Neuronal Systems
Neuronale Strukturen
Lineare Algebra

A human lifespan is not enough to reproduce  the facts about nature, which have been aquired so far. Therefore it is neccessary to publish  the results of scientific work.

After more than 2500 years of well documented human thinking there are a lot of different opinions about the role of mathematics around.

If we are going to look to applications  taking profit of the mathematical theory, we discuss, whether individuals having  created the concepts and proved all the theorems and propositions, carry a responsibility with respect to the impact on the following applications.

The  German mathematician Gauss hesitated to publish  a paper on the geometry of the hyperbolic plane.  Some years later the Russian mathematician Lobatschewski published the theory.

Today we know that Einstein's theory of relativity and nuclear weapons have been developed on the basis of the mathematical concepts of the hyperbolic geometry. If  we  constructed a responsibilitity of the Russian mathematician for  civilian victims  in Hiroshima although, this would be a  violation of the rules of logic and arguing.

Modern life is influenced by digital tools and media offering a variety of services. Ordering a product online for example needs some security. Caesar used a very simple code to encrypt his messages switching the letters of the alphabet. Today every boy or girl scout has learned that codes with a fixed translation table can be decrypted easily.

Mathematicians communicate by chains of implications between formal sentences, which may be seen as a kind of personal opinion using a mathematical language. Implications give rise to  transitive relations.

There are different ways to formulate a proof: It can be done directly or indirectly. We will only give hints and referencres to the  mathematical literature. It is left to  the readers to complete the proofs or look into the literature.

We use the following assertions to illustrate this:

1. Every integer number can be decomposed as a product of  prime numbers.
2. There is an infinite number of primes.

Proof 1: (direct)

Hint:  The set of prime numbers, which are less than n is denoted by Primes(n). The Greek mathematician Erathostenes gave an algorithm to determine the prime elements for an arbtritrary number. My readers may rewrite the routine "Filter"  recursively and extent it in order to list the prime factors of a given number n. The proof of correctness can be done  by induction easily.

Filter (pmax) {
  var i,r[1000],p,val,ret,adr;
  const blank=32;

    call Root(pmax,adr);
    while i<pmax begin r[i]:=i;i:=i+1 end;    
    while p<ret+1                          
      while i+p<pmax begin i:=i+p;  r[i]:=0 end;
      while r[p]=0 p:=p+1
    while i<pmax                                 
      if r[i]>0 then begin val:=r[i];write blank; call putN (val,0) end;

A direct proof  is called constructive, if an algorithm is given.

Proof 2: (indirect)

We assume that the proposition is not correct. Then we have a finite list of primes p1, ... , pn. We multiply all these numbers resulting in a product P and add 1. The result  P+1 cannot be a prime by assumption. According to proposition 1 and our assumption we may decompose P+1 as a product of primes. All these new primes must be greater than any member of our prime list,  which is a contradiction to the our assumption. Thus we conclude that the opposite of the assumption is correct.

The preliminary predicate correct is awarded to such a proof, if it has been published and accepted by the community. Published papers can be referenced by other authors. Referencing is an effective way in order to verify a certain result or to construct a contradiction.

The frequency of references allows to identify the main stream of  research.

More than hundred years ago the British philosopher and mathematician Russell communicated the following problem to his German colleague Cantor, who had worked out the theory of sets:

Is it possible to form the set of  all those sets R, which do not contain R as an element.

It does not matter, on which assumption you are starting, you will end up with a contradiction. It took about half a century to fix this problem by establishing rules about the choice of elements in a set..

Topology was the name for  the analysis of invariants in open neighborhoods.

Statistics is a very interesting area to apply mathematics. 

Physicists use mathematics to construct models, which describe aspects of nature. The terms correct and false do not apply to models. Just the potential of such a construction to predict future observations is important.

Einstein's model of space and time is based on a set of four dimensional vectors  called quaternions. He defined  a quaternian vector by a triple of real numbers, which models the euclidean space and an extra coordinate, which models time.  The geometry for the ensemble of space and time is of hyperbolic type.The curvature of Einstein's space is interpreted as gravitation.

The methods and tools, which have been develloped in statistical sciences enhance the ability to gain knowledge about a given set of things on the basis of counting. The number of objects in a set defines a basic measure.

But nobody is ready to count the number of peas, if there are a  lot of them around. The idea to use a scale marks the next step of rational acting.

Scales feature a pair of bowls, which carry the objects to be compared. The task to find an equivalent in terms of coins for the peas contained in a sack can only be done within an interval of uncertainty..

The principle of uncertainty has been established in quantum theory:

The German physicist M: Planck determined the value h as the minimal amount of uncertainty in nature.

The mathematician H. Witting understands  statistics in especially as mathematical  theory supporting the quality of decisions. He gives in his book "Mathematische Statistik" from 1978 an example:

A new medical treatment  for a disease becomes available. It has been applied to ten persons and eight of them have been cured. It is known that the standard treatment is successful in 65 percent of the cases.

The experts in statistics write down two hypotheses H and K for reasonable answers of the question, which treatment is more adequate. H says that there there is no strong argument yet that the new treatment is the better one, while K states just the opposite of H.

Next it is necessary to define a hurdle h=1-µ in terms of a number near to one, which shall be interpreted as the minimal provable probability that H is wrong. The proof must be given according to the rules of mathematics.

Looking at the example above, with some extra information, that are parameters concerning the trial, the first hypotheses H can be rejected by a probability of 0.95. This means that the decision is encouraged, to apply the new treatment in the future.

Neyman and Pearson called such a method a statistical test at the niveau µ.

In which way can statements on the probability concerning a large group be made on the basis of knowledge of data in a small subgroup only?

The reason in terms of mathematics is the convergency of independend samples of random variables.

We will present a variety of examples and get the reader involved to do some proofs.

Next we look at flat neuronal networks with collateral inhibition. In the book "Neuronale Netze" from 1990 Ritter present a model of neuronal stimulation.on a surface,

We will generalize the scenario to surfaces, which are not orientable. We will show that nonorientable neuronal structures are capabable to reflect signals and can be modelled as normed complex vector space.

The elements of neuronal  networks are are living cells or artificial gadgets, which send an impulse along the axone using the electric energy of polarization.

We will identify the processes, which are  triggered during a depolarisation of a nervous fibre. First we observe that there is a stable state,  which can be characterized in the following way: The outside of a fibre is covered by positive kalium ions, while negative chlorine ions sit at the inside. Such an electrical arrangement is a condensor.

The fibre features several kinds of channels to allow ions to pass from the inside to the outside and vice versa. While the channel for kalium is wide and has a straight form, the natrium ion spirals, while it is passing the wall.

The combination of  the condensor mentioned and the spiral form a resonance circuit.  In this way the condensor gets charged and decharged while natrium ions swing at the same frequency. As soon as a certain amplitude is reached,   natrium ions drop into the fiber. They neutralize chlorine ions inside, and kalium ions outside are set free.

Now, driven by statiscal distribution, more and more kalium ions get inside too, but a small time shift later than the natrium ions. Locally the charge of the  condensor reaches a minimum due to the ion currents resulting  in  a sharp electric impuls, which is sent along the axone. The signal is received by other neurons across neuronal links called synapses.

In terms of mathematics a neuron can be interpreted as a process of summing signals from input fibres. Energy is necessary in order to complete a neuronal cycle and return to the rest state.

We will look at the art to manage projects. In which way mathematics can be used to refine and to optimize existing results.

Another challenge is the task to define a list of substances and keep track of their concentration in different environments. These values form a manifold.

In each nucleus of a living cell you find a molecule known as the double helix DNA.   There are four organic bases, adenine, guanine, thymine and cytosine. The letters A, G,T and C are used to code sequences, which control the processes in a cell. Not much is known about the syntactical rules in general. We have learned that there is some redundancy.

Each piece of information is mirrored  across hydrogen bridges. Adenine fits to guanine and thymine to cytosine by a different number of bridges. The letters A and G feature three of them, while T and C are connected by a double link.

There are biochemical  processes acting  in a cell, which cannot be reversed. Other ones are members of a  cycle. Altogether they make up what we call the life in a cell.

 In terms of mathematics we look at the change of concentration of various substances. In other words this means to approximate the processes linearly. What  we get are a set of matrices forming a ring. We can figure out the spectral values and look at the continuous fuctions defined on them and find out that they are isomorphic images of cell life in terms of concentration of substances.

It seems feasible that nature stores the information inside the cell's DNA in terms of the letters  A, G,T and C.

The absence of life means that everything is constant.

In which way a constant or a steady  state can be described in tems of mathematics?

If we talk about something being in  steady state, we expect that it would not change too much.  Such an arrangement can be used to measure time.
A constant is defined, if we observe no change during a certain time intervall.
Looking to old greek philosophy we learn that the existence of any constant in the  real world may be doubted although. In modern  sciences we find some examples of  observables, which were supposed to be constant in the beginning, but finer methods of examination motivated quite the opposite opinion.

To answer the question of growth is by far not enough. The next challenge originates from the continuing  issue that in  industrial countries a  small  percentage of people  owns the majority of resources. With respect to the distribution of capital  in these so called rich countries their economies seem to be bound within a process of centralization, which may cause  serious threads to the democratic paradigm at the end of the day.

We ask, whether the present approach to handle the streams of money will get the civil world through the crises. Probably a glimpse of smartness  suffices to realize that the believe in the mechanisms of free markets does not surfice.
The meaning of the word believe is quite open in most languages. In mathematics instead of believe the term axiomatic system is used. Such a system must be free of logical contradictions. 

Arrow presented the following four formal axioms, which should be met by a democratic group or society:

1. He postulates that there is a decision process in order to define the preferences of the whole group from the invidual preferences with respect to arbitrary items.

2. It is forbidden that the decisions process mentioned above copies the preferences from a single member only.

3. Preferences, which are in common for all members, are valid for the group also.

4. The process of setting the group preference for  some listed items  must not depend on selections of members regarding other items, which are not on the list.

It can be proved that at least one of the axioms above has to be dropped in order to avoid a logical contradiction.

Another paradoxon was published by Sen saying that minimal, local liberalism in especially with respect to privacy is uncompatible with global preferences, which are common to all members of a society.

The second  axiom does not admit a decision process, which is as simple,  just to copy a member's opinion profile.  Then we say that the opinion profile of member g is dominating  the decisions in G.

If the axioms are interpreted in the context of a fixed number of constant items, the phenomenon of domination of  rational decision making is inevitable. And if a group acts according to the decisions made within a finite set of items, which is not updated  or ammended regularly, domination may result in dictatorship at the end of the day.

But if a group has added a new dimension called time to the process of decision and allows that members of the group may change their opinion with respect to any list of items as well as add  new items, domination can be limited effectively.

From the viewpoint of information theory  each item can be decomposed into a set of statements  allowing binary valued opinions only. We will show that decision making  and acting can be modelled mathematically as  compact or  hermitean operators in the Hilbert space. We give some interesting impulses to answer the question, whether a finite automate is suited for decision making in real terms.

In 2001 S. Keen reminded of mathematical errors in the theory of economics, which have not been fixed yet. If you try to correcte them,  the system produces errors somewhere else. So we are asking, what is wrong in our present democratic systems of finances and liberal marketing.

At first we look at some of the most important paradigms of human cooperation.

Today's economical rules have develloped from  traditional human behavior in smaller hierarchically organized groups.

The paradigm of growth relies on the fact, that growth in nature is exponential, if there is plenty of land or sea to harvest or fish on.

The growth of cooperation is due to the fact, that each human being is individual with respect to feeling, thinking and skills.

Money did appear as soon as there were means of transport and measuring the traditional goods.  It measures the  abstract power to change things and relies on a concept of trust.

The ability to identify faces  is the skill to build a culture of trust in way to add an abstract attribute called creditworthiness to each individual.

But do  these principles still work in our more and more globally influenced economy? 

It is evident that if you compare the same amount of money today and some years later, it is preferred, if the money is available now.  The difference in percent between the value  now and one year later is called interest rate. Paying the interest you buy time to use the money.

Interest rates are a measure of the benefit or profit, which can be triggered by the money borrowed. This part may give rise to financial growth also, while the rest of the interest rate is to compensate for the risk and to earn profit.  

If you run a business with a profit rate, which is greater than the interest rate of a certain loan, the borrowed money brings even more return on the capital invested (ROI).

With a private loan  the  payments of interest must be added to the cost of the financed item of consume. A private person should have the choice  to pay the present value of all the interest rates for the loan  in advance, that means to take a disagio from the bank.

This person would have constant repayment rates, similiar to other expenditures, which are monthly due. The creditability of a person can be calculated by taking the net income for a period subtracting necessary expenditures and committments.

Our present culture of trust depends on financial ratings, which are offfered by serveral institutions.
The more profit in a business is generated the better the rating and the lower the interest rates will be.

Private persons asking a bank for a loan do not only pay higher interest rates, but they are charged with service fees additionally. Generally  it is not possible to compute the return on investment for private expenditures.

Under these circumstances the concentration of money can be relieved a little by legislation calling for more own capital in business, setting rules ensuring the quality of ratings, making private expenditures of investment  type tax deductible and making the cost of private loans more transparent. The option of taking a diasagio in the beginning instead of paying interest rates for each period would preserve quite a few private debtors from becoming insolvant.

These principles are valid for commercial credits to a great extend also.

Traditionally people could buy products as well as service. Since Gutenberg's invention to print  books sellling written text has been limited by licences. They appeared as new  tradable items, even if they were not  made out of material.

At first texts were thought to be written by authors in some natural or formal language. Books and articles were marketed by  publishers in the beginning.

Today even patterns and names  do carry a price tag sometimes.

The turnover of the market places for licences  has become a substantial part of the expenditures of investors and consumers meanwhile.

The orignal concepts of trading immaterial products have been changed and widened by  the internet technology, although civil legislation has not kept pace with that development in most countries yet.

People may wonder, if the right to breathe fresh air could become a subject of a commercial contract on earth in the future.

Living on mars such a procedure could be quite feasable although.

06 / 08 / 2017

 W. Reiwer