**Mathematical
Modeling of the Functions of the Human Brain**

Understanding the functions of the human brain is one of the most important goals of contemporary science. Sciences that engage in studies of brain function include biology, medicine, psychology, psychiatry, chemistry, biochemistry, computer science, mathematics, etc. Most studies about brain function are based on biological and medical sciences using biochemistry. On the other hand there are some studies that are based on computer science and mathematics. There is very little overlap between these two basic groups. Medical books describe human brain without mathematics or they use only a few very simple equations. These equations describe only certain brain parts. Mathematical analyses do not use the actual functionality of neurons, in most examples.

Rather, they simplify neuronal function using simple models that are easy to include in other deductions. These books often attempt to explain brain function using higher neural functions such as in the visual system.

My book Neural Computing and Neural Science (NCNS), begins with the simplest functions of the neuron and neural channels. It then attempts to go beyond these simple functions to include the general functionality of the brain. The analyses in my book required the study of mathematics, computer science, medical psychology and other sciences. In it, I attempt to join all these sciences into one system that can describe function of the brain. The initial idea of the theory described is the theory of movement. Movement is one of the most basic functions of living matter. There is no life without movement. When we talk about any kind of life we must talk about movement. When we talk about movement we have to talk about differential equations.

I cannot expect that everyone will know in any detail what is meant by the term differential equations--especially as applied to a biological system. Differential equations describe movement with the use of the mathematical operation called integration. Integration is the inverse of a derivative. For purposes of understanding the theme it is sufficient to know that there exists something like differential equations and integration.

In short I would explain what differential equations are. The result of differential equations is not a number or set of numbers as is true for other equations. The result of the solution to differential equations is always a set of curves. Only for special cases is the result expressed only as a single curve. However it is always a curve or curves. Differential equations describe solutions to problems including time, and time is an independent variable.

When time is one of the variables in the solution of a differential equation, we always talk about the dynamics of the solution. Thus, since movement involves change in position over time and space, movement and differential equations are related.

There was only one problem. One must ask how the brain can solve differential equations. This is a very basic problem. If the brain were not able to solve differential equations there would be very serious problems with this most basic of functions in living systems--the nervous system, to handle movement, must be functioning, as a differential equation would predict.

So, one must ask:

- How the human brain can control movement if it is not able solve differential equations.
- Or, when we accept that the brain solves differential equations, we must say how the brain solves integration, as integration is basic operation necessary for solution to differential equations.

Brain tissue consists of neurons, their connections and the insulating material we know as glia. Some scientists were thinking about data processing inside of neurons. They thought that neurons are processors having some internal data processing. However, neurons are not able to process data inside; that has been approved. Neurons are only cells having different characteristics than other cells. These differences are not so dramatic that we should suppose that neurons work like integrators. It is not sure that scientists look for function of the brain in the same way as it is depicted in the book NCNS. In spite of that, it seems to be very clear that most deductions can be correct.

The general ideas I would like to discuss can be expressed as follows:

- Movement is the basic rule of life.
- Differential equations describe movement.
- Integration is the basic operation that is used to solve differential equations.
- The human brain must be able to solve integration using neural structures.
- Neurons are cells that are similar to other cells.
- Differences between neurons and other cells are not very substantial.
- So how the brain solves integration?

These seven statements express the controversy between neural structures in the brain and integration. In my book I show how the brain can control movement--and apply the theory as the basis of brain function.

To begin with, we must understand the term movement in an extended sense, meaning that the movement of the body includes also the movement of all substances in the body the fluids and structures of the body. And, that movement is also body movement or change of limbs’ position relative to other objects in the environment. The brain must control all of this. And any analysis--mathematical or otherwise--must take all forms of movement into account.

There is another relevant problem related to brain function. This is our attempt to understand functional aspects of the connections made between neurons. There are two separate problems and they must be treated separately. Differential equations can be solved only when we connect some parts that mutually interact. When we talk about brain function we have to distinguish between dynamic processes and whether these equations are solved using systems with fixed connections or variable connections. Fixed connections in neuroscience result in deterministic behavior, while variable connections behave stochastically. The brain changes connections very quickly. Therefore, the brain functions rather stochastically. Stochastic processes are known in mathematics something like random processes. Such processes have unknown rules of behavior or these rules do not exist at all.

Therefore we cannot know precise equations that could describe function of the brain. However, it is possible that we can find equations that are close to the brain function. Physics is the science that describes nature in simplified form. Physicists always make many compromises to describe processes of nature. Neuroscience must do it the same way because it is very difficult to find precise differential equations describing function of the brain. Neural tissue is many times more complex than the systems that are typically described mathematically in physics. So we have to account with many simplifications when we want to describe function of the brain. However we cannot oversimplify and therein lies the paradox in mathematical modeling of the brain.

On the other hand, even very detailed descriptions of the neuron do not allow us to understand whole brain functioning. We must extend on our knowledge--just as we cannot understand the movement of waves on a beach when we study the movement of one drop of seawater, so can we not understand function in the nervous system by studying the function of an isolated neuron or bit of neuronal tissue. Mathematicians call it a problem of “big numbers”. Problem of big numbers is -- for example -- base for statistics.

We cannot solve function of the brain from the function of each neuron. It is too small part of the brain. Dynamics of fluids or dynamics of weather is not solved in details. Scientists attempt to solve such problems using different terms. They do not say one molecule of water they rather say stream of water. Maps measured from head using electroencephalograph (EEG) are very similar to the maps of atmospheric pressure displayed on TV.

Neuroscience typically a science that studies details. It goes to the details. Therefore neuroscience often loses sight of the scope of whole brain functioning. Why is that so? There is one possible answer.

Neuroscience does not have a device that would allow one to observe or measure whole brain function. There is only one science that has such a device and that science is mathematics. However, mathematics alone cannot solve the problem of whole brain function. Therefore, we must examine the problem by combining the knowledge of brain function gleaned from a number of science disciplines.

This problem of the general view of brain function is studied by two very new logical functions that describe signal spread in the brain. This problem is so complex that it will need more attention in the future. Two basic logical functions are logical-AND and logical-OR. In my work, I study two new functions soft-AND and soft-OR. The word soft here means that the result of the function is not very rigid. The functions were designed to describe the thinking patterns of medical doctors while they are stating diagnoses. So these functions fit the function of the neuron and parts of the dendritic tree where there are some points having processing like threshold for all impulses go through these points. They work similar way as neurons but it is not same. These threshold points help to restore impulses coming to the neuron body. These functions also describe the extended meaning of the threshold point for brain function.

Both “soft” functions show how the brain probably controls a change that includes processing differential equations. These functions control mostly parallel and serial processing in the human brain. Scientists were not sure some years ago whether the brain worked like a serial computer or like a parallel computer. At this time the most accepted theory is that the brain works like a parallel computer with massive parallelism. However, even in the case of the diagnostician, we must postulate that the brain is neither a parallel nor a serial computer. The brain can process both kinds of functions. In one case the brain is able to process language that is serial process. The brain is also able to process visual information. This is parallel process. There are many more examples for parallel or serial processing. There are only few most typical ones.

When we convert these technical terms to biological ones, we can say that parallel processing in the human brain is feeling whereas serial processing is movement. Serial processing in the brain, or movement, is an algorithmic mechanism whereas parallel thinking in the brain, or feeling, is the perception of an environment. Feeling and perception are non-algorithmic mechanisms.

Let us come back to the basic problem. It is integration. The neuron models that are described in some scientific books are very complicated to depict them in this paper. So I would simplify its function to voltage controlled oscillator (VCO) or current controlled oscillator (CCO). Current of ions coming through dendrites to the neuron body controls frequency of impulses going out of the neuron. Excitation increases frequency of generated impulses; inhibition decreases frequency. Characteristics of neuron function are not same for both excitatory and inhibitory inputs (dendrites).

The general rule of neuroscience is that size of each impulse is same regardless of frequency of generated impulses. It means that mean voltage of generated impulses is proportional to the frequency of impulses. The mean voltage is calculated as surface of impulses generated in given time unit. The surface is bigger for bigger number of impulses.

The impulses go out of neuron and they come back to the same neuron in the special case or in other cases they go the other neurons. The communication lines going from neuron output to the neuron input are called axons.

The axons are build from very small segments. Impulses jump from one segment to the following one. This process was used in some type of memories called in computer science “bucket brigade device” (BBD). One segment transfers its content of ions to the other segment. This idea suggests that axons are in fact dynamic memories. It is true but memorizing of this type need deeper explanation that will be added somewhere to this site in the future.

The impulses have left neuron outputs after the input voltage coming to the voltage has been converted to the impulse frequency. Therefore the brain must process decoding of impulses back to the voltage. The simplest decoder that fits to such function is capacitor. Where we can find capacitor in the brain? The brain contains much more capacitors than neurons. The brain’s capacitors are synapses and neuron bodies. The each synapse has very thin gap that is typical capacitor. Neuron body is capacitor for ion currents. Such capacitor work works like battery in a car. However it is bit smaller.

These capacitors serve as decoders of impulses coming to neurons. They change variable voltage to the mean voltage that is again converted to the frequency of outgoing impulses. So the “loop” is closed.

There is a question why nature creates so complicated solution? Why axons do not lead electric current only? The answer is simple. They are not able to lead electric current for longer distances. The impulse travelling through axon is not able to go more than 0.1mm distance. Then impulse loses its size; it disappears. The segments in axons restore impulse size. Biological conductors are very bad comparing to metal wires. However they have other advantages.

The process that works as it has been depicted herein compares to analog computers. Therefore I assume that brain works like analog computer. Analog computers have been designed to process differential equations. The circle in our explanation is closed as well system of coding and decoding in the brain. The brain must solve differential equations and analog computers solve differential equations. Analog computer use integration to solve differential equations. So how it is done in the brain?

The answer is not very simple. We must realize that that coding and decoding process in the brain creates feedback system that we can describe using matrix. Matrix multiplication is basic operation that can describe function of the brain. Matrix multiplication can be described using simple equation

y=M*x

Where **x** is output value in the
all neurons, **y** is internal memory and **M** is connection matrix.

To make feedback system we have to close loop using the following equation.

x=y

Both equations show processing in the brain. My book evidences that both equations can solve integration although there is no integration in these two equations.

The evidence is based on the numerical integrating method using Taylor series as its base formula. The numerical method allows filling all cells of matrix M so the given differential equation is solved. Both equations can solve differential equations in all cases. However they will not be stable or we will not be able to check the correct solution comparing to mathematical equations derived using other methods. Brook Taylor designed its formula about 1700 year. Since that time was his formula used for design of numerous numerical methods that can solve differential equations. Only the method depicted in my book can suggest how the brain can work solving differential equations. Function of the human brain is not only solution to the differential equations as we know them from school books. Numerical methods solving differential equations are based on idea of permanent connections among all parts of the system solved using differential equations. However it is not valid for the brain and especially not for the human brain. Connection of the human brain dynamically changes. So the numerical methods solving function of the human brain using differential equations is valid only in the one concrete instance. After very short instance later is connection changed. The very short instance is as short as we can only imagine. The every time interval longer than zero gives changes in the brain connections.

We have to ask some questions:

- Are these changes temporary or they are permanent?
- What are principles of switching?
- Is there only one principal or there are more principles?

There are some answers to the questions above. These answers are not divided into items according to the numbers of questions.

The neural tissue can process both, permanent and temporary changes. The every impulse can go through synapse or neuron or the impulse does no go through synapse. The decision is done in synapse according to the threshold level. The decision is also made on the neuron inputs. Most of neurons have more input dendrites than only one. For example Purkinje neurons can have up to 100,000 input dendrites or more. As we know that only one impulse is generated we must define rule for generation of one impulse or one set of impulses o the base of huge number of input values that decided about it.

These functions must be simple enough. The nature is not able to create complicated solutions, as we know them from computers' design. People can design very complex and complicated solutions. They are mostly non-effective, as we know very well. The nature creates everything effectively using very simple functions.

I have designed functions that I call
** soft logical functions**. These functions are good candidates
to describe switching in the neural network. They are very simple and their
function respects the rules of human thinking. There in my book are defined
two soft logical functions. The first is

The logical function AND is based on rule that when all inputs have value of 1 the output is also 1. Otherwise the output has value 0. The logical function OR is based on rule that when at least one input has value of 1 the output is also 1. Otherwise the output has value 0. The soft logical functions cannot work so strictly.

The result of soft-AND can be 1 although not all inputs are 1. On the other hand the result of soft-OR can be zero although one of inputs is set to 1.

Definitions of both soft logical functions are depicted in my book in details with full explanation why is that so true. All rules joined with soft logical function deals with temporary connection changes. The permanent changes are related to the memory so I will describe them in another paper.

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*Copyright 1997 by Ales Grmela *