Store:Khrennikov15

10. Connecting electrochemical processes in neural networks with quantum informational processing

As was emphasized in introduction, quantum-like models are formal operational models describing information processing in biosystems. (in contrast to studies in quantum biology — the science about the genuine quantum physical processes in biosystems). Nevertheless, it is interesting to connect the structure quantum information processing in a biosystem with physical and chemical processes in it. This is a problem of high complexity. Paper (Khrennikov et al., 2018) presents an attempt to proceed in this direction for the human brain — the most complicated biosystem (and at the same time the most interesting for scientists). In the framework of quantum information theory, there was modeled information processing by brain’s neural networks. The quantum information formalization of the states of neural networks is coupled with the electrochemical processes in the brain. The key-point is representation of uncertainty generated by the action potential of a neuron as quantum(-like) superposition of the basic mental states corresponding to a neural code, see Fig. 1 for illustration.

Consider information processing by a single neuron; this is the system  $S$ (see Section 8.2). Its quantum information state corresponding to the neural code quiescent and firing, $0/1$, can be represented in the two dimensional complex ${\mathcal {H}}_{neuron}$ Hilbert space  (qubit space). At a concrete instant of time neuron’s state can be mathematically described by superposition of two states, labeled by  $|0\rangle$,$|1\rangle$: $|\psi _{neuron}\rangle =c_{0}|0\rangle +c_{1}|1\rangle$. It is assumed that these states are orthogonal and normalized, i.e., $\langle 0|1\rangle =0$ and$\langle \alpha |\alpha \rangle =1$, $\alpha =0,1$. The coordinates  $c_{0}$ and  $c_{1}$ with respect to the quiescent-firing basis are complex amplitudes representing potentialities for the neuron $S$  to be quiescent or firing. Superposition represents uncertainty in action potential, “to fire” or “not to fire”. This superposition is quantum information representation of physical, electrochemical uncertainty.

Let $F$ be some psychological (cognitive) function realized by this neuron. (Of course, this is oversimplification, considered, e.g., in the paradigm “grandmother neuron”; see Section 11.3 for modeling of $F$ based on a neural network). We assume that $F=0,1$ is dichotomous. Say $F$ represents some instinct, e.g., aggression: “attack” $=1$, “not attack” $=0$.

A psychological function can represent answering to some question (or class of questions), solving problems, performing tasks. Mathematically $F$ is represented by the Hermitian operator ${\widehat {F}}$  that is diagonal in the basis $|0\rangle$,$|1\rangle$. The neuron $S$ interacts with the surrounding electrochemical environment $\varepsilon$. This interaction generates the evolution of neuron’s state and realization of the psychological function $F$. We model dynamics with the quantum master equation (24). Decoherence transforms the pure state $|\psi _{neuron}\rangle$ into the classical statistical mixture (30), a steady state of this dynamics. This is resolution of the original electrochemical uncertainty in neuron’s action potential.

The diagonal elements of ${\widehat {\rho }}_{steady}$ give the probabilities with the statistical interpretation: in a large ensemble of neurons (individually) interacting with the same environment $\varepsilon$ , say  $M$ neurons,$M\gg 1$ , the number of neurons which take the decision  $F=1$ equals to the diagonal element $p_{1}$.

We also point to the advantage of the quantum-like dynamics of the interaction of a neuron with its environment — dynamics’ linearity implying exponential speed up of the process of neuron’s state evolution towards a “decision-matrix” given by a steady state (Section 8.4).