This study examined how dynamic visual backgrounds affect brain activity during a working memory task. Sixteen healthy adults performed a 2-back task while viewing either a stereoscopic dynamic scene containing depth cues and lateral motion (background-on) or a static background (background-off). EEG was recorded from the frontal pole sites (Fp1/Fp2). In the linear analysis, power spectral density (PSD) was estimated, but no clear differences between the two background conditions were observed in any frequency band. In contrast, the nonlinear analysis, which was based on Hilbert-derived amplitude envelopes, showed that the difference measures of Permutation Entropy, Sample Entropy, and Lempel–Ziv Complexity exhibited small, localized increases in complexity under the background-on condition, predominantly within the high-gamma range (101–130 Hz). The effect sizes were small, and the temporal and spectral patterns were not consistent across participants. These findings indicate that dynamic visual backgrounds exert little influence on frontal EEG amplitude but may induce subtle changes in the temporal complexity of the signal.

In this paper, we consider a simple kinematic model, which is a rotating disc on the edge of another fixed disc without slipping, and study the rotation angle of the rotating disc. The rotation angle consists of two parts, the dynamical phase Δ_d and the geometric phase Δ_g. The former is a dynamical rotation of the disc itself, and the geometric motion of the disc characterizes the latter. In fact, Δ_g is regarded as the geometric phase appearing in several important contexts in physics. The clue to finding the explicit form of Δ_g is the Baumkuchen lemma, which we called. Due to the Gauss-Bonnet theorem, in the case that the rotating disc comes back to the initial position, Δ_g is interpreted as the signed area of a two-sphere enclosed by the trajectory of the Gauss vector, which is a unit normal vector on the moving disc. We also comment on typical models sharing the common underlying structure, which include Foucault’s pendulum, Dirac’s monopole potentials, and Berry phase. Hence, our model is a very simple but distinguished one in the sense that it embodies the essential concepts in differential geometry and theoretical physics such as the Gauss-Bonnet theorem, the geometric phase, and the fiber bundles.