-enable it. So for instance in the rabbit, the olfactory bulb (smell brain) and hippocampus (A region involved in memory) oscillate at the theta frequency range ~5-6 htz. When we look at rabbits close up then we can usually see that they are constantly sniffing around very rapidly. This is actually occurring at roughly 5 to 6 htz and is synchronized with the theta brain waves in the olfactory bulb and hippocampus of rabbits. Experiments were performed to try and illicit synaptic LTP (Long term potentiation) in the rabbit hippocampus. LTP is an experimentally induced modification in the synaptic strength of the connection between two brain cells and is believed to be the physiological substrate of our memories. This LTP in rabbit brain components could only be induced when the training stimuli, electrical currents coming from the experimenters micro-electrodes, was timed to coincide with the peaks of the hippocampal theta waves. Otherwise there was no effect and no ‘learning’. We can imagine from these results the idea of the rabbit taking rapid discretized temporal snapshots of the olfactory environment. So therefore in this way time is being partitioned into separate compartments.

In another set of experiments this time performed on humans and relating to the ~40 htz gamma wave which is expressed all over the brain, a similar discontinuity of temporal processing was discovered. Here the human test subject was shown some video footage but it had the special property of being shown only 50% of the time and in an oscillating succession of flashes, timed 25 milliseconds apart and therefore shown at ~40 htz. In fact the display of this visual input was synchronized with the phase of the human test subjects visual cortex gamma waves recorded using sensitive non-invasive surface electrodes. When the showing of the bursts of visual input was made to perfectly correspond with the peaks of the visual cortex gamma waves then everything was fine and the subject saw the video perfectly. It was perceived as a continuous stream with no apparent discontinuities. But when the same visual input was merely phase shifted 180 degrees, so that the exact same input was shown but this time synchronized to correspond with the troughs of the visual gamma waves then the subjects saw nothing. This again demonstrates the slicing up and demarcating of time using brain waves.

Moving on, when we consider the relationship between the main brain waves that have been classified and given labels then again we see a doubling, octaving, powers of two pattern in their relative frequencies. This is not by design, but merely the frequencies around which the power spectra from the EEG (electroencephalogram) recordings were grouped and so were given specific names. First to be discovered in the early 20th century are the alpha waves at around 10 htz. Then we have the theta waves at ~5-6 htz, the beta waves at ~20 htz and gamma waves at ~40 htz. When we arrange these waves in the diagram below then once again we see this octaving frequency doubling pattern. Neuropsychologist now believe that these waves interact with one another and that the lower frequencies harmonically nest the progressively higher ones. In music the most harmonic interval is the octave, and it is easy to see how a harmonic nested temporal representation scheme would most natural employ this octaving spacing of relative brain wave frequencies. Neuroscientists believe that these brain waves are in turn nested in even lower frequency brain waves, i.e. delta waves, and these may even have fractional frequencies or wavelengths extending to several seconds. It would be interesting to see if these lower frequency waves likewise fit into our binary doubling patterning scheme of things.

While we are still waiting for clarification on the exact nature of these lower frequency oscillations with wavelengths spanning even a whole second or more; at the same time we do know a lot about another rhythmic oscillatory phenomenon which the brain processes and which features regular periodic behaviour spanning into the minute range and more. This is music and again we discover an underlying binary doubling structure to it. In the forms of music which are most accessible, easy to listen, easy to remember, most popular and arguably easiest for the brain to process then we find a certain doubling pattern in the grouping of the beat patterns behind it. The kinds of music this particular grouping pattern would apply to are pop music, folk, rock and also importantly what’s know as ‘common time’ classical music with the 4/4 time signature denoting 4 beats to a measure made of quarter notes. When we look at the overall structure of these kinds of music then usually there will a certain grouping of the beats in a doubling, powers of two binary structure. So 4 beats will be grouped into 8, then this will make longer agglomerations of 16 beats, which in turn form composite structures of 32 beats etc. These structures might form the verses and choruses of a pop song say, which are then linked up like modular building blocks. This gives us a very regular patterned rectangular or boxy structure to the way most music is constructed. See diagram below.

While there also exist a variety of funny time signatures and other less ordered musical arrangements, in the kinds of music which we most instantly appreciate there does seem to be this doubling structure behind it. After all, one or two music theorists over the years have said then any collection of arbitrary and even random series of sounds will if, listened to enough, start to sound like a sort of music to the ear. There may be some truth in this. It is certainly true that when sounds are grouped together in a certain sort of way, i.e. in our binary grouping structure, then it can immediate sound like music and we are able to process and also remember it with ease. Sometimes we have that feeling of, ‘not being able to get that tune out of my head’. And sometimes we will hear some popular song or melody just once and have almost perfect or at least very good recall of it afterwards. It would make perfect sense that this was happening because the music was structured in such a way, as to make it as easy as possible for the brain to process. And if, as we believe, the underlying structuring of the brains spatial and temporal representations are inherently binary then those kinds of music which we would find the most accessible and easy to process would be exactly those kinds of music which follow an underlying binary doubling structuring pattern. And we know that this is indeed the case.

We can also apply this sort of reasoning to simple nursery rhymes for children, limericks and some common structures used in poetry. Again there’s a certain meter and patterning of time which again fits into the same sort of organization as the popular sorts of music described earlier. And of course song lyrics are words set to music and also have this same basic accessibility even when read as poetry without the music behind it.

When we further consider the nature of language itself then we discover more binary structure behind it. In fact all human languages as well as computer programming languages and even the language of logic, can be broken down and analyzed in terms of binary trees. This follows from the work of people like Noam Chomsky and a host of others. So in the diagrams below we see simple sentences structured as simple binary trees forming more complex ones which are also binary structured. We can take this process to ever increasing degrees of complexity to form extremely complex linguistic constructs, but still they will obey this binary structuring.

This binary view of language, be it human language or computer code, is very interesting especially when we relate it to the preceding discussing of spatial and temporal representation, which we have shown to both have an inherently binary structure underlying it. Furthermore we demonstrated this binary order in all the kinds of spatial mapping, i.e. egocentric, allocentric and topographic; and also showed how this binary structuring can be seen in many temporal scales, from some various millisecond ranges, right up to the time periods spanned by sections of music. This spatial and temporal aspect of what it is our brains and minds process is sometimes referred to as the ‘sub-symbolic’ level, by artificial intelligence researchers and cognitive scientists. And of course language processing would make up the corresponding ‘symbolic’ level. What all this means is that we have then a single unifying ‘language’ or formalism for describing both the ‘symbolic’ and ‘subsymbolic’ levels. As we will demonstrate later on, this allows us to fully explain how the two levels inter-operate and also to show that they really exist on the same symmetric, self-similar and recursive continuum. So that the spatial, temporal subsymbolic and symbolic and really just manifestations of a deeper underlying symmetry or invariant factor.

Having a common conceptual language to think about the symbolic together with the spatial and temporal sub-symbolic allows us to better understand how these different aspects interact in our brains and minds. And obviously they do. For instance we symbolize, i.e. we describe with words and sentences, the complex spatial and temporal input we see with our eyes and hear with our ears. And vice-versa we may register a stream of words while reading from a book and from them reconstruct in our imagination sights, sounds and even complex social situations happening in specific circumstances or locations. So the spatial and temporal, i.e the so called sub-symbolic becomes symbolized and the symbolic becomes spatialized and temporalized.

Within the sub-symbolic there is also an important interchangeability between time and space that goes on in the way we process and perceive the world. A complex spatial scene is serially broken down in time by scanning our gaze repeated over the image over a certain extended time period in order to register all the salient details contained within it. It is only the simplest of images, a road sign say, that we are able to process in one shot or in very few sequential operations. And in our actions a complex task is carried out over several different physical locations. It is serialized and carried out one specific sub-task at a time. Also something which is so obvious we never think about it. When going from two spatial locations, i.e. from ‘a’ to ‘b’ as we say, it is never spontaneous but involves a journey that occurs over a certain length of time so that a spatial distance becomes expressed as a time period.

Going the other way the temporal is also spatialized. Things or sensory patterns that are spread out over time may be represented spatially. So music and spoken language which are phenomena occurring spread out over time, may be representing on a sheet of paper spatially as music notation or the written word. Sometimes it is said that a composer will ‘see’ the entire score of music in his or her head, and the construction of sentences is sometimes visualized in our minds as we formulate them. Also we represent time in terms of numbers, i.e. hours, minutes and seconds, this is obvious; but what is less well known is that in our minds we spatialize enumerations. Neuropsychologists have discovered a ‘number line’ by which we represent numbers spatially as if arranged in order along a line going from left to right in our minds. They discovered that this ‘number line’ was processed by exactly those same brain regions which handle physical spatial processing. So this is another way that time becomes space.

This interoperability and interchangeability between space and time, i.e. the parallel and the serial becomes very important when it comes to implementing the brain theory in software and eventually directly in hardware. These ideas in relation to the quest to create artificial intelligence will be discussed in a later chapter. The reconciling of the symbolic and subsymbolic which is known as the ‘symbol grounding problem’ in the field of artificial intelligence has not been solved satisfactorily. And a universal and general way to represent time is an important current issue in AI as is the issue of integrating time into existing hierarchical spatial representation schemes. A way of showing the total inter-relatedness of the symbolic, spatial and temporal would obviously be very useful towards these ends. The fact that this integration is also grounded in actual brain physiology and neuropsychological evidence is a bonus. Furthermore once we adopt this way of 'doing’ AI, the reverse engineering the brain approach, then it is also much easier to integrate new findings from the brain and mind sciences into our inventions.

The Binary Neural Substrate of the Brain

Thus far in our survey of the binary brain and mind, we have mainly considered quite high level functioning and emergent phenomena. So we have talked about various spatial and temporal representations. Also we has discussed language and brain waves. Now we show that our binary perspective extends all the way down to the underlying physical substrate of brain organization and physiology. The binary nature of the brain even extends to the very process by which brains come into being i.e. neurogenesis, which is a subset of the process by which our bodies come into being i.e. ontogenesis. This is because every cell that makes up our brains and our bodies, is created in the binary cell divisioning process called mitosis, whereby one cell becomes two, becomes four, becomes eight, then to sixteen, thirty two, sixty four etc. in powers of two or doubling increments. This is what we shall discuss next because it can provide the means by which brains can contain so many inter-operating sub-components that are so exactly laid out and intricately interconnected in a very specific way to produce the detailed hardwiring of our brains. It is very reasonable to conjecture that the binary cell sub-divisioning process of mitosis may be the key to understanding not just how our brains come into being, but also how it comes to be so precisely organized.

So we start life from being a fertilized egg. This singularity from which the human form begins, starts to recursively subdivide as we have already described. And from this process of mitosis, every cell in our body is produced. Along the way we assume a variety of intermediate forms. A few weeks into this process of ontogenesis starting from a fertilized egg, we become something like a flat disc already made of many millions of cells. Then this flatted form rolls up and folds up in intricate ways to start generating our body layouts and more complex morphological forms. See diagrams below.

Sometimes in order to conceptualize things better, physiologists imagine what it would be like if this complex folding of our initial flat disc like form didn’t occur and we stayed as a flat sheet. This is called a flat-map. So we can depict a flat map for our brains with all its diverse regions and sub-components. So in diagram below left we have a flat map of the cerebral cortex and in diagram below right we have a flat map of the hypothalamus. We could potentially specify flat maps for every region and all the structures of the brain.

The question is, how do the different cells in the demarcated regions of the flat map ‘know’ what to be? How do we come to be able to define all the many different regions in the first place? There are two answers to this question and both of them play a role in this process of specifying the layout and functioning of the cells contained in our bodies and brains. The two ways are known as intrinsic and extrinsic. An analogy will help us to understand these concepts more easily. We may ask how do we as people come to be the way we are. And the answer is that it is a combination of nature and nurture. i.e. our behaviour and the way we look is determined by our genetic endowment that we receive from our parents and also from the environment in which we develop and live our lives. The genetic component can be said to be intrinsic and the environmental influences can be called extrinsic. And so it is with the cells in our bodies including brain cells. They know what to be through extrinsic factors, i.e. their surrounding chemical environment and also intrinsic factors which means the cell lineage from which they are derived. As the process of cell division proceeds, then at certain points changes will occur in the DNA of the dividing cells and also what are known as epigenetic factors will also modify the cells. These genetic and epigenetic changes made to cells undergoing mitosis or cell division will be passed on to all future cells which derive from them. So each such modification will start a new lineage with all the future progeny cells containing the change. This will affect ‘intrinsically’ how these cells function and interact with other cells. So this combination of intrinsic and extrinsic influence, determines how cells act and what they are, much the same with people.

Extrinsic factors certainly do play an important role in the development and arrangement of our bodies and brains. This can be through the creation of chemical gradients across certain body regions which may provide relative spatial information to inform cell development. But when it comes to precisely specifying the extremely complex and intricate components and subcomponents of the brain, merely relying on chemical gradients and extrinsic factors to determine cell development becomes limited. The problem with chemical environments is that they diffuse, so if we are to use them for determining exact brain regions then the problem is how do we prevent all the chemical markers for all the different regions from diffusing across to other regions. And we’re still left with the problem of how to specify these myriad sub-regions in the first place. A much simpler solution would be to use intrinsic factors to determine cell functioning and also in order to exactly specify all the different regions of the brain. Because the process of cell division is binary then each new specification of lineage and genetic or epigenetic modification will be occurring in a binary diverging way. With each ‘lineage’ beginning at the branching of a binary tree and all future progeny belonging to this lineage may be represented as nodes of a binary tree corresponding to the mitosis binary cell divisioning process emanating from it. These sub-branchings may in turn contain more genetic and epigenetic modifications specifying sub-lineages and sub-sub-lineages and so on and so forth to create a binary map of all the cells in our bodies and brains. This binary tree of cell divisioning and genetic/epigenetic specification of lineage may be flattened out and represented as a 2-dimensional plane using a special 2-d fractal representation of a binary tree. The diagrams below illustrate this idea.

A simple computer program helps us visualize this process of specifying cell lineages in order to specify different brain regions. It simply involves the creation of a 2-d flattened binary tree like the one in the diagram above. Some simple computer code then detects mouse clicks on any of the nodes of the binary tree depicted in the flattened tree diagram as T junctions. This selects that node. The program then translates this into the shading and marker slightly darker blue of all the enclosing boxes of the subnodes or progeny nodes below the selected node. So for instance by clicking on the very centre node which corresponds to the top node of the entire binary tree, then the entire grid is colored a slightly darker blue. If we do the same with any node then a griddy sub-region corresponding to the space containing all its progeny nodes will likewise be shaded a slightly darker shade of blue. If a picture can be worth a thousand words then a short video more so. This link to a short screen capture video shows the program action...

We can use this program and binary demarcation process to create a variety of patterns to arbitrary levels of depth and complexity. So the diagram below left is an example and immediately to the right of it is another example but created in such a way that it is symmetrical along the midline y-axis.

In the diagram below is the same diagram as above right but juxtaposed with some flat maps of the cerebral cortex and the hypothalamus. What this is meant to show graphically is the idea that we may specify a pattern of demarcations using our computer program that may potentially exactly correspond with the specification of all the different regions of real brains. And because we are proposing that real brains are specified in detail with exactly the same binary demarcation process then this is not merely an analogy but rather a demonstration of a direct correspondence between the workings of the computer program and the way that real brains are specified and formed. Of course there are complications to this picture, in that in real brain cells will migrate from one region to another but we are mainly interested in the specifying of detailed regional cortical maps and the myriad main sub-components of brain, not in the specification of every neuron or support cell in the brain. What would be the case in real brains would be a small set of cells which would play this precise regional specifying role acting as markers and demarcators. All the other cells would operate within the guidelines and signposts provided by this special subset of cells. This would be akin to the operation of hox-genes and the exact specifying of overall body plans in ontogenesis.

So in this way with our computer program we have created a sort of ‘digitized’ brain which is completely specified in the language of binary trees. Through a sub-divisioning and demarcating of the different regions of real brains, myriad spatial contexts are created into which real neurons and support cells may be located. In the creation of our digitized brain map and computer likeness of a brain we are able to place digital neurons and digital spatial/temporal representations. This is where we discover a fantastic set of insights. Because we have already shown that the way the brain represents symbol, space and time; also has an inherent binary digital structure underlying it. All the various spatial maps, topographic maps and representations of temporal sequential information, together with our linguistic symbolic facilities also are inherently binary and digital in nature. Therefore this content of mind fits perfectly together with these contexts of brain, for they are expressed in the same binary language. What this means is that it is very straight forward to place within our digitized brain, all our digitized topographic maps, all our digitized representations of space and time and all our digital symbolic structures of language. Once we have the physical substrate of brain and the representational structures of mind expressed in the same underlying language then this opens up the possibility for us to conceptualize brain/mind and all the things contained within it, i.e. our thoughts, memories, skills together with all the subcomponents of perception and action, as a single integrated whole and single all encompassing structure. We’ll discuss this much more later on.

We’re starting to see a gradual blurring and intersection between the worlds of neuroscience and neuropsychology on the one hand, and the world of computer science on the other. Perhaps it can seem a little odd, the juxtaposition of curvy rounded organic flat map structures with our griddy computer generated likeness of a brain map. However very recent research seems to have demonstrated at least in the wiring of the brain, a systematic grid like arrangement governing how brains are interconnected. Through new very high resolution brain scanning technology and some complex mathematical analysis, it was discovered as recently as 2012, that the nerve fibres connecting up the different regions and subregions of the cerebral cortex are routed in an almost perfectly grid structured way, much like the square road layout of American cities. So in our brains and brains of monkeys also studied, the fibre tracts connecting the cerebral cortex can go across and up but never diagonally. Therefore the fibres may run parallel to each like computer ribbon cables or else intersect only at perfect 90 degree right angles. See diagram below of an actual scan of this fibre arrangement. The researchers found that this pattern holds true at all scales analyzed from an overall whole brain scale to the smaller various levels of cerebral sub-patches. Furthermore this griddy arrangement also seems to hold true for many subcortical structures and even the spinal cord. This griddy wiring pattern found in reals brains would be a perfect fit for connecting up the right angled orthogonally organized structures of our digitized brain.

We may add to our current discussion of the binary brain substrate and griddy nerve fibre wiring patterns, the fact that there is also a ubiquitous pattern to how nerve fibres branch and which is always binary. The branching of the axons or nerve fibres which project output signals forwards from neurons, always branch in a bifurcating binary manner, never in trifurcations, quad-furcations or quint-furcations. And the same for every dendrite branching backwards from a neuron making up the complex dendritic ‘arbor’ or tree-like profusion collecting signals from other neurons to relay to the cell body. Essentially all the axonal and dendritic ‘trees’ existing in our brains actually are binary trees. When we consider the signals that travel along axons, i.e. action potentials, then this too can be considered as essentially binary or digital in nature that is on or off. And the release of neurotransmitters at the end of an axon relaying information to the dendrite of another neuron is quantized. This rounds off our digital, binary, quantized, boxy and griddy picture of physical brain structure and all the interconnections between all its various sub-structures.

Later on we show how our boxy binary digitized brain is an even better fit for modelling real brains than superficial inspection would suggest. Not only does our digitized brain and real brains have in their formation an underlying binary digital code for the exact specification of their respective sub-components. Furthermore, In the same way that there is an orthogonal or right angled boxy structure to our digital brain, likewise there is an orthogonal organizing principle to be found in the neocortex itself to go along with the griddy wiring underlying it. This paves the way for a complete digitization of real brains through a process whereby our theoretical computer generated binary brain is able to map systematically to all the relevant information process components of real brain. This opens up a whole new way to do what is know as ‘Whole Brain Emulation’ or WBE for short. It really takes the Artificial Intelligence theorist Ray Kurzweil’s idea of abstracting away all the non essential details for brain emulation to the absolute maximum. Things don’t get any more abstract or minimal than binary trees and binary combinatorial spaces in computer science. Whole brain emulation using this approach will produce many orders of magnitude improvements in computational efficiency and performance advantage over projects such as the Waterloo University Spaun project and Henry Markram’s whole brain simulation plans. It also has the advantage of being based on a comprehensive brain theory at the outset. When we add fractal time and dynamic recursive processes to our fractal digitized brain then we will be able to show how it can be made to come to life. Amazingly in the operation of this digitized brain will be the subsumption of many of the best ideas and algorithms in artificial intelligence and computer science which we’ll explain in some final sections. What we think is about to happen is a complete unification of neuroscience and psychology together with some of the most important ideas in computer science and artificial intelligence, into a single overarching conception.

But for now we will continue this section on the binary neural substrate of the brain by examining how our binary tree way of looking at time and temporal representation may be implemented by actual neural physiological mechanism. We’ve covered the physiology of the binary structural spatial. Now we consider how binary time may also be given its physiological basis.

The Neurophysiological Basis of Binary Time

Earlier we examined some time dependent phenomena from what may be described as a higher level neuropsychological level. That is we talked about some of the properties of surface brain waves, music and language in order to demonstrate a common binary doubling pattern underlying them all. What we do now is to show how and why this binary patterning comes about. We seek to explain the neurophysiological basis for time representation and time processing.

Time is such an important dimension in our lives and in the operation of our minds. Without it nothing much would happen. It is central, fundamental and all pervading and we would expect that it is implemented by some aspect of our brain architecture which is likewise all pervading. The perfect candidate for this role would be the recurrent loops found throughout the brain, in every structure and at every scale. Cortical columns feedback upon themselves and the individual layers within columns do likewise. The entire brain contains a massive recurrent loop projecting back upon itself and importantly the cortical areas of the prefrontal cortex along with some special posterior areas form recurrent closed loops with the cerebellum and striatum. Here we have a lot of empirical evidence to suggest that these two important structures, i.e. the cerebellum and striatum, are heavily involved in the temporal processing and the representation of ordered sequential information. I believe we can perfectly extrapolate this time handling property of these cerebellar and striatal closed loops to all the other kinds of recurrent loops found in all the other areas of the brain and at all scales.

But how would a recurrent looping back arrangement of neural connectivity enable us to represent, recognize and reconstruct temporal information? The answer is that recurrent loops allows us to represent two consecutive moments in time in the simplest way imaginable, that is as the two diverging branches of a binary tree. This is because one branch can represent ‘past’ and the other branch can represent ‘future’. Taken together these two branches have represented a demarcation in time. These two branches may further divide to produce four branches in total which may represent ‘distant past’, ‘near past’, near future’ and ‘distant future’. This process can be recursive iterated to produce representations of time to any resolution required. But how does a recurrent loop come to represent a temporal binary tree of the sort we have just described? This comes about due to two reasons. They are firstly the fact which we explained earlier, that time is discretized into separate compartments by the action of brain waves. Secondly there is a delay in the time it takes for the signal coming from any brain component travelling along one of these recurrent paths to loop back upon itself. As long as this delayed looping back signal is integrated with the set of signals coming in at the next time compartment, then this allows us to represent the conjugation or joining of the information relayed by the current ‘fresh’ signal together with the information carried by the recurrent signal which relates to the previous time compartment. So in a very simple manner we may register the succession of two time steps i.e.( t ) & ( t + 1 ) ,which we might also refer to as past and future and make correspond with the two nodes of a binary branching. We can iterate this process to produce tree structures encoding more than 2 time steps. See diagram below. And if we can register temporal sequences we can also reconstruct them to form temporal combinatorial codes. The orange arrows in diagram below right, show this reconstructing of the next step in the sequence based on the previous registration of a particular sequence in a particular order.

The finest temporal resolution that this proposed physiological mechanism will be operating at would be ~40 htz or the gamma range. The gamma wave is ubiquitous all over the brain. This is due to its arising from the properties of the gamma-aminobutyric acid A receptor subtype which has a refractory period of ~25ms or a fortieth of a second. Gamma phase locking has been proposed as a mechanism by which representations spread out all over the brain are coordinated and synchronized with each other. This gamma phase locking across distant different regions of the brain has been observed. However it is now thought that the lower frequency brain waves are also involved in the synchronizing of the information processing in different regions of the brain. This brain wave phase locking and in particular with respect to gamma waves, has even been proposed as a candidate for the neural correlate of consciousness. What is also interesting for our discussion of temporal looping struc