Add grains of sand to a pile, one at a time. For a long while, nothing dramatic happens. Then, at some unpredictable moment, a single grain triggers an avalanche. The pile was always close to collapse. The last grain gets the credit it did not deserve.
This is threshold behaviour, and it appears everywhere: a rumour that suddenly goes viral, ice that holds firm and then fractures, a neuron that fires only when input crosses a minimum. In each case, input and output are not proportional. Small changes accumulate invisibly until the threshold is crossed, and then the system snaps.
The sigmoid function is the mathematician’s version: nearly flat on both sides, steep in the middle. It describes the dose-response curve of a drug, the probability of a binary outcome in logistic regression, and the activation of a neuron. Nothing is special about the threshold. It is simply the midpoint of a curve that was always going to be steep somewhere.
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