We prove a convergence theorem in linear dynamic approximation theory which yields estimates of certain series of errors. In [5 and 6] we have developed a model for the function of the cerebral cortex, which is based upon a theorem of linear dynamic approximation theory. Here we extend that theorem from the case of functional to the case of operators. This allows new variants of the models developed in [6] . Instead of looking at the errors of each learning line separately, we can use now the

more »

... rors of bunches of n learning lines. But the theorem which is proved below has a straightforward motivation which does not require the knowledge of [5, 6] . A sequence of points foiiii---S Rfc represent the consecutive inputs to a system S. Other inputs ya-iVii-■ ■ € Rn represent the desired outputs of S but they reach S a little later so that S must compute its guess yt S Rn on the basis of £o,..., £t and yoi•■•> Vt-i-, before getting yt-Of course yt -yt is the error vector and this error is known to S at the time when yt+i is to be computed. We want to design S such that the errors will be minimized in some sense, although this will require of course certain assumptions about the sequences (£t) and (yt). (In our model in [6] the neocortex is represented as a system of millions of overlapping thin columns perpendicular to the cortical layers. Each such column contains a system S. At the bottom of each of them the inputs £t and yt are received and in that particular model yt is one of the coordinates of ft+i.) We specify S as follows. First S has a fixed (nonlinear) preparatory map : Rfc -► Rm -{0}. For example, .) This map 0 does not depend on t. Then S seeks a linear map Mt : Rm -► R" such that the error vector yt -Mt(4>(Çt)) be small in a sense. So in our system S, we have yt = Mt{(£t))-The linear operator Mt is called the long term memory of S while <fi could be called the instinctive memory of S. Mt is computed from <j)(^t-i), Mt-\ and yt-i only, and this computation is very simple so that it can be accomplished in fractions of a second by the tissues of the central nervous system.