Anyway, apple bloom and spike were shocked. Then spike said something. “Hold on, wouldn’t something this orderly be bad for you?” spike asked. Discord shook his head and waved him off.
They reappeared in discord’s house, butttttttttttt, it looked normal. Like in that one episode. Here’s a picture in case you have forgotten dear readers.
Anyway, apple bloom and spike were shocked. Then spike said something. “Hold on, wouldn’t something this orderly be bad for you?” spike asked. Discord shook his head and waved him off.
“Oh no, it won’t, as long as I’m, me, myself,” he then snapped his fingers and another him appeared. “And I.” the other discord blinked then looked at apple bloom and spike. “Hello!” he waved at them. They both looked confused but waved back.
“Howdy.” “hi.” then discord snapped again and sent him away.
“Are chaotic, then I’ll be fine.” discord finished. Apple bloom smiled.
“That’s good to hear!” she excitedly said. Just then, her belly started growling. She looked at her belly and back at discord. He rolled his eyes and snapped his fingers. An apple fell on top of apple bloom’s head, at first she looked at him annoyed but then ate the apple. Discord looked at spike.
“Spike, are you hungry as well?” discord asked. Spike looked at him.
“Well now that you mention it, yeah I kinda am.” discord snapped his fingers and a big bag full of gems appeared. When apple bloom saw this, she was angry.
“Hey! How come I only get one thing while he gets a whole bag filled with things, I WANT ONE NOW!!!!!” she whined. Discord pulled his ears off, and put them into a jar
“Sheesh girl, relax, you almost made my ears explode!” just then his ears inside the jar exploded, and regrew on the sides of his head. he then snapped his fingers, and a big bag full of apples appeared. apple bloom squeed in excitement and ran over to the bag and hugged it.
“thank ya kindly discord.” apple bloom said. he snapped his fingers and a top hat appeared, he Bowed Down to her, and took off his hat.
“anytime my dear sweet, somewhat annoying apple bloom.” he said. suddenly apple bloom trotted by his face and kissed him on the cheek. he blinked and looked at her. “what was that for?” he asked. she looked at him and smiled.
“for givin me what I want.” she said. he blushed. and she just galloped away with her big bag of apples. Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist’s toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available. A remarkable diversity of natural phenomena are thought to be chaotic. Formally, a system is chaotic if it is bounded (meaning that, like a planet circling a star, its dynamics stay inside an orbit rather than escaping off to infinity), and if it is deterministic (meaning that, with the exact same initial conditions, it will always evolve over time in the same way), and if tiny perturbations to the system get exponentially amplified (Glossary (Supplementary Information), Supplementary Figs. 1, 2). The meteorologist Edward Lorenz famously described this phenomenon as the butterfly effect: in a chaotic system, something as small as the flapping of a butterfly’s wings can cause an effect as big as a tornado. This conceptually simple phenomenon—i.e., extreme sensitivity to small perturbations—is thought to appear everywhere in nature, from cosmic inflation1, to the orbit of Hyperion2, to the Belousov–Zhabotinskii chemical reaction3, to the electrodynamics of the stimulated squid giant axon4. These are only a few examples of the many places in nature where chaos has been found.
It is relatively simple to determine if a simulated system is chaotic: just run the simulation a few times, with very slightly different initial conditions, and see how quickly the simulations diverge (Supplementary Fig. 1). But, if all that is available are measurements of how a real, non-simulated system evolves over time—for e.g., how a neuron’s membrane potential changes over time, or how the brightness of a star changes over time—how can it be determined if those observations come from a chaotic system? Or if they are just noise? Or if the system is in fact periodic (Glossary, Supplementary Figs. 1, 2), meaning that, like a clock, small perturbations do not appreciably influence its dynamics?
While a reliable method for detecting chaos using empirical recordings should be an essential part of any scientist’s toolbox, such a tool might be especially helpful to biologists, as chaos is predicted to play an important functional role in a wide variety of biological processes (that said, we note that real biological systems cannot be purely chaotic in the strict mathematical since, since they certainly contain some level of dynamic noise—see Glossary—but that researchers have long speculated that many biological processes are still predominantly deterministic, but chaotic5). For example, following early speculations about the presence of chaos in the electrodynamics of both cardiac6 and neural7 tissue, the science writer Robert Pool posited in 1989 that “chaos may provide a healthy flexibility for the heart, brain, and other parts of the body.”8 Though this point has been intensely debated since the 1980s5,9, a range of more specific possible biological functions for chaos have since been proposed, including potentially maximizing the information processing capacity of both neural systems10 and gene regulatory networks11, enabling multistable perception12, allowing neural systems to flexibly transition between different activity patterns13, and boosting cellular survival rates through the promotion of heterogeneous gene expression14. And there is good reason to expect chaos to exist in biological systems, as a large range of simulations of biological processes15, and in particular of neural systems9, show clear evidence of chaos. Moreover, unambiguous evidence of biological chaos has been found in a very small number of real cases that were amenable to comparison to good theoretical models; these include periodically stimulated squid giant axons4 and cardiac cells16, as well as the discharges of the Onchidium pacemaker neuron17 and the Nitella flexillis internodal cell18. But, beyond simulations and these select empirical cases, most attempts to test the presence or predicted functions of chaos in biology have fallen short due to high levels of measurement noise (Glossary) in biological recordings. For this reason, it has long been recognized that biologists need a noise-robust tool for detecting the presence (or absence) of chaos in their noisy empirical data9,15.
Researchers also need a tool that can detect varying degrees of chaos (Glossary) in noisy recordings. In strongly chaotic systems, initially similar system states diverge faster than they do in weakly chaotic systems. And such varying degrees of chaos are predicted to occur in biology, with functional consequences. For example, a model of white blood cell concentrations in chronic granulocytic leukemia can display varying levels of chaos, and knowing how chaotic those concentrations are in actual leukemia patients could have important implications for health outcomes19. As another example, models of the human cortex predict that macro-scale cortical electrodynamics should be weakly chaotic during waking states and should be strongly chaotic under propofol anesthesia20; if this prediction is true, then detecting changing levels of chaos in large-scale brain activity could be useful for monitoring depth of anesthesia and for basic anesthesia research. Thus, it is imperative to develop tools that can not only determine that an experimental system is chaotic, but also tools to assess changing levels of chaos in a system.
Although classic tools for detecting the presence and degree of chaos in data are slow, require large amounts of data, are highly sensitive to measurement noise, and break down for common edge cases, more recent mathematical research has provided new, more robust tools for detecting chaos or a lack thereof in noisy time-series recordings. Here, for the first time (to our knowledge), we combine several key mathematical tools into a single, fully automated Matlab processing pipeline, which we call the Chaos Decision Tree Algorithm21 (Fig. 1). The Chaos Decision Tree Algorithm takes a single time-series of any type—be it recordings of neural spikes, time-varying transcription levels of a particular gene, fluctuating oil prices, or recordings of stellar flux - and classifies those recorded data as coming from a system that is predominantly (or “operationally”22) stochastic, periodic, or chaotic. The algorithm requires no input from the user other than a time-series recording, though we have structured our code such that users can also select from among a number of alternative subroutines (see Methods section, Fig. 1).
Fig. 1: The Chaos Decision Tree Algorithm21.