From Pixels to Tensors, Part 4: McCulloch–Pitts

Intro

Part 3 ended on an exit. Symbolic AI hit the limit of what a human can foresee and keep patched, and the way out was to stop authoring the behavior and fit parameters from data instead. This part goes back to the beginning of that second road.

The starting points could not be further apart. Newell and Simon's Physical Symbol System Hypothesis says intelligence is the manipulation of symbols according to formal rules, with knowledge written down explicitly by someone who knows it. Connectionism says the opposite: intelligence is not encoded, it is fitted. Nobody writes the rule that a cat has pointed ears and slit pupils. You supply enough pictures of cats and an objective, and the system adjusts the connection strengths among a large number of very simple units until it approximates a function that can tell cats apart.

A net with no loops and a temporal propositional expression turn out to be two notations for the same thing, and the paper converts each into the other. Almost any sentence about which neuron fired when can be built as such a net; the exceptions are the sentences that come out true when nothing fired. Learning is not in scope. The wiring is fixed before the net runs, by assumption, and the rest of this part is about what that assumption costs.

An hour of looking at water

Earlier this year I went whale watching off Perth. It was my first time, and for the first stretch I was useless. People around me kept calling out where to look, and I could not see what they were seeing. The ocean was just a moving surface. I had no rule to apply, and nobody could give me one that helped — "look for the blow" assumes you already know what counts as a blow and what is just chop.

After about an hour, something changed. I started catching the humpbacks' flukeprints: the smooth, oily-looking circles a tail leaves on the surface when the animal goes down, and the particular way the ripples spread. Once I could see those, I could look ahead of them and be watching the right patch of water when the whale came up to breathe.

I still cannot write down the rule. I could not tell you the threshold on ripple diameter, or how I separate a flukeprint from a wind slick. Nothing was explicitly taught to me and nothing was explicitly encoded. An hour of examples reorganized something, and afterwards a perception was available to me that had not been there before.

Two men and a question

In 1943 Warren McCulloch was a psychiatrist and neurophysiologist in his mid-forties who had been preoccupied with logic for two decades. Walter Pitts was a largely self-taught logician, barely twenty, who had turned up at the University of Chicago after reading Principia Mathematica and writing to Bertrand Russell about errors he had found in it. They collaborated on a paper called "A Logical Calculus of the Ideas Immanent in Nervous Activity."¹

McCulloch spent his career circling a single question, which he later used as the title of a lecture:²

What is a number, that a man may know it, and a man, that he may know a number?

The sentence has two halves. The first — what is a number, that a man may know it — asks what numbers and logical relations would have to be like for a physical object to represent and operate on them. The 1943 paper is a large part of the answer.

The second — a man, that he may know a number — asks what a knower has to be for knowing to happen in it. The 1943 paper does not touch it.

What neurophysiology looked like in 1943

The paper opens with two pages of physiology rather than mathematics. Every formal assumption that follows is introduced as a compression of something observed in a lab.

The picture available at the time went roughly like this. The nervous system is a net of neurons, each with a cell body (soma) and an axon. Junctions — synapses — run from the axon of one neuron to the soma of another. Each neuron has a threshold, and excitation has to exceed it to start an impulse. Once started, the impulse is determined by the neuron rather than by the stimulus that triggered it, and it propagates through the whole neuron. Conduction velocity scales with axon diameter, and the thin-and-short versus thick-and-long trade-off roughly cancels, so axonal travel time contributes little to the order in which impulses arrive. No single synapse had ever been observed to fire a neuron on its own; firing required enough neighbouring synapses active together inside a latent addition period of well under a millisecond. After firing, the neuron is briefly refractory, then recovers.

Inhibition is one group of neurons shutting down another. The older explanation routed it through interneurons whose thresholds got raised, but newer measurements showed some inhibition completing in under a millisecond — too fast for an extra cell in the path. So it had to be a synapse acting directly on the neuron being excited. Whether that action was absolute or merely raised the threshold, the experiments of the day could not say. The authors state the choice they make:

As yet experiment has not shown whether the refractoriness is relative or absolute. We will assume the latter and demonstrate that the difference is immaterial to our argument.

The claim is not that the simplification is true. The claim is that any net built on one assumption has an equivalent built on the other, so conclusions invariant under that equivalence hold either way.

Five assumptions

The physical assumptions:¹

(1) The activity of the neuron is an "all-or-none" process.

(2) A certain fixed number of synapses must be excited within the period of latent addition in order to excite a neuron at any time, and this number is independent of previous activity and position on the neuron.

(3) The only significant delay within the nervous system is synaptic delay.

(4) The activity of any inhibitory synapse absolutely prevents excitation of the neuron at that time.

(5) The structure of the net does not change with time.

Each supplies one piece of machinery:

• (1) makes "this neuron fired" a proposition. A neuron is either firing or not, with no intermediate value, which is the two-valuedness propositional logic requires. Continuous firing intensity would not admit this treatment.
• (2) turns the firing decision into a counting-against-a-threshold test, which is where conjunction and disjunction come from.
• (3) discretizes time. If synaptic delay is the only delay that matters, time is an integer counter of delay steps and \(t-1\) means one step earlier.
• (4) gives negation.
• (5) freezes the structure, which terminates the substitution argument in Theorem 1.

Assumption (5) also fixes the scope. A net whose structure never changes cannot learn, and the paper states this at the outset rather than arriving at it.

Four primitives

The paper writes its logic in Carnap's Language II, augmented with the dot conventions from Russell and Whitehead's Principia Mathematica, stated on p. 102: "the most appropriate symbolism is that of Language II of Carnap (1938), augmented with various notations drawn from Russell and Whitehead (1927), including the Principia conventions for dots."¹ Reading it gave me a headache, and getting comfortable enough to follow the proofs took me over two weeks. Everything below is in modern notation.

The dot carries two unrelated jobs depending on position. Flanking a connective, as in .≡., the dots are punctuation: brackets marking how far the equivalence reaches. Standing alone between two propositions, as in N₁(t−1) . N₂(t−1), the same character is conjunction. Position is the only disambiguator, and dropping one dot silently changes how far the connective reaches.

Paper	Modern	What it means
.≡.	↔	equivalence — the dots are brackets, not "and"
. between terms	∧	conjunction
∨	∨	disjunction, unchanged
∼	¬	negation
S(P)	one-step delay	P, one tick earlier — the z⁻¹ of DSP
z₁	t	the free time variable
Π, Σ	iterated ∧, ∨	AND as product, OR as sum, from Boole

Before any theorem, the paper shows four minimal circuits. Everything afterwards is a matter of composing them.

Circuits (b) and (c) differ in one number. At θ = 1 the neuron fires when either input fired; at θ = 2 only when both did. Disjunction and conjunction are one mechanism — count the active excitatory synapses, compare against θ — at two parameter settings.

The diagrams above are my redrawing. McCulloch and Pitts hold the threshold at two throughout and vary the number of endbulbs each axon terminates in: in Figure 1(b) each input arrives with two, so either alone reaches threshold; in 1(c) each arrives with one, so both are needed. I have held the synapse count at one and moved the threshold instead. What decides the neuron either way is the ratio between what the active inputs contribute and what the threshold demands.

Negation comes from assumption (4). One active inhibitory synapse vetoes the neuron regardless of what the excitatory side accumulated. In circuit (d), c2 firing silences c3 whatever c1 does, giving \(N_1 \wedge \neg N_2\). The veto is absolute rather than a large negative weight competing against the excitatory total.

Circuit (a) carries no logic at all. It brings two signal paths of unequal length into temporal alignment, which Theorem 2 requires.

Theorem 1: any net can be written as a formula

The first theorem goes from wiring to logic.¹

Every net of order 0 can be solved in terms of temporal propositional expressions.

A net has order 0 when it contains no circles — no chain of neurons looping back on itself. The paper defines order generally as the size of the smallest set of neurons whose removal leaves the net circle-free, so order 0 means the net was already acyclic. The peripheral afferents are the neurons with no axons synapsing onto them: the net's raw inputs. To solve a net is to write, for every non-input neuron, a formula fixing its firing in terms of the peripheral afferents alone.

The proof is substitution. Each neuron's firing condition is written in terms of its immediate upstream neighbours. Any neighbour that is not a peripheral afferent has a formula of its own, so substitute it in. Repeat. Because there are no circles, no neuron can appear anywhere upstream of itself, so every path back toward the inputs has finite length. A neuron may well be substituted more than once — one with two downstream consumers is expanded in both — but the depth bounds the recursion, so it terminates and what is left mentions only the inputs.

The class of formulas this produces is what the paper calls a temporal propositional expression, or TPE, defined by a recursion with three clauses: a bare predicate is a TPE; if \(S_1\) and \(S_2\) are TPEs then so are \(S S_1\) (the same expression shifted one step into the past), \(S_1 \vee S_2\), \(S_1 . S_2\), and \(S_1 . \sim S_2\); nothing else is a TPE. The third clause bounds the set, which is what permits the next theorem to quantify over all of them.

The \(S\) is a delay operator, the \(z^{-1}\) of signal processing. \(S(P)\) is a new proposition whose value at time \(t\) is \(P\)'s value at \(t-1\). It takes a proposition and returns a proposition, so it applies to an arbitrarily complicated subexpression without reference to that subexpression's internal structure. For a single named neuron, \(N_1(t-1)\) does the same work without the operator. \(S\) is needed when the thing being shifted is a whole compound expression, or a subnet known only by its behaviour.

Theorem 2: any formula can be built as a net

The second theorem is the converse.¹

Every TPE is realizable by a net of order zero.

The proof is induction over the three clauses defining a TPE. A bare predicate is realized by a single peripheral afferent. For the compound cases, assume the subexpressions already have nets realizing them, and combine those nets using primitives (b), (c), and (d).

Two subnets need not have the same depth. If one produces its answer after three synaptic delays and the other after one, feeding both into a combining neuron compares values from different moments and the combined net computes something else. The paper pads the shallower branch with plain delay stages — \(S^m\) and \(S^n\) "for suitable \(m\) and \(n\)" — until both arrive on the same tick.

Nets of order 0 and TPEs therefore describe the same set of behaviours.

Theorem 3: almost every sentence you can write

TPE is defined with a spare set of connectives: delay, disjunction, conjunction, conjunction-with-negation. Ordinary logical writing uses implication, equivalence, and negation applied anywhere. Theorem 3 covers the difference.

Take any sentence built however you want out of elementary statements of the form \(p(z_1 - zz)\) — "neuron \(p\) fired \(zz\) steps ago" — using negation, disjunction, conjunction, implication, and equivalence freely. Theorem 3 says such a sentence is a TPE, and therefore realizable, if and only if it is false when all its constituents are assumed false. Equivalently: if and only if its Hilbert disjunctive normal form contains no term made exclusively of negated terms.¹

What that rules out is not contradictions but their opposite: sentences that come out true when nothing fired. \(\sim p\) is one, as is any tautology. Each asserts something holds of a net whose peripheral afferents are all silent, and a net of order zero with silent inputs fires nowhere.

Contradictions are realizable. \(p . \sim p\) is the \(S_1 . \sim S_2\) clause with the same expression on both sides, and the net for it is a neuron that never fires. With Theorem 3 the translation runs in both directions and covers everything statable about who fired when.

What is actually being proved here

My reading, after two weeks with it: this is a circuit-synthesis result that happens to be about neurons. I went in expecting the origin of neural networks. What I found was a proof about wiring.

No error signal appears anywhere in this paper. No target output, no correction, nothing that changes in response to experience. The vocabulary does not exist yet.

Theorem 1: hand it a net, get back a formula. Theorem 2: hand it a formula, get back a net. Either way you supply the answer first. That is circuit realizability — given a specification, exhibit a circuit meeting it — and it settles what threshold units can express. It says nothing about how a net would find its own specification.

Assumption (5) rules that out from the start. Whoever builds the net sets the wiring and the thresholds, and they stay set.

McCulloch and Pitts do come back to changing synapses, at the end of the same section, among a group of equivalence theorems. They describe a mechanism for it: an axon terminal that cannot at first excite the next neuron becomes an ordinary synapse if it is active at a moment when that neuron fires. A rule for learning, written down in 1943. Theorem 7 then removes the need for it — Alterable synapses can be replaced by circles. Hand them a net whose synapses change and they hand back an equivalent net of fixed structure with a loop in it. Learning is absorbed into something assumption (5) already allows. That tells you what a fixed arrangement can imitate. It gives a net no way to change itself, and that gap is what the next fifteen years are about.

So the contribution is narrower than "they invented the neural network." A neuron here counts its inputs, compares the count against a threshold, and falls silent if an inhibitory synapse fired. Everything complicated comes from how many of them you have and how you wire them together — computation as a property of the arrangement, not of any individual part.

Nothing here accounts for the flukeprints. If the rule I picked up that afternoon could be written down, this paper proves a net could compute it. How an hour of looking installed it in me is a question about a structure that changes, and assumption (5) forbids that.

What 1943 got right, and what it left out

All-or-none holds for the spike. Cross threshold at the axon hillock and the action potential that follows has the same amplitude and shape every time, however hard you pushed it. Assumption (1) is standard physiology.

Everything before the spike is graded. Synaptic input arrives as postsynaptic potentials that sum across space and time until they reach threshold or fail to, and assumption (2) flattens that into counting active synapses. The output side loses more. Much of what neurons communicate rides on firing rate, which is continuous, not on whether one binary event landed in one tick. Retinal photoreceptors never fire action potentials at all — they signal entirely with graded potentials.

Absolute inhibition is the weakest of the five, and the authors said so. Most inhibition is graded, and enough excitation will outvote it. Two mechanisms come close to a real veto. Shunting inhibition opens chloride channels that sit near the resting potential, so rather than dragging the membrane down it short-circuits the excitatory current — division instead of subtraction. Chandelier cells target the axon initial segment, the exact spot where the spike starts, which lets them gate the output almost regardless of what the dendrites did.³

Each assumption takes something real and specialized and promotes it to a universal rule. The authors made that move explicit for inhibition, where they picked the absolute reading and argued the difference was immaterial to the argument. They did not audit the other four the same way.

Assumption (5) is not on the physiology list, and it is the one that gives way next. Real synapses change.

The second half of the question

If the wiring is fixed, someone had to know the answer before the net ran. For a logic gate that is no obstacle — you know what XOR is, and you can draw it. Nobody can draw a cat that way.

What is missing is a way for the net to change its own connection strengths according to what it got wrong. That means dropping assumption (5), replacing counted synapses with real-valued weights so the changes can be small, and defining the error.

The fifteen years in between are not empty. The sketch McCulloch and Pitts proved away — a terminal that becomes effective once it fires alongside the cell — is what Hebb states as a rule in 1949: a cell that repeatedly takes part in firing another has its efficiency in doing so increased.⁴ Nothing in that rule compares the output against what was wanted; it counts coincidences. The comparison arrives with Rosenblatt's perceptron in 1958.

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From Pixels to Tensors Series:

• Part 1: 2D Rendering Baselines
• Part 2: The 3D Graphics Pipeline
• Part 3: Symbolic AI
• Part 4: McCulloch–Pitts

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Footnotes