The BCM learning rule has been used extensively to model how neurons in the brain cortex respond to stimulus. One reason for the popularity of the BCM learning rule is that, unlike its predecessors which use static thresholds to modulate neuronal activity, the BCM learning rule incorporates a dynamic threshold that serves as a homeostasis mechanism, thereby providing a larger regime of stability.udThis dissertation explores the properties of the BCM learning rule – as a dynamical system– in different time-scale parametric regimes. The main observation is that, under certain stimulus conditions, when homeostasis is at least as fast as synapse, the dynamical system undergoes bifurcations and may trade stability for oscillations, torus dynamics, and chaos. Analytically, it is shown that the conditions for stability are a function of the homeostasis time-scale parameter and the angle between the stimuli coming into the neuron.udWhen the learning rule achieves stability, the BCM neuron becomes selective. This means that it exhibits high-response activities to certain stimuli and very low-response activities to others. With data points as stimuli, this dissertation shows how this property of the BCM learning rule can be used to perform data clustering analysis. The advantages and limitations of this approach are discussed, in comparison to a few other clustering algorithms.
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