The Bienenstock–Cooper–Munro (BCM) learning rule provides a simple setup for synaptic modification that combines a Hebbian product rule with a homeostatic mechanism that keeps the weights bounded. The homeostatic part of the learning rule depends on the time average of the post-synaptic activity and provides a sliding threshold that distinguishes between increasing or decreasing weights. There are, thus, two essential time scales in the BCM rule: a homeostatic time scale, and a synaptic modification time scale. When the dynamics of the stimulus is rapid enough, it is possible to reduce the BCM rule to a simple averaged set of differential equations. In previous analyses of this model, the time scale of the sliding threshold is usually faster than that of the synaptic modification. In this paper, we study the dynamical properties of these averaged equations when the homeostatic time scale is close to the synaptic modification time scale. We show that instabilities arise leading to oscillations and in some cases chaos and other complex dynamics. We consider three cases: one neuron with two weights and two stimuli, one neuron with two weights and three stimuli, and finally a weakly interacting network of neurons.

中文翻译：

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BCM学习规则的动态特性。

Bienenstock-Cooper-Munro（BCM）学习规则为突触修改提供了一个简单的设置，将Hebbian乘积规则与保持权重的稳态机制结合在一起。学习规则的稳态部分取决于突触后活动的平均时间，并提供区分增加或减少权重的滑动阈值。因此，BCM规则中有两个必不可少的时标：稳态时标和突触修饰时标。当刺激的动力学足够快时，可以将BCM规则简化为一组简单的平均微分方程组。在此模型的先前分析中，滑动阈值的时间尺度通常比突触修饰的时间尺度快。在本文中，当稳态时间尺度接近突触修饰时间尺度时，我们研究了这些平均方程的动力学性质。我们表明，不稳定性会导致振荡，在某些情况下会导致混乱和其他复杂的动力学。我们考虑三种情况：一种具有两个权重和两个刺激的神经元，一个具有两个权重和三个刺激的神经元，最后是一个弱相互作用的神经元网络。