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Elucidating principles of the brain mathematically
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| Information processing mechanism of the brain | |||
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Dr. Amari used the following example to explain the remarkable capabilities of the brain. "Suppose you see an acquaintance whom you have not seen for a long time. I am sure you have no difficulty in remembering them, even if that person has put on a little weight. It may seem to be a simple process, but it is no easy matter for a computer to simulate the process, because a huge number of steps are required in processing the data." The brain is very slow in processing data, about one-millionth the speed of current computers. In contrast, the brain needs far fewer steps to instantaneously recognize a person. But what makes that possible? "It is the force of numbers," answers Dr. Amari. In the whole human brain, there are more than 100 billion nerve cells, each of which is connected to another 10 thousand nerve cells. "The brain can process information instantaneously because the vast number of nerve cells it contains exchange information in a parallel manner. In this process, each nerve cell acts in a rather fuzzy manner, and its function is not as specific as the function of a computing device. However, the neural circuits as a whole eventually derive appropriate answers." Dr. Amari has tried to elucidate such a complicated system by using mathematics. "Mathematics is a kind of culture that human beings have created as a means to understand the essence of things. Today, in the face of various life phenomena, especially the brain, we are trying to understand this very complicated system by using mathematics. However, we need to establish a new mathematics because conventional mathematics has no chance. Efforts to understand life phenomena and the brain will surely serve as the impetus for creating a new mathematics and new fields of science." |
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| What is the essence of learning? | |||
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"A new mathematics that will allow an understanding of the brain cannot be established in a day. Thus we have not only relied on conventional mathematical theories but have also drawn on the whole of our "mathematical sense" to move forward our own research," says Dr. Amari. One specific research subject based on this approach is learning. For example, people are able to recognize handwritten characters after receiving training in which various examples of handwritten characters are presented in the manner of "this is A, and this is B." "Then, surprisingly, a 'sensation' is created in the brain that will enable you to recognize A or B even if they are slightly different from the original characters. In other words, human beings have the ability to learn the essence or mechanism behind various examples. This is the very nature of learning." However, what is happening in the brain during the process of learning? As training proceeds, the manner of information exchange in the neural circuit gradually changes at synapses, which are the junctions between nerve cells. For example, some pieces of information will run more smoothly, whereas others won't. Thus it is thought that this is how the learning process of a task proceeds. Then, can we expect that the more complicated the neural circuit is and the higher the number of nerve cells, the faster the learning process proceeds? How many examples do we need for training? "If the neural circuit is too complicated, a large number of synapses will require adjustment. Furthermore, if the complicated neural circuit exerts all its powers but has an insufficient number of examples, a phenomenon will take place in which handwritten characters such as A or B can only be recognized when the original sample characters are presented. This is because the neural circuit has been too meticulous to detect the rule behind the examples. We are trying to analyze mathematically the complexity of the neural circuit and the relationship between the number of examples and the progress of learning in an attempt to derive general rules. I am sure that the abstracted rules can be applied to both humans and computers." |
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| Singularity in the learning process | |||
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Among the many areas of mathematics, Dr. Amari puts the emphasis on geometry, which is the branch of mathematics used to explore the properties of figures and space. He has also used figure space to express the relationship between the neural circuit and learning in an attempt to find the rules. For example, the distance between nerve cells is considered to be wide when they are connected through synapses in which the change in information flow significantly affects the progress of learning, whereas the distance is considered to be narrow when they are connected through synapses in which the change has an insignificant effect on the progress of learning. Thus the whole neural circuit can be expressed as a "curved (deformed)" space with the same number of dimensions as the synapses. This space can change into various shapes depending on the complexity of the neural circuit and the number of samples. "We are dealing with the space in order to explore the relationship between the neural circuit and learning. The research subject I am currently fascinated with is the occurrence of a singularity phenomenon, where a point in the space is extremely contracted (the lower panel on cover page). This means that the learning process does not proceed, regardless of any measures taken to change the flow of information through the synapses. In some cases of handwritten character recognition, the discriminating power was found to stay constant regardless of any increase or decrease in the flow rate of information. This leads to the discovery of a phenomenon in which the learning process comes to a halt because no way is found to proceed. In fact, such phenomena were reported in computer image recognition systems, but the cause has remained unknown. "The culprit was found to be the existence of singularities. We have successfully derived the conditions for singularities not to form in order to develop measures to prevent them." |
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| Proposed "Information Geometry" | |||
A nerve cell can generate electrical signals, receive signals from many other cells, and generate electrical signals when the intensity of signals from other cells exceeds a threshold (Figure 1). This is called "firing." How is this firing involved in processing information?"For example, in an experiment where the same figures are repeatedly presented, each firing seems to occur quite randomly. However, it is not completely random, and the phenomenon is governed by rules. The key to analyzing these rules, we think, is 'probability.' For example, once a firing occurs, the probability of occurrence of a subsequent firing, or the probability of occurrence of a firing in a certain period of time increases." It is thought that the number of firings and their timing are a method to express information. For example, when information is sent to a muscle to tell it to "exert a strong force," many nerve cells fire repeatedly in a short period of time. In contrast, when information is sent to a muscle to tell it to "exert a weak force," the number of firings is reduced. The regularity of the firings also conveys different information with different meanings. Furthermore, the comparison of two of three nerve cells may show that their activities are quite different. However, simultaneous consideration of all three nerve cells may lead to the discovery of some relationship among them. To analyze these types of activities in nerve cells, mathematical approaches involving probability and statistics are very useful. In this area, Dr. Amari has also advanced the research by applying geometry to the analysis of probability and statistical distributions. (Figure 2) "In the 1980s, I put forward a proposal that we establish an 'information geometry,' that is, a branch of geometry that can deal with information based on integrated mathematical theories." But why geometry? "Geometry, which can express phenomena in figure space, is the most intuitive of all the fields of mathematics, and helps us to visualize a concept," says Dr. Amari, who has held prominent positions such as Founding Board Member and President of the International Neural Networks Society, and who is thus a global leader in the field of theoretical research on the brain.
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| Thinking process that goes between the conscious and unconscious | |||
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Dr. Amari tells us about his future goals. "For example, even simple organisms can make their own choice whether to get away from or come closer to an external stimulus when it is given. However, how is the information expressed during the thinking process, or how is the final decision made? We want to create a mathematical theory that can describe the dynamics of this process. Furthermore, I am sure that we cannot get by with avoiding the issues of consciousness and mind if we try to explore the thinking process of human beings." The thinking process of human beings is divided into two categories: information processing in the realm of the conscious, and in the realm of the unconscious. For example, handwritten character recognition is dealt with in the realm of the unconscious, and words alone can never describe how the recognition is performed. "When we try to solve mathematical problems, many things pop up in the realm of the conscious such as 'this approach may contribute to solving the problem," or "we are coming to a dead end at this point." In other words, the brain processes information in the unconscious realm and brings the results into the conscious realm. However, what we write on a test paper is limited to what we think logically in the conscious realm. Thus we extract some parts and express them logically to make up a program for processing information. Of course, human beings can process information in a logical way, but we need to pay special attention to the information that is processed in the unconscious realm. |
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| Future of brain research | |||
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The advance of research into a mathematical model of the brain will clarify what can be done or what cannot be done in a system like the brain. This knowledge will serve as an important means to understand what a human being is, and can also be applied to using technology to create a system like the brain. For example, it will be able to contribute to building a robot that, like a human being, can make the right decisions and act on them in various situations. However, how will research proceed in the future? "Physics, including the theory of relativity and quantum theory, has finally resulted in a single set of equations. However this does not hold true for a mathematical model of the brain. I believe it is necessary to establish multi-strata theories that can explain various aspects of the brain from the microscopic world to the macroscopic world, and to systematize them." Finally Dr. Amari talked confidently about prospects for the future. "I think a mathematical model of the brain will be completed by the middle of this century at the latest. I would say that we have reached the level of 30 to 40%. In the rapidly-advancing field of brain research, we need more breakthroughs because we are facing a difficult period. However, I think the future is bright because of the recent trend toward collaboration between theoretical and experimental research. I am optimistic for the future because I believe that human beings have the ability to see into the essence of complex phenomena, even though they are immensely complicated. In physics and chemistry, people have succeeded in mathematically systematizing complicated phenomena even though they seemed extremely complex. I am sure that this approach will hold true for the brain." 
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No brain-measurement equipments, no chemical agents, and no laboratory mice can be found in the laboratory led by Shun-ichi Amari, Unit Leader and Director of Brain Science Institute. Instead, he uses mathematics as his research tool. "Experimentation has revealed in sequence the activities of nerve cells and the functions of genes and proteins in the brain. However, experiment-based data is not all that is necessary to elucidate the mechanism of the brain. Science, like physics or chemistry, starts with the discovery of facts based on experiments, discloses the essence behind a number of facts, and finally expresses that essence in mathematical form. Thus I am sure that the mechanism of the brain can be expressed mathematically," says Dr. Amari, who has been pursuing a mathematical model of the brain for the last 40 years. However, I wonder if the day will ever come when we can elucidate the mechanism of the brain and understand what the human being is in mathematical form?