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Now add more inputs. More neurons. More layers.
The dog also has memories: this smell last Tuesday turned out to be nothing. It has social instincts: other dogs nearby aren't barking. It has experience: barking at this hour never leads anywhere. Each of those is another signal, feeding into another layer, combining with everything else.
By the time the final neuron fires, or doesn't, the signal has passed through dozens of small decisions. Each one built on the last.
No neuron in the middle knows what a dog is. None of them know what barking means. They just add, multiply, and pass the result forward.
The understanding lives in the whole thing, not in any of the parts.