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Cellular Neural Networks (CNN) is a massive parallel computing paradigm defined in discrete N-dimensional spaces. Following the Chua-Yang definition: CNN main characteristic is the locality of the connections between the units: in fact the main difference between CNN and other Neural Networks paradigms is the fact that informations are directly exchanged just between neighboring units. Of course this characteristic allows also to obtain global processing. Communications between non directly ( remote ) connected units are obtained passing through other units. It is possible to consider the CNN paradigm as an evolution of Cellular Automata paradigm. Moreover it has been demonstrated that CNN paradigm is universal, being equivalent to the Turing Machine. A mathematical formal description of the discrete time case is contained in the following equations.

x(t+1) = g(x(t)) + I(t) + SUM( A(yk(t),PA(j)) ) + SUM( B(uk(t),PB(j)) )

y(t) = f (x(t))

where x is the internal state of a cell, y its output, u its external input and I a local value called bias . A and B are two generic parametric functionals, PA(j) and PB(j) are the parameters arrays (typically the inter-cell connection weights). The neighbor yk and uk values are collected from the cells present in the two neighborhood Nr , for the feedback functional A , and Ns , for the control functional B . The two neighborhoods are potentially different. The functionals A and B are also called templates . The instantaneous local feedback function g express the possibility of an immediate feedback effect. This function is typically not used. f is the function that gives cell output from the internal state. Generally is used the chessboard distance convention, expressed by the equation

d(i,j) = max(|x(i) - x(j)|,|y(i)-y(j)|)

In most of cases the system is non-markovian , i.e. the future internal state depends also from the past history of the system. In the special case of time-variant CNN all the above functions, neighborhoods and parameters can be also function of time. In figure is depicted a block-scheme of a generical CNN iteration.

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Cellular Neural Networks Ftp Site at University of Trento, Italy.

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