Relevance of Quantum Mechanics in Circuit Implementation of Ion channels   in Brain Dynamics
  With an increasing amount of experimental evidence pouring in from neurobiological investigations, it is quite appropriate to study viable reductionist models which may explain some of the features of brain activities. It is now quite well known that the Hodgkin-Huxley (HH) Model has been quite successful in explaining the neural phenomena. The idea of circuit equivalents and the membrane voltages corresponding to neurons have been remarkable which is essentially a classical result. In view of some recent results which show that quantum mechanics may be important at suitable length scales inside the brain, the question which becomes quite important is to find out a proper quantum analogue of the HH scheme which will reduce to the well known HH model in a suitable limit. From the ideas of neuro-manifold and the relevance of quantum mechanics at some length scales in the ion channels, we investigate this situation in this paper by taking into consideration the Schr\"odinger equation in an arbitrary manifold with a metric, which is in some sense a special case of the heat kernel equation. The next important approach we have taken in order to bring about it's relevance in brain studies and to make connection with HH models is to find out a plausible circuit equivalents of it. What we do realize is that for a proper quantum mechanical description and it's circuit implementation of the same we need to incorporate the non commutativity inside the circuit model. It has been realized here that the metric is a dynamical entity governing space time and for considering equivalent circuits it plays a very distinct role. We have used the methods of stochastic quantization and have constructed a specific case here and see that HH model inductances gets renormalized in the quantum limit. 
Indranil Mitra, Sisir Roy