In this work, we introduce and investigate the properties of the “relativistic” Hopfield model endowed with temporally correlated patterns. First, we review the “relativistic” Hopfield model and we briefly describe the experimental evidence underlying correlation among patterns. Then, we face the study of the resulting model exploiting statistical-mechanics tools in a low-load regime. More precisely, we prove the existence of the thermodynamic limit of the related free energy and we derive the self-consistence equations for its order parameters. These equations are solved numerically to get a phase diagram describing the performance of the system as an associative memory as a function of its intrinsic parameters (i.e., the degree of noise and of correlation among patterns). We find that beyond the standard retrieval and ergodic phases, the relativistic system exhibits correlated and symmetric regions, which are genuine effects of temporal correlation, whose width is, respectively, reduced and increased with respect to the classical case.

中文翻译：

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具有相关模式的相对论Hopfield模型

在这项工作中，我们介绍并研究了具有时间相关性的“相对论” Hopfield模型的性质模式。首先，我们回顾“相对论”霍普菲尔德模型，并简要描述模式之间相关性的实验证据。然后，我们将研究在低负载情况下利用统计力学工具对结果模型进行的研究。更准确地说，我们证明了相关自由能的热力学极限的存在，并且推导了其阶数参数的自洽方程。对这些方程进行数值求解，得到一个相图，该相图描述了作为关联存储器的系统的性能，该性能是其固有参数（即噪声的程度和模式之间的相关性）的函数。我们发现，除了标准的检索和遍历阶段外，相对论系统还表现出相关和对称的关系 区域，它们是时间相关性的真正影响，其宽度相对于经典情况分别减小和增大。