Quantum mechanics provides the most accurate description of natural phenomena at and below the scale of small molecules, but classical mechanics adequately describes natural phenomena in larger scale systems, such as the scale of a neuron.
The integrated circuit is such a system: the behaviour of individual semiconductors are most accurately and precisely described using quantum mechanics, but the behaviour of the integrated circuit as a whole can be described--and predicted--by some relatively simple rules. While human neurons may be more difficult to describe due to their relative complexity, the mere existence of entangled phosphates does not make neurons unpredictable.
There are systems that generally abide by classical mechanics that have QM foundations like with your example. And then there are systems, like Schrodinger's cat, that have QM foundations but also have QM outcomes (non-classical systems).
Schrödinger's cat is a thought experiment, not a real experiment.
The possibility that neurons firing depend on entangled superposition states of other molecules is an example of a QM process (non-classical system).
Even though the neuron as a whole is at a large enough scale not to be in a QM state itself, the relevant function of it is in a probabilistic state. Whether the neuron fires or not is ultimately up to the unknown nature of QM.
In Fisher's proposed mechanism, when the first neuron fires, the connection is triggered and then the second neuron fires. One doesn't need QM to describe that system, just as one doesn't need QM to describe the function of an integrated circuit.
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