PERCOLATION THEORY AND BRAIN STRUCTURE
Iosim M.1, Popov K. 2
1Search Line Inc., USA.,
27, Adams st. Old Bridge, NJ 08857, USA
2Syktyvkar state
university, Russia., 55 Octyabrsky pr. 167001, Syktyvkar, Russia,
E-mail: popov@ssu.edu.komi.ru
Keywords: percolation, infinite cluster, brain structure, neuron net.
The structure and functional purpose of different parts of the head brain core is investigated in detail [1]. What about the mechanisms of realization of such brain functions as thinking, memory, recognition of images and so on, there is no clear conception about it as on the nuclei, as on the macroscopic level. That is why, any model description of the brain processes, which give the real information about object characteristics, is interesting.
In this paper we suggest to use the percolation theory [2] for description of realization mechanism of some head brain functions. The theory of percolation cluster was developed in physics as a system approach for description of the space structures connectedness on a macroscopic scale, when we can neglect the elementary organization of the object. Without state here the main thesis of this theory, we'll try to argue the principle opportunity of application of this approach to brain description. Being a statistical in its foundation, there in percolation cluster theory were got the significant results even for the systems which consists of 104 elements. If state the value of neuron number in gray matter of the head brain core as 1.5 1010, and its volume - as 700 cm3, the mean concentration of neurons come to 2.107 cm-3, which form a value high enough for averaging procedure and application of the idea of "infinite cluster". If we take a volume of 1 cm3 as a lower limit of an individual part of core, responsible for certain function, the number of neurons within this region will be high enough for validity of suggested model.
Lets consider the core of heart brain or some its part, responsible for certain function, as a three-dimensional structure (net) of neurons, distributed accidentally, which connected with each other by dendrites and axons through synapses. As those regions, as the whole core have the space border, through which the information can penetrate in active zone and leave it.
We can consider the incoming information as a sensor signals (excitations) or signals from the other parts of a brain. The output information is impulses which move to the executive physiological organs or to the other parts of brain. Lets consider the sign of the observed activity evince or function execute as the existence of structure bond from one to another (opposite) borders of separated part of brain core. This bond is supplied with some part of active neurons, which realized the function. Another, passive in those act, neurons can either fulfill parallel functions, or exist in waiting or rehabilitation mode. Those structure, which contain the group of neurons and fulfill the functional bond in the limit of zone, including the border, can be defined as an "infinite cluster". In another words, "infinite cluster" is corresponded to the brain function. It is possible in the limit of some zone the appearance of a "finite clusters", which sizes do not allow the information about its structure to leave the borders of zone. For example, an "infinite cluster" can be corresponded to the realized thought, while a "finite clusters" to the subconscious one. The "finite clusters", while forming groups and rebuilding, can grow up to the borders, but not certainly
There are some structure independent invariants in Percolation theory, which gave us an interesting opportunity for quantitative valuation of a number of neurons taking part in forming of an "infinite cluster" and a number of bonds between active neurons of an "infinite cluster". First of all, it is a part of functional zone volume I1, occupied by active neurons, which belong to the "infinite cluster":
I1 = f Xc , (1)
here: f - packing coefficient, Xc - critical (baffle) part of active neurons necessary for appearance of an "infinite cluster". In the second place, it is an average number of neurons I2, with which every neuron of an "infinite cluster" is connected:
I2 = Z Yc , (2)
here: Z - average number of bonds of neuron with neighbors, Yc - critical (baffle) part of active bonds necessary for appearance of an "infinite cluster". Calculations made for the large number of known three-dimensional greets gave the following values of introduced invariants: I1 = 0.16 and I2 =1.5. If, for example, we take in formula (1) the packing coefficient as for accidentally packed up balls f=0,637, that the part of active neurons will be equal to 0.25. Certainly, it is only illustration and it needs to take into account the special properties of bonds, orientated character of excitation transfer and some other features of biological object. But it is the problem of the future investigations in the frames of suggested approach.
In conclusion, we would like to note that this model can be put in foundation of computer experiments, which can provide a new information about heart brain functioning.