What Sets the Neural Resting Spectrum - Spectral computation from screened pair interactions

Abstract

A resting cortical spectrum contains a narrow alpha peak, a weaker and narrow beta peak, and a broad component with no resolvable peak above them. Mechanistic accounts locate each rhythm in a time — a loop transit, or the decay of perisomatic inhibition — so one loop delivers one band, a second band needs a second mechanism, and the relative amplitude of the two is predicted by nothing. This note computes the spectrum instead. Alpha and beta are taken to be two internal states of a single bound object: a pair consisting of one excitatory and one inhibitory population transient, held together because each regenerates the other as it decays.Solving the two-body problem for that pair gives discrete states below a dissociation edge and a scattering sector above it, and the whole spectrum follows. Four numbers go in — two relaxation times and two frequencies, all calibrated in earlier work. The rest comes out: the ratio of the two line amplitudes, |Ψ2(0)|^2/|Ψ1(0)|^2= 0.11, so that beta must be at least five times weaker than alpha in power; a common absolute linewidth of 3.98 Hz for both lines; a continuum edge at 20.5 Hz, against 2ω0= 20.4 Hz from an independent route; and a scattering sector that is flat to one per cent from 25 to 90 Hz, so that within the model the shape of the broadband component — including the exponent of its 1/fχ fall-off — is carried by the drive rather than by the tissue, a reading that is testable and not yet tested. Against a reference decomposition of adult resting EEG the computed beta-to-alpha ratio of 0.11 meets a reconstructed 0.09, and a power-law drive reproduces the measured background from the edge to 90 Hz; below the edge the model has no broadband weight at all. Pursuing that shortfall is what produces the note’s largest claim. The one-body response of the transport framework cannot supply it — normalised to the background at 1 Hz it overshoots alpha by a factor of 144, because it is resonant rather than flat — but a second, slower pair species can. Taking the dendrite-targeting inhibitory time scale, an order of magnitude above the perisomatic one, and changing nothing else, the same eigenvalue problem returns bound states at 1.96 and 5.51 Hz with an edge at 5.70 Hz: delta and theta. The bands of the resting spectrum then appear as one pair problem indexed by one time, with each class of inhibitory interneuron supplying two discrete lines and a continuum above them — perisomatic giving alpha, beta and gamma, dendrite-targeting giving delta, theta and a continuum running up through the alpha range. Below about 1.5 Hz the account is still empty. The amplitude prediction presumes a contact measurement operator, and is shown to survive smearing in the relative coordinate only up to about 0.2 σ — a margin the note needs and cannot presently discharge, though the dark-state selection rule is immune to the same objection. What makes this computation possible is that the pair interaction is no longer posited. Screening the cortical connectivity kernel by the response of the surrounding tissue — Debye’s algebra, run with the sign an excitable driven medium demands — yields V (r) = −V0 K0(r/σ) with σ = ℓ(1 − y)^(−1/2), y being the ratio of correlation generation to one-body relaxation. Three consequences for tissue follow. The reach σ is not the basket-cell arborisation width but a state variable set by gain over adaptation, so the manipulation that unbinds beta is reversible within a session rather than structural. The binding scale contains no length at all, which makes the invariance of alpha across brain sizes a consequence of conserved biophysics rather than of conserved geometry. And the collapse of screening at y = 1 is a description of the loss of local containment, with a spectral signature — bands proliferating beneath a fixed edge — that pre-ictal recordings can be checked against. Two standing objections are untouched: there is no small parameter at g ≈ 42, and the framework still does not compute an absolute frequency. Keywords: computational neuroscience, neural oscillations, resting-state EEG, alpha rhythm, aperiodic component, excitation–inhibition balance, bound states; two-body problem, screening, correlation hierarchy, threshold networks, non-equilibrium physics.