New principles of computer processing

NEW PRINCIPLES OF COMPUTER PROCESSING

Entropy and information in biological systems
In spite of enormous complexity of biological systems in general and human in particular, the notion of entropy turns out to be very useful in description of certain aspects of such objects.

As first example we consider the following psychological experiment: Suppose we have N lamps; and lamp number j flashes with probability pj. The human should point out as quickly as possible which lamp flashed. What is the mean time of human reaction if the experiment is repeated many times?

The answer turns out to be rather unexpected: the average reaction time is proportional to the entropy (2.2) of our experiment (and not to number of lamps N as one would naively think) [Hyman R.,1953].

Moreover, we can repeat the experiment under another conditions: we can ask the human to point to the flashed lamp as soon very quickly, such that sometimes he can even make mistakes. Then we get two probabilistic experiments: experiment a, whose outcomes are flashes of lamps, and experiment ß, whose outcomes are human reactions. It turns out that the average time of human reaction in this case is proportional to information I(, ß)!

The conclusion we can make from these experiments is that the speed of signal propagation in human nerves is proportional to mathematically well-defined amount of information contained in the signal!

Another indication of the relevance of entropy notion in biological system is a well known fact that in any population of biological species most of their physical characteristics (like say weight, or length) have gaussian distribution density. As we know, the gaussian distribution (2.18) maximises the entropy if the dispersion is fixed. Therefore, we conclude that the principle of maximal entropy (and also the constancy of dispersion) has direct application to evolution of biological population!

Of course, it seems very tempting to generalise the notion of entropy and information to the level of biological objects, or, say, mankind as a whole, looking at evolution of our civilisation, or even the whole universe as huge random experiment. Such attempts were made my many authors (see, for example, book of Stonier [Stonier T., 1990]). Many authors these days speak about "information field" as a fundamental field of nature and so on. However, if we speak about total amount of information (or entropy) contained in, say, human body, or our whole civilisation, it is very difficult to explain what do we mean. How to formulate the experiment explicitly? Or, in thermodynamical terms, how to count all possible microscopic states of the system? Which states should be considered to be different? The problem is that, since the gap between the level of civilisation and fundamental quantum level is very big (in between we have chemical level, biological level, social level...) it does not seem possible to give satisfactory theoretical answer to this question at the moment.

Nevertheless, taking mankind as a whole, or choosing one single human, it is clear, as we can see from above examples, that the notion of entropy can really be used for effective description of certain aspects of their life. So maybe it is possible to measure some characteristic of human experimentally, which can be then satisfactorily interpreted as the "total entropy of human being"? In the sequel we introduce such variable. Then, based on the results of calculation of this variable for different types of human, argue, that this variable can really pretend to be a natural candidate for experimentally measured human entropy, or "measure of human uncertainty". We will referrer to the results of GDV experiments.

Statistical interpretation of BEO-grams. BEO-entropy
To put the essentially qualitative discussion of the previous section to more solid background one should choose one or another mathematical approach for analysis of BEO-gram structure. Ideally, we would like to have an algorithm which would allow to distinguish between images which visually belong to essentially different classes. It was demonstrated by numerous experiments that [Korotkov K., 1998] these different classes, in turn, describe essentially different human states. It was proposed [Proceedings, 1999] the analyse of BEO-grams using so-called "analysis of fractal dynamics", which allows to encode the essential information carried by the image to relatively small number of parameters (eight). This allows develop an algorithm which effectively attribute images to different classes.

Here we present another approach to the same problem, based on simplest possible probabilistic interpretation of BEO-gram image. Namely, let us present the BEO-gram as a circular picture having certain distribution of density along the radii. This image may be presented as some function F(x) where x is an angle within the interval [0,2]. Parameter F{x) may be different: for example, it may be the maximum distance from the weight center of the BEO-gram to the most distant point - max radii, or the average brightness along the radii, or the distance to the median. In practice we present calculations for a set of parameters.

As we can see from many examples of BEO-grams, as a rule, function F(x) is highly discontinuous. This suggests the idea to consider function F(x) as random variable, and calculate associate statistical parameters. Let us introduce the total brightness parameter

and define new normalised function

Denote maximal and minimal values of function f(x) by fmin and fmax respectively (see fig.???). Let us build a graph P(f) of density distribution of values of function f(x) on the interval [fmin, fmax] o Let us also introduce the normalised distribution p(f) by the formula

Obviously function g(f) satisfies the normalisation condition

Now we are in position to introduce standard statistical characteristics: the mean value fo, dispersion m1 and higher moments mj by the formulas (2.15), (2.16).

The entropy definition arising in this context is given by the formula (2.17):

We call entropy BEO - the BEO-entropy.
Of course, the proposed statistical interpretation of function F is some approximation to the real process only: in practice there may be certain correlations between the amplitude at different points (probably, there exist both short-range and long-range correlations). Nevertheless, for us the main role of BEO-entropy notion will be just practical: it will allow to introduce natural classification of BEO-gram by "degree of misbalance". Namely, for highly inhomogeneous BEO-grams (which, according to experimental data in turn correspond to unstable human state) the random variable F(x) has high degree of uncertainty, which should lead to relatively high value of BEO-entropy BEO. In contrary, homogenous "quiet" BEO-grams correspond to random variable F(x) with low degree of uncertainty, which should give rise to relatively low values of BEO.

Results of calculations of BEO-entropy eBEO presented in the next section for different types of BEO-grams confirm importance of this characteristic of BEO-gram. Moreover, we shall see strong indications that variable eBEO is probably related to the entropy of biological object discussed in above. Namely, values of BEO turn our to vary depending on age and state of human in exactly the same way as we would expect.

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