Derivation of a Schr\"{o}dinger Equation for Single Neurons Through Stochastic Neural Dynamics

Despite the prevalent view that quantum mechanics is irrelevant to macroscopic biological systems because of inherent noise and decoherence, this paper demonstrates that the electrical noise (Brownian motion) in neuron membranes gives rise to an `emergent' Schr\"{o}dinger equation involving a new neuronal constant $\hat{\hbar}$, fundamentally challenging the standard view of neuronal behaviour. This result could provide new insights into the underlying mechanisms of brain function, thus challenging existing paradigms in both quantum physics and neuroscience. A possible empirical test of this emergent quantum behaviour would be to look for quantum fluctuations in subthreshold neural oscillations.