About my background
In the middle of the 60-ties, I worked at a Swedish telephone company with analysis and optimisations of signal processing systems. Formerly such systems consisted of interconnected components such as resistors, inductors and capacitors. I retired in 1993.
In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.
If we have only two components - each having a parameter value – the problem is very simple. Let the first parameter value be the shortest distance to the left edge of a picture (below) while the second value is the distance to the bottom edge. Then, if the interconnection is given, a point in the picture represents the system unambiguously.
Suppose now that all points inside a certain triangle (region of acceptability, marked by red edge) will meet all requirements according to the specification of the system, while all other points does not, and that the spread of parameter values is uniformly distributed over a circle (green). Then, if the circle touches the three sides of the triangle, the centre of the circle would be a perfect solution to the problem.
But if we have 10 or 100 parameters, then the number of possible parameter combinations becomes super-astronomical and the region of acceptability will not possibly be surveyed. I begun to think that the man was not all there.
The problem was almost forgotten until a system designer entered my room about half a year later. He wanted to maximize the manufacturing yield of his system that was able to meet all requirements according to the specification, but with a very poor yield.
Oh, dear! I would not like to get fired immediately. So, we wrote a computer program in a hurry, using a random number generator giving Gaussian (normally) distributed numbers according to the bell curve to the left in the figure below. A cluster of points in two dimensions - where each pair of two normally distributed parameters is represented by a point - is seen to the right.
The system functions of each randomly chosen system were calculated and compared with the requirements. In this way we got a population (generation) of about 1000 systems from which a certain fraction of approved systems was selected. For the next generation the centre of gravity of the normal distribution was moved to the centre of gravity of the approved systems and this process was repeated for many generations.
After about 100 generations the centres of gravity reached a state of equilibrium. Then the designer said “but this looks very god”. And we were both astonished, because we had only put some things together by chance. A closer look revealed that there is a mathematical theorem valid for normal distributions only stating:
If the centre of gravity of the approved systems coincides with the centre of gravity of the normal distribution in a state of selective equilibrium, then the yield is maximal (theorem 6.2.2).
This gave an almost religious experience. Here a mathematical theorem solved a difficult problem without our knowledge and independently of the structure of the region of acceptability. But in order to fulfill the theorem exactly, infinitely many random points must be generated, which is of course impossible. Nevertheless the solution was good enough for our technical purposes. Our very simple process was also similar to the natural evolution in the sense that it worked with random variation and selection.
Darwinian evolution: Later it turned out that this is not very far from the Darwinian evolution of natural systems, which is my main concern today. The analogue to manufacturing yield was the mean fitness determined as a mean over the set of individuals in a large population. Already here a connection between mean fitness and the spread in parameter values is clearly seen. More generally the spread in parameter values is an analogue to the disorder in morphological characters.
Looking at the triangle and the circle above it is clear that a small arbitrary displacement of the circle causes mean fitness to decrease, but may be taken back again if the radius of the circle is decreased, i. e. if the disorder of the morphological characters is decreased. This means that mean fitness and disorder may be simultaneously maximal even if the distribution of parameters deviates from normal.
More generally the theorem of normal adaptation may be proved in two different ways leading to the following more general formulation of the theorem: A normal distribution may always be adapted for maximum mean fitness and a corresponding maximum disorder (average information) to any region of acceptability (theorem 6.2.3). The condition of optimality is that the centre of gravity of the normal distribution coincides with the centre of gravity of the survivors, i. e. parents to offspring in the next generation.
Note, that this is in contrast to the mean fitness earlier defined as a mean over the set of genes in the gene pool, which led to the dubious “fundamental theorem of biology” due to Fisher (1930) because it does not consider the simultaneous maximization of disorder.
Neural networks: Because nerve cells may in principle add and multiply signal values and because many researchers agree that an evolution of signal patterns is going on in our brains, digital circuits (neural networks) would perhaps simulate an evolution of signal patterns in certain parts of the central nervous system. In fact, I have also proposed a very simple digital circuit as a model of the evolution in the brain.