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The dog's two-neuron setup was simple. But a real dog's decision to bark isn't.
Smell and sound are just the start. Is the thing moving? How close is it? Has this smell appeared before? Is it night? Is the owner home?
A single neuron can't weigh all of that at once. But a group of neurons arranged in layers can.
The first layer takes the raw inputs: smell, sound, movement, distance. Each neuron combines a few of them and fires or doesn't.
The second layer doesn't see the raw inputs at all. It only sees what the first layer decided. Maybe one neuron in that layer combines something alive out there with it's close and loud and produces something like: this might be a threat.
The third layer takes that, weighs it against the time of day and whether the owner is home, and produces the final signal.
Bark.
No single neuron said that. No single neuron knew what a threat was. Each one just did its small calculation and passed the result to the next layer in line.