FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
1997, VOLUME 3, NUMBER 3, PAGES 903-923
A. Yu. Plakhov
Abstract
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In the paper the differential equation obtained by
V. A. Marchenko and L. A. Pastur \cite{MP} is examined which describes
spectral distribution in some ensembles of large random
matrices. The solution of this equation is explicitly found,
as well as the rule proposed in \cite{MP} for finding out
intervals on real axis complement to spectrum is proven.
The methods of V. A. Marchenko and L. A. Pastur
are applied in the neural networks theory for studying
evolution of spectrum of interneuron connection matrix
describing REM sleep. Asymptotic behavior of spectrum is
investigated; it is proven to differ qualitatively in cases
where a parameter $\alpha$ corresponding to memory loading
with memorizing patterns is less than some critical
value $\alpha_c$ , and where $\alpha > \alpha_c$ . From
viewpoint of associative memory in neural networks,
all the patterns are memorized as a result of sleeping
in the first case, and are not in the second one.
All articles are published in Russian.
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