 Solution

The input vector $x$ contains the activities of presynaptic units or sensory features, and $w$ contains the corresponding synaptic weights. The linear output $y=w^{T}x$ is their weighted sum; it may be a centered activity proxy rather than a literal nonnegative spike rate. The learning time scale is $\tau >0$, and take $\alpha >0$ for normalization. The term $yx$ is a Hebbian learning increment that strengthens weights correlated with the postsynaptic output. The term $-\alpha y^{2}w$ opposes unlimited growth.

For slow learning under stationary input statistics, average over inputs while holding the current weights fixed. Define $C=⟨x{x}^T⟩$. The averaged Oja's rule isIts squared norm obeysWhenever the output second moment is positive, this drives the norm toward $1/\sqrt{\alpha }$, preventing the unbounded growth of an unconstrained Hebbian linear neuron. The direction becomes sensitive to dominant input correlations. Such a unit can learn a feature detector without supplied target labels, providing a simple mechanism for unsupervised learning and principal component analysis. This averaged analysis assumes input fluctuations are fast relative to learning; it is not a claim that every individual stochastic update follows the same deterministic trajectory.