Store:QLMit18

11.3. Psychological functions

Now, we turn to the model presented in Section 10. A neural network is modeled as a compound quantum system; its state is presented in tensor product of single-neuron state spaces. Brain’s functions perform self-measurements modeled within theory of open quantum systems. (There is no need to consider state’s collapse.) State’s dynamics of some brain’s function (psychological function) $F$ is described by the quantum master equation. Its steady states represent classical statistical mixtures of possible outputs of $F$ (decisions). Thus through interaction with electrochemical environment, $F$ (considered as an open system) resolves uncertainty that was originally encoded in entangled state representing uncertainties in action potentials of neurons and correlations between them.

Entanglement plays the crucial role in generating consistency in neurons’ dynamics. As in Section 11.1, suppose that the quantum information representation is based on 0–1 $0-1$ code. Consider a network of $n$ neurons interacting with the surrounding electrochemical environment $\varepsilon$, including signaling from other neural networks. The information state is given by (32). Entanglement encodes correlations between firing of individual neurons. For example, the state (33) is associated with two neurons firing synchronically and the state (34) with two neurons firing asynchronically.

Outputs of the psychological function $F$ based biophysically on a neural network are resulted from consistent state dynamics of individual neurons belonging to this network. As was already emphasized, state’s evolution toward a steady state is very rapid, as a consequence of linearity of the open system dynamics; the off-diagonal elements of the density matrix decrease exponentially quickly.

12. Concluding remarks

Since 1990th (Khrennikov, 1999), quantum-like modeling outside of physics, especially modeling of cognition and decision making, flowered worldwide. Quantum information theory (coupled to measurement and open quantum systems theories) is fertile ground for quantum-like flowers. The basic hypothesis presented in this paper is that functioning of biosystems is based on the quantum information representation of their states. This representation is the output of the biological evolution. The latter is considered as the evolution in the information space. So, biosystems react not only to material or energy constraints imposed by the environment, but also to the information constraints. In this paper, biological functions are considered as open information systems interacting with information environment.

The quantum-like representation of information provides the possibility to process superpositions. This way of information processing is advantageous as saving computational resources: a biological function $F$ need not to resolve uncertainties encoded in superpositions and to calculate JPDs of all compatible variables involved in the performance of $F$.

Another advantageous feature of quantum-like information processing is its linearity. Transition from nonlinear dynamics of electrochemical states to linear quantum-like dynamics tremendously speeds up state-processing (for gene-expression, epimutations, and generally decision making). In this framework, decision makers are genes, proteins, cells, brains, ecological systems.

Biological functions developed the ability to perform self-measurements, to generate outputs of their functioning. We model this ability in the framework of open quantum systems, as decision making through decoherence. We emphasize that this model is free from the ambiguous notion of collapse of the wave function.

Correlations inside a biological function as well as between different biological functions and environment are represented linearly by entangled quantum states.

We hope that this paper would be useful for biologists (especially working on mathematical modeling) as an introduction to the quantum-like approach to model functioning of biosystems. We also hope that it can attract attention of experts in quantum information theory to the possibility to use its formalism and methodology in biological studies.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was partially supported by JSPS, Japan KAKENHI, Nos. 26247016and 17K19970. M.O. acknowledges the support of the IRI-NU collaboration, Japan .