10. Triple Pulse & the Biology of Sleep



A specific Math/Fact matrix spawned my Theory of Attention. The express intention of this book is to establish the validity of this matrix. The Living Algorithm (LA) generates the mathematical system, i.e. Data Stream Dynamics (DSD), that provides the underlying logical structure of the theory’s womb. Empirical data, experiential, observational and experience, provides the evidence that is the substance.

Prior articles established patterns of correspondence between the LA’s Triple Pulse and various types of human behavior, i.e. some sleep-related phenomena and creative sessions. The current article investigates the correspondence between the Triple Pulse and the biological systems related to sleep.

Dement’s opponent process model is the standard. It states that our bodies have a far-reaching biological network that forces us to sleep. It consists of two processes; Process S works to keeps us awake, while Process C works to put us to sleep. Scientists employ a military metaphor, i.e. two battling armies, to describe the system. The esteemed scientific community remains mystified as to why natural selection would choose this bizarre system that forces us into a vulnerable state of mini-hibernation for nearly one third of every daily cycle.

We suggest that a mathematical metaphor based upon the Living Algorithm’s (LA) system of information digestion provides more explanatory power. LA mathematics indicates that alternating number strings maximize the dimensions of data stream acceleration. Multiple studies, e.g. Sleep Deprivation & Naps, have exhibited that there is positive correlation between data stream acceleration and our cognitive performance. Alternating number strings provide a plausible model for Processes C and S. The inference is straightforward. Evolutionary forces chose this peculiar biological alternation because it maximizes our cognitive performance, which would certainly facilitate survival.

This explanation, of course, raises some major questions. Why is data stream acceleration related to cognitive performance? And what is the relationship between evolution and the LA’s mathematical system?


Significance of correspondence between TPulse & Sleep Biology?

Are there any patterns of correspondence between the biology of sleep and the Living Algorithm’s Triple Pulse? Why is this significant?

Prior: TP ≈ Sleep Interruptions

Prior chapters established that the mathematical behavior of the Triple Pulse effectively modeled certain key aspects of our sleep related behavior. For instance, just as interrupted or shortened sleep exerts a negative impact upon our cognitive abilities, interrupting or shortening the Triple Pulse’s 2nd (Rest) Pulse also exerts a negative impact upon the ideal dimensions of the following 3rd (Active) Pulse.

Basis of Correspondence? Need more evidence

What is the basis of these correspondences? Are they due to a confounding variable or an unidentified mathematical artifact with no explanatory power? Or could there be some kind of peculiar causal relationship that links the LA’s mathematical process with human behavior? To validate or dispel these alternative possibilities, let us examine some more evidence.

Article intent: TP ≈ Sleep Biology

To that end, the current chapter explores the relationship between the biology of sleep and the Triple Pulse. There seems to be some kind of linkage between the Living Algorithm's mathematical system and our mental behavior (cognitive skills). Are there also patterns of correspondence between mathematical and biological behavior? To answer these questions, let us examine the current understanding regarding the biology of sleep.

Dement's 'Opponent Process' Model: the Biological Mechanisms behind Sleep

Sleep Biology: Our Source (once again): Medina’s brain rules

Dement’s ‘opponent process’ model provides a widely accepted mechanism for the biology of sleep. What does this ‘opponent process’ model entail? To avoid the distortion that comes from interpretation we will again quote directly from Dr. Medina’s brain rules – the highly respected book that transmits recent discoveries of mainstream science to the educated public.

Dement’s “Opponent Process Model”

According to Medina, Dement, the father of sleep research, and Kleitman, a fellow researcher, made significant discoveries that introduced a new paradigm into the field. It is called the ‘opponent process’ model. Their model indicates that our waking and sleep states are “locked in vicious, biological combat”. There is an ongoing

 “pitched battle between two powerful and opposing drives, each made of legions of brain cells and biochemicals with very different agendas. Though localized in the head, the theater of operations for these armies engulfs every corner of the body.”

Mysterious ongoing Conflict – day and night

The conflict has a few strange features.

“First, these two opposing forces are not only engaged during the night, while we sleep, but also during the day, while we are awake. Second, they are doomed to a combat schedule in which each army sequentially wins one battle, then promptly loses the next battle, then quickly wins the next and so on, cycling through this win/loss column every day and every night. The third strange thing is that no one army ever claims final victory in this war. This incessant engagement results in the cyclical waking and sleeping experiences all humans encounter every day (and night) of our lives.”

Counterintuitive results: Need for Rest does not drive Biology of Sleep

This conclusion seems counterintuitive. Common sense would instead simply suggest that our system gets tired and needs sleep to rest from the day’s activities. Clearly the body craves rest after sustained physical activity. It is easy to assume that same reasoning applied to sustained mental activity. Or perhaps we might reason that the light of the sun wakes us up; and the dark of the night puts us to sleep.

Dement’s model suggests that our intuitions are misguided. Rather than a reaction to either the day/night or rest cycles, the experience of exhaustion at the end of a long day is actually a byproduct of an internal biological battle. In other words, the sleep wake cycle is not a cooperative system triggered by external factors. This cycle is best represented as an ongoing struggle between two competing systems. Once again, the findings of science seem to violate our common sense.

Continuous Conflict between 2 biological forces

What is the mechanism behind this biological conflict?

“One of Kleitman’s and Dement’s greatest contributions was to show that this nearly automatic rhythm occurs as a result of the continuous conflict between two opposing forces.”

Composition of Opposing Armies

What are these ‘opposing forces’ composed of? And what are they called?

“One army is composed of neurons, hormones, and various other chemicals that do everything in their power to keep you awake. This army is called the circadian arousal system ‘process C'. If this army had its way, it would make you stay up all the time. Fortunately it is opposed by an equally powerful army. Also made up of brain cells, hormones, and various chemicals. These combatants do everything in their power to put you to sleep. They are termed the homeostatic sleep drive 'process S'.  If this army had its way, you would go to sleep and never wake up. It is a strange and paradoxical war. The longer one army controls the field, the more likely it is to lose the battle.” (brain rules, pp  153-55)

Current Understanding

The following diagram represents the prevailing paradigm of the scientific community regarding the biology of sleep.

Model raises 2 mysteries: 1) Why behind what?

There are two mysteries associated with this model.

1) What is the reason behind this mysterious process? Scientists know how sleep biology works, but they can’t explain why. Although scientists accept Dement‘s opponent process model as an accurate description of the biology of sleep, we are no closer to understanding why it works the way it does. Why does it appear that opposing armies fight for supremacy merely to retreat at the moment of victory?

2) Why did Evolution choose bizarre mechanism?

2) What evolutionary advantage would lead to the biological development of the ‘opponent process’ model? Why did natural selection choose this bizarre biological scheme to force us into a vulnerable state of mini-hibernation called sleep every day? Clearly the sleeping portion of the cycle leaves the sleeper in a quite vulnerable state. Why would nature expose itself to such risks in order to engage in ongoing systemic warfare?

Mechanism behind opponent process model remains a scientific mystery.

Scientists in many disciplines have employed all the sophisticated technology at their disposal – brain scanners, electronic imaging, and such – to uncover the reasons behind sleep's biological component. Despite decades of research, the powerful scientific community remains mystified by this mysterious counter-intuitive biological process. Why hasn’t the scientific community been able to come up with some answers despite extensive research? Could it be because scientists are searching for a physical explanation, when actually an explanation based upon the digestion of information would serve us better?

The Living Average & the Triple Pulse

TP ≈ Dement?

How does Dement’s 'opponent process' model, a widely accepted scientific theory regarding the biological sleep cycle, corresponds with the mathematics of the Triple Pulse? Are there any similarities whatsoever, or just a mish-mash of suggestive patterns with no real substance? Just an imaginary dragon that we might see in swirling clouds?

LA generates data stream derivatives

To answer these questions, we must evoke yet another innate mathematical feature of the Living Algorithm (LA). Recall that the LA’s sole function is digesting data streams. This digestive process reveals the data stream's ongoing rates of change (the derivatives). Similarly, a car's velocity and acceleration are also rates of change.

Triple Pulse is 2nd derivative: Liminals higher derivatives

Thus far we have only examined the 2nd derivative (acceleration) of a specific data stream. When the Living Algorithm digests a specific data stream consisting of a number string of 120 1s, followed by a number string of 120 0s, and then another number string of 120 1s, the Triple Pulse is one of the results. The Triple Pulse is the visualization of the 2nd derivative of this data stream. The Liminals (the other colored areas) are the higher derivatives (the rates of change) of the same data stream that generated the Triple Pulse. The graph at right provides a visualization of these data stream derivatives.

X-axis, the number of iterations; Y-axis, relative values

The horizontal x-axis indicates the number of iterations (repetitions) of the Living Algorithm process. As a purely mathematical process, the vertical y-axis indicates relative values, i.e. more or less1. Because the number strings consists of 0s and 1s, these relative values are between 0 and 1. From visual inspection, we can see that the data stream's acceleration is greatest at the peak of the Pulse and becomes smaller with each subsequent iteration.

Living Average: the 1st Derivative of a Data Stream

We have introduced the higher derivatives of a data stream. But we have yet to meet the data stream's 1st derivative. It is called the Living Average, a.k.a. the Decaying Average.

The Rise of the Triple Pulse's Living Average

Let's inspect the Living Average (the 1st derivative) of the same data stream that produces the Triple Pulse. In this particular case, the data stream begins with a string of 1s. The Living Average starts at 0. As each successive 1 enters the system the Living Average, the data stream's 1st derivative, begins to rise. Similarly, the temperature rises as heat enters a room. In such a manner, we can liken the Living Average to the temperature gauge of our information digestion system. As the number of 1s in the system increases, the Living Average approaches 1. We call this value 'practical' one. The Living Average’s progress from 0 to 1 is shown in the graph at right.

The Fall of the Triple Pulse's Living Average

After this point as long as 1s are added to the system, the Living Average remains in the state of 1 indefinitely. In like fashion, a poorly insulated room might reach an equilibrium point where additional heat does not increase the temperature, but just maintains the temperature. If, however, the composition of the data stream shifts from 1s, to a string of 0s, the result is quite different. The Living Average immediately begins falling from the peak. After a sufficient number of 0s have been added to this mathematical system, the Living Average reaches practical 0. In parallel fashion, after the heat is turned off, a room's temperature falls, eventually reaching equilibrium with the outside.

Alternation of number strings of '1s' and 0s maximizes changes in Living Average.

At this point, the data stream's composition shifts again, from 0s to a string of 1s. With this shift, the Living Average’s cycle repeats itself, growing from 0 to 1. Similarly, when the heat is turned on again, the temperature rises again. It is evident that the alternation of a string of 1s with a string of 0s keeps the Living Average in a constant state of change. Without the alternation, the Living Average 'stagnates' at either 1 or 0.

Living Average & TP based upon same data stream

Despite their differences in appearance, the Living Average shown here and the classic Triple Pulse are derivatives of the same exact data stream. The former is the data stream's 1st derivative – shown above, while the latter is the 2nd derivative – shown at right.

A Mathematical Battle between 1s & 0s?

Triple Pulse battle between an army of 1s and an army of 0s.

Repetition for retention, hence assimilation: these graphs visualize processes that are pure mathematics. As such, they represent mathematical patterns generated when the Living Algorithm digests the Triple Pulse data stream. To exhibit the parallels between Dement's sleep biology model and our mathematical model, let us employ Dement's military metaphor to characterize the relationship between the Triple Pulse data stream (the alternation of 1 strings and 0 strings) and the Living Average (the data stream's 1st derivative). In other words, we can describe this particular mathematical process as a battle between the 1s and 0s.

Army of 0s (System S) and Army of 1s (System C); Living Average, gauge of battle's progress.

The stream of 1s corresponds with Process C (the multitude of biological processes that want to keep us Conscious). The stream of 0s corresponds with process S (the multitude of biological processes that drive us to Sleep). The Living Average indicates the ongoing progress of the battle between these two opposing forces. This value is a gauge of the state in the battle. For instance, when the Living Average is closer to 1, the army of 1s is winning. When the Living Average reaches 1, the army of 1s has won the battle, but not the war, as we shall see.

After the Army of 1s wins the battle, the Army of 0s takes over.

The Living Average (the data stream's 1st derivative) begins at 0 (the beginning of the day). This indicates that the 0s have temporarily won the ongoing battle. The army of 1s (Process C) assert themselves in the attempt to change the Living Average from 0 to 1. Eventually the 1s win the battle – as they have changed the Living Average to 1 (the end of the day). At this point the army of 0s (Process S) takes over and begin attempting to change the Living Average back to 0. Eventually the 0s win the battle with the 1s, as the Living Average returns to 0. Now, the 1s reawaken from their sleep and begin reasserting control again. At the point that the 1s completely win the battle, the 0s reassert themselves again. This process continues ad infinitum

Simplistic Battle makes no sense.

From this conceptual perspective, this is a mysterious process. Why would the 1s give up control after completely annihilating the 0s and vice versa? The 1s are on the verge of total victory, when they relinquish control to the vanquished opponent. In terms of winning and losing, this strategy makes no sense. The simplistic view of this mathematical process as a battle between 1s and 0s doesn't explain anything and leads to greater confusion.

If either Process C or S wins the war, the LA System 'stagnates'

If Process Sleep wins the war, we enter a coma and die.

We’ve established that Dement's ‘opponent process’ model and the Living Algorithm's mathematical model exhibit symmetrical patterns. For a deeper understanding of this mysterious process, let us tweak the mathematical model to see what it has to reveal about the mechanisms behind Dement's model. What would happen if one of Dement's armies were able to win the war – vanquish his opponent forever? It is obvious what would happen if Process Sleep (the army of 0s) won the war. We would remain in the unconscious state of sleep permanently. Inevitably we would enter a coma and die.

What's wrong with Process Consciousness winning the war?

It is not so obvious what would happen if Process Consciousness (the army of 1s) won the war. Process C, our circadian arousal system, wants to keep us up awake forever. What's wrong with that? Most of us would love to enjoy continuous consciousness rather than enter a state akin to death for one third of every day. In fact, this is one of the mysteries of cognitive science. Why must humans enter a state of mini-hibernation every 24 hours, especially since it makes us vulnerable to our predators? What evolutionary mechanism conjured up this bizarre behavior?

Endless string of 1s to model what happens if Process C won the war

Let us employ a data stream consisting of an endless string of 1s to represent/symbolize what happens when the army of 1s (Process C) wins the war. In other words, there is no more army of 0s (Process S) to fight for supremacy. Process C (the army of 1s) reigns supreme and we never have to sleep.

If Process C wins the war, Data Stream Derivatives flatline

When the Living Algorithm digests a data stream consisting solely of 1s, the 1st derivative (the Living Average) rises from 0 to 1 and then flatlines. Similarly, the higher derivatives generate pulses and then flatline at 0. In other words, the rates of change (the derivatives) are minimized when the content of the data stream remains the same. In like fashion, a data stream's derivatives are minimized when its content is random. The following graph illustrates this mathematical phenomenon.

Graph: Data Stream of 1s à Stagnant Derivatives

This is a visual representation of the derivatives of a data stream consisting of 360 1's. The horizontal x-axis indicates the number of times the Living Algorithm process is repeated (the iterations). The vertical y-axis indicates the relative sizes of these rates of change (the data stream derivatives). After the initial 120 1's, all the derivatives fade to 0. In other words, the Living Algorithm's information system 'stagnates' if either of the armies wins the war, not just the battle. Stagnates might be a mild descriptor, as the flatlining is more reminiscent of what we see on a hospital's machines after the signs of vitality shut down and the patient dies.

Graph: Triple Pulse DS > Vibrant Derivatives

Alternately, the regular alternation of a string of 1s and a string of 0s refreshes the successive pulses. This refreshment maximizes the dimensions of the data stream derivatives.


Alternation of Number Strings maximizes dimensions of DS Derivatives

The benefit of alternating numerical strings is not winning or losing the battle (as visualized by the Living Average). Rather than vanquish the opponent, the benefit of the alternation between 1s and 0s is simply to maximize the dimensions of the data stream's rates of change. Rather than adversarial, the 1s and 0s are in a cooperative relationship to maximize change in the Living Algorithm's mathematical system. Another way of stating the same phenomenon, the alternation of number strings prevents 'stagnation' in the information system.

Implications of Mathematical Model?

What are the implications of this curious mathematical model regarding the biology of sleep? It is evident that the alternation of Process C and Process S from Dement's ‘opponent process’ model would maximize change in the data stream derivatives related to the LA info digestion process. Does this alternation prevent mental stagnation? If so, what kind of change are we talking about?

Integrating Sleep Deprivation Correspondences

Review Correspondences between Triple Pulse and Sleep Deprivation for understanding

To gain insight into the potential meaning of the connection between Dement’s biology of sleep and LA’s mathematical system, let us examine some prior results, specifically the patterns of correspondence between the Triple Pulse and Sleep Deprivation.

Active Pulse generated by string of 1’s

As implied by the name, the Triple Pulse consists of 3 pulses (green). The positive pulses are called Active Pulses and the negative pulse is called the Rest Pulse. The Active Pulse is generated when the Living Algorithm digests a string of 120 1s.

Process C's Army of 1s generates Active Pulse;

In the Sleep Deprivation Study, the Active Pulse is associated with our cognitive skills. In the current Sleep Biology Study, the same ‘army’ (string of 1s) generates a Living Average that corresponds with Process Consciousness. In other words, the same army of 1s is associated with the Active Pulse (cognitive skills) is also with Dement’s biological Process Consciousness.

Process S's Army of 0s generates Rest Pulse.

The Rest Pulse is generated by a string of 120 0s. In our Sleep Deprivation Study, the Rest Pulse is associated the state of sleep. The army of 0s also generates a Living Average that corresponds with Process Sleep. The same army of 0s that the Rest Pulse (sleep) is associated with Dement’s biological Process Sleep.

Living Average applies to Sleep Biology, TP to Sleep Deprivation à TP to Sleep Biology?

More specifically and more significantly, the mathematical relationship between the army of 1s and the army of 0s mimics the warring relationship between Processes C and S, as well as the mysterious relationship between our cognitive abilities (thinking) and sleep. LA mathematics reveals the logic behind these relationships. The Living Algorithm digests the same data stream composed of alternative strings of 1s and 0s. The data stream's 1st derivative, the Living Average, mimics the warring relationship between Processes C and S, while the 2nd derivative, the Triple Pulse, mimics the relationship between sleep and our cognitive skills. Could it be that the underlying logic of the data stream's 2nd derivative applies equally to the relationship between Processes C and S?

Periodic Alternation of Active & Rest Pulses to avoid stagnation & maximize dimensions (refresh potentials)

As previously illustrated, the alternation of numerical values (the army of 1s and the army of 0s) refreshes the potentials, i.e. maximizes the dimensions, of the data stream derivatives of the Living Algorithm's mathematical system. Without the alternation, both the Pulses and the Liminals, i.e. the higher derivatives, disappear. If the content of the data stream doesn't change periodically, the data stream derivatives flat-line. The information system stagnates. How does this strictly mathematical process apply to Dement's model for the biology of sleep?

Active Pulse associated with cognitive abilities in Sleep Deprivation study

In our Sleep Deprivation study, we saw that the Rest Pulse of 0s refreshes the Active Pulse of 1s, just as sufficient sleep refreshes our testable cognitive abilities. As such, the Rest Pulse is associated with the state of sleep, while the Active Pulse is associated with our ability to think - our cognitive abilities. Without the Rest Pulse, the Data Stream Derivatives flat-line - stagnate. Similarly without sleep, humans die.

Evolution of Process S to avoid cognitive stagnation?

Could it be that Process Sleep evolved to prevent cognitive stagnation? Is it possible that Process S is necessary to refresh or even maximize our cognitive abilities? Without Process S, does Process C lead inevitably to mental stagnation? Diminished cognitive skills would certainly minimize the chances of any organism’s survival.

Number String Alternation parallels Process C & S Alternation

It is evident that the alternation of number strings prevents mathematical stagnation in the LA system. In like fashion, is it possible that the alternation of Processes C & S prevents mental stagnation? If so, the mathematical logic of the LA system certainly parallels the biological logic of Dement’s opponent process model.

Amazing Implications of Math/Fact Matrix regarding Sleep Biology

DSD provides answers to questions surrounding Dement’s model

It seems that the LA’s mathematical system (Data Stream Dynamics) supplies some plausible, yet tentative, answers to the 2 questions raised by the biology of sleep. If our Math/Fact matrix holds water, then evolutionary forces chose Dement’s opponent process model to maximize our cognitive performance. Just as the alternation of the Active and Rest Pulse acts to refresh the dimensions of each pulse, the alternation of the C and S processes acts to refresh our cognitive abilities.

These answers, of course, raise new questions.

? LA system ≈ Sleep Biology

Why is there a striking pattern of correspondence between the LA’s mathematical processes and the biology of sleep? Why is the biological logic of sleep similar to the Triple Pulse’s mathematical logic? What is the reason behind this unusual pairing? While the connection could be random, an artifact or based upon a confounding variable, let us assume that there is a causal relationship at the heart of this Math/Fact Matrix. What are the implications of this premise?

Evolution had reason for choosing Dement model

Let us commence this investigation with the connection between biology forms and evolution. The scientific community operates under the common assumption that biological forms have an evolutionary basis. In other words, natural selection chose, stabilized and passed on the specific forms associated with sleep biology for at least a few hundred thousand years for a specific reason. This process was not accidental.

As noted entologist Dr. Michael Gershon states in his book The Second Brain:

“Evolution is a powerful editor. Body parts that are frivolous or not absolutely necessary have little chance of making it through the rigors of natural selection.”2

Why did Evolution choose Triple Pulse model?

On the assumption that the mathematical correspondence is real, why did evolution choose the Triple Pulse’s mathematical process? Could it be that the LA provided an evolutionary advantage? Is it possible that the processes associated with sleep biology evolved to take advantage of the mathematical processes of the LA, specifically those associated with the Triple Pulse?

Must show other Evolutionary connections to substantiate hypothesis.

We are putting forth the somewhat outrageous proposition that Dement’s opponent process model for biological sleep evolved to take advantage of the LA’s mathematical system – Data Stream Dynamics (DSD). Yet the current evidence remains flimsy. If we could demonstrate the existence of other examples of a similar phenomenon, this would help to substantiate our bold hypothesis. For instance, are there any other biological processes that seem to take advantage of the LA’s mathematical system?3 Is there any direct evidence linking the LA’s mathematical processes to an evolutionary leap?4

Do evolutionary forces exploit our method of digesting info?

Another set of questions surround the relationship between living systems and the Living Algorithm. How would evolutionary forces take advantage of the LA’s special properties? Do living systems employ this mathematical system in some way? If so, in what way? Is it conceivable that living systems employ the LA to digest environmental data streams?

LA as Life’s computational interface. Plausible Hypothesis? Why not other math systems?

Is this even a plausible proposition? Why would Life choose LA as her computational interface with sensory data streams? What are her advantages? Is there any reason that living systems wouldn’t employ the myriad equations of Physics or Probability or even computer algorithms for this task?5

We must hold these questions and speculations in abeyance until we gather more evidence.


1 The values associated with an ellipse are also relative. The same mathematical ellipse can be applied to planetary motion, where the values are enormous, or to a student's homework assignment, where the picture fits upon a page.

2 Dr. Michael Gershon, Second Brain, p. xiii.

3 In a subsequent chapter, we will examine the connections between the universally accepted Posner’s Attention Model and the LA’s mathematical processes.

4 In the Boredom Principle treatise, we develop the notion that the Big Leap in human evolution (when we developed modern language etc) was due at least in part to the association of pain with shrinking data stream derivatives. In brief mental stagnation evoked the pain of boredom.

5 My treatise Living Algorithm: Mathematics of the Moment provides a detailed analysis of these topics.


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