The coexistence of the broad-band fluctuation and alpha rhythm of the brain dynamics is studied based on the zero-crossing property of the local electroencephalographic (EEG) recording in eyes closed and eyes open. A two-component zero-crossing scenario, consisting of a broad-band fractal and narrow-band rhythm components, is assumed. Scaling is found in the power law distribution p(tau) approximately tau(-nu) of the crossing time interval tau of the broad-band fluctuation. In alpha dominant brain state, the alpha rhythm interval L also exhibits scaling in the form of power law distribution: p(L) approximately L(phi). Our main result is the relationship nu + phi approximately 3 that characterizes the organization of these two prominent features of the brain dynamics. The possible role of self-organized criticality of punctuated equilibrium in this organization is discussed.