the manifold
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We think with pictures, not numbers.

Computer software is designed to work with bits, characters and other discrete symbols. However, the sights and sounds of the world around us are not discrete; they are continuous. Everything we see, everything we hear, and everything we touch is a physical quantity defined on a continuum of space, time, or frequency. Since digital computers work with discrete numbers, great effort has been spent on trying to convert the continuous images found in nature into discrete symbols so they can be processed with a machine. This conversion is unnecessary and invariably destroys important information contained in the images. It is possible to perform any logical operation on continuous images without ever having to convert them to discrete symbols.

When a musician reads or writes the notes on a page of sheet music, they are using a visual image to express a sound. The sequence of notes is plotted along the horizontal axis to signify their position in time, and pitch or frequency is plotted on the vertical axis. Thus, images, in a general sense, may represent audio sounds. The pressure or temperature of the surface of the skin can also be described as an image. Images can be three-dimensional, for example a representation of the physical matter in the space around us. Internally, an image may represent the forces on the muscles in the body. The ability to associate arbitrary images allows an organism to learn cause-and-effect relationships and thereby estimate probable future events. In this context, image associations are a fundamental cognitive function. A child learning new words is learning to associate visual images with sound images. Image associations allow us to memorize the multiplication tables, the addition tables, or any logical relationship.


Psymap

Psymap Image Attractors


Fig. 1.  A dynamical system with multiple image associations implemented using image attractors. When started from an image that is near one to the image attractors the system converges towards it. This behavior demonstrates that the attractor is stable and allows it to be used as a continuous symbol.


If a person changes their appearance slightly, we still recognize them. We classify many different individual cats or dogs as all belonging to the same species or breed. Different people with very different vocal characteristics may say the same word, which we recognize and spell the same way. In all of these cases, many different images are grouped together as a symbol. Continuous symbols can be formed by building dynamical systems with image attractors that “pull” nearby images towards them. The ability of an image attractor to pull similar images toward it gives the attractor the stability necessary to be used as a symbol in a cognitive system. A system that contains four image attractors is demonstrated in Fig. 1. The same system is started from different initial images, each of which is close to one of the attractors, but with the black regions removed. The dynamical system is constructed from two image association processors, which are shown on the left side of Fig. 1 and labeled with the Greek letter lambda. Each of the image association processors accepts two input images and produces one output image. The output image is fed back into one of the inputs of the opposite processor forming a "figure eight" shaped recurrent loop.

The vast majority of neurons in the human brain are organized into layers on the surface of the cerebral cortex. The thickness of the layers remains constant over large regions, several square centimeters in size, and then suddenly change to a different thickness in neighboring regions. Based on the thickness of the neuron layers, Korbinian Brodmann divided the cortex into approximately fifty areas, which are shown in Fig. 2. Each of the Brodmann areas corresponds to the dynamical system similar to the one shown in Fig. 1. The two image association processors are formed from the external and internal cellular layers of the cortex. Since each Brodmann area corresponds to a system with a fixed number of image associations, the layers in that area will have the same thickness. However, the thickness will suddenly change between Brodmann areas since different areas contains a different number of image associations. The set of all Brodmann areas, which together form the cerebral cortex, can be modeled as a computational manifold automata.


Brodmann Areas


Fig. 2.  Brodmann’s  illustration of the human cerebral cortex shows areas where the neuronal layers have the same thickness. Each Brodmann area corresponds to a dynamical system such as the one shown in Fig 1, which is capable of recognizing and associating continuous symbols. The pair of output images arise from the external and internal pyramidal layer of neurons in each area of the cerebral cortex.


The images are chemical, not electrical.

Computation does not require electricity. Mathematics is abstract and a variety of mechanical computing devices were in widespread use before the discovery of electric and magnetic fields. If the limb of a tree is removed, the tree will grow a new one with the all the detailed structure necessary to create the leaves and branches. This massive computation takes place without electricity. It is performed by cells signaling to each other with chemical messengers. These plant hormones are closely related to the chemical neurotransmitters in the brain.

Neurons are computing elements that process neurotransmitter data. The neurotransmitters acts like ink molecules on a sheet of paper, where each synapse is a single “dot” on the page. The dendritic tree of each neuron reads the concentration of neurotransmitter “ink” in the input image. The axonal tree writes and erases the neurotransmitter ink on the output image. This computation is illustrated in Fig. 3.


Neuron as a chemical computation


Fig 3.  The dendritic trees of two neurons sense the concentration of “neurotransmitter ink” in the input image and write neurotransmitter to an output image based on their internal calculations.


A neuron may be sensitive to a particular pattern of neurotransmitter concentration in the input image. When the neuron detects that pattern, it fires, sending an electrical action potential down the axon and causing the release of neurotransmitter in the output image. Chemical models of computation have a much higher resolution and are much more powerful than electrical artificial neural network models. For example, as shown in Fig 3, only a small number of neural cells are required to remember an association between two high-resolution neurotransmitter images. The computations are described mathematically by the theory of neurotransmitter fields.

Even the electrical action potentials are the result of chemical ions flowing across the neuron's cell membrane. The actual computation is chemical, only the communication about the movement of the ions is electrical. Moreover, action potentials occur within the confines of an individual neuron. Neurons detect the activity of other neurons chemically through neurotransmitters.

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