20 — Mechanistic models Retrospective · exploratory

Five equations for one oscillation: neural masses and mean fields

Analysis basis 42 recordings, 1,630 two-minute windows, 22,820 electrode-windows per model; 4 flicker sweeps · dataset release 2026.09.16 · left out: electrode-windows with under 40 s of clean signal · last recomputed in analysis release 4.24 · current

The sections above describe the alpha oscillation; the learned models of §21 will represent it without being told what it is. This section sits between the two. A mechanistic model is a small set of differential equations for the neural populations believed to produce the oscillation — populations of excitatory and inhibitory neurons in cortex and thalamus, their coupling strengths, delays and synaptic time constants. Fit one to a stretch of EEG and its fitted constants — the gains, delays and time constants that give its equations their shape — become a description of that stretch in the vocabulary of physiology rather than of signal processing: not “the peak is 10 Hz and 8 dB tall” but “the resonance rings for about five cycles before it dies”. A model that fits the observed dynamics does not thereby identify the underlying mechanism: several different population models can produce the same spectrum, so throughout this section a constant is read physiologically only where the recovery test shows the data constrain it. Five such models — two linear, three nonlinear — were fitted to each of the fourteen electrodes of every one of the 1,630 two-minute windows of this archive: 22,820 electrode-windows per model.

The claim, in five lines

Observation: Fitted to half of a recording at one electrode, the two linear models predict the spectrum of the other half with R² 0.94 and 0.96 (archive-mean spectrum: 0.74; medians over 532 recording-electrodes), and the flicker resonance to within 0.29–0.34 Hz from unstimulated EEG alone at the electrodes the resonance was measured on. The alpha envelope does not require a limit cycle at any electrode of any recording: where the test has power (152 recording-electrodes with a peak of 12 dB or more) all 152 sit at the linear limit, and the Hopf density predicts held-out envelope power better than the linear one in only 37 of them, by a median gain of -0.0004 nats per sample. That test pools each recording; window by window with the eyes closed the answer changes (§22).
Status: 38 recordings, 532 recording-electrodes with enough windows; 4 flicker sweeps with a measured resonance and enough unstimulated windows; reference-table ABC with 60,000 and 30,000 prior simulations for the nonlinear models. A recovery test for every constant (simulate with known values, refit with the same code): all six L1 constants are identified; in L2 the delay and the combined loop gains x, y, z are partially constrained (and, barely, the intrathalamic gain and one rate constant), while the cortical and corticothalamic gains are not; in N2 one rate constant is identified, and the other rate constant and the connectivity are partially constrained; in N3 no individual coupling is recovered.
Interpretation: At this temporal resolution and under this envelope model, the data do not require a limit cycle; a lightly damped, noise-driven linear resonator is sufficient. Weak self-sustained dynamics obscured by filtering, by a weak alpha peak or by the chosen summary statistic are not excluded. Physiology at the level of single constants is limited to the coordinates the recovery test supports. Over whole recordings, nonlinearity shows in the harmonic, which Jansen–Rit predicts from spectral shape alone, and not in the amplitude statistics; window by window with the eyes closed it does (§22).
Alternatives: Differences in the fitted constants between session labels track eye closure, which sharpens the peak; several models fit equally well, so a good fit does not single out a mechanism, and within a model many combinations of constants reproduce the same spectrum.
Discriminating test: Drive it: the models disagree about the response to flicker at f₀/2 and about how the resonance moves with drive amplitude, and the rig of §18 can deliver both.

What is new here

None of the five models is new; three are decades old. What has not been done, as far as we could find, is the combination in this section.

Density. These models are ordinarily fitted once per subject or once per condition, to an averaged spectrum. Here each is fitted to every two-minute window of 42 recordings, which makes it possible to ask how reliable a constant is (does it belong to the session or wander from window to window?), how it moves through a session, and how it differs between states in one person.
Held-out prediction. Each model is judged on what it predicts about data it did not see — the other half of the recording, the resonance a flicker later measured, the harmonic it was never shown — rather than on how well it fits.
Five models, one signal. Linear and nonlinear, lumped and spatially extended, fitted to the same windows, so their answers can be compared.
Physiology and representation joined. The fitted constants are then used as an embedding, put through every test the learned models face in §21, and built into those models as inputs — a bridge that the two literatures rarely cross.

How the constants are estimated from these recordings

Every model is fitted to each electrode separately: the clean samples of one electrode over a window, with runs shorter than four seconds discarded, so a window yields fourteen vectors of constants per model and nothing is fitted to an averaged signal. All five models are scale-free, because the headset's gain is unknown; only the shape of the signal is fitted. Three estimation routes are used, each matched to what the model can produce.

Linear models (L1, L2): fit the formula to the spectrum. A linear model has a closed-form power spectrum, so its constants are found by least squares on the log spectrum (Welch, 4-s segments, 2–30 Hz; above 30 Hz the headset's roll-off and muscle dominate). The fit is robust to outlying bins and started from several initial guesses; the best is kept. L1 has five free constants, L2 ten.
Hopf model (N1): fit the distribution of the envelope. The question this model answers is whether alpha is self-sustained or noise-driven, and the answer lives in the statistics of the alpha envelope, not in the spectrum. The signal is band-passed ±2 Hz around its alpha peak, the envelope taken by Hilbert transform, and the envelope's power fitted by maximum likelihood to the model's stationary distribution — a normal truncated at zero whose single constant η says which side of the bifurcation the system sits on.
Nonlinear neural-mass models (N2, N3): simulate, then match. Jansen–Rit and the nonlinear corticothalamic model have no formula for their output, so they are fitted by approximate Bayesian computation. Sixty thousand (N2) and thirty thousand (N3) sets of constants were drawn from broad priors, each simulated for seventy seconds, and each simulation reduced to the same ten scale-free statistics the archive uses for real data (peak frequency, height and width, the 1/f exponent, α–2α bicoherence, 1:2 phase locking, harmonic ratio, subharmonic height, spectral entropy, theta/alpha). For each window, the 300 simulations nearest to it in those statistics are kept and their constants regression-adjusted toward the observed values, $θ^{*} = θ_{i} - {\hat{β}}^{⊤} (s_{i} - s_{o b s})$ . A recovery test — fitting simulated data with known constants — says which constants the data can actually determine. It is now run for all five models, and every table of constants carries the result as a status: identified (recovery ρ ≥ 0.8), partially constrained (0.5–0.8), model-coordinate only (the fit returns a number, but other combinations reproduce the data about as well) or prior-dominated (the estimate stays near its prior or a fit bound). Only identified and partially constrained constants are interpreted below.

Every electrode of every window yields one vector of constants per model (41 numbers in all, 574 per window); the electrode-by-electrode contrasts and scalp maps are under Electrode by electrode below. The tables below summarize the per-electrode fits by recording median over all electrode-windows (22,820; L1 fits the spectrum with a median R² of 0.89, L2 0.91). Reliability is the share of a constant's window-to-window variance that belongs to the recording (an intraclass correlation): near zero the estimate is noise, near one it is a property of the session. The last three columns compare recording medians between psychedelic-labeled and ordinary sessions (Mann–Whitney, false-discovery-rate corrected over all 41 constants); these contrasts inherit the eye-state confound of §16 in full.

The five models, and what their constants can and cannot say about this brain

A neural mass lumps a population into one average membrane potential and one firing rate; a mean field lets that average vary over the cortical sheet and propagate. “Linear” means the firing-rate sigmoid has been replaced by its slope at the operating point, which makes the spectrum a formula; “nonlinear” keeps the sigmoid and has to be simulated.

L1 · Linear damped oscillator linear · mass

The smallest model with a resonance: a second-order system driven by white noise, on a power-law background, the linear description of alpha that goes back to Lopes da Silva and colleagues⁶⁹.

\ddot{x} + 2 ζ ω_{0} \dot{x} + ω_{0}^{2} x = σ ξ (t) + g u (t)

P (f) = b f^{- χ} + \frac{A ω_{0}^{4}}{(ω_{0}^{2} - ω^{2})^{2} + (2 ζ ω_{0} ω)^{2}}, Q = \frac{1}{2 ζ}, τ = \frac{1}{ζ ω_{0}}

Its impulse response is a decaying sinusoid, $h (t) \propto e^{- ζ ω_{0} t} \sin (ω_{0} \sqrt{1 - ζ^{2}} t)$ — the perceptual echo of §19 — and its response to a periodic drive $u (t)$ is the Lorentzian resonance curve of §18.

Fitted constantsL1: six numbers per window

Constant	Meaning	Median [IQR] over recordings	Reliability	Recovery ρ	Status	Psychedelic-labeled	Ordinary	q
f0_hz	Natural frequency of the resonator, Hz	10.1 [9.69, 10.3]	0.23	0.96	identified	10.3	9.81	0.001
zeta	Damping ratio ζ; smaller means a sharper, longer-ringing oscillation	0.091 [0.043, 0.139]	0.20	0.98	identified	0.040	0.126	0.000
Q	Quality factor 1/2ζ: about how many cycles the oscillation rings for	5.46 [3.59, 11.6]	0.22	0.98	identified	12.4	3.96	0.000
decay_s	Time for a perturbation to decay by 1/e, 1/(ζω₀)	0.179 [0.122, 0.368]	0.18	0.98	identified	0.388	0.141	0.000
peak_gain_db	Height of the resonant peak over the background at f₀	5.04 [1.22, 11.3]	0.18	0.98	identified	12.3	1.45	0.000
aperiodic_exponent	Slope χ of the f^−χ background	0.701 [0.619, 0.819]	0.17	0.96	identified	0.633	0.785	0.013

What it says. At a typical electrode this alpha oscillation is well fit by a lightly damped resonator: median Q 5.46, so a perturbation rings for about 5 cycles before it dies, with a decay time of 0.18 s. These are the most reliable constants in the section (reliability 0.17–0.23), and the only model whose every constant is identified in the recovery test: the sharpness of the resonance is a property of a session, not of a moment. In psychedelic-labeled sessions the resonance is sharper — Q 12.39 against 3.96, damping 0.040 against 0.126, peak 12.3 dB against 1.5 dB over the background — which is also what closing the eyes does, so the contrast describes the sessions, not the psychedelic; the medians are over all fourteen electrodes, and the posterior electrodes' own values are in the scalp maps below.

L2 · Linear corticothalamic mean field linear · field

Robinson's model⁶⁵: excitatory and inhibitory cortex, thalamic relay and reticular nuclei, a loop delay. Each population low-pass filters its input through its dendrites, fires through a sigmoid, and cortical activity spreads as a damped wave:

[\frac{1}{α β} \frac{d^{2}}{d t^{2}} + (\frac{1}{α} + \frac{1}{β}) \frac{d}{d t} + 1] V_{a} (t) = \sum_{b} ν_{a b} ϕ_{b} (t - τ_{a b}), Q_{a} = S (V_{a})

[\frac{1}{γ^{2}} \frac{\partial^{2}}{\partial t^{2}} + \frac{2}{γ} \frac{\partial}{\partial t} + 1 - r_{e}^{2} \nabla^{2}] ϕ_{e} = Q_{e}, γ = v / r_{e}

Linearized about the resting state, with gains $G_{a b} = ρ_{a} ν_{a b}$ and $L (ω) = [(1 - i ω / α) (1 - i ω / β)]^{- 1}$ , the response to thalamic input is

\frac{ϕ_{e} (k, ω)}{ϕ_{n} (k, ω)} = \frac{G_{e s} G_{s n} L^{2} e^{i ω t_{0} / 2}}{(1 - G_{e i} L) (1 - G_{s r s} L^{2}) [k^{2} r_{e}^{2} + q^{2} r_{e}^{2}]}

q^{2} r_{e}^{2} = {(1 - \frac{i ω}{γ})}^{2} - \frac{1}{1 - G_{e i} L} [G_{e e} L + \frac{(G_{e s e} L^{2} + G_{e s r e} L^{3}) e^{i ω t_{0}}}{1 - G_{s r s} L^{2}}]

and the electrode spectrum is $P (ω) = A \int | ϕ_{e} |^{2} d^{2} k$ plus a floor. Alpha arises where the delayed loop $G_{e s e} L^{2} e^{i ω t_{0}}$ reinforces itself, near $f \approx 1 / (t_{0} + 2 / α + 2 / β)$ . Axonal range and velocity are fixed ( $r_{e}$ = 86 mm, $v$ = 8.6 m/s, γ = 100 s⁻¹); eight constants are fitted.

Fitted constantsL2: the corticothalamic loop

Constant	Meaning	Median [IQR] over recordings	Reliability	Recovery ρ	Status	Psychedelic-labeled	Ordinary	q
Gee	Cortical excitatory gain onto excitatory cells	11.1 [4.17, 19.7]	0.11	0.55	prior-dominated	6.58	17.8	0.003
Gei	Cortical inhibitory gain (negative)	-8.37 [-12.2, -6.18]	0.12	0.28	model-coordinate only	-6.06	-10.8	0.000
Gese	Loop gain cortex → relay → cortex	10.4 [6.92, 13.3]	0.06	0.42	model-coordinate only	13.5	7.50	0.000
Gesre	Loop gain cortex → reticular → relay → cortex (negative)	-7.62 [-8.64, -6.57]	0.05	0.41	model-coordinate only	-9.03	-7.07	0.007
Gsrs	Intrathalamic loop gain relay ↔ reticular (negative)	-0.626 [-0.780, -0.490]	0.09	0.51	partially constrained	-0.655	-0.626	0.629
t0_ms	Round-trip corticothalamic delay	75.5 [72.1, 83.8]	0.18	0.76	partially constrained	72.4	82.9	0.002
alpha_per_s	Dendritic decay rate α	89.7 [86.1, 97.5]	0.07	0.41	model-coordinate only	94.3	88.8	0.090
beta_per_s	Dendritic rise rate β	111 [97.7, 131]	0.07	0.55	partially constrained	116	109	0.331
x	Cortical stability coordinate G_ee / (1 − G_ei); instability at 1	0.902 [0.702, 1.01]	0.03	0.73	partially constrained	0.862	0.902	0.806
y	Corticothalamic stability coordinate (G_ese + G_esre) / [(1 − G_srs)(1 − G_ei)]	0.144 [0.017, 0.222]	0.03	0.79	partially constrained	0.220	0.034	0.000
z	Intrathalamic stability coordinate −G_srs αβ / (α + β)²	0.118 [0.095, 0.159]	0.09	0.71	partially constrained	0.137	0.114	0.607

What it says. Within this corticothalamic model the best-fitting round-trip delay is 75.5 ms (72.4 in psychedelic-labeled sessions, 82.9 in ordinary ones, q = 0.002), and the delay is only partially constrained by the data (recovery ρ 0.76), so the shorter loop in the labeled solutions is read with that caution, and it inherits the eye-state confound like every contrast here. The individual gains are not individually constrained: in simulation, spectra generated with known gains return the cortical and corticothalamic gains with ρ 0.28–0.42 (Gei, Gese, Gesre), because many combinations reproduce the same spectrum. So the inhibitory-gain constant in the labeled solutions (Gei -6.1 against -10.8; recovery ρ 0.28) is a model coordinate, not a measured inhibition. What the data do constrain are Robinson's combined loop gains (x, y, z: ρ 0.73, 0.79, 0.71); the cortical coordinate x sits at 0.86 in the labeled sessions against 0.90 (1 is the edge of instability). Reliability is lower than for L1 (0.03–0.18). The fit sets no condition on stability: 50 % of these fitted sets of constants describe an unstable system, and §22 (Result 1) refits the model inside its stable zone.

N1 · Hopf normal form nonlinear · mass

Every oscillator close to the point where an oscillation is born reduces to this one equation (Stuart–Landau), used as a canonical model of alpha by Freyer and colleagues⁷⁰, which makes it the minimal test of whether alpha is a self-sustained oscillation or a noise-driven one:

\dot{z} = (λ + i ω_{0}) z - | z |^{2} z + σ η (t)

Its stationary envelope power $s = | z |^{2}$ is a normal distribution cut off at zero,

p (s) \propto \exp [- \frac{(s - λ)^{2}}{2 σ^{2}}], s > 0; η = \sqrt{2} λ / σ

For $η ≪ 0$ this is an exponential — exactly what a linear resonator driven by noise gives — and for $η > 0$ a hump at $s = λ$ : a limit cycle. One number tells the two apart.

Fitted constantsN1: which side of the bifurcation

Constant	Meaning	Median [IQR] over recordings	Reliability	Recovery ρ	Status	Psychedelic-labeled	Ordinary	q
eta	Distance from the Hopf bifurcation in noise units, √2 λ/σ; the fit is floored at −6, the linear limit	-6.00 [-6.00, -6.00]	0.22	–	partially constrained: detected at ≥14 dB peak, not at 5.6 dB	-6.00	-6.00	0.013
f0_hz	Oscillation frequency	10.1 [9.69, 10.3]	0.23	–	identified	10.3	9.83	0.001
env_tau_s	Correlation time of the alpha envelope	0.190 [0.164, 0.217]	0.03	–	descriptive	0.215	0.168	0.000
amp_cv	Coefficient of variation of the envelope (0.52 for filtered noise)	0.564 [0.549, 0.575]	0.04	–	descriptive	0.561	0.564	0.397

What it says. η sits at or near the linear floor at almost every electrode of every recording, psychedelic-labeled or not (median -6.00). How much that shows depends on the strength of the alpha peak, which a recovery test measures: Stuart–Landau simulations with a known η, buried in 1/f background and fitted with the same pipeline, are detected as limit cycles (η ≥ 0) in every run when the peak stands 14 dB above the background, but in at most 17% of runs at 5.6 dB. Among the 152 recording-electrodes with a peak of 12 dB or more, 152 sit at the linear floor and the Hopf density predicts held-out envelope power better than the exponential in 37 (median gain -0.0004 nats per sample). At this temporal resolution and under this envelope model the data therefore do not require a limit cycle, and a linear noise-driven resonator is sufficient — the regime in which L1 and L2 are read. Weak self-sustained dynamics that filtering, a weak peak or this summary statistic would hide are not excluded. These are statements about whole recordings, which pool their eyes-open and eyes-closed windows; window by window with the eyes closed, the amplitude is held steadier than a linear resonator allows in 45 % of the electrode-windows with a strong alpha peak (§22, Result 9).

N2 · Jansen–Rit neural mass nonlinear · mass

One cortical column⁶⁶: pyramidal cells, excitatory and inhibitory interneurons. Each synapse turns firing rate into potential through a second-order kernel; each population turns potential into rate through a sigmoid $S (v) = 2 e_{0} / [1 + e^{r (v_{0} - v)}]$ :

{\ddot{y}}_{0} = A a S (y_{1} - y_{2}) - 2 a {\dot{y}}_{0} - a^{2} y_{0}

{\ddot{y}}_{1} = A a [p (t) + C_{2} S (C_{1} y_{0})] - 2 a {\dot{y}}_{1} - a^{2} y_{1}

{\ddot{y}}_{2} = B b C_{4} S (C_{3} y_{0}) - 2 b {\dot{y}}_{2} - b^{2} y_{2}

with $C_{1} = C, C_{2} = 0.8 C, C_{3} = C_{4} = 0.25 C$ and the EEG read as $y_{1} - y_{2}$ . Fitted by approximate Bayesian computation, as described above.

Fitted constantsN2: synaptic gains and rate constants

Constant	Meaning	Median [IQR] over recordings	Reliability	Recovery ρ	Status	Psychedelic-labeled	Ordinary	q
A_mV	Excitatory synaptic gain	6.48 [6.06, 7.37]	0.14	0.41	model-coordinate only	7.43	6.16	0.000
B_mV	Inhibitory synaptic gain	40.6 [34.0, 49.1]	0.11	0.60	prior-dominated	33.7	48.2	0.000
a_per_s	Excitatory rate constant (identified)	128 [124, 133]	0.06	0.94	identified	130	128	0.377
b_per_s	Inhibitory rate constant (identified)	51.8 [49.6, 55.5]	0.04	0.67	partially constrained	55.7	50.1	0.000
C	Connectivity constant scaling C₁…C₄	105 [101, 116]	0.04	0.50	partially constrained	102	105	0.237
p_mean	Mean external input, pulses/s	258 [221, 272]	0.11	0.49	prior-dominated	266	240	0.049
p_sd	Standard deviation of the input	121 [109, 127]	0.05	0.29	prior-dominated	107	124	0.011

What it says. In the recovery test only the excitatory rate constant a is identified (ρ 0.94) and the inhibitory rate constant b partially constrained (ρ 0.67); the connectivity C is borderline (ρ 0.50), and the gains and the input are model coordinates or prior-dominated. The inhibitory rate constant b is higher in psychedelic-labeled sessions (55.7 against 50.1 s⁻¹, q < 0.001): faster inhibitory synapses, which in this model sharpen the oscillation. The inhibitory gain B (33.7 against 48.2 mV) is prior-dominated and should not be read. L2 moves along an inhibition-related coordinate too, but there the coordinate is not individually recoverable, so the agreement is suggestive rather than confirmatory.

N3 · Nonlinear corticothalamic mean field nonlinear · field

The equations of L2 with the sigmoid kept and the delay explicit, the form used to study corticothalamic bifurcations³⁵, integrated numerically (thirteen constants: eight couplings ν, the delay, two dendritic rates, input mean and noise) and fitted by the same ABC procedure against 30,000 simulations. It shares L2's diagram. This is the model whose bifurcation set contains a period-doubling route, which is why it was built: the 2:1 head-movement locking of §11 asked for it.

Fitted constantsN3: thirteen couplings and constants

Constant	Meaning	Median [IQR] over recordings	Reliability	Recovery ρ	Status	Psychedelic-labeled	Ordinary	q
nu_ee	Coupling e → e	1.33 [1.25, 1.45]	0.04	0.22	prior-dominated	1.37	1.30	0.264
nu_ei	Coupling i → e	-6.39 [-7.33, -5.90]	0.18	0.37	model-coordinate only	-5.92	-7.06	0.000
nu_es	Coupling relay → cortex	4.47 [4.35, 4.64]	0.10	0.25	prior-dominated	4.40	4.55	0.065
nu_se	Coupling cortex → relay	2.95 [2.72, 3.18]	0.09	0.40	prior-dominated	2.66	3.08	0.000
nu_sr	Coupling reticular → relay	-1.74 [-1.81, -1.59]	0.07	0.44	prior-dominated	-1.81	-1.66	0.001
nu_re	Coupling cortex → reticular	1.22 [1.17, 1.25]	0.05	0.20	prior-dominated	1.24	1.21	0.092
nu_rs	Coupling relay → reticular	1.08 [0.879, 1.23]	0.16	0.39	prior-dominated	1.23	0.900	0.000
nu_sn	Coupling input → relay	2.57 [2.46, 2.80]	0.10	0.42	prior-dominated	2.72	2.50	0.007
t0_s	Round-trip delay, s	0.093 [0.092, 0.097]	0.06	0.74	partially constrained	0.093	0.094	0.464
alpha_per_s	Dendritic decay rate	63.9 [58.3, 69.4]	0.08	0.62	partially constrained	69.7	61.8	0.007
beta_per_s	Dendritic rise rate	528 [504, 553]	0.09	0.28	prior-dominated	506	537	0.007
phi_n	Mean input	3.32 [2.83, 3.54]	0.13	0.33	prior-dominated	2.89	3.47	0.004
phi_sd	Input noise	1.53 [1.40, 1.70]	0.11	0.36	prior-dominated	1.66	1.44	0.007

What it says. The fitted N3 solutions differ most between states along the relay-to-reticular coupling coordinate ν_rs (1.23 against 0.90, q < 0.001), the link that closes the intrathalamic loop. That constant is not uniquely recoverable: in the recovery test on held-out simulations near the data it returns ρ 0.39, and its posterior is barely narrower than its prior (contraction 0.12), so it is prior-dominated and the contrast is a statement about this model's coordinates, not about thalamic physiology. Only the delay and one synaptic rate constant are partially constrained; no individual coupling is. All fitted constants (CSV) By electrode (CSV)

What each model predicts, and whether it was right

A fit is cheap. Each model was therefore asked for the prediction it is built to make, on data it had not been fitted to.

Prediction 1The other half of the recording

Windows of each recording were split alternately; at every electrode the linear models were fitted to the mean spectrum of one half and scored on the other (R² of log power, 2–30 Hz; medians over 532 recording-electrodes, 38 recordings; the posterior electrodes P7, O1, O2, P8 in the last column).

Predictor	Fitted half	Held-out half	Held-out, posterior electrodes
L1 linear oscillator (5 constants)	0.959	0.945	0.940
L2 corticothalamic (10 constants)	0.972	0.960	0.966
Archive-mean spectrum, rescaled	—	0.744	0.749

Result. Both generalize within a recording almost as well as they fit, and the corticothalamic model's five extra constants buy one hundredth of R². Why it matters. A good fit is cheap here; it does not select a mechanism. Any claim about mechanism has to come from a prediction the models disagree about.

Prediction 2The flicker resonance, from unstimulated EEG

If alpha is a resonance, the frequency at which flicker drives it hardest is fixed by the same constants that shape the resting spectrum. Each model was fitted to a session's unstimulated windows at each electrode and asked where the driven resonance would be; the resonance was measured on the aggregate of O1, O2 and P8, so those electrodes are the fair test.

Sweep	min	Electrode	Measured f₀	L1	L2	Spectral peak
LSD experiment 016	11	O1	9.93	10.38	10.00	10.75
LSD experiment 016	11	O2	9.93	10.30	10.30	10.50
LSD experiment 016	11	P8	9.93	10.31	10.35	10.50
LSD experiment 016	40	O1	10.20	10.38	10.00	10.75
LSD experiment 016	40	O2	10.20	10.30	10.30	10.50
LSD experiment 016	40	P8	10.20	10.31	10.35	10.50
LSD experiment 016	70	O1	10.91	10.38	10.00	10.75
LSD experiment 016	70	O2	10.91	10.30	10.30	10.50
LSD experiment 016	70	P8	10.91	10.31	10.35	10.50
Workday recording 015	6	O1	9.72	10.02	9.65	10.00
Workday recording 015	6	O2	9.72	9.93	9.70	10.00
Workday recording 015	6	P8	9.72	9.91	9.70	10.00
Mean absolute error, Hz (O1, O2, P8)				0.34	0.29	0.41
Mean absolute error, Hz (all fourteen electrodes)				0.36	0.35	0.49

Result. Both models beat reading the peak off the spectrum at the electrodes the resonance was measured on, and the error grows away from them (all electrodes: 0.36 and 0.35 Hz), as it should if the resonance is a posterior property. Why it matters. This is the test that ties the passive archive to the driven experiments of §18; only 4 sweeps have both a measured resonance and enough unstimulated windows, so it is an illustration of the test rather than a result.

Prediction 3Limit cycle or filtered noise?

The Hopf model was fitted to the alpha envelope of one half of each recording at each electrode and scored, against the linear prediction, on the other half.

Result. The fitted η sits at the linear limit almost everywhere (median -6.00; the floor of the fit is −6); the nonlinear density predicts held-out envelope power better than an exponential in 50 of 531 recording-electrodes, by a median of -0.0004 nats per sample — nothing. Psychedelic-labeled and ordinary sessions both sit at that floor (median −6.00 in each), so a test between them is uninformative. Why it matters. Where the test has power — the 152 recording-electrodes with a strong alpha peak — the amplitude statistics are those of a lightly damped resonator rung by noise, not of a self-sustained oscillator; elsewhere the test cannot tell the two apart (recovery test above). That is the regime in which L1 and L2 are valid, and it is why the driven experiments of §18–19 see a linear system.

Prediction 4The harmonic, from spectral shape alone

A linear model cannot produce a harmonic. The nonlinear models were fitted, at every electrode-window, using six spectral-shape statistics only, and asked to predict four signatures of nonlinearity they had not been shown (rank correlation with the measured value across 42 recordings, on recording medians over electrode-windows / skill over predicting the archive mean):

Held-out signature	N2 Jansen–Rit	N3 corticothalamic
α–2α bicoherence	0.76 / +0.31	0.76 / +0.07
2α / α power ratio	0.88 / +0.12	0.63 / -4.72
1:2 phase locking	0.72 / -1.47	0.86 / +0.10
Subharmonic height (α/2)	-0.08 / -18.40	-0.27 / -8.28

Result. Jansen–Rit gets the harmonic right: told only how tall, wide and fast the alpha peak is, it predicts how strongly alpha is coupled to its own second harmonic (rank correlation 0.76, with positive skill). Neither model predicts the subharmonic. Why it matters. A sigmoid acting on a large oscillation produces exactly this, which supports reading the 2α peak of §11 as waveform shape rather than a separate oscillation. A rank correlation also arises if tall peaks simply have more of everything, so the skill column is the one to read.

Electrode by electrode

Everything above is summarized by recording medians over all electrode-windows. The same five models were also fitted to each of the fourteen electrodes separately, window by window: 22,820 electrode-windows, 22,617 of which yield all 41 constants (the rest have too little clean signal on that electrode). The estimation is the same at every electrode, so per-electrode values differ only in the input signal. One caution applies to L2: it is a model of the whole corticothalamic loop, so fitted to one electrode it describes that electrode's spectrum in the model's terms; it does not claim that each electrode sees a loop of its own.

Electrode	Alpha frequency f0 (Hz, L1)	Sharpness Q (L1)	Peak height above background (dB, L1)	L1 fit R²	L2 fit R²	Predicted flicker resonance (Hz, L2)	Cortical stability x (L2; instability at 1)	Distance from the bifurcation η (N1)	Distance to nearest Jansen–Rit simulation (N2)	Distance to nearest corticothalamic simulation (N3)
AF3	10.11	10.6	4.6	0.92	0.93	9.20	1.02	-6.0	0.91	1.77
F7	10.31	11.2	4.7	0.91	0.92	9.40	0.89	-6.0	0.93	1.79
F3	9.99	8.4	6.1	0.91	0.92	9.35	1.05	-6.0	0.91	1.61
FC5	10.19	11.8	5.0	0.88	0.90	9.50	0.88	-6.0	1.01	1.67
T7	9.84	6.2	2.1	0.56	0.71	9.65	0.77	-6.0	2.09	1.92
P7	10.52	5.4	2.3	0.89	0.91	9.50	0.94	-6.0	0.86	1.37
O1	10.32	6.0	5.3	0.91	0.92	9.60	0.86	-6.0	1.01	1.58
O2	10.33	5.3	7.0	0.91	0.93	9.85	0.83	-6.0	0.99	1.45
P8	10.34	4.9	6.1	0.90	0.93	9.80	0.87	-6.0	1.00	1.34
T8	10.23	4.8	4.9	0.89	0.91	9.50	0.71	-6.0	1.11	1.39
FC6	10.06	8.1	4.5	0.87	0.89	9.20	0.71	-6.0	1.13	1.68
F4	10.01	7.3	6.2	0.90	0.92	9.30	0.77	-6.0	1.06	1.59
F8	10.31	9.7	3.7	0.90	0.91	9.05	0.84	-6.0	1.05	1.89
AF4	10.03	10.1	4.6	0.90	0.91	9.25	0.88	-6.0	1.03	1.83

Medians over all windows. Sharpness Q is how many cycles a disturbance rings for; the peak height is the resonance above the 1/f background; the predicted flicker resonance is the frequency at which the fitted L2 loop responds most strongly to a drive entering the thalamic relay; the two distances measure how well the nearest simulated brain matches the electrode's statistics (smaller is better). Data (CSV)

The models hold across the scalp, with one exception. L1 fits every electrode well (median R² 0.87–0.92), except T7, the contact flagged as degraded in about a third of all windows (§04), where it falls to 0.56 and the nearest Jansen–Rit simulation is about twice as far away. The Hopf constant η, the distance from the bifurcation, sits at the linear limit on every electrode, so over a whole recording no electrode shows the signature of a self-sustained limit cycle; the conclusion of Prediction 3 holds at every electrode, with the window-level qualification of §22. Sharpness is highest over the front of the head (Q about 11) and lowest over the back (about 5), while the peak stands highest at O2, P8, F3 and F4. The four posterior electrodes agree with one another on sharpness (median window-by-window rank correlation between electrode pairs 0.80) and alpha frequency (0.53), and only weakly on the Jansen–Rit and corticothalamic constants (0.50 for the excitatory gain A, 0.27 for the loop delay t₀), in line with how poorly those constants are identified.

Psychedelic-labeled minus ordinary sessions, eyes-closed windows only, session medians. Red: higher in psychedelic-labeled sessions; blue: lower. A bold ring marks a difference that survives the false-discovery rate over every constant and electrode (q < 0.05).

Psychedelic-labeled against ordinary sessions, electrode by electrode. Session medians were compared separately for eyes-open and eyes-closed windows (14 psychedelic-labeled and 9 ordinary sessions have eyes-closed time), with the false-discovery rate controlled over every constant and electrode (574 tests per eye state). With the eyes closed, 107 differences survive; by model: L1 38, N3 28, L2 21, N2 13, N1 7. The clearest is sharpness: Q is higher in the psychedelic-labeled sessions at every electrode (by 3.2 to 10.1), significantly at AF3, F7, F3, P7, O1, O2, P8, FC6, F4, F8 and AF4. Alpha frequency is also higher at every electrode with the eyes closed, but by little (+0.04 to +0.29 Hz), and significantly only at FC6. With the eyes open, 220 of 574 differences survive; there the frontal electrodes show faster alpha (AF3, F3, FC5, FC6, F4, F8 and AF4) while the posterior ones do not, which most likely reflects a weak, noisy frontal peak in ordinary eyes-open windows rather than a frontal effect.

What this does not license. The eye-state split uses the classifier of §16, which reads alpha itself, so matching on it partly matches on alpha strength, and sharpness depends on how clearly the peak stands out. Sessions under protocol P2, whose eyes-open and eyes-closed blocks are timed by the clock, are the clean test. The constants most sensitive to eye state are also the ones that differ most, so the psychedelic contrast here is a description, not an effect.

The fitted constants as a representation

A vector of fitted constants is an embedding with named axes: 41 constants at each of fourteen electrodes, 574 numbers per window. Put through the read-out of §21, with whole sessions held out, they place 38 of 40 sessions on the correct side of the label (AUC 0.99, log loss 0.21), against 39 (0.15) for the best foundation model and 35 (0.26) for sixteen hand-built features. On the three open-ended targets of §21 they are the best single representation for the long-range memory of the alpha envelope (R² 0.29) and for alpha phase coupling with alpha strength removed (0.30), and a weak one for signal complexity (0.53). Why it matters. Five hundred interpretable numbers do nearly as well as the best foundation models on the label and better than all of them on two quantities that look beyond the spectrum, and they carry a name for what is read. They are also given to the learned models of §21 as inputs, with a consistent result: they add little to a model pretrained on tens of thousands of hours, which already contains what a resonator measures, and a great deal to a weak network or one that starts from nothing.

What the fitted constants say about LSD, DMT and ayahuasca

The fitted constants can be asked to tell the compounds apart directly. Each compound gets its own CatBoost read-out over all 574 constants — that compound against every other session, ordinary days included — with whole sessions held out, and the read-out's SHAP values on the compound's own windows say which constants, and which electrodes, it leans on. Nothing learned is involved: the numbers come straight from the per-electrode fits described above.

Read-out	Sessions	Held-out AUC	Log loss / base rate	Called at 0.5	Leans on (constants, summed over electrodes)	Electrodes
LSD	6	0.97	0.19 / 0.42	4 of 6	L2 · x, L1 · peak gain (dB), L1 · aperiodic exponent	T7, F4, O1
DMT	7	0.80	0.36 / 0.42	0 of 7	L1 · aperiodic exponent, L1 · peak gain (dB), N2 · b (per s)	F8, FC5, T7
ayahuasca	2	not estimable: a held-out fold has no session of this class			L1 · aperiodic exponent, L1 · peak gain (dB), N2 · a (per s)	T7, P7, O1

Where they differ. The LSD read-out ranks its sessions well (AUC 0.97) and leans most on L2 · x; the DMT read-out ranks less well (AUC 0.80), leans most on L1 · aperiodic exponent, and with seven sessions in forty its calibrated probabilities stay below 0.5, so at equal costs it calls none of them. Ayahuasca rests on two sessions, so its attributions are shown and its score is not estimable. With 15 sessions split 6/7/2, no blinding, and dose, timing and eye closure differing between compounds, these describe what each read-out leans on, not pharmacology.

Where they agree. A read-out that sets one compound against the rest spends whatever the compounds share on telling them apart, so the table shows similarities poorly. The fits themselves show them (recording medians over all electrode-windows):

Sessions	n	Sharpness (Q)	Peak height above background (dB)
ordinary days	26	4.0	1.5
LSD	6	15.0	13.0
DMT	7	7.2	8.6
ayahuasca	2	14.9	14.3

All three compounds sit above ordinary days on both: what the compounds share, in this model's terms, is a sharper and taller alpha resonance, and it is also what closing the eyes produces. The learned models' account of the same question is Result 15 of §21. Data (CSV)

Fifteen recordings after the freeze

The two psychedelic sessions retained for this section and the thirteen sessions of the prospective test (§21) were all put through the four predictions of this section with the models as published. Nothing was refitted to them.

Prediction 1, the held-out half of the spectrum. The L2 corticothalamic model predicts the other half of each recording with R² 0.80 to 0.98 in fourteen of the fifteen recordings (published median 0.960), against 0.42 to 0.87 for the archive-mean spectrum; the L1 linear oscillator is a few hundredths behind it, as on the cohort. The exception is Workday recording 024, where both linear models give R² of about zero at every electrode: its in-sample fits are poor too, its alpha is weak and it carries ten stimulation windows, so it is a recording the linear resonator does not describe rather than a failed prediction of a described one. The psilocybin session's post-ingestion half on its own gives 0.962 and 0.970.

Recording	Windows	R² L1	R² L2	R² archive mean	Hopf better (electrodes)	… with strong alpha	Hopf η	Peak gain (dB)	Exponent	L2 loop y
Psilocybin experiment 017	59	0.971	0.979	0.85	0 / 14	0 / 0	-6.0	14.3	0.60	0.42
Ayahuasca experiment 018	52	0.944	0.959	0.64	2 / 14	1 / 11	-6.0	9.0	0.58	0.29
Ayahuasca experiment 019	33	0.933	0.929	0.66	7 / 14	6 / 9	-2.8	7.8	0.72	0.16
Workday recording 027	21	0.954	0.970	0.87	0 / 13	0 / 1	-6.0	1.6	0.76	0.06
Daily-life recording 005	30	0.939	0.956	0.64	0 / 14	0 / 0	-6.0	1.1	0.71	0.08
Daily-life recording 004	19	0.943	0.956	0.87	0 / 14	0 / 2	-6.0	-0.2	0.73	0.03
Workday recording 020	20	0.937	0.944	0.61	0 / 14	0 / 0	-6.0	4.7	0.81	0.18
Workday recording 021	26	0.931	0.940	0.70	0 / 14	0 / 0	-6.0	4.8	0.76	0.15
Workday recording 025	30	0.924	0.939	0.65	0 / 14	0 / 12	-6.0	3.9	0.82	0.14
Workday recording 022	31	0.913	0.936	0.65	0 / 14	0 / 0	-6.0	0.8	0.82	-0.03
Workday recording 009	39	0.905	0.935	0.56	0 / 14	0 / 0	-6.0	1.2	0.81	-0.05
Workday recording 026	33	0.903	0.925	0.86	0 / 13	0 / 4	-6.0	2.8	0.83	0.03
Workday recording 028	16	0.871	0.923	0.42	12 / 14	12 / 14	-4.9	0.8	0.63	0.01
Workday recording 023	17	0.760	0.801	0.54	11 / 14	8 / 9	-2.0	2.3	0.76	-0.05
Workday recording 024	8	-0.018	-0.001	0.54	1 / 14	0 / 0	-6.0	0.9	0.76	0.01

Held-out R² is the median over the fourteen electrodes of the log-power R² (2–30 Hz) of the other half of the recording. Hopf better: electrodes at which the Hopf model (N1) beats the linear prediction of the held-out envelope power; strong alpha: electrodes with an alpha peak of at least 12 dB above the background. η, peak gain, exponent and loop y are recording medians over electrode-windows. Data (CSV)

Prediction 2, the flicker resonance, scored against the sweeps the sessions carried. Ten of the new sweeps belong to eight of these recordings, so the resonance frequency predicted from unstimulated EEG can be set against a measurement the models never saw. On the seven well-fitted October sweeps (r² ≥ 0.6) the L2 model predicts the driven f₀ to a mean absolute error of 0.39 Hz (L1 0.54, the spectral peak 0.37), against 0.29 Hz on the five published sweeps; over all ten, poorly fitted sweeps included, 0.60 Hz. The two post-dose psilocybin sweeps are predicted to 0.11 and 0.01 Hz. The misses are the sweeps that did not fit: the pre-dose psilocybin sweep (r² 0.26) and Workday recordings 025 and 027 (r² 0.35 and 0.36, §18). The predictions sit on average 0.33 Hz below the measured values.

Predicted against measured resonance for the ten October sweeps (filled; faded where the sweep's own fit has r² below 0.6), by the three estimators; the dashed line is equality. Hover for the sweep. Data (CSV)

Prediction 3, the Hopf limit cycle against the linear resonator. At the recording level none of the three psychedelic sessions needs a limit cycle: the Hopf model wins at 0 of 14, 2 of 14, 7 of 14 electrodes in Psilocybin experiment 017, Ayahuasca experiment 018, Ayahuasca experiment 019 (in the second ayahuasca session 1 of its 11 strong-alpha electrodes). The two recordings where the Hopf model wins almost everywhere are ordinary office sessions, Workday recording 028 (12 of 14, η -4.9) and Workday recording 023 (11 of 14, η -2.0): the late office recordings whose alpha swings widely across the session, the same recordings that withdrew the amplitude-wander claim of §08. A limit cycle at the desk is the regime of §22 appearing in ordinary eyes-closed pauses, not a psychedelic signature.

The constants. On the constants that separated the classes above, the three new psychedelic sessions fall inside their classes: alpha peak gain 14.3, 9.0, 7.8 dB against an ordinary median of 1.5; the L2 loop constant y 0.42, 0.29, 0.16 against 0.03. Across the four classes, with 6, 7, 4 and 2 sessions, no constant separates them (Kruskal–Wallis p from 0.11 to 0.82); LSD against DMT differs on the quality factor Q (16.1 against 7.2, p 0.05) and the loop constant z (p 0.04), nothing else. Psychedelic against ordinary separates on 8 of the 9 constants shown.

Six model constants, recording medians over electrode-windows, by class; bar, the class median; black rings, the fifteen post-freeze recordings. Hover for the recording.

Within the psilocybin session, before and after ingestion, the quality factor rose from 9.4 to 13.5, the peak gain from 9.7 to 14.9 dB and the Hopf margin η from −6.0 to −3.5, while the exponent fell from 0.68 to 0.59 (p ≤ 0.002 for each, electrode-windows of one session not being independent); the post-ingestion part also holds more eyes-closed time, which is the obvious alternative reading.