A memory you do not look up — you fall into. Twenty-five neurons hold three patterns as the low points of an energy landscape; hand the net a corrupted fragment and it rolls downhill, flipping neuron by neuron, until it comes to rest in the nearest stored memory. Down the center the data flows: a broken probe goes in, the net descends, the recalled pattern comes out. The blue team builds and defends it; the red team tries to break it.
source J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, PNAS 79(8):2554–2558 (1982) — scanned issue at archive.org/details/sim_proceedings-of-the-national-academy-of-sciences-usa_1982-04_79_8. Rendered, not quoted.
Every neuron is a bit si = ±1. Store patterns with the outer-product (Hebbian) rule, no self-connection:
W = Σp xp xpT, wii = 0. The matrix is symmetric by construction.
Update one neuron at a time, asynchronously, by the sign of its input:
si ← sign( Σj wij sj ).
Define an energy E = −½ sT W s. Each single flip that the rule makes can only lower E or hold it — so the state slides monotonically into a local minimum, and the stored patterns are dug in as those minima. Recall is gradient descent.
Live values for the panel state — energy and overlap with each stored memory:
| memory | overlap m | is minimum |
|---|
The storage rule is not new here: it is the Hebbian rule — cells that fire together wire together — written as an outer product. That is the sphere before this one; its correlations are W.
Hopfield's move is what comes after: read that same W as the coupling of an Ising spin glass and hand it an energy function. Suddenly memory is physics — a stored pattern is a valley, recall is a ball rolling down. The next sphere takes the descent further: from discrete flips to a smooth energy gradient. Each sphere is the next one's premise.
The blue team's live re-check, recomputed from the weights right now: all three memories are fixed points, a corrupted probe converges to the correct one, and the energy never rises along the descent. If red breaks symmetry, this is where it shows.
N = 25 neurons on a 5×5 grid. Three patterns are stored by the Hebbian outer product into a single 25×25 weight matrix — PLUS, X, and RING. Because they are well separated, all three sit safely below the network's capacity (about 0.138 N ≈ 3.5 patterns).
The input you feed the panel is a probe: one of these memories with a handful of bits flipped — the fragment to be completed. That is the whole input; the descent below does the rest.
| memory | overlap | bits wrong |
|---|
Every energy and overlap is computed on the spot from W and the live state — nothing is looked up. Click cells to break a memory by hand, then run and watch it heal.
What the net produces, proven live: each stored pattern is an exact fixed point (a full sweep changes nothing), and a corrupted probe descends to the correct one. The energies of the three wells are computed, not asserted — PLUS = −300, X = −444, RING = −420. The standard probe (a PLUS with 3 bits flipped, E = −168) rolls down to E = −300 and stops.
The blue team's witness (left) recomputes these live; the red team (right) breaks the symmetry that makes descent possible.
Worse, the true minima are not the only ones: the net also has spurious states — the inverse of every memory (overlap −1), and odd mixtures of three memories — and it will happily "recall" one of those instead. The modern successors (Krotov–Hopfield dense associative memory, and the softmax attention of transformers) store exponentially more and are what actually scaled. The 1982 net is the seed, not the tree.
"The net always recalls the stored pattern." Cut. It can settle on a memory's inverse or a spurious mixture — the panel labels those honestly when they happen.
"Hopfield invented associative memory." Corrected. Amari (1972), Little (1974) and Willshaw came earlier; Hopfield's contribution was the energy function that made recall a physics of descent, and it is that framing rendered here.
"Energy always decreases." Kept, conditioned. Only for symmetric weights under asynchronous updates. Break the symmetry (window 6) and E can rise and the net can cycle forever — which is exactly the tamper.
The red team's move: make one connection one-way — set w7,12 ≠ w12,7. Symmetry was the only thing guaranteeing descent. The blue team's witness (window 7) is watching.
Break the symmetry and E is no longer a downhill-only function: the stored patterns stop being fixed points, the standard probe never settles, and its energy oscillates up and down forever. The witness recomputes, disagrees, and turns red. Nothing is faked — the void is real and it is caught.