1. The Cellular Systems
The main features of cellular systems are regularity and homogeneity. In fact a cellular system can be defined as a structured collection of identical elements called cells. The structure is given by the choice of a lattice. Such lattices are 1-dimentional, 2-dimentional and, less used, 3 or more dimensional.
Here are a few examples of common used 2-dimentional lattices:
Fig. 1 0 Cells, lattices, neighborhoods and indexes
neighborhood - a set of cells that are directly interacting with the central cell;
cell - the basic computational unit in a cellular automata(CA) ; they are nonlinear dynamic systems.
The cell dynamics can be:
- continuous in time - they are mathematically described by ODE (Ordinary Differential Equations)
- discrete in time - they are described by a difference equation.
2. An example for a cell dynamics discrete in time
defines the dynamics of the output 
of a discrete time autonomous cell, defined by a weighted summation of all neighborhood cell outputs at the previous time clock. The neighborhood is represented by the set N and a unique index k is chosen to identify the neighboring cell in a particular neighborhood.
The above cell is one of the simplest possible. It is autonomous since there is no external input to drive the dynamics of the cell. In the most general case a cell is described by the following variables:
- inputs - usually denoted by a variable u (scalar) or u (vector of inputs) ;
- states - usually denoted by variable x (scalar) or x (vector of states) ;
- initial states - a particular state variable at the initial moment, t=0 ;
- outputs - usually denoted by variable y (scalar) or y (vector of outputs) .
3. The "Standard Cellular Neural Network" cell
The ODE defining the standard cellular neural networks [Chua and Yang, 1988] is:
where 
and the initial state 
are specified for all cells in the neighborhood N.
This is a more complex, continuous time, dynamic system with a nonlinearity induced by the relationship between states and outputs.
In general (non-standard CNN cell) equation ( 1 ) can be replaced with
where f is a nonlinear function and G is a gene.
In all previous examples the dynamics of the cell for the same input excitation and the same initial state is significantly influenced by the values given to certain parameters ( 
, 
and z ). It was proposed to pack all these parameters in a unique vector 
called a gene, since it determines the overall function of the cellular system much like the DNA.
4. The Cellular Neural Network (CNN) model
The CNN model was proposed by Chua as a practical circuit alternative to Hopfield and other type of recurrent networks. The CNN cell is a continuous time and continuous state dynamical system with some saturated nonlinearity (see equation (1)) which is well suited for implementation using analog circuits.
An important step was the introduction in 1993 of the concept of CNN Universal Machine (CNN-UM) [Roska & Chua, 1993]. Within the framework of the CNN-UM a CNN kernel is employed to perform sequentially various information processing tasks. Recent electronic implementations of the CNN-UM are the sensor computers [Roska, 2000], having the capability to sense and to process an image on the same chip.
Several generation of microelectronic chips were reported so far, as well as development tools which allow an user to program the CNN as a visual microprocessor. There is a wide range of applications, mostly in the area of image processing. Such application include image segmentation, image compression, fast halftoning, contour tracking, image fusion, pattern recognition, etc.
Note: Most of the content is taken from the "Universality and emergent computation in cellular neural networks" book, written by Radu Dogaru.
Professor Radu Dogaru is with the Applied Electronics and Information Engineering Department of "Politehnica" University of Bucharest, where he teaches courses in Neural Networks, Computational Intelligence, Cellular Systems, Numerical Methods for Bioengineering.
He is the recipient of the Fullbright award (1996) and a co-recipient of the Romanian Academy Award - Tudor Tanasescu for research in computational intelligence for signal processing (received in 1997).
