A two-body eigenvalue problem for the resting cortical spectrum: line positions, weights, a dissociation edge, and the speed the screening has to have

Abstract

A resting cortical spectrum carries a narrow peak near 10 Hz, a weaker and equally narrow peak near 20 Hz, and a broad 1/f background running beneath and above them. Mechanistic accounts locate each rhythm in a relaxation time, so a second band requires a second circuit, the ratio of the two amplitudes is predicted by nothing, and the broadband component is a residual. We compute all of it from one object. Retaining the pair correlation in the first two rungs of the hierarchy and closing at the third gives a memory kernel and a quadratic pole equation whose two time constants are fixed by the measured centre frequency and linewidth. Dressing the connectivity kernel by the response of the surrounding tissue—the screening calculation, with the sign an excitable and metabolically driven medium demands, so that the reach is extended rather than shortened—gives a finite-range attractionV(r) =−V0K0(r/σ), and closing the pair operator on the same kernel fixes the coupling. The neurobiology this yields. Alpha and beta are not two rhythms but one bound pair of population transients—one excitatory, one inhibitory—in its ground and first excited internal state. Their amplitude ratio is then a property of one eigenfunction of the pair problem, 0.105 against 0.09 read off a resting decomposition, and both lines carry the same absolute linewidth: beta is weaker than alpha but not broader, a combination no two-circuit account produces. The named band boundaries are an artefact of where interneuron classes cluster; a second, slower inhibitory class carries its own pair with its own two lines in the delta and theta range. The screening that makes the binding possible cannot, however, be carried by the population relaxation that sets the rhythm: evaluated at finite frequency the loop gain rolls off at 3.2 Hz, far below the 10.7 Hz binding energy of the lower state, and the second bound state is lost. The level structure survives only if the polarisation runs through a channel withτs≤1 ms—the sub-millisecond AMPA transmission onto interneurons, or connexin-36 electrical coupling, and not the membrane time constant. The screening speed is then readable from the spectrum, because the amplitude ratio is a monotone function of it, and gap-junction manipulation should move the lower line upward in frequency rather than merely weaken it. Reported connexin-36 knockout spectra lose beta and spare theta, which is the pattern the account requires. Finally, the 1/ fbackground, which the pair sector cannot produce below the dissociation edge, is the coarse-graining spectrum of cortical patches: two-dimensional diffusive washout with the area weighting a sensor imposes givesχ= 3/2 with no free parameter. The same patch scale supplies the small parameter that controls the selection of the ring class. Five of the six predictions are testable on data already collected. Keywords: computational neuroscience, neural oscillations, resting-state EEG, alpha rhythm, aperiodic component, excitation–inhibition balance, bound states, Wilson-Cowan system, two-body problem, screening, correlation hierarchy, threshold networks, non-equilibrium physics, BBGKY hierarchy, cluster expansion.