Part one of a three part series on nonlinear dynamical systems, self regulation and NeurOptimal neurofeedback.
Living systems are nonlinear dynamical systems, meaning they are defined by how their state evolves over time under rules where small changes can have large effects and the parts do not simply add up. Unlike a thermostat, which defends a single setpoint, healthy biological systems maintain flexibility by moving between states. Research on heart rhythms, gait and brain activity suggests that this structured variability is a marker of health, and that both excessive rigidity and disorganized randomness are associated with illness and aging.
There is a thermostat on the wall of the clinic. It has one job, a fixed opinion about the correct temperature, and no capacity for growth. When the room drifts above its setpoint it turns the heat off. When the room drifts below, it turns the heat on. That is the whole personality.
For about a century, this device has quietly shaped how we talk about the human body. We say the body maintains balance. We describe stress as something that knocks us off balance and recovery as returning to baseline. The metaphor is so comfortable that most of us never examine it. It is also, in an interesting and useful way, wrong.
Your heart is not defending a setpoint. Neither is your sleep, your attention, your mood, or the electrical activity humming across your cortex right now as you read this sentence. These systems are doing something considerably more sophisticated, and considerably weirder, than a thermostat. Understanding what they are actually doing turns out to matter a great deal for how we think about anxiety, trauma, sleep, and what it means to help a nervous system settle.
This article is the first of three. It covers the ground rules: what a dynamical system is, what makes one nonlinear, why nature is full of them, why the healthiest systems are the variable ones, and how living things regulate themselves with nobody in charge. The second article applies all of this to the brain. The third looks at what NeurOptimal neurofeedback actually does with it. If you have ever wondered why the phrase dynamical neurofeedback contains the word dynamical, you will have your answer by the end of part three. Please pace yourself.
Dynamical systems, nonlinearity, and why both words matter
The field this series draws on is called nonlinear dynamical systems, and the phrase is worth taking apart, because each word is doing work.
A dynamical system is anything described by two things: a set of state variables saying what condition it is in right now, and a rule saying how that condition changes over the next instant. Position and velocity, plus a law of motion. Predator and prey numbers, plus birth and death rates. Membrane voltage, plus the equations governing ion flow. The defining ingredient is time. A dynamical system is not a photograph of a thing, it is a description of how the thing moves.
Because time is built into the definition, the object of study is not a value but a trajectory, the path a system’s state traces as it evolves. Ask a dynamical systems researcher what a heart is doing and they will not be satisfied with a heart rate. They will want the sequence, because the sequence is where the information lives.
Then there is nonlinear, which describes the rule.
A linear system has two convenient properties. The output is proportional to the input, and the parts add up. Push twice as hard, get twice the result. Understand the pieces separately and you understand the whole. Most of the engineering that built the modern world runs on this assumption, and it works beautifully for bridges, circuits, and springs.
A nonlinear system has neither property. Output is not proportional to input, so a small nudge can produce an enormous effect and a large shove can produce almost nothing, depending on when it arrives and what state the system is in. The parts do not add up, so studying a neuron in isolation tells you surprisingly little about what eighty six billion of them do together. And history matters. Where the system has been changes how it responds to what happens next.
If you have ever tried to have a productive conversation with your partner about the dishwasher at the wrong moment, you have direct experimental evidence that human systems are nonlinear. The same words, the same volume, entirely different result. The input did not change. The state of the system did.
Nature runs on nonlinearity. Weather, heart rhythms, ecosystems, neurons, immune responses, family dynamics, and the behaviour of a room full of teenagers are all nonlinear dynamical systems. Which means the thermostat model, which is a linear control loop, is a poor description of nearly everything alive.
In plain language: a dynamical system is described by how it changes over time. When the rule governing that change is nonlinear, timing and context matter as much as force. The same input can produce completely different outcomes depending on the system’s current state.
A short and mostly painless tour of chaos
In 1961, a meteorologist named Edward Lorenz was rerunning a weather simulation and, to save time, typed in a rounded version of a number from an earlier printout. He entered 0.506 instead of 0.506127. The forecast that came out was completely different from the original.
This was not a computer error. It was a discovery. Lorenz had found that in certain systems, differences too small to measure grow, and grow, until the system’s future becomes practically unpredictable. He called it sensitive dependence on initial conditions, and the world eventually called it the butterfly effect, which is a lovely phrase that has since been used to justify an enormous number of poor decisions in time travel films.
Two things about chaos are routinely misunderstood, and both matter here.
The first is that chaotic does not mean random. Lorenz’s weather model was completely deterministic. Every step followed exactly from the one before according to fixed rules. There was no dice roll anywhere in it. Chaos is what happens when simple, rigid rules produce behaviour that never quite repeats and cannot be predicted far ahead. It is order that looks like disorder from the outside.
The second is that chaotic systems are not formless. They have structure. If you plot a chaotic system’s behaviour over time in what mathematicians call state space, a kind of map where every possible condition of the system gets its own location, the system traces out a trajectory. It never visits the same point twice, but it stays within a bounded region, and nearby trajectories pull apart from each other at a characteristic rate. That rate is what mathematicians actually measure when they call a system chaotic.
The technical name for the region trajectories tend to settle into is an attractor, and attractors are the central objects of dynamical systems theory. There are a few flavours worth knowing.
A fixed point attractor is a marble in a bowl. Roll it anywhere and it ends up at the bottom. A thermostat lives here.
A cycle attractor is a pendulum, or a heartbeat described very crudely, or the roughly annual cycle of hockey optimism and hockey grief. The system repeats.
A strange attractor is the interesting one. The system stays within a bounded region, never exactly repeats, and remains sensitive to small differences. Weather does this. So, it appears, does a healthy brain.
Hold on to the marble and bowl image. It is going to do a lot of work in the next article, when we start describing worry loops and shutdown responses as bowls that are unhelpfully deep.
In plain language: chaos is not randomness. It is what you get when simple rules produce patterns that never quite repeat and cannot be predicted far in advance. Living systems use this.
The healthiest heart is not the steadiest one
Here is the finding that reorganized how a lot of clinicians think about the body.
If you record a healthy young adult’s heartbeat and measure the interval between each beat and the next, you do not get a metronome. You get a fluctuating series that speeds up and slows down continuously, on timescales from seconds to hours. The variation is substantial. It is also not noise.
Notice what kind of measurement this is. It is not a snapshot of a value, it is the shape of a trajectory over time. This is the point at which physiology becomes a dynamical systems question, because no single heart rate reading can tell you whether the system producing it is healthy. Only the dynamics can.
Now record someone with congestive heart failure. The beat to beat intervals become strikingly regular. The heart becomes more metronomic, and the more metronomic it becomes, the worse the prognosis. A heart that has lost its variability is a heart in trouble.
This is genuinely counterintuitive. We are trained to think of steadiness as health and variation as instability. In physiology it is often the reverse. Ary Goldberger and colleagues spent years documenting this pattern across systems, and the picture that emerged is now known as the loss of complexity hypothesis: aging and disease are associated with a breakdown in the structured variability of physiological signals, in one of two directions.
Systems can become too rigid, locking into narrow, repetitive, overly regular behaviour. Or they can become too erratic, producing variation that has lost its internal structure and become closer to true randomness. Health sits between these, in a place with a lot of variation that is nonetheless organized.
The same pattern shows up in gait. Young healthy walking has subtle, structured fluctuation in stride timing. In aging and in Parkinson’s disease, that structure degrades, and the degradation predicts falls. It shows up in postural sway, in breathing, in hormone release, and in EEG.
So the goal for a nervous system is not stability. Stability, taken to its logical conclusion, is a very bad outcome. The goal is adaptability, and structured variability is what adaptability looks like on a graph.
In plain language: healthy bodies fluctuate in organized ways. Too little variation and too much disorganized variation are both signs of trouble. Flexible beats steady.
Fractals, or why your body keeps repeating itself at every scale
There is a specific kind of structure in that variability, and it has a name most people have met on a poster in a physics classroom.
A fractal is a pattern that looks similar at different scales. Zoom in on a coastline and you get shapes that resemble the coastline you started with. Zoom in on a fern and each frond resembles the whole. Zoom in on your lungs and the branching of the bronchi resembles the branching of the bronchioles, which resembles the branching below that, for roughly twenty three generations of branching, all of it packing an absurd amount of surface area into a container the size of your chest.
Your circulatory system does this. Your kidneys do this. Your neurons do this.
More surprisingly, your heartbeat does it in time rather than in space. The pattern of fluctuation over a few seconds statistically resembles the pattern over a few minutes, which resembles the pattern over an hour. There is no single characteristic timescale. Physicists call this scale free behaviour, and it is a signature of systems that are organized across many levels at once rather than controlled from one level.
Why does the body build itself this way? Because fractal organization is efficient and robust. It distributes function across scales, so damage at one scale does not take down the system. It packs enormous capacity into small volumes. And it means the system can respond to demands arriving at any timescale, which is a useful property for an organism that must handle both a sudden loud noise and a gradual change of season.
Brain activity shows scale free structure too. This becomes important in the next article.
In plain language: healthy biological signals show similar patterns whether you look at them over seconds or hours. That layered organization is part of what makes bodies robust.
How nature regulates itself with nobody in charge
If there is no thermostat, what is doing the regulating?
Feedback. Enormous amounts of it, in loops, arranged in layers.
Negative feedback is the stabilizing kind. A change triggers a response that opposes the change. Blood glucose rises, insulin is released, glucose falls. This is what the thermostat does, and bodies do plenty of it. Negative feedback is why you are not currently boiling.
Positive feedback amplifies. A change triggers a response that increases the change. Blood clotting works this way, with each activated platelet recruiting more. So does labour, where contractions trigger hormone release that triggers stronger contractions. So does a neuron reaching threshold, where a small depolarization triggers a cascade that becomes an all or nothing spike.
Textbooks tend to treat positive feedback as the dangerous one, which is understandable given that runaway amplification is how you get avalanches and panic attacks and comment sections. But a system with only negative feedback cannot change states. It can only defend where it is. Positive feedback is what lets a system commit, switch, and move decisively from one mode to another. You need both. Health lives in the relationship between them.
This is where the word homeostasis starts to show its age. Walter Cannon coined it in the 1920s and it has served us well, but the newer concept is allostasis, which means stability through change. The idea is that the body does not defend fixed setpoints so much as it predicts what will be needed and adjusts in advance. Cortisol rises before you wake, not because something has gone wrong, but because your system anticipates that getting out of bed is coming. Blood pressure shifts before you stand.
Allostasis is a better fit for a nonlinear dynamical system, and it comes with a clinical companion concept: allostatic load, Bruce McEwen’s term for the cumulative cost of running those adjustments too hard for too long. Chronic stress, in this framing, is not simply unpleasant. It is a system spending its adaptive capacity, and the sign of that spending is often a loss of flexibility. Fewer states available. Slower returns. Narrower range.
Now the last piece, and the most philosophically interesting one. Much of biological order arises with no controller at all. This is self organization.
An ant colony builds sophisticated foraging networks and no ant is managing the project. Each ant follows local rules about pheromone concentration. The network is what happens. A flock of starlings produces those extraordinary rolling shapes over a winter field, and there is no lead bird; each bird tracks roughly seven neighbours and the murmuration emerges. Heat a shallow pan of oil evenly and past a certain threshold the fluid spontaneously organizes into neat hexagonal convection cells. Nobody assigned the hexagons.
Order without a manager, produced by many local interactions and feedback. That is what a brain is doing, at a scale that makes the ant colony look like a small business.
In plain language: bodies regulate through layered feedback loops rather than a central controller, and much of their organization emerges from local interactions rather than being directed.
Timing, delay, and why feedback can go wrong
One more property, because it will matter enormously in the third article.
Feedback loops have delays. The signal takes time to travel, the response takes time to develop, and the effect takes time to register. When the delay is short relative to how fast the system changes, the loop is stable and corrective. When the delay is long, the correction arrives after the situation has already moved on, and the system starts to overshoot, then overcorrect, then overshoot again.
Michael Mackey and Leon Glass showed in the 1970s that adding delay to a simple model of blood cell production could push it from steady output into wild oscillation, without changing anything else. They went on to argue that a whole family of illnesses might be understood as dynamical diseases, where the underlying components are fine but the system’s timing has shifted into a pathological rhythm. Certain arrhythmias, some respiratory disorders, and some neurological conditions fit this description.
The clinical implication is worth sitting with. Sometimes nothing is broken. The timing is off.
And the corollary, which is the seed of everything in article three: if timing is what determines whether feedback helps a system or destabilizes it, then feedback delivered with the right timing, at the right moment, in the right amount, is a very powerful thing. Not because it is forceful, but because it is well placed.
In plain language: feedback that arrives too late can make a system oscillate. Feedback that arrives at the right moment can help it settle.
What all of this has to do with anxiety, sleep, and mood
We have been talking about hearts and ants and convection cells. Here is why it matters for the reasons people actually call a counselling clinic.
If arousal, attention, mood, and sleep behave like states in a nonlinear dynamical system rather than dials set at a level, then several familiar experiences look different.
Feeling stuck is not a character flaw or a lack of effort. It is what a system looks like when it has settled into a deep attractor and small nudges are not enough to move it out. Anyone who has been told to just relax during a panic response has experienced the inadequacy of small nudges applied to a deep attractor.
Sudden shifts are normal, not alarming. Nonlinear dynamical systems change abruptly at thresholds. Falling asleep is not a gradual dimming; it is a transition. So is the moment a difficult conversation tips from tense to genuinely hard.
Small things can matter more than they look. If a system is sitting near a transition point, a modest change in sleep, movement, connection, or breathing can shift it. This is not wishful thinking; it is a property of nonlinear dynamical systems near thresholds. It also explains why the same small change did nothing last month.
And flexibility, not calm, is the useful target. A nervous system that can move into activation when activation is warranted and back out again afterward is doing well. One that is stuck anywhere, including stuck in calm, is not.
In plain language: treating mood and arousal as states a system moves between, rather than levels to be set, changes what progress looks like.
Where neurofeedback fits into this picture
Here is the bridge to the rest of the series, and then we will stop.
Most traditional neurofeedback approaches are, structurally, thermostats. An assessment identifies activity that appears too high or too low in a particular frequency band at a particular location, a target is set, and the system rewards the brain for moving toward that target. This is a legitimate and well studied approach, and there are good reasons to use it. But the underlying model is a linear control loop: identify the correct value, train toward it. Sojourn offers assessment informed approaches like this, along with HEG biofeedback, which works with blood flow rather than electrical activity.
NeurOptimal neurofeedback at Sojourn starts from the other model. It does not set a target for your brain, because the premise is that a self organizing nonlinear dynamical system does not need to be told where to go. It needs accurate, well timed information about what it is doing, at the moments it is doing it. Given that, the system reorganizes on its own, the way an ant colony reroutes or a flock changes shape, without any bird receiving instructions.
Whether that premise holds is a fair question, and one we will treat honestly rather than promotionally in article three. The science of brain dynamics is robust; the research specifically on NeurOptimal is much thinner, and we will say so plainly. If you want a practical overview in the meantime, we have written an overview of how NeurOptimal works, its benefits, and the options available.
What we hope you take from this first article is smaller and more durable than any claim about neurofeedback. Your body is not a machine holding a setpoint. It is a layered, self organizing, feedback driven dynamical system whose health shows up as flexibility, which is a property of how it moves over time rather than of where it sits at any given moment. When people struggle, it is very often not because something is broken but because the system has lost some of its range.
That reframe is worth having whether or not you ever put a sensor on your head.
Coming next in this series
Part two, The Brain on the Edge, takes everything here and applies it to neurons: what brain rhythms actually are, why cortical networks appear to sit near a critical point between order and chaos, what happens to those dynamics in anxiety and trauma, and how complex systems change when they do change.
Part three, Inside a NeurOptimal Session, describes exactly what the system measures, what those brief interruptions in the audio are, why a brain might be able to use that information, what the evidence does and does not support, and what change tends to look like over time.
A short glossary for this series
Dynamical system: a system described by state variables plus a rule for how those variables change over time. Its subject matter is trajectories, not single values.
Nonlinear dynamical system: a dynamical system whose governing rule is nonlinear, so output is not proportional to input, the parts do not simply add up, and history matters.
Trajectory: the path a system’s state traces through state space as it evolves.
Control parameter: a quantity that, when moved past a threshold, changes the qualitative behaviour a dynamical system is capable of.
Transient: the behaviour a system shows on its way to an attractor, rather than once it has arrived.
State space: a map on which every possible condition of a system has its own location. Behaviour over time traces a path across it.
Attractor: a region of state space a system tends to settle into. A fixed point is a marble at the bottom of a bowl; a cycle repeats; a strange attractor stays within bounds without ever repeating.
Sensitive dependence on initial conditions: the property whereby tiny differences grow into large ones, making long range prediction impossible even in a fully determined system.
Fractal or scale free: showing similar structure at many different scales, in space or in time.
Negative feedback: a response that opposes a change and stabilizes the system.
Positive feedback: a response that amplifies a change and lets a system switch states.
Allostasis: achieving stability through anticipatory change rather than by defending a fixed setpoint.
Self organization: the emergence of coordinated order from local interactions, with no central controller.
Frequently asked questions
What is a nonlinear dynamical system?
A dynamical system is anything defined by a set of state variables plus a rule for how they change over time, which means its subject matter is trajectories rather than static values. It is nonlinear when that rule is not proportional, so effects do not scale with causes and the components cannot simply be added together. Hearts, neurons, ecosystems, weather and families all qualify. The practical consequence is that such systems have thresholds, abrupt transitions, and behaviour that depends heavily on their current state and their history.
Is chaos theory the same as saying the brain is random?
No, and the distinction matters. Chaotic systems are deterministic, meaning each state follows from the last according to fixed rules. What makes them chaotic is that tiny differences grow, so the future becomes practically unpredictable even though nothing random is happening. A chaotic system also has structure and stays within a bounded region of possible states. Describing brain activity as nonlinear or chaotic is a statement about the kind of order it has, not a claim that it is disorganized.
Why is high heart rate variability considered a good sign?
Beat to beat variation reflects the ongoing adjustment of the autonomic nervous system to internal and external demands. When that variation is present and structured, it suggests the system has range and can respond. When it flattens toward metronomic regularity, that flexibility has usually been reduced. Heart rate variability is a general marker of adaptability rather than a diagnostic test, and individual readings are affected by age, fitness, medication, sleep, alcohol, illness and measurement method, so it is best interpreted as a trend rather than a score.
Does this mean stability is bad?
Not at all. It means the useful kind of stability is dynamic rather than fixed. A healthy system holds together while moving through a wide range of states, the way a cyclist stays upright through continuous small corrections rather than by freezing. What the research associates with poorer outcomes is not stability but rigidity, meaning a loss of the ability to move between states.
Is neurofeedback based on chaos theory?
Some approaches draw explicitly on nonlinear dynamical systems thinking, including dynamical neurofeedback systems like NeurOptimal, and there is a peer reviewed literature framing neurofeedback in terms of self organized criticality. It is important to be precise here: the research on nonlinear brain dynamics is well established, while the research on any specific neurofeedback system is more limited. The science described in this article explains a plausible model of how feedback could be useful. It is not evidence that a particular device produces particular outcomes.
Do I need to understand any of this to benefit from counselling or neurofeedback?
No. This series exists for people who find the mechanism interesting or reassuring, and plenty of people do well without ever reading a word about attractors. If you would rather skip the physics and talk about what is actually going on for you, that is an entirely reasonable place to start.
Where to start
If this way of thinking is useful, part two applies it directly to the brain. You can also read more about our neurofeedback services and how they are used at the clinic.
Curious whether counselling or neurofeedback might fit what you are dealing with? View therapist profiles, approaches and availability, and book online through JaneApp. If you are not sure who to start with, contact us and we can help you find a good fit.
