Direct experimentation on the brain is often costly, time-consuming, or impossible. Increasingly, neuroscientists have been experimenting on models of the brain, programmed into computers

UNDERSTANDING BRAIN SYSTEMS

One step beyond the study of individual brain cells is the study of how whole systems work. The complex symmetrical movements of the eyes that track moving images are being studied to reveal how the brain transforms sensory input into motor commands. Another complex brain system puts together information from many individual sensors in the hand that respond to different sorts of touch.

The external world must be represented within the brain in a reference frame for the brain to combine sensory and motor signals to achieve coordinated movement. Natural reference frames are suggested by sensory structures in the body, such as the ears' semicircular canals that sense movement.

A solid theoretical and mathematical foundation for understanding brain function is being developed and will be useful in guiding future research. Computer models have been built of a cerebellar neuron and of the entire cerebellar cortex, which contains hundreds of thousands of neurons.

MATHEMATICAL THEORY AND COMPUTER MODELING OP BRAIN FUNCTION

Andras J. Pellionisz, Ph.D.

Computer modeling of complex neuron networks can piece together templates of the way our brains mightwork. Developing mathematical brain theories and computer models of neuronal function is an unusual (perhaps unheard-of) activity in most medical centers.

However, less than three generations ago, the use of blood transfusions and ambulances were also unheard of. Today, progress dictates that providing "state of the art" medical care is not enough; it must be coupled with biomedical research and innovations to continuously improve clinical standards. Similarly, it is becoming clear, experimental biomedical research must be supported by mathematical theories..

In fields other than healthcare, such a wide spectrum ranging from routine service to basic research and development is not new. For instance, running computer centers and producing computers is incomplete without supporting basic research on semiconductor technology, which must be backed by sophisticated theories of solid-state physics.

Surprisingly, a similar realization in the neurosciences - that laying down a solid theoretical and mathematical foundation for ex erimentation is essential - is not equally obvious. Professor and physiology and biophysics chairman Rodolfo Llinas' research group is therefore pioneering in this regard. For more than a decade, its investigations of the cerebellum have been banking on quantitative mathematical modeling and theoretical interpretation. Lately the use of computers and mathematical rriodeling have become more common, but NYU holds its lead in theoretical and modeling studies as a basis of experimental biomedical research.

Why is it so important to forge a theoretical basis for brain function? Simply, mounting economical, rational, and emotional pressures from society make the exclusively phenomenological, serendipitous experimentation less and less feasible. It is no longer easy to find the funds to support the investigation of every possible interesting experimental project. Increasingly harder choices must be made regarding avenues of study to pursue. In order to spend wisely on biomedical research, one should be able to theoretically weigh the relative importance of particular research projects.

And, even if cost were not a problem, the tremendous amount of knowledge accumulated in the neurosciences during the past decades should be organized in a rational framework before scientists are overwhelmed by the next deluge of information. A theoretical synthesis of our knowledge is also desirable because experimental results are achieved at the sacrifice of animal lives, an ethical problem that attracts increasing public attention.

For these reasons, the strategy at NYU Medical Center has been to integrate experimental physiology with theoretical biophysics. Such an approach already gave us a mathematical model of cerebellar function. It will, we hope, open up avenues for understanding of the rest of the brain, while keeping research costs reasonable and reducing the number of animal experiments needed.

What have we achieved so far to demonstrate the success of this combined experimental-theoretical approach? The best example is to summarize a theoretical conclusion drawn from about two decades of experimentation on the cerebellum.

The cerebellum, in the back of the skull, coordinates movements. Like any other part of the brain, the cerebellum has been analyzed at many levels simultaneously (cf. articles in this issue by Drs. Llinas, Baker, Hillman, and Simpson). Clinical observations have been made of the motor deficiencies that occur when large parts or the whole of the cerebellum have been destroyed. Studies have revealed the structure of the strictly regimented arrays of myriads of nerve cells within the cerebellar cortex and its nuclei. Electrical activities of individual nerve cells within such extensive neuronal networks have been carefully measured. The extracellular ionic milieu that surrounds these nerve cells has been analyzed. Changes in ionic permeability of the neuronal membrane that separates the inside of individual neurons from the common environment have been studied, with emphasis on the tiny synapses where neurons transmit messages. These synaptic regions, only a few millionths of a millimeter in size, consist of a presynaptic site on one neuron that releases neurotransmitter chemicals to carry information across the gap - the actual synapse - to the postsynaptic receiving site on the other neuron. Finally, the biochemical properties of the neurotransmitter molecules themselves have been investigated.

The cerebellum is the organ that coordinates sophisticated movements, such as those necessary for playing music or being active in sports. But how can the widely scattered discipline of neuroscience coherently account for this global function on the basis of different levels of data, in a rigorous scientific manner?

That was the challenge to investigators of neuronal modeling (computer simulation studies) and mathematical brain theory.

Starting with computer modeling of the electrical activity of a single cerebellar neuron, the "patchwork" of the functioning ofwhole neurons emerged. Computers proved essential for such a holistic representation, although 10 years ago few neuroscience laboratories other than NYU ran their own computer centers. Today, because of computer modeling ofcerebellar function, medical students can"see" firing of neurons by animated computer graphics.

Our research, however, moved on to build a computer model of the entire cerebellar cortex, which contains hundreds of thousands of neurons. A synthesis of available functional and structural data has paved the way to an axiomatic, mathematical tensor-theory of CNS function, with special regard to the function of the cerebellum.

An example may help to explain this work. To maintain a steady gaze upon an object while the head is passively moved, compensatory head and/or eye movements have to be generated. To do this, the cerebellum must gather sensory information from the six semicircular canals of the vestibular apparatus of the inner ear (three on each side) and calculate how much each of the numerous neck muscles or 12 eye muscles (six per eye) must contract or relax in order to generate a compensatory movement of the head or eye. Finally, it has to convey these instructions to the muscles.

The fact that the cerebellum must convert head-position information into muscle-position information can be put in mathematical expressions. In mathematics, information on direction and speed of movement is expressed by vectors. The mathematics used to geometrically represent such a transformation from one vector to another (that is, by neuronal networks from sensory to motor systems) is tensor analysis. This sophisticated mathematical discipline was used by Albert Einstein in developing his theory of relativity and was thus shown to be suitable for yielding powerful interpretations of natural geometrical phenomena. Today, we have made it a tool that permits us to generate hypotheses about brain function that can then be tested experimentally.

Tensor network theory of the central nervous system is still a very new development in neuroscience. Nevertheless, it offers the exciting possibility that the extensive neuron networks of the brain will soon become mathematically explorable. How theory and modeling will facilitate our understanding of our brains, our ability to treat them, and to aid them with instruments, are questions that are likely to occupy neuroscientists for a long time to come.

AndrasJ. Pellionisz, Ph.D., is Research Associate Professor of Physiology and Biophysics

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