Zwanzig-Mori projection operators and EEG dynamics: deriving a simple equation of motion

We present a macroscopic theory of electroencephalogram (EEG) dynamics based on the laws of motion that govern atomic and molecular motion. The theory is an application of Zwanzig-Mori projection operators. The result is a simple equation of motion that has the form of a generalized Langevin equation (GLE), which requires knowledge only of macroscopic properties. The macroscopic properties can be extracted from experimental data by one of two possible variational principles. These variational principles are our principal contribution to the formalism. Potential applications are discussed, including applications to the theory of critical phenomena in the brain, Granger causality and Kalman filters.

💡 Research Summary

The paper introduces a macroscopic theory of electroencephalogram (EEG) dynamics that is grounded in the fundamental laws governing atomic and molecular motion. By employing the Zwanzig‑Mori projection‑operator formalism, the authors systematically separate the full microscopic Liouville dynamics into a subspace spanned by the observable EEG variables and a complementary subspace that contains all remaining degrees of freedom. This separation yields an exact equation of motion for the EEG signal that takes the form of a generalized Langevin equation (GLE):

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