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Remanent Overlaps in the Hopfield Model with Zero Temperature Sequential Dynamics

K. Tokita

1993
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Progress of theoretical physics
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329 Remanent overlaps (ms) and their basins of attraction are numerically studied for the Hopfield model with zero temperature sequential dynamics. Relationships between the ms's and initial overlaps are obtained for the relatively large a's, the rate of memory-loading. The asymptotic dependence of ms on a is also reported. A family of globally coupled Ising spin systems has recently attracted much attention. The models include the SK model of spin glasses!) and the Hopfield model of neural
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... orks_ 2 ),3) Their most significant characteristic is the existence of a large number of (meta)stable states and multivalley structures in phase space, key words often associated with "complex systems". Indeed one expects that an initial state evolves until the bottom of a valley is reached where the system is permanently trapped. In fact, many theoretical studies have reported relationships between remanent overlaps and initial overlaps in the Hopfield model with sequential 3 )-5) or synchronous 6 )-8) dynamics. Although these contain some numerical analysis, the system size and the number of samples were rather smalL Moreover, a systematic numerical analysis which includes finite size effects on the zero temperature (T=O) sequential dynamics has never been performed yet. This paper reports remanent overlaps and the basins of attraction for the Hopfield model with zero temperature sequential dynamics, in order to extract some information for the valley structure_ Here the remanent overlap defined later refers to a macroscopic order parameter, which corresponds either to the overlap with a memorized pattern or to a generalized remanent magnetization_ According to finite size scaling analysis, the distribution of the remanent overlap ms approaches a delta function as the system size becomes larger-This implies that at T =0 and N -7(X) the value of ms is determined by the initial overlap and the parameter a(==p/N), the rate of memory-loading, where p is the number of random patterns for memories and N the system size. Hence the dependence of the value of ms on the initial overlap and on a parameter a is obtained. Here we concentrate on relatively large values of a. The above dependence of ms enables us to get the relationship between remanent overlaps and initial overlaps, which leads to a distribution function of the remanent overlap. We have also studied the asymptotic dependence of ms on a and have obtained the power-law decay of ms to the value corresponding to the remanent magnetization obtained in the SK modeL This supports the Downloaded from https://academic.oup.com

doi:10.1143/ptp/90.2.329
fatcat:jf54igprpzghbbwejlfxw3ptqq