= Solution
Unmodified <Hebbian learning> reinforces correlated input and output but has no mechanism limiting synaptic strength. The second rule adds activity-dependent negative feedback and stabilizes the <norm> at an input-scale-independent value, provided the input keeps exciting the output. The third has a cubic weight penalty independent of the instantaneous output; its equilibrium <norm> instead depends on the strength of the input <second moment>.
For the third averaged rule, an equilibrium along <eigenvector> $i$ has radial linearized <eigenvalue> $-2\lambda_i/\tau$ and transverse <eigenvalues> $(\lambda_j-\lambda_i)/\tau$. Thus a positive simple largest <eigenvalue> gives stable equilibria in its two opposite directions, whereas lower eigen-directions are unstable to larger components. With generic initial weights, it learns the dominant correlation direction, linking the rule to <principal component analysis>. These are idealized normalization and feature-extraction mechanisms, not claims that biological synapses implement these exact equations.
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