Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 6E (Mathematical Biology)

Set $s=|w{|}^2$. The exact Hebbian learning rule givesThus the norm never decreases, and increases whenever the output is nonzero. Because learning is slow relative to input fluctuations, its averaged equation is $\tau \dot{w}=Cw$, where $C=⟨x{x}^T⟩$ is the input second-moment matrix. It is a covariance matrix only if the inputs have zero mean. Diagonalizing the positive semidefinite matrix $C$ gives components $w_j(t)=w_j(0)e^{{\lambda }_{j}t/\tau }$. Hence weights grow without bound whenever the initial vector has a component in an excited positive-eigenvalue direction. A vector entirely in $\ker C$, or identically zero input, is an important stationary exception.

For the modified Hebbian learning rule,It increases below $s_{\ast }={\alpha }^{-1/2}$ and decreases above $s_{\ast }$, without crossing that equilibrium. More precisely, for a nonzero initial norm,Under persistent excitation, meaning the integral tends to infinity, this givesWithout persistent excitation the limiting norm need not reach this value; zero weights remain zero. The quartic norm stabilization of Hebbian learning bounds the norm and supplies negative feedback at large amplitudes.

Averaging the fast fluctuations in $yx=x{x}^{T}w$ givesThus a nonzero steady vector satisfiesIts direction is an eigenvector of the input second-moment matrix, and its magnitude is the square root of that eigenvalue. The zero vector is also a steady state and satisfies the displayed relation with $\lambda =0$. This is an averaged steady state; the microscopic fluctuating update need not vanish at each instant.

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.