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I'm afraid a neuron may not be a logic gate with synapses serving as inputs/outputs and behaving exactly the same on every activation.

It's well known it in fact isn't, otherwise learning would be impossible. Learning still isn't perfectly understood but one key characteristic is likely modulating synaptic strength (the weights mentioned). Also, yes, every cell and in particular neurons are very complex systems, although synapses themselves have various simplifying properties (specially along the axon, electrical communication really is the main method of communication).

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