In artificial neural networks, "neurons" in each layer are generally each a linear combination of neurons from the previous layer, followed by a nonlinear function to avoid multiple layers being equivalent to a single matrix multiplication.
Cells have complex signalling pathways, where some protein binds to or phosphorylates another protein, which in turn acts on another protein, and so on. If protein X is activated by proteins A and B, then the activation of X can be considered a nonlinear function (because only unactivated X can be activated) of a linear combination of A and B. Nonlinearity can also come from proteins that require multiple types of activation, or proteins causing more production of other proteins. All these things are very common in cells.
There is no backpropagation in cell signalling, but it's well known that evolution without backpropagation is sufficient for artificial neural networks. This "evolution" is not just the evolution of entire organisms — it's also common for death or reproduction of individual cells to be triggered by external signals, and this can be considered a type of evolution of cellular signalling "neural networks".
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