22 — Brain states, transitions and phase diagrams Retrospective · exploratory

Where the equations change character, and whether the recordings go there

Analysis basis 42 recordings, 9,047 sixty-second windows every 20 s; two-minute windows for the measures that need them; eye state from blink waveforms · dataset release 2026.09.16 · left out: windows that overlap flicker stimulation; LSD experiment 016 from minute 95 (head-motion episode) · last recomputed in analysis release 4.26 · current

A model made of differential equations does more than fit a spectrum: it has places where its behavior changes in kind. A state that returns to rest after every kick can lose its stability and become an oscillation that sustains itself, a Hopf bifurcation; two states can come to exist side by side, with jumps between them; an oscillation can double its period. This section treats the recordings as a dynamical system of the whole head. It finds the collective variables (the wave modes of each band), asks which of them approaches an instability and names the instability, draws the phase portraits, identifies the brain states as regimes of a phase diagram and follows the transitions between them, builds a mechanistic network that crosses the same bifurcation, and asks the learned representations of §21 whether they can read the state and foresee the transition. The psychedelic sessions are the reason for asking. Each begins before the effects and ends near their peak, when the headset comes off, so if a compound carries the system across a boundary, the crossing should lie inside the recording.

What is asked, and of what

Four words carry the section. A collective variable is a pattern over the whole head whose amplitude is followed as one number: here the spatial wave modes of each band. An order parameter is a quantity that is zero, or small, on one side of a transition and not on the other: here the amplitude of a mode and, above all, how steadily that amplitude is held, which separates a noise-driven damped mode from a self-sustained one with the same spectrum. A control variable is what is turned to cross the transition: visual input, which the participant controls by closing the eyes, and whatever a compound changes, for which time into the session is the only stand-in. A bifurcation is a value of the control variable at which the attractor of the equations changes kind, and a phase diagram is the map of the regimes over the control variables. The equations in question are the five of §20, first one electrode at a time (Results 1 and 2), then coupled over the head (Results 14 to 16); what is new is that the fitted quantities are no longer read one at a time but through where they place the system relative to the places its behavior changes.

Result 1The five models as dynamical systems: where each one changes character

A differential equation has places where its behavior changes in kind, and a fitted model can be asked how far it sits from them. For each of the five models of §20 the equilibria, their linear stability and the attractor of the noise-free equations were computed, so that every fitted electrode-window reduces to one coordinate per model: a distance from that model's oscillatory instability. The table says what each model can do and whether its coordinate is constrained by these recordings; the tests are simulations with known answers.

Half of the published L2 fits are unstable systems. The L2 fits of §20 minimize the distance between the model spectrum and the measured one, with no condition on stability. The spectrum formula stays finite when a mode has crossed into growth and cannot tell a mode that decays at a given rate from one that grows at it, and the fit starts at 2 Hz, below which the slow mode lives. Of 22,662 fitted electrode-windows 49.5 % have at least one growing mode (46.8 % the zero-frequency mode, 6.5 % the alpha mode): a linear model in that state has no stationary spectrum to fit. Refitted inside the stable zone (the same objective, a penalty on the zero-frequency condition and a count of right-half-plane modes that rejects any step into an oscillatory instability) the median R² is 0.907 against 0.914, and 5.6 % of fits end on the constraint, where the unconstrained optimum is unstable. The stable fits are the ones used in this section; in them the alpha mode decays at a median of 5.5 per second (interquartile range 1.0 to 11.9).

Neither nonlinear model has a period-doubled regime to offer. Of the 60,000 Jansen–Rit simulations of the reference table 61 % have a limit cycle in their noise-free equations and the rest a fixed point; of the 30,000 corticothalamic ones 1.9 % oscillate. After an 80-second transient and a strict test (two return levels that alternate) 2 corticothalamic draws are period-doubled and no Jansen–Rit draw with a single maximum per cycle is. A line at half the alpha frequency therefore has no counterpart in the regimes these models visit for physiological constants.

Result 2The reference transition: closing the eyes

One transition in the archive has a known time. Seven sessions open with a block of about two minutes with the eyes open and one with them closed, their boundaries located from the record itself. Closing the eyes raises posterior alpha power 12-fold, and in the 6 sessions where both blocks lie in one file the rise from 10 % to 90 % of the step takes 1.6 s at the median (0.6 to 11.7 s). Every model's coordinate moves the same way, from deep inside the stable regime to the edge of the oscillatory instability: the L1 relaxation rate falls from 13 to 2.0 per second; the Hopf coordinate η rises from the linear floor to −0.7; Jansen–Rit goes from a probability of 0.00 that its noise-free equations oscillate to 0.71, with the growth rate of its alpha focus rising from −11.4 to −1.0 per second; the corticothalamic loop from −8.3 to −2.6. This is the yardstick for everything that follows: a change of regime in this system looks like a step of an order of magnitude within seconds, read the same way by five models with different equations.

Result 3The collective variables: wave modes of the whole head, and their names

Everything before this point describes one electrode at a time. A phase transition is a property of the whole system, so the collective variables come first. For each band the cross-spectral matrix of the thirteen electrodes (F4 left out: its contact fails in every session; common average reference) was pooled over the archive and decomposed into its eigenvectors. These are the spatial wave modes of the band: each is a pattern of amplitudes and phases over the scalp, and the connectivity between electrodes is the same object seen from the other side, since the cross-spectrum is the sum of the modes weighted by their power. A mode's pattern can be named by projecting it on the spherical harmonics of the head: front–back, left–right and up–down are the three dipolar patterns of degree one, and the literature has names for each.

Which modes are near an instability, by a linear model of the whole head. A linear model with delays (order 12, 94 ms) fitted to all electrodes at once in every sixty-second window has collective modes of its own, each with a frequency, a growth rate and a scalp pattern, and the pattern assigns each to a family. With the eyes closed in LSD sessions the front–back alpha family decays at 0.9 per second (a mode at zero would sustain itself), the left–right at 4.9 and the up–down at 7.3; in the other psychedelic sessions 1.4, 7.1 and 8.2. With the eyes open in ordinary sessions the order reverses: the front–back family is the most damped of the three (21.2 per second) and the up–down the least (11.8). The front–back mode is also the fastest with the eyes closed (10.66 Hz against 10.05 for the left–right mode). The theta and beta families are far from zero in every condition. So the only collective mode of any band that comes close to an instability is the front–back alpha mode, and the state of the eyes decides which of the three alpha modes is the least damped: a fact that Result 15 will turn against the homogeneous field model.

Result 4Which instability: a supercritical Hopf bifurcation of the alpha front–back mode

Each mode's complex amplitude was followed sixteen times a second through every recording, and in every sixty-second window the distribution of its squared size was fitted with the stationary density of the Hopf normal form driven by noise, which has one constant, η: the distance of the mode from its Hopf point, in units of the noise. Below the point the density is a decaying exponential (a noise-driven damped mode); above it the density has a hump at a finite amplitude, which is the limit cycle. The border is not chosen by eye. 1,302 windows were simulated from the linear model with delays fitted to recorded windows and sent through the same code: a linear system never reaches η above zero, and crosses −2.1 (the threshold for "near") in 5 % of windows. A mode is therefore oscillating where η > 0, near its Hopf point between the two, and at rest otherwise.

Which instability, and of which mode. Of the six modes followed, one leaves rest: the alpha front–back mode. With the eyes closed it is oscillating in 26 % of windows in LSD sessions, 37 % in the other psychedelic sessions and 13 % in ordinary ones, and near its point in a further 24, 18 and 8 %; with the eyes open it oscillates in 2.0, 1.1 and 0.1 % of windows. The alpha up–down mode comes near its point in psychedelic sessions (17 % of LSD eyes-closed windows, 20 % in the other psychedelic sessions, against 2 % in ordinary ones) and rarely crosses it. The alpha left–right mode and the theta and beta modes stay at the linear level everywhere (at most 1.8 % oscillating in any condition). With sessions as the units, the share of a session's eyes-closed windows in which the front–back mode oscillates is 0.52 at the median of the LSD-labeled sessions with enough such windows (five of the six), 0.20 over the 10 other psychedelic sessions and 0.02 over the five ordinary ones (psychedelic against ordinary, p = 0.190: with so few ordinary sessions the test has little power, and two ordinary recordings reach the regime too).

The instability has a name. The amplitude grows continuously from zero as η crosses zero, the density has a single hump above the point, and there is no coexistence of rest with a large oscillation that survives the linear yardstick (the fifth-order fit that allows it reports "coexistence" in 9 to 21 % of the simulated linear windows, so that reading is not used). That is a supercritical Hopf bifurcation of the alpha front–back mode: not a saddle-node, not a saddle-node on an invariant circle, not a period doubling, and not a double Hopf, since no second mode crosses with it. And it is a bifurcation of a whole-head pattern: the frontal-pole electrodes carry as much of the mode as the occipital ones (Result 3), and the phase portraits of Result 5 show the same cycle at every electrode.

What else happens as the mode crosses its point. Read against η of the front–back mode, the other order parameters trace a bifurcation diagram. The number of alpha modes occupied at once (a participation ratio) falls from 3.7 at the linear floor to 1.5 above the point: the band condenses into one pattern, whose share of alpha power rises from 32 to 75 %. The alpha harmonic near 21 Hz locks 2:1 to the mode (locking 0.10 at the floor, 0.41 above the point; a linear system gives 0.06), and it is locked in 99 % of oscillating windows against 5 % of linear ones: the harmonic is the waveform of the cycle, not a mode of its own. The front–back and left–right modes change from moving together to competing: their size correlation beyond what their coherence implies goes from +0.11 at rest (above the linear range of ±0.09) to −0.13 above the point, the sign of two modes drawing on one pool of gain, as in the theory of coupled laser modes. The frequency of the mode falls from 10.41 Hz near the point to 9.85 Hz well above it, an amplitude-dependent frequency, and the lagged (traveling) share of alpha rises from 3 to 22 %. Theta and beta power do not move with η (0.86 to 0.92 and 0.72 to 0.83 log units): the transition belongs to the alpha band and takes the other bands with it only through the harmonic. Two earlier results are the same facts seen differently: the front–back and sideways waves that §13 found phase-locked under LSD are the front–back and left–right modes locked beyond the point, and the spatial reconfigurations that §17 counts between minutes sit beside the condensation within a minute described here.

Result 5Phase portraits of the modes: a limit cycle of the whole head

A phase portrait draws a variable against its own rate of change, so that an oscillation becomes a closed curve. Here the variable is a mode's coordinate (the band-limited signal of the thirteen electrodes projected on the mode's pattern), the rate of change is divided by 2π times the band's center frequency so that a pure oscillation is a circle whose radius is the mode's amplitude, and each panel is four seconds of trajectory. A noise-driven damped mode wanders in and out of the origin; a limit cycle keeps a steady radius and leaves the center empty. In LSD experiment 011 at minute 20 (rest) the front–back mode is a tangle around the origin with a mean radius of 5.7 μV; at minute 75 (eyes closed, oscillating) it is an annulus of 42 μV with an empty center. The left–right and up–down modes grow in the same stretch (12.2 and 6.6 μV) but keep crossing the center: excited by the cycle, not on one. Drawn electrode by electrode (the mode's share of each electrode's signal), the same cycle appears at AF3 and AF4 as at O1 and O2, with radii that follow the pattern's loadings; T7 and T8, which carry little of the pattern, show it smallest. The theta and beta modes stay tangles in every stretch.

Result 6The unstable minutes: where they are, and what they look like in the raw EEG

"Unstable" needs saying carefully. In these windows it is the rest state that is unstable: η of the alpha front–back mode is above zero, so a small departure from quiet does not decay but grows until the mode settles on its limit cycle. The EEG is not erratic there; it is the opposite, a self-sustained oscillation holding a steady amplitude. In the stable windows the quiet state is the attractor and alpha is noise ringing a damped resonator. There are 917 such sixty-second windows (every 20 s, so they overlap) in 23 of the 42 recordings, forming 218 episodes that cover 451.0 minutes of recording: 423 minutes in the psychedelic sessions and 28 in eight ordinary ones. Ayahuasca experiment 006 alone holds 158 minutes in 51 episodes, the longest 18.3 minutes; outside it no episode lasts more than 6.7 minutes. Every window with its start and end minute, η, envelope coefficient of variation, front–back share and harmonic locking is in the downloads.

What the raw EEG looks like. The chart shows eight clean seconds of the thirteen electrodes in the window with the largest η of two recordings, beside the nearest eyes-closed window at the linear floor of the same recording, with the front–back mode's coordinate and envelope and the distribution of its envelope power over the whole minute. The honest thing to say first is that in the raw traces alone the two states can look much the same: in LSD experiment 011 the resting window at minute 84 carries as much alpha as the oscillating one at minute 79 (front–back share 0.75 against 0.80). In Resting baseline 002 the contrast is easier because the resting window at minute 29 is a low-alpha one, and the oscillating window at minute 17 is 10 Hz waves on every row from AF3 to AF4, front and back in opposite sign, with the slightly peaked waveform that produces the harmonic near 21 Hz.

How to tell them apart. The difference is in the amplitude statistics over tens of seconds, not in the waveform of any single cycle. At rest the envelope waxes and wanes all the way to near zero, the spindle-like bursts of a damped oscillator driven by noise; past the point it never visits the origin. The number for this is the coefficient of variation of the mode's envelope: a linear mode driven by noise gives 0.523 whatever its spectrum, the resting windows give 0.58 at the median, the oscillating ones 0.36. The same fact as a distribution: the squared envelope of a damped mode is exponential, most probable value zero; a limit cycle gives a hump at a finite power with the origin depleted, and η is the fitted position of that hump in units of the noise. Two more signs travel with it: the band condenses into the front–back pattern (1.5 alpha modes occupied against 3.6 at rest, which in the raw traces is the same wave on every electrode rather than alpha confined to the occipital pair), and the harmonic locks to the fundamental. So the rule for a new sixty-second stretch is: project the alpha-band signal on the front–back pattern, take the envelope, and ask whether its coefficient of variation is well below 0.52 and its squared distribution has a hump away from zero. The height of the alpha peak on its own does not decide it.

Result 7Brain states, transitions between them, and the phase diagram

A brain state, in the sense this section can give the words, is a regime of the phase diagram: which modes are at rest, which are near their Hopf point, which oscillate, and which relations between modes are locked. With one mode leaving rest, the states of this archive are three: the front–back alpha mode at rest, near its point, or oscillating, with the harmonic locked to it in the last. The regime of one minute predicts the next: a window at rest is followed by rest with probability 0.92, an oscillating window by oscillation with probability 0.63 (0.69 in the other psychedelic sessions, 0.43 in LSD sessions, 0.39 in ordinary ones); "near" is the least stable, going to rest in 46 % of cases and on to oscillation in 25 %. Of the 305 onsets of oscillation, 172 happen with the eyes closed both before and after, and of the 285 ends, 151: the system switches between rest and the cycle at the scale of minutes without any change at the eyes. The slow dynamics of all 29 order parameters together (a transfer operator built from time-lagged correlations, with a lag of one minute) give the same picture from the other side: three long-lived sets of windows holding 80, 16 and 4 % of the archive, the second being the eyes-closed psychedelic stretches, with timescales that grow with the lag, which is drift rather than stationary switching. This does not contradict §21, which finds no discrete clusters in the learned spaces: the states here are regimes of one mode's amplitude, while the embeddings are organized by eye state and recording.

The phase diagram against the control variables. Two things are known to control the regime: the state of the eyes, which the participant controls, and whatever changes over a psychedelic session, for which time into the session is the stand-in (every such session begins before the effects and ends near their peak). With the eyes open the oscillating regime is all but absent in every kind of session and every stretch of time: eye state is the first control variable. With the eyes closed the regime is present from the first 25 minutes of the LSD sessions (median share 0.67 over two sessions), that is, from that day's baseline, before the effects; in the ayahuasca and DMT sessions it grows with time, from 0.11 in the first 25 minutes (8 sessions) to 0.53 after two hours (four sessions); in ordinary sessions it is 0.00 at the median, with two recordings as the exceptions. So the compounds do not create the regime, which exists on ordinary days with the eyes closed; what they change is how much of the eyes-closed time is spent in it, and in the long ayahuasca and DMT sessions that share rises over hours.

Result 8The shape of a transition: a drift through the point, not a jump

The 118 transitions into the oscillating regime that happen with the eyes closed before and after, and that show a clear change of level, were aligned on the moment the smoothed size of the mode crosses the midpoint between its levels before and after, and set against 374 control moments drawn inside eyes-closed stretches with no transition within two minutes. The rise from 10 % to 90 % of the step takes 7.8 s at the median, and runs from under a second to half a minute (quartiles 0.56 to 27.1 s; 1.3 s in LSD sessions, 12.6 s in the other psychedelic sessions, 3.2 s in ordinary ones). Before the moment, the variance of the log size falls (Kendall's τ against time negative in 74 % of transitions, against 36 % of control moments; p < 0.001) and its one-second autocorrelation does not rise (42 % against 51 %; p = 0.246). Critical slowing down, the recovery from kicks slowing as a system approaches a bifurcation, would raise both the variance and the autocorrelation; flickering between two coexisting states would raise the variance. Neither is seen. A falling variance with no change of autocorrelation is what a mode shows when it drifts through a supercritical Hopf point under a slowly moving control: the noise-driven excursions shrink as the orbit takes over the amplitude. The shape agrees with Result 4's naming of the instability, and it says that the transition is not announced minutes ahead by the quantities that would announce a fold or a bistable switch.

Result 9Where the alpha system sits: far below the instability with the eyes open, at its edge with them closed

Eye state here is read from blinks, not from alpha, because the archive's eye-state estimate is itself built on posterior alpha and would make every statement about alpha circular. A blink is counted when it has the waveform of one (positive at the frontal pole, at least 60 μV, 50 to 140 ms wide, the same at both frontal-pole electrodes); a sixty-second window with three or more is eyes open, a window with none in itself or its two neighbors is eyes closed, and the rest are left unclassified. That gives 2,969 eyes-open and 2,309 eyes-closed windows outside flicker stimulation. The two classes are not equally sure: a blink shows that the eyes were open, and a hundred seconds without one are taken to mean that they were closed, which is safe at rest and less so at a screen, where a stretch of open eyes can pass without a blink that meets the criteria.

Eyes open, the alpha system is far below its instability; eyes closed, it is at the edge. With the eyes open 86 % of windows sit at the linear floor of η and 2 % lie beyond the Jansen–Rit Hopf point; the line is 13 per second wide. With the eyes closed the line is 1.9 per second wide, 39 % of windows lie beyond the Jansen–Rit Hopf point, and 12 % have η above zero at the median posterior electrode. η above zero means that the amplitude of the oscillation is held steadier than a linear oscillator driven by noise can hold it, whatever its spectrum. The test of that is a copy of the same signal with its phases randomized, which keeps the spectrum and removes everything else. In 600 windows drawn evenly across the range of alpha peak heights, eight such copies of each posterior electrode's signal (14,384 in all) reach η above zero in 0.06 %. Of the 386 electrode-windows of that sample with an alpha peak of 20 dB or more, 45 % of the real signals do, with an envelope coefficient of variation of 0.42 against 0.53 for the copies (a linear oscillator gives 0.523).

It depends on the session. Across the 20 sessions with at least ten eyes-closed windows, the share with η above zero runs from 0 % to 37 %. It is highest in Workday recording 013 (37 %), LSD experiment 016 (31 %), Ayahuasca experiment 006 (31 %) and Resting baseline 002 (23 %), and it is 1 % or less in five sessions whose alpha peak stands as high, 15 dB or more above the background: Ayahuasca experiment 005 (1 %), DMT experiment 007 (0 %), LSD experiment 015 (0 %), DMT experiment 003 (0 %) and Resting baseline 001 (0 %). The height of the peak therefore does not decide it.

A change of damping, not of drive. Between the eyes-open and the eyes-closed windows of one session, alpha power scales as the inverse of the relaxation rate with an exponent of 1.13 (median of 21 sessions; interquartile range 0.38 to 1.49). A noise-driven oscillator whose damping changes while its drive stays the same has an exponent of 1, and that is what the approach to a Hopf bifurcation from below looks like.

What this changes in §20. There the alpha envelope was tested once per recording and electrode, fitting on one half and scoring on the other, and required no limit cycle anywhere. That test pools the eyes-open and eyes-closed windows of a recording, and a mixture of two amplitudes has a broader envelope distribution than either. Taken window by window, the linear resonator remains the description of the eyes-open minutes and of most eyes-closed ones, and in a substantial minority of eyes-closed minutes the data do require an oscillation that regulates its own amplitude.

Result 10Two states, and switches between them that the eyes do not explain

If alpha sits at an edge it should be seen on both sides of it. Posterior alpha power, smoothed over one second, is bimodal in 24 of the 39 recordings with enough clean signal for the test (two Gaussian modes on a log scale more than two pooled standard deviations apart), and it switches between its low and high state in two ways. 762 switches straddle an eyelid or large-amplitude artifact: the eyes closing or opening. 1,591 happen inside an unbroken clean run with no such artifact within 3 s, and the peak-to-peak deflection at the frontal pole in the five seconds around them is 52.4 μV against 52.5 μV at control moments of the same runs (p = 0.33): the eyes did nothing the frontal electrodes saw. A linear process also wanders across a threshold, so each clean run was replaced twenty times by a phase-randomized copy with its own spectrum: the observed rate exceeds that of the copies in 35 of 39 recordings (beyond three standard deviations in 23), and the slow variance of log power exceeds theirs in every recording, by a factor of 1.03 to 1.55. The excess is 0.10 switches per clean minute in ordinary recordings, 0.32 in LSD sessions and 0.42 in the other psychedelic-labeled ones (labeled against ordinary, p = 0.001); the labeled sessions also hold most of the archive's eyes-closed time, where both states are within reach. Two states visited without any change at the eyes is what the bistable range of the quintic Hopf form produces, and it is also what slow changes of vigilance produce; the recordings do not tell the two apart.

Result 11What moves in the course of a session

Every psychedelic-labeled recording starts before the effects and ends near their peak, when the headset comes off, so time into the session stands in for the quantity that would control a transition. Each coordinate was followed every 20 s through every recording, and its slope taken within each session separately for eyes-open and eyes-closed windows, so that a change of eye behavior cannot pass for a drift. Sessions are the units of every test.

Eyes closed, every psychedelic-labeled session with enough data. The alpha line narrows in 14 of 14 sessions (sign test p < 0.001); where alpha is strong enough for its coherence time to be measured, the decay rate of that coherence falls in 11 of 14, so the narrowing is a slower relaxation and not only a steadier frequency, though frequency wander falls too (12 of 14). The phase lag between the posterior and the frontal electrodes turns negative in 12 of 14, the Kuramoto order parameter falls in 12 of 14 and the weighted phase-lag index rises in 12 of 14. The corticothalamic loop's growth rate moves toward zero in 11 of 14. Only four ordinary sessions have enough eyes-closed time for a slope, they are short, and their line narrows in two of them, so none of this can be attributed to the compounds as against time spent with the eyes closed.

LSD against the other compounds, eyes closed. In the LSD sessions with enough eyes-closed windows alpha frequency rises in 5 of 5 (+0.41 Hz per hour), alpha power falls in 4 of 5, Lempel–Ziv complexity rises in 5 of 5 and η falls in 4 of 5: a faster, weaker, narrower line whose amplitude is less regulated. In the other psychedelic-labeled sessions power rises in 7 of 9 and the Jansen–Rit probability of a limit cycle in 7 of 9: toward the instability, not away from it.

LSD, eyes open, against ordinary sessions. Here there is a reference: 16 ordinary sessions with enough eyes-open windows. Their line width does not drift (−0.03 log10 units per hour; 7 of 16 rise). In the LSD sessions it narrows by 0.47 log10 units per hour, a factor of 3.0 (5 of 6 fall; against ordinary p = 0.002, q = 0.032); the Kuramoto order parameter falls in 6 of 6 (q = 0.032); the posterior–frontal lag turns negative in 6 of 6 (q = 0.032); and the correlation length grows by 3.3 cm per hour against 0.4 (q = 0.042). A relaxation that slows while the correlation length grows is the textbook signature of a system moving toward a critical point, and with the eyes open the alpha system has room to move: it starts far below its instability. Over the eyes-open time of the LSD sessions the line narrows from 7.4 per second in the first third to 2.6 in the last, close to the 1.9 of the archive's eyes-closed windows, and the share of windows beyond the Jansen–Rit Hopf point goes from 6 % to 15 % (ordinary sessions: 14.9 and 15.4 per second, 0.1 % and 0.3 %). The amplitude does not become regulated on the way: η is above zero in 2.6 % of the eyes-open windows of the LSD sessions. In the other psychedelic-labeled sessions the eyes-open line narrows in 4 of 7 (−0.06 log10 units per hour at the median).

Not a matter of half-closed minutes. A minute counted as eyes open could hold a stretch with the eyes closed, and more of them late in a session. Two checks say the drift is not that. With the longest blink-free stretch of each window and its number of blinks in the model, the line width still falls by 0.44 log10 units per hour in the LSD sessions (6 of 6 fall) against 0.01 in the ordinary ones (p < 0.001). And in the windows with blinks spread throughout (at least six, and no stretch of more than 20 s without one), which leave two LSD sessions with enough data, the line narrows in both (−0.26 in LSD experiment 015 and −1.01 in LSD experiment 016) while the ten ordinary sessions with enough such windows do not drift (+0.00 at the median).

Ramps and steps. A logistic course was fitted to each drifting coordinate in each psychedelic-labeled session, with the blink class as a covariate. In LSD experiment 016 the line width takes 43 minutes to go from 10 % to 90 % of its change, centered on minute 47 of the 94 analyzed, and the phase lag 54 minutes centered on minute 39; in LSD experiment 011 the lag takes 101 minutes. Of the 86 session-coordinates that drift, 42 change over twenty minutes or more (the median is 19 minutes). 24 are fitted as a step of under five minutes, 18 of them better than a straight line; twelve of those 18 fall where the eye class changes or within three minutes of one of the jumps of Result 12, and the other six are single coordinates in single sessions (the Kuramoto order parameter in DMT experiment 012 at minute 119; alpha power in LSD experiment 002 at minute 77; the line width in LSD experiment 004 at minute 15; the posterior–frontal lag in LSD experiment 004 at minute 59; the growth rate of the N3 mode in LSD experiment 016 at minute 35; the posterior–frontal lag in Psilocybin experiment 014 at minute 58).

Two clocks. Part of the drift runs on a second clock: time since the eyes closed. In the seven sessions with at least two eyes-closed stretches, one of ten minutes or more, the posterior–frontal lag falls inside a stretch in 7 of 7 (−0.24 per hour, p = 0.016) and comes back when the eyes open: with that clock in the model the slope left over the session is −0.019 per hour at the median. It is largest in LSD experiment 015 (−0.12), LSD experiment 011 (−0.10) and DMT experiment 012 (−0.09). The sawtooth is plainest in Ayahuasca experiment 006, whose longest eyes-closed stretch lasts 62 minutes. The line width narrows on both clocks (−0.32 log10 units per hour inside a stretch, in 5 of 7 sessions; −0.08 over the session). A coordinate that relaxes over tens of minutes after each eye closure and resets at the next opening is a slow variable of the eyes-closed state itself, and any account of the psychedelic sessions has to subtract it first.

Result 12Jumps: the eyes, alpha by itself, and the candidates

A penalized search for moments at which the nine main coordinates change level together finds at least one in 12 of 15 psychedelic-labeled sessions and 3 of 21 ordinary ones (p < 0.001). Of the 22 jumps, 16 come with a change of 0.3 or more in the archive's estimate of eye closure. That estimate is built on alpha and moves whenever alpha does, so blinks are the independent witness. In eight of the 16, blinking changes in the matching direction, falling to two thirds or less of its rate as alpha rises or rising by half or more as alpha falls: an eye closing or opening, the transition of Result 2. In the other eight it does not: alpha changes level for minutes while blinking carries on as before. The remaining six jumps are changes of other coordinates.

• One session spends tens of minutes at a time with a self-regulated amplitude. Every recording was searched for stretches of ten minutes or more in which most eyes-closed windows have η above zero. There are three in 32 recordings, two of them in Ayahuasca experiment 006: minutes 200 to 235 (81 % of 98 windows, median η +0.67, coefficient of variation 0.37) and minutes 275 to 300 (84 % of 43 windows, median η +0.62, coefficient of variation 0.37); the other is ten minutes of Workday recording 013. In the episodes of Ayahuasca experiment 006 alpha is slower than in the rest of that session's eyes-closed time (9.70 and 9.90 Hz against 10.22), its peak is no higher (20.8 against 20.6 dB), and the posterior–frontal lag is at its largest (−0.49 and −0.47 against −0.27). The first begins within ten minutes: the share of such windows is 7 % and 13 % in the two five-minute stretches before it and 80 % in its first. At minute 231 the record is interrupted (only 33 s of that minute are clean); alpha comes back 0.55 Hz faster and η falls from +1.7 to −0.8 before recovering. The same session has stretches without a blink in which alpha is absent (minutes 155 to 160 and 165 to 170) or sits at the linear floor (minutes 25 to 30 and 75 to 90). So within one recording, with no blink to mark a change at the eyes, alpha is found on both sides of the Hopf point for tens of minutes at a time. What moved it is not recorded; over 5 hours with the eyes closed, vigilance is the first candidate.
• Elsewhere the regulated amplitude fades with time. In the LSD sessions the share of eyes-closed windows with η above zero is 12 % in the first third of the eyes-closed time, 5 % in the second and 3 % in the last, falling in three of the five sessions and rising in none; in ordinary recordings 20, 13 and 4 %; in the other psychedelic-labeled sessions without the one above, 5, 0 and 3 %. Under LSD the line narrows while the amplitude becomes less regulated, not more: nearer the instability by one coordinate and no nearer by the other, which is not what one oscillator moving along one axis toward a Hopf point would do.
• The line at half the alpha frequency in LSD experiment 016 is not a period doubling. §11 traced this line to a head movement. A period doubling would be a bifurcation of the alpha oscillation itself, so the line was tested again as one. It stands 0.12 log10 units above the background before minute 68, 0.39 between minutes 80 and 95 and 0.72 in the head-motion episode that follows. Its 2:1 phase locking to alpha at O1 is 0.08 while it rises (the 95th percentile of time-shifted copies is 0.04), no stronger than the 0.07 between the same two bands in minutes 20 to 38, before the line existed, and 0.02 in the episode, where the line is largest and the locking is gone. The coherence of that band with the headset's accelerometer goes from 0.06 to 0.26 to 0.35 (a neighboring band stays at 0.05). A period-doubled oscillation is locked to its fundamental by construction, so its line cannot be largest where the locking is absent; this line follows the head. Result 1 adds that neither nonlinear model has a period-doubled regime to explain it with.
• Alpha drops out and returns in an ordinary recording. In Resting baseline 002 the alpha peak stands 22 dB above the background. At minute 40 it falls (one of the jumps of the table above) and stays under 8 dB until minute 60; in the longest stretch of that time without a single blink, minutes 48 to 57, it stands 4 dB; afterwards it returns to 24 dB. Nine minutes at rest without a blink are unlikely with the eyes open, so this reads as alpha vanishing with the eyes closed and coming back, the ordinary signature of the edge of sleep, and a change as large as that of Result 2.

Result 13The complex-systems repertoire: near-critical throughout, nonlinear with the eyes closed, not chaotic

The standard instruments of the field were run on every two-minute window with at least a minute of clean signal (1,486 windows), each referred to surrogates of the same window where the measure allows it.

Criticality. The branching parameter m says how many events one event leads to; at m = 1 activity neither dies nor explodes. The conventional estimate from avalanches is biased upward when a few sensors see a large system and one source reaches several of them at once (it gave about 1.3 in §07, which set it aside), so the multistep-regression estimator, built for exactly that case, is used here. It gives m = 0.958 at the median (interquartile range 0.938 to 0.974; 4.5 % of windows at or above 1), an intrinsic timescale of 686 ms: close to critical and below it, the reverberating regime reported for cortex in vivo. It is the same with the eyes open and closed (0.955 and 0.961) and the same in psychedelic-labeled and ordinary sessions (0.955 and 0.957, p = 0.867). The long-range memory of the alpha envelope does differ: its detrended-fluctuation exponent is 0.71 in psychedelic-labeled sessions against 0.60 in ordinary ones with the eyes open (p < 0.001) and 0.73 against 0.69 with them closed (p = 0.008). So one marker of criticality moves with the session label and the other does not move at all.

Chaos. The 0–1 test returns 0.992 for the slow part of the signal and 0.997 for the whole of it, the value of a stochastic or chaotic signal, and returns 0.992 for surrogates with the same spectrum and nothing else: it does not tell these signals from filtered noise. The largest Lyapunov exponent differs from its surrogates in 4 % of eyes-open windows, about what chance gives, and in 13 % of eyes-closed ones. There is no evidence of low-dimensional chaos.

Nonlinearity. Time-reversal asymmetry, which a linear Gaussian process cannot have, differs from surrogates in 51 % of eyes-closed windows and 26 % of eyes-open ones (median z 1.88 against −0.28; higher with the eyes closed in 11 of 16 sessions), and a nonlinear forecast beats the linear surrogates in 13 % against 6 %. The eyes-closed signal is measurably nonlinear, in agreement with Result 9; the eyes-open signal much less so.

A phase diagram of the electrodes as spins. The maximum-entropy (Ising) model is the standard test of criticality in neural populations: the thirteen electrodes' band-limited signals are made binary (above or below their median every 250 ms), the pairwise model that reproduces every mean and every correlation is fitted exactly over the 8,192 patterns, and the model's heat capacity is computed as its temperature is varied. A recording whose heat capacity peaks at temperature 1 sits at the critical point of its own model. For the alpha band the peak lies at 0.88 in the psychedelic sessions (16 recordings) and 0.56 in the ordinary ones (24; p < 0.001), and the temperature follows the share of the recording spent with the eyes closed (rank correlation 0.63) and the share spent in the oscillating regime (0.74): the psychedelic recordings are nearer the critical point of this model because they are oscillating more of the time, the same fact seen through a different instrument. Theta gives 0.47 against 0.49 and beta 0.48 against 0.41: far from critical in every class. The energy landscapes have 6 to 5 basins in every band and class, and within the long recordings the temperature moves in no consistent direction between the first 30 minutes and after the first hour (6 of 11 psychedelic recordings up, 3 down).

Result 14A mechanistic network: a ring of Jansen–Rit columns with a gradient crosses the same bifurcation

The five models of §20 describe one electrode each. The question of this section is about a pattern over the whole head, so the models have to be coupled. The most direct way is a network of Jansen–Rit columns, the N2 model, placed on a ring around the head (72 columns, standard constants), coupled through their delayed pyramidal firing rates with a coupling that falls off with distance (range 4.3 cm, conduction speed 4.3 m/s on the scalp), and read through the same thirteen electrodes, the same common average reference and the same estimators as the recordings. The control is the columns' connectivity constant C, with the input held at 130 pulses per second, above the fold of the single column, so that only the alpha branch exists (with the input as the control the noisy columns fall onto the quiescent branch that coexists below the fold at 114 per second, which is why that version failed). A column alone crosses its Hopf point at C ≈ 133.

What the coupling and a gradient do. With coupling but no gradient the front–back and left–right families of the ring are degenerate: they cross their Hopf points together and the band does not condense into one pattern. A front-to-back gradient of the constant (C larger at the back by 10) separates them: the front–back family crosses at C ≈ 125 and the left–right at 130, so there is a range of C in which only the front–back mode oscillates, and the front–back mode is the faster of the two, as in the recordings. Taken across its line with noise and read as the recordings are, the ring reproduces the bifurcation diagram of Result 4: occupied alpha modes 4.1 → 1.8 (recordings 3.7 → 1.5), 2:1 locking of the harmonic 0.08 → 0.47 (recordings 0.10 → 0.41), competition between the two modes +0.00 → −0.09 (recordings +0.11 → −0.13), and a lagged share that rises. The uncoupled columns do not condense (occupied modes 5.9 → 5.3). The harmonic locking the ring produces is the waveform of the Jansen–Rit cycle, which is not sinusoidal; the Hopf-form ring tried first had no harmonic and put the front–back mode slower than the left–right one, and was set aside for that reason. What the ring does not reproduce is the co-movement of the two modes at rest (the positive excess correlation at the linear floor), which needs a shared drive the model does not have.

Result 15The field model with spatial modes: a homogeneous field cannot order the modes as the recordings do

The other way to couple is a field. In the linear corticothalamic field (L2) every spatial mode of wavenumber k has the same corticothalamic constants and a spectrum that depends on k only through the term (k r_e)², where r_e is the axonal range; k is geometry, and on a sphere the three dipolar modes share one value. The pooled spectra of the three alpha modes (0.125 Hz resolution, 2 to 30 Hz, 3,092 windows by class and eye state) were fitted with shared constants and one κ = k r_e per mode, inside the stable zone. The fits are good (R² 0.92 to 0.99) and return the family damping of Result 3 (LSD, eyes closed: −1.5, −5.8 and −6.8 per second). But the κ each mode needs changes with the state: with the eyes closed the front–back mode needs the smallest κ² (3.5 against 6.2 and 7.0 in LSD sessions, 2.2 against 4.3 in the other psychedelic sessions), with the eyes open in ordinary sessions the largest (11.5 against 2.4). Forced to one κ per mode across all conditions, as geometry requires, the fit makes the three modes degenerate (growth rates within 0.6 per second of each other in every condition) and the LSD eyes-closed spectra are fitted worse (R² 0.915 → 0.880). The reason is structural: for every fitted set of constants the alpha mode's damping rises monotonically with κ², so a homogeneous field orders its modes by wavenumber in every state, while the recordings reverse the order between eye states. Per session the same thing shows as the fit pushing the front–back κ² to its bound in 35 % of the ordinary eyes-open fits and never with the eyes closed. The splitting of the three alpha modes, and its reversal when the eyes open, is spatial inhomogeneity, which is what the ring of Result 14 needed its gradient for; the field with spatial modes is the model that says why a gradient is necessary.

Result 16The ring inverted onto the recordings: a phase diagram in the model's control space

With a second gradient, sides against ends (δ₂; positive where the sides of the head are more connected than the front and back), the ring has three controls and three measured quantities to match: the growth rates of the front–back, left–right and up–down families in each sixty-second window (Result 3). A positive δ₂ makes the left–right family the least damped and the front–back the most damped, which is the eyes-open order; a front–back gradient with a negative δ₂ gives the eyes-closed order. 3,904 eyes-classified windows were placed at the point of the ring's grid (2,016 linearizations) whose three rates are closest to their own, and the ring's Hopf surface became the phase boundary of a phase diagram in the model's own control space.

Eyes-closed psychedelic windows sit 2 to 3 units of C below the front–back Hopf line (medians), with a front-to-back gradient of 8 and a sides-against-ends gradient of -8; δ₂ is negative in 83 % of the LSD and 97 % of the other psychedelic eyes-closed windows, and the ring matches their three rates within 0.18 to 0.31 per second. Eyes-open ordinary windows sit 44 units below the line with δ₂ positive in 78 % of them, and the ring matches them poorly (median mismatch 4.3 per second; at its lowest connectivity the ring's damping does not reach the 21 per second of eyes-open alpha). Ordinary eyes-closed windows split: the resting baselines lie close to the line, the workday windows counted as eyes closed lie 26 or more below it, which is the earlier finding that a workday stretch without blinks is usually open eyes without blinking. Because a fitted linear model is stable by construction no window lands above the line; the oscillating regime shows as a distance of one or two units from it. Read as a phase diagram, the recordings occupy a thin shell just inside the ring's boundary when the eyes are closed, and a region far from it, with the opposite gradient, when they are open; the psychedelic sessions move minutes from the second region into the first, and along the shell, without crossing it by the ring's account.

Result 17Can the learned representations read the dynamical state?

The learned representations of §21 were trained without any of this in view. Whether they carry it is tested the way §21 tests everything else: each coordinate of this section becomes a target, a CatBoost read-out with fixed hyperparameters is fitted on the embeddings of four fifths of the session groups and scores the rest, and every window is scored by a read-out that never saw its session. Two numbers are kept. R² over all windows rewards telling sessions apart; R² within sessions, with the target and the prediction each centered on their session mean, rewards following the course of a session, which is what a transition is. The second is the one reported here.

The coordinates that are functions of the spectrum. The line width, the Jansen–Rit probability of a limit cycle and the growth rates of the L2 and N3 modes are computed from the power spectrum of the window, and they are read accordingly. Among the networks that were never given the fitted constants the best within-session R² is 0.83 (LaBraM, fine-tuned) for the Jansen–Rit probability, 0.64 (LaBraM, fine-tuned) for the line width, 0.66 (EEGPT, frozen) for the growth rate of the L2 alpha mode and 0.48 (EEGPT, fine-tuned) for that of the N3 mode; sixteen spectral features give 0.82, 0.62, 0.73 and 0.50. No network reads these appreciably better than the spectral features do: to a read-out, where a window sits relative to these instabilities is a statement about the height and sharpness of the alpha peak.

The coordinates that are not in the spectrum. The Hopf coordinate η and the coefficient of variation of the envelope say how steadily the amplitude is held, in which two signals with one spectrum can differ. Here the networks read more than the spectral features: 0.42 (EEGPT, frozen) against 0.17 for the coefficient of variation, and 0.54 (EEGPT, fine-tuned) against 0.39 for η (η leaves its floor only where the alpha peak is high, which is the part of it that spectral features can follow). The fitted constants contain both and return them at 0.67 and 0.74; the networks that were given the constants as inputs reach 0.38 to 0.60 and 0.39 to 0.67. Result 19 isolates this property with signals built for the purpose.

The coordinates between electrodes. The Kuramoto order parameter and the posterior–frontal phase lag are relations between electrodes, which neither a spectrum nor a description fitted to one electrode at a time can hold. The networks read them from the fourteen channels together: 0.68 (EEGPT, fine-tuned) for the order parameter and 0.46 (EEGPT, frozen) for the lag, against 0.15 and 0.11 for the spectral features and 0.22 and 0.31 for the fitted constants. The graph transformers trained on this archive reach 0.48 to 0.56 and 0.20 to 0.39. These are the coordinates that moved most consistently in the LSD sessions of Result 11.

What is not read. The line at half the alpha frequency (best within-session R² 0.10 without the constants, 0.20 with everything allowed); the branching parameter m (best within-session R² −0.01 without the constants, 0.01 with everything allowed); the fluctuation exponent of the alpha envelope (best within-session R² 0.06 without the constants, 0.12 with everything allowed); the coherence decay rate (best within-session R² 0.03 without the constants, 0.32 with everything allowed). For the branching parameter, the fluctuation exponent and the coherence decay rate the value in one window also says nothing about the value four minutes later (Result 18), so within a session they behave as estimation noise around a session level and there is little to follow. The half-frequency line is the one that has to be looked for directly, as Result 12 did.

Eye state without alpha. The eye class defined from blink waveforms, which owes nothing to alpha, is read with an area under the ROC curve of 0.88 to 0.97 by the networks trained on this archive, 0.89 to 0.93 by the foundation models, 0.91 by the fitted constants and 0.88 by the spectral features: every representation places a window on the right side of the archive's largest transition.

Result 18Can they forecast it?

Reading the present state is not prediction. For that the read-out is asked for the coordinate two and five windows later (4 and 10 minutes) from the embedding of the present window alone, and compared with persistence, the present value of the coordinate carried forward, which uses no embedding at all. A representation forecasts only if its squared error is lower than that of persistence beyond the session-bootstrap interval.

4 minutes ahead. Four coordinates are carried by persistence alone with an R² of 0.52 or more (the line width, the posterior–frontal phase lag, the Kuramoto order parameter and the eye class), and on these no representation beats persistence. For the other eleven persistence explains less (R² −0.71 to 0.30); on nine of them at least one representation does better, and what it returns is close to the level of the session: the highest within-session R² of any representation on any of them is 0.13. The network trained with neighboring windows as positives has the largest gain on five of the 15 coordinates.

10 minutes ahead. Two coordinates are carried by persistence alone with an R² of 0.61 or more (the Kuramoto order parameter and the eye class), and on these no representation beats persistence. For the other 13 persistence explains less (R² −0.67 to 0.42); on ten of them at least one representation does better, and what it returns is close to the level of the session: the highest within-session R² of any representation on any of them is 0.06. The network trained with neighboring windows as positives has the largest gain on seven of the 15 coordinates.

What it says. The embedding of one two-minute window is a better estimate of where a session stands than one noisy value of a coordinate, and no better than the value itself for the coordinates that are measured well: that is smoothing, not anticipation, and the network whose training pulls neighboring windows together does the most of it. No representation announces a change of regime before the coordinate has begun to move. The archive could hardly show one doing so: its transitions with a known time are eye closures made by choice, which nothing in the preceding EEG needs to announce, and its drifts are ramps that persistence already follows.

The brain states of Results 4 to 8, read by the same representations. Four targets per two-minute window: the share of the window in which the alpha front–back mode oscillates, its median distance η from the Hopf point, whether the window is oscillating, and whether a transition into the regime begins within the next two minutes. Reading the present state, the networks never given the constants reach a within-session R² of 0.35 to 0.45 for the oscillating share (foundation models; the best is EEGPT, fine-tuned) and 0.58 to 0.66 for η, against 0.22 and 0.36 for the sixteen spectral features; the fitted constants, which contain η, return 0.70 and 0.78. The oscillating regime is classified with an area under the curve of 0.91 to 0.95 by the networks and 0.81 by the spectral features. Whether a transition begins within the next two minutes is read with an area under the curve of 0.77 to 0.81, but the spectral features reach 0.81 and the fitted constants 0.80: what every representation reads is the present nearness to the point (a window near its Hopf point is the one that goes on to cross it), not an announcement that the spectrum does not carry. Forecasting the oscillating share four and ten minutes ahead beats persistence by at most +0.07 and +0.16 of the variance, and no gain's session-bootstrap interval excludes zero. The conclusion of Result 18 holds for the states themselves: recognition, not anticipation.

Result 19Same spectrum, different regime: a benchmark with known answers

The eyes-closing transition changes the spectrum, so a representation could read it from the spectrum alone. A bifurcation is a change in the kind of dynamics, and the clean test of whether a representation carries that is a pair of signals with the same spectrum and different dynamics. Such pairs were built and added to real windows: 102 eyes-open windows from 38 session groups, each given an injected oscillation at 10.3 Hz with the scalp pattern and phase lags of this archive's eyes-closed alpha, in two versions. A limit cycle (the Hopf normal form above its bifurcation) against its phase-randomized twin, which is a noise-driven linear oscillator with exactly that spectrum; an oscillator switching between rest and a large cycle (the bistable range of the quintic form) against its twin; a period-doubled oscillation against the same two spectral lines with unrelated phases; and as a control two phase-randomized copies of one signal, where there is nothing to find. Every representation that can embed a new window embeds both versions; a read-out fitted on four fifths of the session groups says, for each held-out background, which version is which. Chance is 0.5, and the background is identical within a pair.

A limit cycle against a noise-driven oscillator with the same spectrum. With the limit cycle well above its bifurcation (η = 3), 16 of the 16 representations that were never given the fitted constants tell the two apart beyond chance: the foundation models score 0.70 to 1.00, the graph transformers 0.94 to 0.97 and the two transformers trained on this archive 0.72 to 0.92. Close to the bifurcation (η = 1.5) 14 of 16 still do: the foundation models 0.46 to 1.00, the graph transformers 0.73 to 0.88, the transformers 0.52 to 0.65; the best is EEGPT, fine-tuned, at 1.00. The sixteen spectral features score 0.48 and 0.49, which is chance, as the construction requires. With the linear probe the same networks do somewhat better (close to the bifurcation the graph transformers reach 0.85 to 0.99), so the information is there in a simple form.

Bistable switching. 10 of 16 beat chance: the foundation models 0.54 to 0.86, the graph transformers 0.53 to 0.79, the transformers 0.47 to 0.63.

Period doubling. Telling a period-doubled oscillation from two spectral lines at the same frequencies with unrelated phases takes the phase relation between two frequencies. With the half-frequency component at half the amplitude of alpha, 5 of 16 representations beat chance with the CatBoost read-out (EEGPT, fine-tuned: 1.00; EEGPT, frozen: 0.98; REVE, frozen: 0.63; BIOT, fine-tuned: 0.63; and one more) and 11 with the linear probe (EEGPT, frozen: 1.00; EEGPT, fine-tuned: 1.00; REVE, frozen: 0.96; LaBraM, frozen: 0.81; and seven more); at a quarter of the amplitude, 2 (EEGPT, fine-tuned: 0.87; EEGPT, frozen: 0.82) and 7 (EEGPT, frozen: 1.00; EEGPT, fine-tuned: 1.00; REVE, frozen: 0.74; Graph transformer (nested), static graph: 0.70; and three more). The best network trained on this archive alone reaches 0.57 (Transformer (REVE's design)) with the CatBoost read-out and 0.72 (Transformer (REVE's design)) with the linear probe, against 1.00 and 1.00 for the best pretrained one, so a period doubling in a new recording would be looked for with a pretrained network or with the direct test of Result 12.

With the fitted constants. The fitted constants alone separate the limit-cycle and bistable pairs (1.00, 1.00 and 0.96), because the Hopf coordinate and the statistics of the envelope are among them, and the networks that were given the constants as inputs follow (close to the bifurcation 0.97 to 1.00; bistable 0.72 to 0.94). On the period-doubled pairs the constants are at chance (0.54 and 0.49): none of the five models describes a phase relation between alpha and half its frequency.

The controls. On the control pair the paired accuracies of the 28 representations run from 0.35 to 0.61 and one of them has an interval that excludes chance: that spread is the noise of the benchmark, and a difference smaller than it means nothing. The relaxation rate of a linear oscillator at fixed power does change the spectrum (it is the line width), and every representation reads it (R² 0.71 to 0.96), the spectral features included (0.92).

What it says. A network that reads the waveform can tell a self-sustained oscillation from a noise-driven one with the identical spectrum, which no spectral summary can. Among the networks never given the constants the best pretrained foundation models (EEGPT and BIOT) do it almost without error, the graph transformers trained on this archive come next, and the plain transformers trained on this archive fall behind near the bifurcation. That is the property a detector of a Hopf crossing needs. It is recognition of a regime that is present, not anticipation of one that is coming.

Result 20The regimes of the sessions recorded after the freeze

The statement written down before the sessions were read had three parts: with the eyes open every session should sit at rest; the ayahuasca session should fall inside the psychedelic sessions' eyes-closed range of oscillating shares; the ordinary sessions inside the ordinary range. The first version of this test, in the October update, used the alpha-based eye-state read-out, which labels the strongest-alpha minutes as closed and so selects for the oscillating regime; the blink witness has since been rebuilt for all fifteen recordings, and this result is scored on it. Two parts hold. With the eyes open every session is at rest: the oscillating share of blink-classified eyes-open windows is at most 0.02 in any session. Ayahuasca experiment 019 holds the limit cycle in 0.12 of its blink-closed minutes, inside the published psychedelic range of 0.00 to 0.57 (median 0.20); the two retained sessions sit above the published psychedelic maximum, Ayahuasca experiment 018 at 0.72 and Psilocybin experiment 017 at 0.64, with the highest Hopf margins in the archive (0.85 and 0.57) and stay probabilities of 0.93 and 0.89. The third part fails on the blink witness as it did on the read-out: the eight ordinary sessions with at least five blink-closed windows hold the limit cycle in 0.06 to 0.54 of them (median 0.39), against a published ordinary median of 0.00 (range 0.00 to 0.47, 10 sessions). The reason is no longer the eye class. The sessions recorded since September open with a deliberate eyes-closed block, which most published workday sessions lacked, and in that block an ordinary brain at a desk reaches the oscillating regime as readily as a resting one: the published ordinary reference was narrower than ordinary eyes-closed EEG is. The regime is recognized again in the new sessions, and the ordinary range of this section must be read as 0 to about 0.5 rather than near zero.

Windows of 60 s every 20 s; closed = no canonical blink in the window or its neighbors, open = three or more canonical blinks (Result 7). Sessions with fewer than five closed windows carry no closed-eye statistics. Data (CSV)

Result 21The dynamics of the alpha front–back mode by psychedelic class

The classes are told apart on several whole-session measures elsewhere on this site, the alpha frequency ladder of §17 and the wave-vector magnitude of §12 among them. If those differences are real they should show in the dynamics of the mode that carries the instability. Per session, from the eyes-closed windows, this result compares the mode's frequency in the limit cycle and at rest, its Hopf margin η, cubic gain and oscillating share, the locking of the first two alpha modes, the alpha–beta power competition, and the regime sequence (stay probabilities, dwell times, onsets) across the four classes.

The frequency ladder lives inside the limit cycle. The mode's frequency in its oscillating windows is 10.54 Hz under LSD, 10.43 under psilocybin, 10.26 under DMT and 10.09 under ayahuasca (Kruskal–Wallis p 0.005; LSD against DMT p 0.02), and it tracks the posterior alpha center frequency across the psychedelic sessions (Spearman ρ 0.75). LSD is the one class whose limit cycle runs faster than its own rest windows (rest minus oscillating -0.21 Hz against +0.18 for DMT and +0.20 for ayahuasca; p 0.27 across classes, so a direction and not yet a result). Two further measures separate the classes, both pointing at LSD: the locking of the first two alpha modes (0.32 against 0.15 for DMT, p 0.03; four classes p 0.03) and the alpha–beta power competition, weakest under LSD (0.18 against 0.25, p 0.03).

The Hopf margin and the oscillating share do not separate the classes. η is 0.14 for LSD against -6.0 for DMT (p 0.07; the DMT sessions are short, with about fifty closed windows each) and the oscillating share 0.52 against 0.18 (p 0.14); across the four classes p 0.28 and 0.49. η does not follow the frequency ladder, |k| or the harmonic locking across sessions (|ρ| below 0.3). Against the ordinary sessions the psychedelic sessions differ on the condensation of alpha onto its first mode (0.74 to 0.75 against 0.54, p 0.003), the cubic gain (p 0.006), the 1–2 locking (p 0.005) and the dwell time of oscillating episodes (1.00 min against 0.33, p 0.005).

Once oscillating, every psychedelic class stays. The regime transition matrices below are alike across the classes: from the oscillating state the next window is oscillating with probability 0.80 (LSD), 0.84 (DMT), 0.89 (ayahuasca) and 0.88 (psilocybin) against 0.73 for the ordinary sessions, and ayahuasca almost never falls straight back to rest (0.01). From rest, every class leaves through the near state rather than jumping.

P(next window is rest · near · oscillating | this window), consecutive blink-closed windows pooled over the sessions of a class; in brackets the number of transitions. Data (CSV)

The three new sessions, drawn. The portraits below show the mode in each regime of each new psychedelic session: a wandering radius at rest, a steady radius in the oscillating windows (33, 27 and 43 µV), the limit cycle of Result 4 in three further sessions and two further compounds.

What this result cannot do is carry a class-level claim on two psilocybin and four ayahuasca sessions; the LSD-against-DMT contrasts are leads at their stated p-values, and the one finding that survives the counts is that the frequency the classes differ on is the frequency of the limit cycle itself.

What the section says

The recordings contain one transition whose cause is known, made by the eyelids, and it carries the alpha system within seconds from a heavily damped state to the edge of a self-sustained oscillation. Looked at as a whole-head system, that edge belongs to one collective mode, the front–back alpha pattern, and it is a supercritical Hopf point that the mode crosses and recrosses at the scale of minutes while the eyes stay closed, drifting through rather than jumping. Everything else in the alpha band organizes around that crossing: the other modes fall silent, the harmonic locks, the two leading modes begin to compete. The psychedelic sessions do not create the regime, which the ordinary days also reach with the eyes closed; they spend more of their eyes-closed time in it, and in the long ayahuasca and DMT sessions that share grows over hours. Against that slow motion the coordinates of single electrodes drift the way Results 11 and 12 describe.

A mechanistic network that reproduces the crossing exists and is simple: columns of the Jansen–Rit model on a ring, coupled through delayed firing with a front-to-back gradient of their connectivity. A homogeneous field with fixed geometry cannot do it, which says that the gradient is the physics and not a convenience. Inverted onto the recordings, the ring turns every minute into a point in its control space and the recordings into paths just inside its boundary. The learned representations read where a minute sits and tell a self-sustained oscillation from a noise-driven one; none of them announces the crossing before it begins. What would make this more than description is in the last line of the panel above: a control variable that is measured rather than inferred, an ordinary reference for the eyes-closed hours, and a kick.