Perceptron theory · the substrate jump · the fifth body
The perceptron in a crossbar
If the others are clever, this one is almost insolent. Lay weights down as conductances on a grid of memristors, drive the rows with input voltages, and the physics does the rest: Ohm's law multiplies (I = G·V at every crosspoint), Kirchhoff's law sums (the currents pour down each column and add themselves). A full dot product, computed in one electrical instant, inside the memory — no fetch, no clock, no von Neumann round trip. The weight and the multiplier are the same physical thing.
weights = memristor conductances Gᵢⱼ · inputs = row voltages Vᵢ · multiply = Ohm (I=GV) · sum = Kirchhoff (column current) · threshold = the sense amp
✓ STRONG
The in-memory MAC. A crossbar computing Σ GᵢVᵢ by Ohm + Kirchhoff is bedrock circuit physics, demonstrated as analog dot-product engines (HP, Ielmini, Yang). One step, no data movement.
◐ MIDDLING
A real crossbar accelerator. It works, but analog: device variability, conductance drift, limited bit-precision, IR drop, sneak paths, write endurance. Useful in the right regime, fiddly everywhere else.
◔ FRONTIER
Reliable in-situ training at scale. Programming millions of analog conductances precisely, online, with yield — and multi-bit weights that hold — is open engineering.
I · One crosspoint — the weight that multiplies itself
A memristor is a resistor that remembers: drive it hard and its conductance latches to a new value (SET higher, RESET lower) and stays there with the power off. That stored conductance Gis a weight. Read it gently with a small voltage and I = G·V — the weight multiplied by the input, in a single device. Its fingerprint is the pinched hysteresis loop: an I–V curve that always passes through the origin (no voltage, no current) and opens into a loop because the conductance depends on history.
conductance is non-volatile — it holds with power off
the pinched-hysteresis I–V loop (Chua 1971 · TiO₂ device, HP 2008) · the dot = your read point I = G·V · loop pinches at the origin
0.50
stored conductance G (the weight)
+0.20
read current I = G·V (the product)
multiply= Ohm's law, in the deviceweight store= non-volatile conductancesigned w= a G⁺,G⁻ pair (next module)
SET and RESET are how you write the weight — the same pulses that store a bit store a synapse. The memory cell and the multiplier collapsed into one component is the whole trick.
II · The crossbar — a dot product in one electrical instant
Tile the cells into a grid. Each row is an input wire held at voltage Vᵢ; each column is an output wire. Every crosspoint dumps a current Gᵢⱼ·Vᵢ onto its column, and the column simply adds them — Kirchhoff has no choice. The current at the foot of a column isIⱼ = Σᵢ Gᵢⱼ Vᵢ the dot product, for free, in parallel, the moment you apply the voltages. Signed weights use a differential pair of columns: w = G⁺ − G⁻. A sense amp on the difference is the threshold.
x₁ x₂
rows = input voltages · cells colored by conductance · currents pour down the G⁺ / G⁻ columns and sum · differential sense amp = the neuron
—
I⁺ − I⁻ = b + Σ xᵢwᵢ
DARK
fires when I⁺ > I⁻
No instruction ran. No number moved to a processor and back. The array is the computation — that's why it sidesteps the von Neumann bottleneck, and why a crossbar can do a thousand-wide dot product as fast as a one-wide one.
III · It learns — program the conductances, then stack for depth
Training is just writing conductances. Run the perceptron rule and each weight's G⁺/G⁻ pair nudges up or down until the cuts line up — AND and OR resolve in a few sweeps. And the old wall stands: one crossbar is one linear cut, so XOR jams at 3/4 (verified). The escape is the franchise ending — a second crossbar fed by a nonlinear sense layer. Stacked crossbars are deep nets; the body changed, the lesson never does.
target:
input plane · copper = the neuron fires · corners ring green when correct for the target
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cases correct
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verdict
Pick AND or OR, hit train, and the conductance map settles into a clean cut. Switch to XOR: one corner stays crossed however long it programs — a single crossbar can't fold the plane. Add a 2nd crossbar and a hidden layer of conductances bends the boundary; XOR snaps to 4/4. Five bodies, one moral: depth beats the line.