Molecular Computing [Electronic resources] نسخه متنی

section 4.5 of this chapter).

The second fundamental that gives the system the ability to solve computationally complex problems is its multilevel organization. Nicolis (see, for instance, 1986) has discussed the main principles and details of this organization in detail.

From these considerations, it can be said that both the behavioral complexity of the system and its ability to solve problems of high computational complexity are determined by the same fundamentals as that of a reaction-diffusion system. Therefore the degree of behavioral complexity could be a decisive point in determining the choice of a system capable of solving computationally complex problems.

It is known (Field and Burger 1985) that the dynamics of distributed nonlinear chemical systems that display sufficiently complicated behavior can be described by a system of nonlinear differential equations of the type:

where U_i(r, t) is the concentration of the ith component of reaction proceeding in the system, A is a control parameter, D_ij are diffusion coefficients, and N = 1, 2, 3, … , N.

The behavior of this system is determined by the complicated nonlinear kinetics of reactions at each spatial point r_k, described by functions

and also by diffusion of reaction components.

At the same time, such an excitable system can be considered as a realization of a neural network where:

Each point of the medium is a primitive microprocessor.

The dynamics of microprocessors can be characterized by complicated chemical reactions produced by external excitations.

Short-range interactions between primitive processors occur (in principle, each microvolume is coupled with all others by diffusion, but because of a rather low spreading speed, these interactions proceed with a delay proportional to the distance between microvolumes).

In the general form, homogeneous neural nets can be described by a system of integrodifferential equations (Masterov et al. 1988):

where G[−T_i − A + Z_i] is the response function for elements of ith type upon activation by Zⁱ, T_i is the shift in function G, and Φ_m is the function of spatial coupling between active elements.

These integrodifferential equations cannot generally be represented by the above system of kinetic differential equations (see Vasilev et al. 1987). Under some assumptions, however, both of these models prove to be adequate.

Based on these considerations, it is natural to broaden the boundaries of the pseudobiological paradigm in comparison with McCulloch and Pitts's original approach (1943) and in particular:

To pass from discrete neural networks to distributed information processing media

To pass to systems with much more complicated nonlinear dynamics than in the case of neural networks usually under discussion

To look for systems possessing multilevel organization

Different separate attempts to do the above are known. We feel the most effective way is to use the unique properties of reaction-diffusion media that comply with the above-mentioned demands.

During recent decades, two basic implementations of the reaction-diffusion paradigm have been developing. The first of these is numerical simulations. In this case, solving systems of reaction-diffusion equations presents an opportunity to perform image processing operations—to generate textures and so on. The second is an attempt to design "hardware" information processing means capable of performing different operations of high computational complexity. Both of these strategies are discussed below.