Invented scenario Everything in this story is invented. It is here to give intuition, never evidence.

Dr Kabir Menon is three months into his anaesthesia training at a teaching hospital in Bengaluru. His patient this morning is Joseph D'Souza, a retired bus conductor of 58, who is having a knee replaced. The drug is propofol, given by a pump that estimates how much of it has reached the brain. The pump shows that estimate as a number, the effect-site concentration, in micrograms per millilitre.

Kabir raises the target slowly, in small steps, and asks Mr D'Souza to squeeze his hand after each one. At 2.5 he squeezes. At 2.8 he squeezes, more slowly. At 3.0 he does not. Kabir writes the number down.

Two hours later the operation is over, and Kabir lets the concentration fall. He asks again for a squeeze at each step. At 2.5 nothing happens. At 2.0 nothing. At 1.4 the hand closes, and Mr D'Souza opens his eyes and asks whether they have started.

Kabir shows the two numbers to his consultant, Dr Anjali Rao. He lost response at 3.0 and found it again at 1.4. Going down, he stayed under well past the point where he went under on the way up.

"His brain has two stable states," Kabir says. "Awake and anaesthetised. It is an attractor system with hysteresis. I read about it last week. It is called neural inertia."

Anjali has run this hospital's theatre lists for eleven years. She likes the idea and she does not trust the word. "Or the pump's estimate of what is in his brain is wrong," she says. "The number on the screen is a model of an average patient. If the drug reaches his brain more slowly than the model thinks, you would see the same gap with no attractor at all. What would tell the two stories apart?"

That question is the subject of this paper. Dynamical systems theory gives mind science a rich vocabulary: attractors, bifurcations, hysteresis, critical slowing down, metastability and criticality. Each word names something exact in mathematics. In papers about the brain and consciousness, each word is also used loosely, as if naming a pattern explained it.

Here is the answer in plain words, before the argument. A dynamical concept carries explanatory weight only when it predicts a signature that a rival explanation does not. An attractor predicts that a perturbed state returns, and that time spent in a state follows a particular law. A bifurcation with hysteresis predicts a gap between the way in and the way out that does not shrink when the change is made slowly. Critical slowing down predicts that recovery from small disturbances gets slower before a transition, but the same slowing can appear for other reasons. Metastability and criticality are the hardest to test, because the measurements that seem to show them are easy to produce in other ways.

The consequence for Kabir is practical. Neural inertia is a real hypothesis with real evidence behind it, but his two numbers do not test it. The consequence for mind science is broader. The vocabulary of dynamics should be used as a set of predictions, and each prediction comes with a control that could defeat it.

[AUTHOR NOTE: if you have watched an induction or emergence from anaesthesia, or have been anaesthetised yourself, a two-sentence first-person account of what the transition looked or felt like would ground this scene.]

The paper builds the case in steps. Part 1 says what an attractor is and what it predicts. Part 2 turns to bifurcations and hysteresis, and builds an instrument that separates a true bistable switch from a slow lag. Part 3 tests the early-warning signals that are meant to precede a transition. Parts 4 and 5 take on metastability and criticality, the two concepts most often linked with consciousness. Part 6 asks what the Kashmiri Śaiva doctrine of spanda, a pulse at the root of every state, can and cannot add.

Start with a ball in a bowl. Push it and it rolls back to the bottom. Push it harder, over the rim, and it rolls into the next bowl instead. That picture is most of what people mean when they say a brain state is an attractor.

The mathematics makes the picture exact. A dynamical system is a rule that says how a state changes from one moment to the next. The state can be anything that can be written as a list of numbers: the positions of a ball, the firing rates of a thousand neurons or the concentrations of chemicals in a cell. The set of all possible states is called the state space. The rule turns every point in that space into a direction of motion, so the whole space is filled with arrows, like wind on a weather map.

An attractor is a region of the state space that nearby states flow into and do not leave. The simplest is a fixed point, a single state where all motion stops, like the bottom of the bowl. The set of states that flow into a given attractor is its basin. A system with two attractors has two basins, divided by a boundary. On the boundary sits a balance point, unstable in the way a pencil on its tip is unstable.

The double well#

The standard model of a two-state system is called the double well. Its state is one number, x. The rule says x moves downhill on a curve with two valleys, one at x = −1 and one at x = +1, separated by a hill at x = 0. Add random kicks, noise, and the state jiggles around in one valley. Occasionally a run of kicks in the same direction carries it over the hill, and it settles in the other valley.

Two predictions follow. The first concerns how long the state stays in one valley before it hops. In 1940 the Dutch physicist Hendrik Kramers worked out the answer for a particle jostled by heat (Kramers 1940). The mean time spent in a valley grows exponentially with the height of the hill divided by the strength of the noise. Halve the noise and the wait does not double. It squares, roughly.

The second prediction concerns the shape of the waiting times. Each hop is set off by a lucky run of kicks, and the system has no memory of how long it has waited. That makes the waiting times follow an exponential distribution, the same law as the time between radioactive decays. Short waits are the most common and long ones fall off steadily. One mark of an exponential is that the standard deviation of the waits equals their mean.

A minimal worked case: measuring the hill#

I ran the double well with six levels of noise, using fixed seeds so the runs can be repeated. For each level I recorded how long the state stayed in a valley before crossing to the other. At a noise strength of 0.35 the mean stay was 283 time units. At 0.5 it was 36. At 0.6 it was 19.

Kramers' law says the logarithm of the mean stay should rise in a straight line against one over the noise strength squared. The slope of that line should be twice the height of the hill, which in this model is a quarter. So the slope should be 0.5. The fitted slope from my runs was 0.49. At the noise level of 0.45, I also checked the shape of the waits. Over 124 stays the standard deviation was 1.17 times the mean, close to the value of one that an exponential predicts.

This is a small result, and that is the point. The concept of an attractor has done work here. It predicted two things in advance: a straight line on a particular plot, with a slope fixed by the hill, and a particular shape of waiting times. Both could have failed. A system that switched on a timetable would give the wrong shape. A system with no hill would give no straight line.

What the brain data would need to show#

For a brain state to count as an attractor in this exact sense, three things need to be seen. Perturbations should decay: a brief push, from a sound or a magnetic pulse, should be followed by a return to the same pattern. Time in the state should follow a law that depends on noise and on the depth of the basin, and changing either should change it in the predicted direction. And the transition should be quick compared with the time spent in each state, because the system spends little time on the hilltop.

Each of these can be measured. Something close to the first has been tried: in 2005 Marcello Massimini and colleagues gave the brain brief magnetic pulses during waking and deep sleep, and found that in sleep the response stayed local and short-lived (Massimini et al. 2005). Whether that reflects a return to an attractor is a matter of interpretation. The second and third are rarely tested at all. Most papers that call a brain state an attractor show only that the state is stable for a while. Stability for a while is what any slowly changing system shows, and it is not evidence of an attractor.

A bifurcation is a change in the number or kind of attractors as some setting of the system is turned. The setting is called a control parameter. For the brain under propofol, the obvious candidate is the drug concentration. For a pair of moving fingers, it might be the speed of a metronome.

The simplest kind is called a saddle-node or fold bifurcation. Take the double well and tilt it, by adding a constant push c to the rule. As c grows, the left valley gets shallower and the right one deeper. At a critical tilt, the left valley and the hilltop meet and vanish together. A state that was sitting in the left valley now has nowhere to rest, and it runs to the right. For the double well this happens at a tilt of about 0.385, or 2 divided by 3 times the square root of 3.

Now reverse. Tilt the other way, slowly. The state sits in the right valley, which is now the only valley at first. Even when the left valley reappears, the state stays where it is, because the right valley is still there and deep. It drops back to the left only when the right valley vanishes, at a tilt of −0.385.

So the system's state depends on the path taken. Going up, it switches at +0.385. Going down, it switches at −0.385. Between those values, two stable states exist side by side, and which one you see depends on where you came from. This is hysteresis, from the Greek for lagging behind.

The confound: lag looks like hysteresis#

Back to Anjali's objection. A gap between the way in and the way out is the signature of hysteresis. But a gap also appears in any system that responds slowly to a change, if the change is made faster than the system can follow.

Picture a heavy door on a stiff spring, pushed open by a rising wind. If the wind rises quickly, the door lags behind. When the wind falls, the door lags again, on the other side. Plot door against wind and you get a loop, even though the door has only one resting position for every wind speed. There is no second valley, only slowness.

For propofol, the slowness has a name. The drug has to travel from the blood to the brain, and the pump estimates that delay from a model of an average patient. If Mr D'Souza's brain takes up the drug more slowly than the model assumes, the estimate on the screen will run ahead of his brain on the way up and behind it on the way down. A loop appears, from the pump's error alone.

The two explanations make different predictions about speed. A loop caused by lag shrinks as the change is made more slowly, because a slow change gives the system time to catch up. A loop caused by bistability does not shrink to zero, because the two valleys are there however slowly you go. Its width settles at the distance between the two fold points.

The instrument: a sweep-rate test#

That difference is the basis of the instrument for this paper. It is a test anyone with a model or an experiment can run. Sweep the control parameter up and back down at several speeds. Measure the width of the loop at each speed. Plot width against slowness.

I ran the test on two systems. The first is the tilted double well from above, which is truly bistable. The second is a monostable system with a single resting state that moves smoothly with the control parameter, but which relaxes slowly, like the heavy door. Both were swept from −0.8 to +0.8 and back, with the same small noise and fixed seeds. The speed was set by how many time steps the system was given at each of 81 settings, from 10 to 3,000.

The results separate cleanly. At the fastest sweep, both systems show wide loops: 1.14 for the bistable system and 1.32 for the slow monostable one. The monostable loop has the larger width, so a single fast sweep would point you the wrong way. As the sweep slows, the monostable loop collapses: 0.52, then 0.16, then 0.02 and finally zero. The bistable loop narrows only a little and then stops, at 0.76 to 0.78. The theory predicts a width of 0.77, twice the fold point.

The negative control. The monostable system is the control. It has no attractor structure at all, beyond its single resting state. A test that called it bistable would be broken. At fast sweeps, a test based on one loop would call it bistable, and that is exactly the mistake Kabir is at risk of making. Slowing down is what exposes it.

Where the instrument fails. The test has limits, and they matter for the brain. It needs sweeps slow enough for a lag to die out, and in a patient or an animal there may be no such speed that is also safe and practical. It assumes that noise is small. With large noise, the state can hop between the two valleys before either fold is reached, and the loop shrinks for that reason too, which mimics a lag. And it assumes that the system itself does not change during the sweep, which a living brain under a drug for hours may well do.

Back to theatre four#

The sweep-rate test shows what Kabir would need. A patient cannot be swept a dozen times. But the same logic was applied, with more care, in animals. In 2010 Eliot Friedman, Max Kelz and colleagues measured the drug doses at which mice and fruit flies lost and regained the righting reflex, which is a standard sign of anaesthesia in animals (Friedman et al. 2010). They found that the doses differed on the way in and the way out. In mice they measured drug concentrations in the brain itself, and the brain concentration at loss of the reflex was higher than at its return. That is the measurement that rules out the pump error.

They added a second test. Flies with a mutation in a gene called Shaker, which affects sleep, showed a collapsed loop, without obvious changes in how they took up or cleared the drug. In mice unable to make the signalling chemical noradrenaline, the loop was altered, and restoring that signalling in the nervous system brought it back to normal. The authors proposed a barrier between the waking and anaesthetised states, a tendency of the nervous system to resist a change of state, and called it neural inertia.

This is a case where the hysteresis concept carries weight. It predicted a loop that survives the removal of drug lag, and the loop survived. It predicted that changing the barrier, here by a gene, would change the loop, and it did.

Kabir's two numbers are not evidence of this. They are what both stories predict. Anjali's suggestion, when he brings it to her, is to look at the patient's brain activity on the monitor at both crossings, and to see whether the pattern in each state is the same whichever way it was entered. That is a second signature of two attractors, and it costs nothing to check.

The finger experiment found that recovery slows before a switch. That observation has a general form. As a valley grows shallow, the force pulling the state back to the bottom grows weak. A small push takes longer to die away. The system becomes sluggish, and it also wanders further, because the weak pull lets noise carry it further from the bottom.

This is called critical slowing down. It can be measured without pushing the system at all, from its natural fluctuations. Take a record of the state over time. Measure how similar each moment is to the one before, a quantity called the lag-one autocorrelation. Near a stable bottom, the state forgets quickly and the autocorrelation is low. As the valley flattens, the state forgets slowly and the autocorrelation rises towards one. The variance, the spread of the fluctuations, rises too.

In 2009 Marten Scheffer and colleagues gathered evidence that this rising autocorrelation and variance appear before sudden shifts in many systems: lakes turning murky, climate changes in the past, fish populations collapsing and epileptic seizures (Scheffer et al. 2009). The idea spread quickly to neuroscience and psychiatry, as a possible warning before a seizure, a relapse of depression or a change in conscious state.

A worked case: three ways to see a warning#

I ran three versions of the double well, each for 6,000 steps with fixed seeds. In the first, the tilt rises slowly towards the fold point, so that the valley really does flatten. In the second, the tilt stays fixed, far from the fold, but the noise becomes more correlated over time, like the changing weather. In the third, the tilt stays fixed and the noise is large, so that the state can be kicked over the hill without warning.

For each run I measured the lag-one autocorrelation and the variance in a moving window, up to the moment the state crossed the hill or the run ended. I summarised the trend with a standard rank measure called Kendall's tau, which is +1 if the measure only ever rises and 0 if it has no trend. Ecologists usually treat a tau above about 0.5 as a warning.

The approaching fold gave a clear warning. In the run shown, the autocorrelation trend was 0.69 and the variance trend 0.68, and the state tipped. That is the textbook case.

The changing noise gave a stronger warning and no transition. In the run shown, the autocorrelation trend was 0.89, higher than for the real fold, and the state never tipped. This is the false alarm. The measure cannot tell a flattening valley from a change in the noise that drives the system.

The large-noise run tipped with no warning. Its autocorrelation trend was 0.11. This is the counterexample: a real transition between two attractors with no slowing down beforehand. Peter Ditlevsen and Sigfus Johnsen made the same point for climate records in 2010 (Ditlevsen and Johnsen 2010). Slowing down is expected only when a transition is caused by a valley vanishing. Noise-driven jumps between stable valleys give no warning, and many brain transitions may be of that kind.

To check that these runs were typical, I repeated each case 40 times with different seeds. The approaching fold gave a trend above 0.5 in 95 per cent of runs, and tipped in all of them. The changing noise gave a trend above 0.5 in every run, and tipped in only 12 per cent. The large noise gave a trend above 0.5 in 5 per cent of runs, and tipped in 53 per cent.

In 2012 Carl Boettiger and Alan Hastings pointed to a further trap (Boettiger and Hastings 2012). If you pick out, from many records, the ones that ended in a transition, and then look for warnings before them, you will find warnings more often than chance, even in systems that give none. The selection itself manufactures the signal. Their fix is to state in advance what you will measure and on which records.

What this means for anaesthesia and sleep#

The question for consciousness research is which kind of transition the brain makes. If the loss of consciousness under a drug is a fold, warnings should appear as the drug rises. It could instead be a noise-driven hop between valleys that both exist across a wide range of doses. Then warnings should be absent, and the moment of the transition should vary from one induction to the next in a way that looks random.

Neither of these is known in general. Studies report changes in the brain's rhythms before loss of consciousness, but few are designed to separate a flattening valley from a change in the drive. For the drug case the drive is the drug itself, so a change in the noise is hard to rule out. The early-warning concept carries weight only when the noise is known to be steady, or when the warning is checked against a record where no transition followed.

The active test: push and time the recovery#

There is a better version of the warning, and the finger experiments used it. Instead of reading slowness from the system's own fluctuations, push it and time how long it takes to recover. Scholz, Kelso and Schöner did this by briefly disturbing one finger and measuring how long the pattern took to return (Scholz, Kelso and Schöner 1987). As the metronome sped up towards the switch, recovery took longer.

The active test has an advantage over the passive one. Its result does not depend on the colour of the background noise, because the push is a known, separate event. A lake whose weather has changed will show rising autocorrelation, but a known disturbance to it will still be absorbed at the old speed if its valley has not flattened. Recovery time after a deliberate push measures the depth of the valley directly, where autocorrelation measures it only through the noise.

In the brain, the push can be a sound, a flash or a magnetic pulse, and the recovery is the brain's response returning to its baseline. Massimini's pulses were a step in that direction. A study that delivered the same push at several drug doses on the way into anaesthesia, and timed the recovery, would test for a fold far more cleanly than any passive measure.

An attractor is a state the system goes to and stays in. Much of what a brain does seems different. Thoughts do not settle. Patterns of activity form, hold for a fraction of a second, and give way to others. Kelso, who began with the fingers, later gave this a name in the brain: metastability, a regime in which the parts of a system tend towards coordination without becoming locked in it (Tognoli and Kelso 2014).

The word has two meanings, and this causes trouble. In physics, a metastable state is a shallow valley, like the supercooled water: stable for a while, but not the deepest state. In brain dynamics, it often means something else: a system with no stable states at all, which passes near a series of almost-stable patterns, lingering at each one before moving on. Mikhail Rabinovich and colleagues described one version of this, a chain of saddle points, each of which the state approaches and then leaves along a particular route (Rabinovich, Huerta and Laurent 2008).

A saddle point is a balance point that attracts from some directions and repels in others, like a mountain pass. A state can travel towards a pass along the valley floor, slow down near it, and then leave along a different valley. A chain of such passes, each leading to the next, gives a sequence of patterns that repeats in a fixed order with dwell times at each one. Rabinovich's group proposed this as a model for sequences in the insect sense of smell and in other neural processes that unfold in a fixed order.

The two meanings make different predictions. Shallow valleys predict memoryless, exponential dwell times, as in Part 1, and a random order of visits. A chain of passes predicts a fixed order and dwell times that grow longer, run by run, as noise falls. A study that calls the brain metastable should say which kind it means, because the data can tell them apart.

A worked case: oscillators that half agree#

The standard model is a set of oscillators, each with its own natural rhythm, pulling on one another. It was introduced by the Japanese physicist Yoshiki Kuramoto in 1975, and Steven Strogatz later wrote a clear history of it (Strogatz 2000). Each oscillator is a point going round a circle at its own speed. The coupling strength, K, sets how strongly each one is pulled towards the average of the others.

With weak coupling, the oscillators run independently and their average is close to zero. With strong coupling, they lock together and move as one. Between these, they half agree. Clusters form and dissolve, and the degree of agreement rises and falls over time.

The degree of agreement is measured by a number called the order parameter, R. It is 0 when the oscillators are scattered round the circle and 1 when they move as one. A common measure of metastability is how much R varies over time. A locked system has a steady R near 1. A scattered one has a steady R near 0. A metastable one has an R that swings.

I ran 64 oscillators with natural speeds drawn from a bell curve, at eleven coupling strengths. Metastability, the standard deviation of R over time, peaked at 0.11 near K = 1.4, and fell to zero once the oscillators locked at K = 3 and above. The theory for an infinite population puts the onset of synchrony at K = 1.6 for this spread of speeds. The peak sits just below it, where clusters keep forming and breaking.

The control. With no coupling at all, R still varied, with a standard deviation of 0.05. That variation comes only from having a finite number of oscillators, which drift into and out of chance alignment. So half of the peak value is what you would get from a system that does nothing. A metastability index means little until it is compared with the value for the same number of uncoupled parts.

A true story of an attractor seen directly#

Most claims of attractors in the brain are inferred from behaviour or from large-scale recordings. One is close to direct. In 2015 Johannes Seelig and Vivek Jayaraman recorded from a ring of neurons in the brain of the fruit fly, in a structure called the ellipsoid body (Seelig and Jayaraman 2015). They found a single bump of activity that moved round the ring as the fly turned, tracking its heading like the needle of a compass. The bump persisted in darkness, when the fly had no view to follow.

Theorists had long proposed a network for exactly this, called a ring attractor. Neurons arranged in a ring excite their neighbours and inhibit everything else, so that one bump of activity can sit anywhere on the ring and stay there. In 2017 Sung Soo Kim, Hervé Rouault, Shaul Druckmann and Jayaraman tested the prediction directly (Kim et al. 2017). They used light to switch on neurons at a different place on the ring, writing in a false heading. The circuit took the new bump and held it, with the old one gone. The network enforced one bump, as a ring attractor must.

This is the strongest kind of evidence for an attractor: a perturbation that moves the state, followed by the predicted persistence. It concerns heading, not consciousness. But it shows that the concept can be tested in neural tissue and pass.

What metastability can and cannot explain#

Gustavo Deco, Viktor Jirsa and Anthony McIntosh argued in 2010 that the brain's resting activity sits near the edge of instability, where it can explore many patterns (Deco, Jirsa and McIntosh 2010). Michael Breakspear's 2017 review set out how models of large-scale brain dynamics produce such behaviour, and how they can be fitted to recordings (Breakspear 2017). Francisco Varela and colleagues had earlier proposed that brief synchrony between distant regions binds them into a single moment of experience, with the synchrony constantly forming and dissolving (Varela et al. 2001).

These ideas share a claim: that a flexible, conscious brain lives between too little coordination and too much. That claim is testable, and some of it has support. Measures of how varied the brain's patterns are tend to drop under deep anaesthesia and in dreamless sleep. But a drop in variety is predicted by many accounts, including ones with no special role for metastability. The concept carries weight only if it predicts something more specific, such as the dwell times of each pattern, or how they change as coupling is altered by a drug, and if those predictions beat the deaf choir.

The last concept is the most ambitious. A system is critical when it sits exactly at the boundary between two kinds of behaviour, such as order and disorder. At that boundary, events of every size occur. A small disturbance can die out at once or spread through the whole system, and no size is typical. In 1987 Per Bak, Chao Tang and Kurt Wiesenfeld proposed that some systems tune themselves to this boundary without any outside help, which they called self-organised criticality, using a pile of sand as the example (Bak, Tang and Wiesenfeld 1987).

The simplest model for the brain is a branching process. Each active neuron triggers, on average, m others in the next moment. If m is below 1, activity dies out. If it is above 1, activity grows until something stops it. At m = 1 the system is critical, and bursts of activity, called avalanches, come in all sizes.

In 2003 John Beggs and Dietmar Plenz recorded from slices of rat cortex on a grid of electrodes and found such avalanches (Beggs and Plenz 2003). Their sizes followed a power law, a straight line on a log-log plot, with the slope expected for a critical branching process. The idea that the brain is critical took off, and so did the claim that criticality is good for computation, because a critical system responds to the widest range of inputs (Chialvo 2010).

The trap: what you do not record#

A recording sees only a tiny fraction of the neurons in a brain. That matters more than it seems. In 2018 Johannes Wilting and Viola Priesemann showed that estimating m from a subsample gives a badly biased answer, and that the bias makes systems look less critical than they are (Wilting and Priesemann 2018). They proposed a fix, which they called a multistep regression estimator.

I ran the test myself. A branching process ran for 100,000 steps with a true m of 0.98, close to critical. I then recorded only a random fraction of its events: all of them, a tenth, a hundredth and a thousandth. For each, I estimated m in two ways. The naive way regresses the activity at each step on the activity at the step before. The multistep way regresses across many lags and reads m from how the relationship decays with the lag.

With every event recorded, both methods gave 0.98. With a tenth recorded, the naive estimate fell to 0.72. With a hundredth, it fell to 0.20, and with a thousandth to 0.02. The multistep estimate stayed at 0.98 throughout, and 0.977 at a thousandth. The same process, recorded at a hundredth, looks to the naive method like one far from criticality. As a negative control, I ran the estimator on activity with no branching at all, subsampled to a hundredth. It gave −0.004, correctly close to zero.

The other trap: power laws without criticality#

The second trap runs the other way. A power law in avalanche sizes is not proof of criticality. In 2017 Jonathan Touboul and Alain Destexhe showed that power laws can arise in models that are not critical at all, including some driven only by noise (Touboul and Destexhe 2017). So can the scaling relationships that were thought to be a stronger sign. A straight line on a log-log plot is what many processes produce over a limited range of sizes.

Criticality and consciousness#

The bold version of the claim links criticality to consciousness directly. In 2014 Robin Carhart-Harris and colleagues proposed that the brain under psychedelic drugs moves towards criticality and greater disorder, and that normal waking consciousness sits just below it (Carhart-Harris et al. 2014). In 2022 Daniel Toker and colleagues analysed cortical recordings from a range of conscious and unconscious states. They argued that during conscious states the brain's slow electrical rhythms sit near the boundary between stability and chaos, and that they move away from it during unconsciousness (Toker et al. 2022). In 2019 Fontenele and colleagues reported that in anaesthetised rats, signs of criticality appeared only in a narrow range of cortical states, between more synchronised and less synchronised regimes (Fontenele et al. 2019).

These are serious studies. A careful review in 2022 by Jordan O'Byrne and Karim Jerbi set out how much remains unsettled (O'Byrne and Jerbi 2022). Different measures of criticality disagree. The subsampling and power-law traps are common. And "near critical" often comes without a stated distance. Criticality carries weight in consciousness research only when three things are in place. The estimator must survive subsampling. The claim must say how far from critical each state is, in numbers. And moving the brain along that axis must change consciousness in the direction predicted, not merely go along with it.

What a good criticality study would report#

It is easy to say what such a study would contain. It would estimate the branching ratio with a method that survives subsampling, and report the naive estimate beside it so that readers can see the size of the correction. It would report the distance from critical as a number with an uncertainty, for each state. It would include a control recording in which the same analysis should find no criticality, such as activity shuffled in time. And it would test a prediction that only a critical account makes, such as a sharp change in the response to small pushes as the drug crosses a particular dose.

Some recent work meets several of these conditions. Few studies meet all of them. Until more do, the claim that consciousness requires criticality is a promising hypothesis, and it should be graded as one.

The Kashmiri Śaiva tradition has a text devoted to change itself. The Spandakārikā, the "Stanzas on Vibration", is a short work of the ninth century, attributed to Vasugupta or to his pupil Bhaṭṭa Kallaṭa (Spandakārikā, GRETIL; Spandakārikā with Kṣemarāja, GRETIL; Dyczkowski 1992). Its subject is spanda, a word that means a slight movement, a throb or a pulse. In the text, spanda is the activity of consciousness itself, the movement by which every state of mind arises and passes.

This paper's working theory, the Recognition Theory of Mind (RTM), borrows the word for its first layer. RTM is a hypothesis under test, with five layers: L1 spanda, the pulse of activity; L2 prakāśa, the shining or showing of something; L3 vimarśa, reflective grasp; L4 pratyabhijñā, recognition; and L5 cit, awareness as such. If any part of RTM should meet dynamical systems theory, it is the first layer.

What the stanzas say#

Three verses bear directly on this paper. The translations are my own close renderings of the Sanskrit on GRETIL, whose numbering I follow, checked against Mark Dyczkowski's translation.

The opening verse praises the god "by whose opening and closing of the eyes the world dissolves and arises" (yasyonmeṣanimeṣābhyāṃ jagataḥ pralayodayau, 1.1). The pair unmeṣa and nimeṣa, opening and closing, is the text's basic image. States of the world, and of mind, come and go as a rhythm of opening and closing.

The third verse turns to states of consciousness. "Even in the differences of waking and the other states, it spreads through them without being divided from them. It never departs from its own nature as the perceiver" (jāgradādivibhede 'pi tadabhinne prasarpati, nivartate nijān naiva svabhāvād upalabdhṛtaḥ, 1.3). The states change. Something does not.

The third section gives a practice: "When one is absorbed in a single thought and another arises, that is unmeṣa; one should observe it for oneself" (ekacintāprasaktasya yataḥ syād aparodayaḥ, unmeṣaḥ sa tu vijñeyaḥ svayaṃ tam upalakṣayet, 3.9). An earlier verse makes the same move with stronger states. In great anger, in delight, in the moment of asking "what shall I do?", or while running, "the state one reaches, there spanda is established" (1.22).

Where the stanzas and the mathematics agree#

There are three points of contact, and I state them as correspondences, not identities.

First, both treat change as primary. Dynamics describes a system by its rule of motion, not by a list of its states. The stanzas describe consciousness by its pulse, and its states as what the pulse produces. Second, both direct attention to the seam between states. The informative moments in a dynamical system are its transitions: the dwell times, the slowing before a fold, the path of escape. Verses 1.22 and 3.9 place spanda at exactly such moments. Third, both distinguish what changes from what persists. A dynamical system has states that change and a rule that does not. Verse 1.3 has states that change and a perceiver that does not.

Where they part#

The third correspondence is also where the bridge breaks. In dynamics, what persists is the rule, a mathematical object with no point of view. In the stanzas, what persists is the perceiver, upalabdhṛ, the one who knows each state. These are different kinds of thing. A rule of motion does not experience the states it generates. The stanzas say the perceiver does.

What this means for RTM#

The tradition earns a place in this paper on one point: it tells you where to look. If RTM's first layer is the brain's activity and its rhythms, the tradition's advice is to study the transitions, not the states, and to study them from the inside as well as from the outside. That advice matches the paper's thesis. The signatures that make dynamical concepts carry weight are signatures of transitions.

It also sets a test for RTM itself. If L1 spanda is the layer at which dynamics applies, then the measured statistics of transitions should constrain the higher layers. A change in dwell times, under a drug or with practice, should show up in what is shown (L2) and in what is grasped (L3). If transition statistics could change freely without any effect on the higher layers, the first layer would be doing no work in the theory.

[AUTHOR NOTE: if your own practice includes the kind of attention to the arising of thoughts described in Spandakārikā 3.9, a short account of what you noticed, and how reliably, would help readers judge the thought experiment.]

Here is the strongest case against my thesis, put as its best defenders would put it. It comes from two directions at once.

The first says I am too generous. Every physical system can be written as a dynamical system. Every system that stays in a state for a while has, in some description, an attractor. Every system that changes suddenly has, in some description, a bifurcation. If the vocabulary fits everything, it explains nothing about consciousness in particular. The signatures I demand, such as exponential dwell times and loops that survive slow sweeps, are generic properties of noisy nonlinear systems. Finding them in a brain tells you that the brain is a noisy nonlinear system, which no one doubted.

The second says I am too strict. Tim van Gelder argued in 1998 that cognition is best understood as the behaviour of a dynamical system, and that asking dynamics to prove itself against rival explanations misreads the situation (van Gelder 1998). On his view, the rivals, such as computational models with stored symbols, are themselves dynamical systems described at a coarser level. There is no neutral test between them. The question is which description is most useful, and dynamics wins on usefulness wherever timing matters.

I take the first objection seriously, and it sharpens my claim. The signatures are generic, and that is why they are useful. A generic signature tells you which class of mechanism you are in, and the class is what the rival explanations deny. The drug-lag story says there is one valley. The two-attractor story says there are two. A loop that survives slow sweeps decides between them, whatever else is true. The vocabulary says nothing about consciousness in particular, I agree. It says something about the shape of the transitions between conscious and unconscious states, and that is a question about consciousness.

The second objection I accept in part. Van Gelder is right that dynamics and computation are not opposites, and Part 4's ring attractor is both: a network that computes heading by being an attractor. He is wrong, I think, that there is no neutral test. In each worked case of this paper there was a rival with different predictions. The rival was not always computational. It was a lag, a change in the noise, a deaf choir or a subsampled recording. The test is not dynamics against computation. It is one dynamical story against another.

Finally, there is a version of the case against that I cannot answer. It says that the brain is so high in dimension, and so far from the simple models, that the clean signatures may never be measurable in it. A double well has one variable. A brain has billions. If that is right, the concepts may be true of the brain and still never carry weight in explaining it, because their predictions cannot be checked. I hold the opposite view on the evidence of the ring attractor and of Friedman's loop, which were checked. But two cases do not settle a field.

I should also say what I concede. The vocabulary of dynamics has shaped good experiments even where the concepts were used loosely. The idea that the brain might sit near a critical point sent people to record avalanches, and those recordings are useful whatever the final verdict on criticality. A loose concept can be a good guide to where to look. My claim is narrower: it cannot be an explanation of what is found until it has passed a test that a rival could fail.

Concrete observations would weaken the thesis, and I list them so they can be checked.

• If a dynamical concept, used without any distinguishing signature, were shown to predict new results about conscious state transitions that rival explanations missed, my demand for signatures would be too strict.
• If the sweep-rate test were run on an anaesthetic in animals with brain concentrations measured, and the loop collapsed to zero at the slowest safe rate, neural inertia would lose its main support and the hysteresis concept would carry no weight there.
• If early-warning signals before loss of consciousness were found to be reliable, with a matched false-alarm rate below one in five in records with no transition, critical slowing down would carry more weight than I give it.
• If subsampling-corrected estimates of the branching ratio were the same in waking, sleep and anaesthesia, the link between criticality and consciousness would fail at the level of that measure.
• For RTM, if the statistics of transitions in brain activity could be changed widely, by drugs or by practice, with no change in reported experience, the first layer would be doing no work in the theory.

What I still don't know#

I don't know whether the loss of consciousness under a drug is a fold or a noise-driven hop, and I don't know of a human study designed to tell them apart. I don't know whether any brain-wide metastability index beats its own uncoupled control by a wide margin in real data, because few studies report the control. I don't know how far the subsampling correction carries over from a simple branching process to brain recordings, where the activity is not a single branching chain. And I don't know whether first-person reports of the arising of thoughts can be made reliable enough to count as data about dwell times.

There is one more gap I should own. My simulations use one-variable and simple many-variable models. They show what each signature looks like and how it can be faked. They do not show that any of these signatures is present in a human brain.

Dynamical systems theory gives mind science a set of exact words, and each one makes a prediction. An attractor predicts that a pushed state returns, and that time in a state follows a law set by noise and by the depth of the basin. A fold with hysteresis predicts a loop that survives slow change. Critical slowing down predicts rising sluggishness before a fold, and it can be faked by changing noise. Metastability predicts swings in coordination that should beat an uncoupled control. Criticality predicts a branching ratio near one, and a naive estimate from a subsample will hide it.

Use each dynamical word as a prediction with a stated control: a slower sweep, a steady-noise check, an uncoupled population or a subsampling correction. Where the prediction is tested and passes, the word explains something. Where it is not tested, it is a description.

The Spandakārikā points to the same place from the other side. It tells the practitioner to watch the seam between one state and the next. The mathematics says that the seam is where the informative events are. Neither tells you, by itself, who is watching. The curve and the verse each stop short of that question, in different places. That is a question for the higher layers of RTM, and for later papers in this series.

Kabir does not get a second patient swept up and down a dozen times. What he gets, from Anjali, is a habit. At his next induction he raises the drug more slowly, and he writes down the brain monitor's pattern at both crossings, as well as the hand squeeze. At emergence he does the same. When the gap shrinks with the slower induction, he says nothing about attractors. When it does not, on a later list, he brings Anjali both sets of numbers.

"Now it might be inertia," she says. "Now you have something to test."

The habit generalises beyond theatre four. Whenever a paper about the brain uses a dynamical word, ask three questions. What signature does the word predict? What rival would produce the same data without it? And was the control run that would tell them apart? If all three have answers, the word is doing work. If not, it is a promising name for something not yet explained, and it should be read as one.

1. Strogatz's account of the Kuramoto model, for the clearest introduction to coupled oscillators (Strogatz 2000).
2. Haken, Kelso and Bunz, and the follow-up tests of critical slowing in finger movements, for a dynamical model that predicted its own signatures (Haken, Kelso and Bunz 1985; Scholz, Kelso and Schöner 1987).
3. Scheffer and colleagues on early-warning signals, read with Ditlevsen and Johnsen on their limits (Scheffer et al. 2009; Ditlevsen and Johnsen 2010).
4. Friedman and colleagues on neural inertia, for hysteresis in anaesthesia with drug lag ruled out (Friedman et al. 2010).
5. Wilting and Priesemann on subsampling, and O'Byrne and Jerbi on the state of the criticality debate (Wilting and Priesemann 2018; O'Byrne and Jerbi 2022).
6. Dyczkowski's translation of the Spandakārikā with its commentaries, for spanda in its own terms (Dyczkowski 1992).

Each source was opened and checked against its publisher page, DOI record, arXiv record or a primary-text edition before it was cited. Anything I could not confirm is marked “to verify”.

Primary research

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3. Haken, H., Kelso, J. A. S., and Bunz, H. (1985). A theoretical model of phase transitions in human hand movements. Biological Cybernetics 51, 347–356. https://doi.org/10.1007/BF00336922
4. Kelso, J. A. S., Scholz, J. P., and Schöner, G. (1986). Nonequilibrium phase transitions in coordinated biological motion: critical fluctuations. Physics Letters A 118(6), 279–284. https://doi.org/10.1016/0375-9601(86)90359-2
5. Scholz, J. P., Kelso, J. A. S., and Schöner, G. (1987). Nonequilibrium phase transitions in coordinated biological motion: critical slowing down and switching time. Physics Letters A 123(8), 390–394. https://doi.org/10.1016/0375-9601(87)90038-7
6. Friedman, E. B., Sun, Y., Moore, J. T., et al. (2010). A Conserved Behavioral State Barrier Impedes Transitions between Anesthetic-Induced Unconsciousness and Wakefulness: Evidence for Neural Inertia. PLoS ONE 5(7), e11903. https://doi.org/10.1371/journal.pone.0011903
7. Massimini, M., Ferrarelli, F., Huber, R., Esser, S. K., Singh, H., and Tononi, G. (2005). Breakdown of Cortical Effective Connectivity During Sleep. Science 309, 2228–2232. https://doi.org/10.1126/science.1117256
8. Scheffer, M., Bascompte, J., Brock, W. A., et al. (2009). Early-warning signals for critical transitions. Nature 461, 53–59. https://doi.org/10.1038/nature08227
9. Ditlevsen, P. D., and Johnsen, S. J. (2010). Tipping points: Early warning and wishful thinking. Geophysical Research Letters 37, L19703. https://doi.org/10.1029/2010GL044486
10. Boettiger, C., and Hastings, A. (2012). Quantifying limits to detection of early warning for critical transitions. Journal of the Royal Society Interface 9, 2527–2539. https://doi.org/10.1098/rsif.2012.0125
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13. Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20. https://doi.org/10.1016/S0167-2789(00)00094-4
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19. Bak, P., Tang, C., and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of the 1/f noise. Physical Review Letters 59, 381–384. https://doi.org/10.1103/PhysRevLett.59.381
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22. Wilting, J., and Priesemann, V. (2018). Inferring collective dynamical states from widely unobserved systems. Nature Communications 9, 2325. https://doi.org/10.1038/s41467-018-04725-4
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Primary texts and translations

1. Vasugupta (or Bhaṭṭa Kallaṭa). Spandakārikā. Sanskrit text on GRETIL, input by Marino Faliero (not proof-read); verse numbers in this paper follow this file. https://gretil.sub.uni-goettingen.de/gretil/corpustei/transformations/html/sa_vasugupta-spandakArikA.htm
2. Vasugupta. Spandakārikā with the Spandanirṇaya of Kṣemarāja. Sanskrit text on GRETIL, based on the edition by M. K. Shastri (Srinagar, 1925), input by Oliver Hellwig. https://gretil.sub.uni-goettingen.de/gretil/corpustei/sa_vasugupta-spandakArikA-comm.xml

Secondary sources and leads

1. Dyczkowski, M. S. G. (1992). The Stanzas on Vibration: The SpandaKārikā with Four Commentaries. Albany: State University of New York Press. https://www.jstor.org/stable/jj.18255324

Mahapatra, R. R. (2026). Dynamical systems for cognition: attractors, bifurcations and metastability. Mind Frontier, paper 13, version 1.0. Rajesh Ranjan Mahapatra. https://rajeshrmahapatra.com/papers/mind-frontier/U0214-dynamical-systems-for-cognition.html (DOI to be assigned.)

@misc{mahapatra2026u0214,
  author       = {Mahapatra, Rajesh Ranjan},
  title        = {Dynamical systems for cognition: attractors, bifurcations and metastability},
  year         = {2026},
  howpublished = {Mind Frontier, paper 13},
  note         = {Version 1.0},
  doi          = {10.XXXX/placeholder},
  url          = {https://rajeshrmahapatra.com/papers/mind-frontier/U0214-dynamical-systems-for-cognition.html}
}