We study the learning of an external signal by a neural network and the time to forget it when this network is submitted to noise. The presentation of an external stimulus to the recurrent network of binary neurons may change the state of the synapses. Multiple presentations of a unique signal lead to its learning. Then, during the forgetting time, the presentation of other signals (noise) may also modify the synaptic weights. We construct an estimator of the initial signal using the synaptic currents and in this way define a probability of error. In our model, these synaptic currents evolve as Markov chains. We study the dynamics of these Markov chains and obtain a lower bound on the number of external stimuli that the network can receive before the initial signal is considered forgotten (probability of error above a given threshold). Our results are based on a finite-time analysis rather than large-time asymptotic. We finally present numerical illustrations of our results.

中文翻译：

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记忆的简单网络模型中记忆寿命的数学分析

我们研究了神经网络对外部信号的学习以及当该网络受到噪声影响时忘记它的时间。向二元神经元循环网络提供外部刺激可能会改变突触的状态。独特信号的多次呈现导致其学习。然后，在遗忘时间内，其他信号（噪声）的呈现也可能会修改突触权重。我们使用突触电流构建初始信号的估计器，并以这种方式定义错误概率。在我们的模型中，这些突触电流演变为马尔可夫链。我们研究了这些马尔可夫链的动力学，并获得了在初始信号被认为被遗忘之前网络可以接收的外部刺激数量的下限（错误概率高于给定阈值）。我们的结果基于有限时间分析而不是大时间渐近。我们最后给出了我们结果的数值说明。