Co-operativity in neurons and the role of noise in brain

In view of some recent results in case of the dopaminergic neurons exhibiting long range correlations in VTA of the limbic brain we are interested to find out whether any stochastic nonlinear response may be reproducible in the nano scales usimg the results of quantum mechanics. We have developed a scheme to investigate this situation in this paper by taking into consideration the Schrodinger equation (SE) in an arbitrary manifold with a metric, which is in some sense a special case of the heat kernel equation. The special case of this heat kernel equation is the diffusion equation, which may reproduce some key phenomena of the neural activities. We make a dual equivalent circuit model of SE and incorporate non commutativity and noise inside the circuit scheme. The behaviour of the circuit elements with interesting limits are investigated. The most bizarre part is the long range response of the model by dint of the Central Limit Theorem, which is responsible for coherent behaviour of a large assembly of neurons.

💡 Research Summary

The paper tackles the puzzling observation that dopaminergic neurons in the ventral tegmental area (VTA) exhibit long‑range correlations that are difficult to explain with conventional neuronal network models. The authors propose a quantum‑mechanical framework in which the dynamics of neuronal assemblies are described by the Schrödinger equation (SE) defined on an arbitrary Riemannian manifold equipped with a metric tensor (g_{\mu\nu}). By recognizing the SE as a special case of the heat‑kernel (diffusion) equation, they reinterpret neuronal activity as a diffusion process on a curved space, where curvature modulates the effective diffusion coefficient and thus the spatial extent of signal propagation.

To make the abstract formalism experimentally tractable, the authors construct a dual equivalent electrical circuit that mirrors the SE. In this mapping, the wavefunction (\psi) corresponds to a node voltage, the Laplacian operator to a network of inductors and capacitors, and the non‑commutative canonical commutation relation (