Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 84 / 1

A binary Hopfield network has $N$ recurrently connected units $s_i\in \{-1,+1\}$, symmetric weights $w_{ij}=w_{ji}$, zero self-weights, and thresholds ${\theta }_i$. A distributed memory is a desired population pattern ${\xi }^{\mu }$. One Hebbian learning prescription isCoactive signs strengthen a connection, and opposite signs weaken it. This is Hebbian memory weights for a Hopfield network, rather than a separate address for each memory.

Initialize the units with the input cue, which can be a corrupted or partially specified pattern. Update one unit at a time by $s_i\leftarrow \mathrm{sign}({\sum }_j{w}_{ij}{s}_j-{\theta }_i)$, retaining its old sign when the field is exactly zero. A fair asynchronous schedule revisits every unit. The energychanges on a single update by $\Delta E=-(s_i^{\prime }-s_i)({\sum }_j{w}_{ij}{s}_j-{\theta }_i)\le 0$. An actual flip with the tie convention strictly reduces energy; finitely many states imply eventual arrival at a fixed point. This proves asynchronous Hopfield energy descent. Synchronous updating lacks this single-unit argument and can cycle.

For one stored pattern and zero thresholds, the field at $s=\xi$ is $(N-1){\xi }_i/N$, so the pattern is a fixed point. With multiple patterns, the desired signal is accompanied by cross-talk from other memories; not every arbitrary set is guaranteed to be stable. Retrieval is attraction from a cue to a stored-pattern fixed point, provided the cue lies in its basin of attraction. Spurious minima and limited basins are part of the model, not errors in its update algorithm.

To an experimental neuroscientist, a Hopfield network offers a concrete hypothesis about associative memory: information is distributed over a population, a partial cue can complete a learned pattern through recurrent interactions, and synaptic changes can alter which persistent activity patterns are accessible. It predicts attraction basins, interference between similar memories and resilience to some corrupted inputs. These are useful qualitative questions for experiments, and the connection between Hebbian learning, collective dynamics and stored patterns is mathematically testable.

Its limitations are equally concrete. Binary units suppress spike timing, neural adaptation, heterogeneous conductances and dendritic computation. Exact symmetric effective interactions, unrestricted signs and the absence of delays are idealizations; individual biological neurons generally do not implement arbitrary positive and negative outgoing synapses in the way the unrestricted weight matrix permits. The simple learning rule supplies neither realistic forgetting nor a complete account of biological long-term synaptic plasticity. Memory capacity and retrieval reliability depend on load and pattern correlations, and spurious attractors occur. Moreover, a stable activity pattern is not by itself an account of consolidation, behavioral readout or the anatomical location of a memory. It is an explanatory model of associative retrieval, not a literal complete circuit diagram of the brain.

Since the derivative of $\tanh (\beta x)$ at zero is $\beta$, the Jacobian matrix of this directed rate network isWith $a=1+\tau \lambda$, its characteristic determinant isThus the eigenvalues satisfyThis establishes three-neuron ring stability from the loop product. The ring is directed, so the symmetric-weight Hopfield network energy argument does not apply. Assume the physical relaxation time $\tau >0$.

An even number of negative weights gives $\Gamma >0$ when all three connections are nonzero. Put $r={\Gamma }^{1/3}$. The three cube roots giveThe complex pair always has negative real part, and the real eigenvalue is negative exactly when $r<1$. Therefore strict linear asymptotic stability requires $\Gamma <1$. The origin is unstable when $\Gamma >1$. A zero connection gives $\Gamma =0$ and all eigenvalues $-1/\tau$, also stable, though the matrix need not be diagonalizable.

At $\Gamma =1$, linearization has a simple zero eigenvalue, so a strict negative-real-part test alone does not settle nonlinear stability. For completeness, take a nonzero real eigenvector $c$ of the weighted loop matrix $M=\beta W$, satisfying $Mc=c$, and set $x_i=c_i{u}_i$. The local equations areThe center direction has all $u_i=a$. Averaging the three equations on the center manifold gives $\tau \dot{a}=-({\beta }^2{\sum }_i{c}_i^2/9)a^3+O(a^5)$. Its leading coefficient is negative, while the other modes decay exponentially. Hence the exact boundary origin is locally asymptotically stable for the specified hyperbolic-tangent nonlinearity, with algebraic rather than exponential decay. This qualification distinguishes actual nonlinear stability from the usual strict Jacobian criterion.

An odd number of inhibitory connections gives $\Gamma <0$. Write $r=|\Gamma {|}^{1/3}>0$. The cube roots of $-r^3$ have arguments $\pi ,\pi /3,-\pi /3$, soThe real mode is always damped. The pair has negative real part precisely when $r\cos (\pi /3)<1$, givingfor strict linear asymptotic stability. For $r>2$ the origin is unstable to an oscillatory mode.

At $r=2=1/\cos (\pi /3)$ the eigenvalues are $-3/\tau$ and $\pm i\sqrt{3}/\tau$. Thus the linear oscillation frequency is $\boxed{{\omega }_H=\sqrt{3}/\tau }$ and its period is $2\pi \tau /\sqrt{3}$. Varying $r$ crosses the imaginary axis transversely, since $d\mathrm{Re}{\lambda }_{+}/dr=1/(2\tau )>0$.

The hyperbolic tangent has a negative cubic saturation and no quadratic term. To check the nonlinear onset, let $v$ be a critical complex eigenvector of $M=\beta W$ and $p$ its normalized adjoint eigenvector, with $p^{\ast }v=1$. For this weighted cycle, ${\overline{p}}_i{v}_i=1/3$. Since $Mv=(1+i\sqrt{3})v$, the cubic derivative evaluated on $(v,v,\overline{v})$ has projectionIts real part is strictly negative. With no quadratic contributions, this gives a negative first Lyapunov coefficient: the generic onset is a supercritical Hopf bifurcation, with a small attracting periodic orbit just beyond the threshold. Exactly at threshold the linear system has undamped oscillations, but nonlinear cubic damping makes sufficiently small oscillations decay; one must not infer a nonzero sustained cycle at the critical parameter solely from the imaginary eigenvalues.