Theory
The basic BCM rule takes the form
where is the synaptic weight of the th synapse, is that synapse's input current, is the weighted presynaptic output vector, is the postsynaptic activation function that changes sign at some output threshold, and is the (often negligible) time constant of uniform decay of all synapses.
This model is merely a modified form of the Hebbian learning rule, and requires a suitable choice of activation function, or rather, the output threshold, to avoid the Hebbian problems of instability. This threshold was derived rigorously in BCM noting that with and the approximation of the average output, for one to have stable learning it is sufficient that
or equivalently, that the threshold, where and are fixed positive constants.
When implemented, the theory is often taken such that
where angle brackets are a time average and is the time constant of selectivity.
The model has drawbacks, as it requires both long-term potentiation and long-term depression, or increases and decreases in synaptic strength, something which has not been observed in all cortical systems. Further, it requires a variable activation threshold and depends strongly on stability of the selected fixed points and . However, the model's strength is that it incorporates all these requirements from independently-derived rules of stability, such as normalizability and a decay function with time proportional to the square of the output.
Read more about this topic: BCM Theory
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