Pink Noise Perfection and the Value of Slowness

Continued from: “Dopamine, ADHD, and Signal to Noise Ratio

It’s not about “raising your frequency,” but rather about getting all your frequencies to communicate, by harmonizing. By breaking “noise” down into its component waves, we can see specific relationships between frequencies, illustrating which timescales get prioritized and how well they integrate with other timescales.

Colors of Noise

People with ADHD tend to have slower and more variable reaction times than people without ADHD, particularly in time-pressured activities. The data for both populations looks noisy, without any pattern to it, until one takes the time to analyze the noise as David Gilden did.[1]

This is what happens when you put and astronomically trained scientist in a psych lab. Astronomers  analyze the frequency spectrum of light to determine the fuel stars burn, how fast they move, and in which direction. These same techniques can analyze any spectrum of frequencies, from electromagnetic waves, to audible frequencies, to the frequencies of actions or reaction times.

Any combination of frequencies without a “signal” is considered noise. People subtract noise from data sets to get a clearer signal. Yet, sometimes the noise itself contains important information, like a signature of memory.

Not all noise is created equal. White noise is random, and thus carries little useful information. Pink noise, on the other hand, although also disregarded as “just noise” and subtracted out of the statistical data. Yet pink noise offers information embedded in the noise if one looks at it in the right way. This information suggests memory[2] and self-organization.

Pink noise appears everywhere in nature, including in humans. When analyzing people’s reaction times, Gilden found while people without ADHD exhibited a significant pink noise signature while those with ADHD and those recovering from alcoholism exhibited 90% white noise and 10% brown noise. Perhaps pink noise holds a clue to ADHD’s difficulty gaining traction with the outside world, in correspondence with dopamine dearth. To understand what this might mean, let’s take a look at what noise is and the difference between white, pink, and brown noise.

Dissecting Noise

A musical notes differ from one another by their frequency. Frequencies, measured in Hertz or beats per second, distinguish between scales of time. Higher frequencies establish fine distinctions in time by having wavelengths of short duration. Lower frequencies correspond to longer wavelength durations and thus larger scales of time. Frequency and wavelength are always inversely proportional, but amplitude, or loudness, varies independently of either. A note’s loudness does not depend on it’s wavelength or frequency.

A combination of notes, or individual frequency waves, interfere with one another adding up to another waveform which you can think of as a chord.

Fourier

Figure 1‑1 This image illustrates a total waveform (bold line) together with the simpler frequencies (fainter lines) that interfere to make it up.

Just as a guitar chord breaks down into it’s component notes, or an orchestra into various instruments, and each instrument into the notes they play, one can break any noise down into its component frequencies. Similarly, white light breaks down into its component frequencies when passed through a prism, revealing a spectrum of colors, spread out according to frequency: violet light with the highest frequency and shortest wavelength and red light with the lowest frequency and longest wavelength. Fourier transform acts like a prism and separates out a wave’s component frequencies in to a spectrum. Using Fourier analysis to analyze a chord reveals which frequencies, or notes, make it up and how loud they are.

Fourier2

Figure 1‑2 On the left you see the total wave function as it plays out through time. The wave function represents the interference pattern that results from the sum of the collection of frequencies illustrated behind it. The graph on the right separates out the different frequencies and illustrates their amplitudes into a power spectrum.

Organizing frequencies from high to low creates a frequency or power spectrum, which illustrates the relationship between frequency and power. Different noises have different shapes in this spectrum. These shapes describe the differing amplitudes of each frequency. Higher amplitudes translate to greater intensities, like brightness or loudness.

1_f_noise

Figure 1‑3 (Popesso, P., Magnelli, B., Buttiglione, S., Lutz, D., Poglitsch, A., Berta, S., … & Gastaud, R. (2012). The effect of the high-pass filter data reduction technique on the Herschel PACS Photometer PSF and noise. arXiv preprint arXiv:1211.4257. http://inspirehep.net/record/1203077/plots#0 )

The most familiar shape of noise is white noise. In white noise, like white light, all the frequencies carry equal intensity, yielding a flat frequency spectrum. When all the frequencies have the same amplitude no signals stand out over the heads of the others. When all the colors have the same intensity the light appears as white, hence “white noise.” White noise is like all those background conversations at a party, all equally loud and thus, impossible to make out. When the longer, slower, redder wavelengths carry greater intensity than shorter, bluer wavelengths, then the noise gains pinkish or reddish hue, hence pink noise, or red noise–as brown noise is sometimes called.

For pink noise, as frequency decreases, energy increases. The energy is the same for proportionally wide frequency intervals. In other words, lots of small waves contain the same amount of energy as a few big waves. Less energy for higher frequencies means the energy between the frequencies of 1000 to 2000 Hz, is the same as the energy between 10 and 20 Hz. The fewer the waves, the more energy they carry. Thus in pink noise, the energy distribution across the scale of frequency illustrates the fractal property of scale invariance, the exact version of self-similarity.

In both brown and pink noise, longer wavelengths have more power than the shorter wavelengths, but power changes more dramatically between frequencies in brown noise than in pink. In pink noise, the power is inversely proportional to frequency, p = 1/f. If frequency doubles, power halves. If frequency triples, power divides by three. Thus power and frequency remain in closer proximity to one another than in brown noise, where p = 1/f2. With brown noise, when frequency doubles, power divides by four. If frequency triples, power divides by nine. Power and frequency diverge. The pink noise power spectrum fits the greatest range of frequencies most efficiently into a finite range of intensity and visa versa.

Noise colors also vary in predictability. When frequencies of the same amplitude add together, as in white noise, the wave they add up to is unpredictable. Brown noise on the other hand, is predictable on short intervals, and random over larger scales.

The short-term predictability of Brown noise results from the much larger amplitude of the longer frequencies. The individual frequencies are predictable. Their combination adds up to an unpredictable wave. In Brownian motion, the strength of the longer wavelengths washes out the tiny amplitude contributions of the smaller wavelengths.

Brown noise gets its name from Brownian motion, the random motion of molecules, which is predictable on short intervals, and random over larger scales. This is also called a random walk pattern. “For example, while touring a city without a guide or plan, we make turns at random at the intersections, but our walk in straight streets is predictable (‘random walk’ pattern).”[3] The random walk pattern is also called the drunkard’s walk: each step might move the same distance, one step, but who knows the next step’s direction. Brown noise is predictable on a local scale, mixed with points of unpredictability, making it unpredictable on global scales.

Pink noise also features a healthy mix of predictability and surprise, offering just enough more surprise than Brown noise to avoid too much predictability. Pink noise, like brown noise has more powerful slow frequencies, but the difference in power between the slow and fast frequencies is less extreme in pink noise than in brown noise. Pink noise holds the sweet spot in scaling.

Pink Noise

Goldilocks

When Goldilocks stumbled into a power spectrum, finding it the home of the three noises: Papa Bear Brown, Mama Bear Pink, and Baby Bear White, she found one fit just right. She found Papa Bear Brown too boring and predictable, Baby Bear White too random and hard to follow, but Mama Bear Pink was just right –enough reliability for comfort and enough novelty to challenge one’s understanding.

Pink noise held the sweet spot between predictability and surprise. Along these lines, most music reveals pink noise signatures. We derive great pleasure from establishing a pattern, which many can follow and learn to anticipate, and then interjecting variations on that theme, which keeps the music surprising, engaging, and delightful. The element of surprise and variation activates the dopamine system and adds an element of challenge to the anticipation. Pink Noise also appears not only in music, but also in the frequency spectra of many natural phenomena including: speech, heart beat rhythms, neural activity, statistics of DNA sequences, tide and river levels, quasar light emissions, flicker noise in electronic devices, financial systems, and resistivity in solid states, and as a signature of memory and self-organization.

This Goldilocks balance between predictability and novelty, between comfort and surprise, emerges as a result of a specific relationship between frequencies. Predictability emerges because the slowest frequencies have larger amplitudes than the fastest frequencies. For pink noise, the longest, slowest waves carry the the most power. Thus the long timescales provide the signals for the shorter timescales to entrain to and nest within. The scale invariance offers the elements of surprise.

Scale invariance gives pink noise both its consistency and its flexibility, both its determinism and its freewill, it’s memory and its ability to make a new choice in the moment. The repetition of patterning across scale and the strength of the long-term frequencies offers predictability. The fluctuations of multiple levels of scale infuse unpredictability, like in the Weierstrass function or the coastline paradox.

Memory and Scale Invariance

The power of those long, slow wavelengths in pink noise, indicate memory.[4] The system “remembers” the intensity of those larger moments in the smaller moments, offering a temporal container for the smaller moments to organize themselves within.

Waves seek to entrain with one another. Waves with higher amplitude carry greater energy[5], thus tend to entrain shorter waves within their patterns. Considering attention as a form of entrainment, the loudest and brightest demand our attention and thus our entrainment. Larger amplitudes drive the entrainment of the frequencies surrounding them. In pink noise, the slower frequencies have the greater amplitude and thus set the bass line for the shorter faster frequencies. Since the most powerful waves extend over the longest time periods, these longer moments organize the smaller moments within them.

If the slowest frequencies also have the greatest intensity, as they do in pink and brown noise then the longest time scales have the greatest intensity. These provide something of a slow, strong bass line rhythm and memory around which the melodies ofreq wvlthf the higher frequencies can dance. Since frequency and wavelength vary in inverse proportion to one another, slower frequencies mean longer wavelength times. The lower the frequency, the larger the moment it brings present. For example, frequencies of .001 Hz have wavelengths of 1000 seconds, or about 17 minutes.
A wave carries all of its information in each moment. Like a color or a pitch appears in an instant, a single moment carries the information of the entire wave. A scale invariant wave, in particular, reveals its long-term patterns in the minutest of microcosms, the pattern of the whole within the smallest of its parts. The self-similarity of pattern provides a timeless link between different scales of time, like Plato’s eternal ideas link disparate objects as they manifest in finite space-time. The relationship between frequency scales establishes the relationship between time frames, the interlocking of seconds, minutes, hours, and longer and shorter timescales as well.line rhythm and memory around which the melodies of the higher frequencies can dance. Since frequency and wavelength vary in inverse proportion to one another, slower frequencies mean longer wavelength times. The lower the frequency, the larger the moment it brings present. For example, frequencies of .001 Hz have wavelengths of 1000 seconds, or about 17 minutes.

Both brown and pink noise are scale invariant, emphasizing the longer slower waves modulating the faster waves, offering some local predictivity. As discussed before, the local predictivity is stronger in brown noise because the slow wavelengths carry much more power than the faster ones do in brown noise than in pink noise. In brown noise, a level of local predictability modulates surprise, while pink noise infuses a greater degree of novelty.

If the slower frequencies’ amplitude is too much higher than the faster frequencies however, as in brown noise, then perhaps the larger moment carries too much power, making things too predictable; the memory is too strong, and the minor fluctuations of the present moment is not given enough voice. Then, perhaps, expectations dictate reality, and personality and generational patterns repeat themselves. For instance a person who experienced abandonment as a child may come to expect it from every relationship, clinging to their partner, until the partner feels smothered and leaves too. In this way a person’s fear can contribute to its fruition, an overriding pattern repeats itself, preventing an infusion of novelty.

In brain modeling, brown noise emerges from increased inhibition of neuronal firing.[6] With the longest wavelengths carrying the most power, brown noise carries with it a certain amount of predetermination, a set of expectations inhibiting the possible degrees of freedom. Predetermination makes sense as an inhibitory factor, offering a long-term trajectory that negates some present possibilities. White noise, on the other hand emerges from increased excitation, where every minute possibility feels important, making discernment and pattern formation difficult.[7]

ADHD and embodiment

Pink noise appears as a signature of health, balancing long-term memory and predictability without negating the smaller influences that arise in the moment. If brown noise is inhibitory and white noise excitatory, then the 10% brown noise 90% white noise combination Gilden found in ADHD adults and recovering alcoholics might indicate both increased inhibition and excitation, in contrast to the pink noise of their non-ADHD peers. This seems to suggest folks with ADHD might have a harder time relating shorter time scales to longer time scales, due to their wide intensity disparity and lack of scale invariance between the fast and slow frequencies. This also speaks to the unique cocktail of depressants and stimulants sought out by folks with ADHD in efforts to self-medicate. Stimulants, like caffeine, sugar, Adderall, Ritalin, may help to break through overriding inhibitions by bringing up the power of the faster frequencies, providing focus. Depressants, like alcohol, may serve to soften the paralyzing inhibitions associated with too strong of memory based inhibitions.

ADHD is creation of the modern world, for three basic reasons: lifestyles involve more sitting, faster paces, and less immersion in the natural world. Before modernity, people, particularly children, were not expected to sit still for 6+ hours a day. Their tasks involved their bodies in ways few do these days. Only with more sedentary, knowledge based lifestyles does embodied engagement become pathologized.

A child is ready to learn to read, only after conquering certain physical milestones, like being able to balance on one foot, jump rope backwards and forwards, skip cross laterally, and walk on a balance beam.[8] Before a person can dissociate from the present, embodied moment, as required for concentrated mental effort like reading, their biology demands that they have mastered locating and manipulating their body in space. The body requires integration, before a child can sit still and concentrate on something non-physical. Our lack of physical activity in the modern world contributes to ADHD. Bodies want to move, and if your body cannot move, your brain cannot settle. Not surprisingly, recent research shows, “physical activity is ADHD medicine.”[9]

As the adult world becomes more urbanized and sedentary, the inhibitions placed on children’s movements increase as well. City living and packed schedules do not often allow for roaming woods, fields, and hillsides. This brings us to the other modern world casualty associated with decreased freedom of physical movement, decreased connection with the natural world. Not surprisingly, time spent in green spaces also decreases ADHD symptoms.[10]

In addition, the modern world is one of human timescales, divided into smaller and smaller segments in an effort to fit in more, quicker, faster, better. Is it surprising that a world of rapid-fire, over-stimulation would produce people who have difficulty focusing for extended periods of time? Is it a measure of health to adapt to such a system?

The modern world lacks integration with the longer timescales which immerse humanity, until the last hundred years. For millennia, before internal combustion and electricity, we cycled with the sun and the seasons, immersed in the timescales of trees, stones, and cosmic cycles. The natural world provided the bass line for us to dance within. Separating ourselves from the earth’s turning and starry reels with electric lights; from the seasons, through year-round produce and climate control; from centuries old forests with chainsaws and strip malls; from ancestors with nursing homes, we dampen the power of those long, slow frequencies, and amplify minutes and milliseconds flattening our power spectrum to the din of white noise, silencing the music of the spheres and our own collective memories.

White noise illustrates a lack of emphasis on longer, slower frequencies, instead pumping up the power of shorter and shorter timescales. The modern world does the same thing, leaving people feeling anxious and scattered. How do we get back to pink noise? Slowing waaaay down opens the present moment to new degrees of freedom. Bodily awareness cultivated through movement brings brain waves into dialog with one’s heartbeat and breathing. Regular time in nature mitigates ADHD symptoms by opening the space for self-direction, reducing human imposed restrictions, and offering a point of connection to the slow frequencies, the wisdom of trees, rocks, stars.

This is an excerpt from my forthcoming book “The Texture of Time”

References:

[1] Gilden, David L, and Hilary Hancock. 2007. “Response variability in attention-deficit disorders.” In Psychological Science 18, no. 9 (2007): 796-802.

[2] Gilden, David. 2001. “Cognitive emissions of 1/f noise.” In Psychological Review. Jan;108(1):33-56.

[3] Buzsaki, Gyorgy. 2006. Rhythms of the brain. New York: Oxford University Press. p 121

[4] Gilden, David. 2001. “Cognitive emissions of 1/f noise.” In Psychological Review. Jan;108(1):33-56.

[5] A2 ∝ E (Amplitude squared is proportional to energy. When amplitude doubles energy quadruples)

[6] Rǎdulescu, Anca, and Lilianne R. Mujica-Parodi. 2013. “Network connectivity modulates power spectrum scale invariance.” In NeuroImage.

[7] Mathematian Anca Rǎdulescu and biomedical engineer Lilianne R. Mujica-Parodi modeled the effects of white noise in two regions involved in the dopamine-driven basal ganglia loop: the amygdala, associated with emotional reactivity, memory, and decision-making, and the prefrontal cortex where planning occurs. Their models show people with increased emotional reactivity and anxiety should have more white noise in the amygdala, in correspondence with weaker inhibition from the planning center, the prefrontal cortex. People with decreased emotional reactivity should show white noise in the prefrontal cortex however, in correspondence with stronger excitation messages from the amygdala. People with average emotional reactivity, on the other hand, should illustrate pink noise in both the amygdala and prefrontal cortex.

[8] Axness, Marcy, 2014. “Is your child ready to read? A checklist.” In Mothering.com

[9] Hamblin, James. 2014. “Physical activity is ADHD medicine.” The Atlantic. http://www.theatlantic.com/health/archive/2014/09/exercise-seems-to-be-beneficial-to-children/380844/

[10] Kuo FE and Taylor AF. 2004. “A potential natural treatment for attention-deficit/hyperactivity disorder: evidence from a national study.” American Journal of Public Health. 2004 Sep;94(9):1580-6.

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One thought on “Pink Noise Perfection and the Value of Slowness

  1. Pingback: Dopamine, ADHD, and Signal to Noise Ratio – The Texture of Time

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