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) =−V0 K0(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 correlation kernel, 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. New in this version, the screening that makes the binding possible cannot 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 with τs≤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. Also new, the 1/fbackground, which the previous version could not produce below the dissociation edge and recorded as a failure, is the coarse-graining spectrum of cortical patches: two-dimensional diffusive washout with the area weighting a sensor imposes gives χ= 3/2 with no free parameter. The same patch scale supplies the small parameter the ring selection previously lacked. And the rhythm a sensor records is not the pair. An object 1.37 mm across cannot produce a scalp signal: 105independently phased patches would sum incoherently and fall two to three orders of magnitude short, so the alpha wave spanning several centimetres is the collective spatial phase of many local pair modes—the pair supplies the frequency, the envelope supplies the macroscopic form. Two consequences are sharp. The speed at which travelling alpha is reported to move is a phase velocity,v=fαλ, which covers the whole reported range of 0.1–1 m s^(−1) over 1–10 cm with no parameter and requires speed to rise linearly with wavelength at fixed frequency, where a conduction-delay account requires the reverse. And the envelope autocorrelation, which the linear theory fixes at 40 ms with no freedom at all through the Siegert relation, is measured to persist over 5–300 s: a discrepancy of 10^2to 10^4 that is the largest in the account and a parameter-free measure of the nonlinearity it does not carry. Seven of the eight tests require no new recording. 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.