Inside an Artificial Neuron

Consider a neuron that scores whether a machine reading deserves inspection; its input vector contains vibration, temperature, age, and a missing-sensor flag.

The familiar formula separates evidence aggregation from the response rule. Inputs x enter as a vector of measured or learned values. Weights w scale each coordinate before combination. The contributions are added into one pre-activation w·x. A learned bias b shifts when the unit responds. Finally an activation φ maps the pre-activation to the emitted output.

The last box is the one beginners read as cosmetic. It is also the one with a measured price tag. In 2012 Krizhevsky and two colleagues changed nothing but φ and timed the result. The Figure 1 caption of their ImageNet paper reports: “A four-layer convolutional neural network with ReLUs (solid line) reaches a 25% training error rate on CIFAR-10 six times faster than an equivalent network with tanh neurons (dashed line).” The learning rates were tuned independently for each of the two networks. No regularization was used. So the six-fold gap belongs to the response function and to nothing else in the pipeline.

The rectifier's advantage was already on record. Glorot and two colleagues had reported it a year earlier, finding that rectifying neurons “yield equal or better performance than hyperbolic tangent networks”. They added a second, stranger measurement about what φ does to a layer's output: “after uniform initialization of the weights, around 50% of hidden units continuous output values are real zeros”. So the choice of φ sets both how fast the unit learns and how often it emits exactly nothing. The fifth box in the diagram is a modelling decision with numbers attached, not a finishing flourish.

Hold every other input fixed. A positive weight then makes the pre-activation rise when its own input rises, and a negative weight pushes it in the opposite direction.

This interpretation is local to the chosen representation and to the surrounding network. Feature scaling, interactions, later layers and correlated inputs can all overturn a simplistic “importance” story. The word doing the work in that sentence is *local*, and in 2017 a group of vision researchers put a number on it.

Take the 256 units of AlexNet's conv5 layer, trained on Places205, and multiply the whole representation by a random orthogonal matrix Q drawn uniformly from SO(256). That is what the Network Dissection paper did. The layers that follow can undo such a change exactly, so “each rotated representation has exactly the same discriminative power as the original layer”. Accuracy cannot tell the two apart. Legibility can. Bau and his coauthors write: “Denoting AlexNet conv5 as f(x), we find that the number of unique detectors in Qf(x) is 80% fewer than the number of unique detectors in f(x).” Intermediate rotations Q^alpha for 0<=alpha<=1 show interpretability declining smoothly as the rotation grows. This is not one brittle threshold. It is a continuous trade of readability for nothing at all.

The paper's conclusion is the sentence to carry into every weight inspection you ever do: “interpretability is neither an inevitable result of discriminative power, nor is it a prerequisite to discriminative power”. A coordinate is readable because of the basis training happened to land in. Rotate the basis and the function is untouched while four out of five readable units vanish. The meaning was never inside the coordinate.

Without a bias term, a linear response boundary must pass through the origin of the input space. The bias is what lets training place that boundary wherever the evidence demands. Drop b and the neuron responds according to w·x alone: the boundary is anchored at the origin, its placement is less flexible, the feature encoding may have to be bent to compensate, and the constraint is worth accepting only when it is intended. Keep b and the neuron responds according to w·x+b: the boundary can shift, the unit carries a baseline tendency, and that flexibility costs exactly one trainable parameter. That is why affine layers keep it by default.

A default is a decision, and two independently built large language models decided the other way. Google's PaLM paper, published in 2023, states it in two sentences: “No biases were used in any of the dense kernels or layer norms. We found this to result in increased training stability for large models.” The Allen Institute for AI made the same call in OLMo in 2024, and gave the same reason: “No biases. Following LLaMA, PaLM, and others, we exclude all bias terms from our architecture in order to improve training stability.”

Neither team forgot the bias. Both removed it on purpose, in production systems, and both named the thing they bought with it. That is the strongest argument for giving b its own name. It is a separable object with a separable cost, something you add for boundary placement or remove for stability at scale. Either way you should be able to say which of the two you chose.

The pre-activation is the affine result before the response function. The activation is the value after that function.

Debugging often requires inspecting both. A healthy pre-activation distribution can still produce poor activations if the response function saturates or clips most values. This is not pedagogical bookkeeping. One of the most widely deployed pieces of training machinery in deep learning is defined by which of the two it touches.

Batch normalization goes in one exact place. Ioffe and Szegedy said where in 2015: “We add the BN transform immediately before the nonlinearity, by normalizing x = Wu + b.” The pre-activation is the object being manipulated. The activation is downstream of the intervention. The numbers attached to that placement are large: BN-x5 reached Inception's 72.2% accuracy in 14 times fewer training steps, and the BN-Inception ensemble reached 4.9% top-5 error on the 50,000-image ImageNet validation set. The same arithmetic quietly retires the term from the previous section. Once Wu + b has been normalised, b can be dropped, its role passing to the learned shift parameter beta.

There is a second lesson underneath the first. In 2018 Santurkar and three colleagues at MIT re-examined the technique. They confirmed the effect and rejected the explanation: “There is a consistent gain in training speed in models with BatchNorm layers”, they report, yet “such distributional stability of layer inputs has little to do with the success of BatchNorm.” You can intervene on the pre-activation, measure that it works, and still be wrong about why. That is one more reason to know precisely which of the two values your chart, your hook or your fix is touching.

A short manual trace makes later matrix notation much easier. Write the input vector out in a fixed coordinate order. Multiply each coordinate by its weight and record the contributions separately rather than as a running total. Add the bias and write down the resulting pre-activation as its own named value. Apply φ exactly once. Then perturb a single input and compare the change in the output with that input's weight.

Run the trace twice more, and the three anchored results in this lesson become things you have felt rather than read. Once with b removed, as PaLM and OLMo removed it: the boundary snaps back through the origin and nothing else in the arithmetic moves. Once with the pre-activation normalised before φ, in the position Ioffe and Szegedy specify — “We add the BN transform immediately before the nonlinearity, by normalizing x = Wu + b.” — and watch b become redundant, its job handed to the learned shift parameter beta. Finally, swap φ from tanh to a rectifier. Notice how many of the units in a layer you have just set to exactly zero: after uniform weight initialization, around 50% of them.