  The study of the human psyche has elucidated a bipartite structure of
cognition reflecting the quantum-classical nature of any process that generates
knowledge and learning governed by brain activity. Acknowledging the importance
of such a finding for modelization, we posit an approach to study brain by
means of the quantum-classical dynamics of a Mixed Weyl symbol. The Mixed Weyl
symbol is used to describe brain processes at the microscopic level and
provides a link to the results of measurements made at the mesoscopic scale.
Within this approach, quantum variables (such as,for example, nuclear and
electron spins, dipole momenta of particles or molecules, tunneling degrees of
freedom, etc may be represented by spinors while the electromagnetic fields and
phonon modes involved in the processes are treated either classically or
semi-classically, by also considering quantum zero-point fluctuations.
Zero-point quantum effects can be incorporated into numerical simulations by
controlling the temperature of each field mode via coupling to a dedicated
Nos\`e-Hoover chain thermostat. The temperature of each thermostat is chosen in
order to reproduce quantum statistics in the canonical ensemble. In this first
paper, we introduce a quantum-classical model of brain dynamics, clarifying its
mathematical strucure and focusing the discussion on its predictive value.
Analytical consequences of the model are not reported in this paper, since they
are left for future work. Our treatment incorporates compatible features of
three well-known quantum approaches to brain dynamics - namely the
electromagnetic field theory approach, the orchestrated objective reduction
theory, and the dissipative quantum model of the brain - and hints at
convincing arguments that sustain the existence of quantum-classical processes
in the brain activity. All three models are reviewed.

Neurons and Cognition;Quantum Physics