The Analytic Fitness™ Dictionary – Adaptation

This inaugural entry of the Analytic Fitness™ Dictionary looks at the single most important law for training theory: the Law of Adaptation. (3.320 words, estimated reading time: 15-17 min)

Adaptation is the most important biological concept for training theory but it is also the most misunderstood.

The trouble starts early, with the word itself: the term ‘adaptation’ is often ambiguous and denotes sometimes a biological process and others times the end-state of that process. I have a habit of following philosopher Rudolf Carnap when it comes to ambiguous concepts and using subscripts to distinguish them, which I’d do here like so:

  • Adaptation1 is the process by which adaptive systems (living organisms mostly but not exclusively) learn to respond to stimuli from their environment.
  • Adaptation2 is the end-state of adaptation1 and is reached when the response of the adaptive system to the stimulus has become minimal.

I can almost picture my readers starting in surprise while reading the last sentence. Confusion is understandable: one would probably expect that an adaptive system is adapted to a stimulus when it’s response is ‘optimal’, or something like that. But adaptation2 need not be optimal and not even good for the adapted2 individual by any relevant standard.

Getting that part right is key to understand how adaptation{1,2} applies to training, so let’s dive into it.

Adapted for what?

The notion that adaptation is a ‘good’ thing comes from a half-baked analogy between physical fitness and reproductive fitness.

Adaptation has a clear-cut sense in evolutionary biology. On the surface, it does not correspond perfectly with either adaptation1 or adaptation2:

[A] biological adaptation is an anatomic structure, a physiological process or a behavior’s trait of an organism that has been selected by the natural evolution under the assumption that such traits increase the probability of reproduction of an organism.

Martin H, De Lope & Maravall (2009)[1]

We won’t need some adaptation3 though because the above is, in fact, adaptation2 in disguise: once the adaptation (structure, process, or behaviors) has been selected by natural evolution in a given species, the response of organisms belonging to that species to their environment becomes minimal: they no longer need to change anything to survive. At least, until a new environmental change.

The half-baked analogy between biological adaptation and adaptation to training shows up in the first pages of the otherwise remarkable textbook by Zatsiorsky & Kraemer, Science and Practice of Strength Training. Under the heading Adaptation as the main law of training (p. 3) Z&S note that athletes adapt to changes in their training in the same way as organisms adapt to changes in the environment and propose a different definition for ‘adaptation’:

In a broad sense, adaptation means the adjustment of an organism to its environment. If the environment changes, the organism changes to better survive the new conditions. In biology, adaptation is considered one of the main features of living organisms.

Zatsiorsky & Kraemer, 2006

The idea that “the organism changes to better survive” is misleading. First, as I mentioned above, the process does not occur at the level of individuals, but at the level of species: that’s where natural selection operates. Thus, and without going into too many details of biological evolution, “better survive” means, in a biological context, survive long enough to spread its genetic material thereby guaranteeing that a biological adaptation2 will persist in the species. And that’s why the male European mantis (mantis religiosa) is superbly adapted: it keeps shagging a female who’s eating its head.

Praying Mantis
Superbly adapted. Still dying.

An in-depth discussion of how incorrect is Z&S’s analogy belongs to armchair philosophy. It would be intellectually stimulating but unlikely to contribute to anything practical. To cut that part of the story short, occurrences of ‘fittest’ in survival of the fittest and in fittest on earth are a case of homonymy rather than synonymy: they have different meaning, which breaks down the analogy. I’m tempted to use subscripts but there’s enough of that already, so let’s stick to the expressions:

  • “Survival of the fittest”. In this context, (reproductive) fitness is the ability of an individual to disseminate their genetic material, thereby guaranteeing that future generations of the species will keep carrying said genetic material.
  • “Fittest on Earth”. In this context, (physical) fitness is the ability of an individual to perform better than others at the CrossFit® games, thereby guaranteeing that sponsor will keep them on the payroll.

Note here that fittest-on-Earth fitness is neither necessary for nor sufficient to reproductive fitness: top CrossFitters athletes are good-looking, which certainly improves their reproductive fitness, but they also take copious amounts of performance-enhancing drugs which tend to disrupt hormonal balance and by the same token reproductive fitness.

Whatever holds for fittest-on-Earth fitness holds mutatis mutandis for other types of athletic fitness, be they measure by performance at the Olympics or at other championships. (Incidentally, the PED situation is at least as bad as with CrossFit ®).

Now, Zatsiorsky & Kraemer essentially let their implied notion of adaptation hanging high and dry, foregoing the discussion of its relation to training. So we might as well leave it be and just put the analogy at rest. But we can try to salvage something out of the analogy because there is a generalized notion of adaptation{1,2} that does a better job for training purposes that the biological notion, as we will soon see.

The Law of Adaptation

The Law of Adaptation (LoA) is first and foremost a law that describes the behavior of adaptive systems.

Adaptive systems, in general, are a fascinating topic but also a rather hairy one, so for all intent and purposes, the LoA will suffice in practice to define them: a system S is adaptive relative to some stimulus if and only if it obeys the LoA relative to that stimulus. If you are familiar with the notion of stimulus already, skip the fine prints below.

Systems and Stimuli: The no-math version. A system S in a given environment is anything that can be considered to be ‘in’ the environment while having its own internal state that can be considered separately from the environment. For instance, a human body has an internal state that determines some changes, e.g. hormonal changes. But hormonal changes can also result from interaction with the environment. We would say that some event E in the environment is a stimulus for hormonal change in a human body B if the odds that B transitions from state B to some other state B’ when E happens (where B and B’ are characterized by a different hormonal balance) are greater than the odds that B transitions from B to B’ in the absence of E, for instance as the result of a natural hormonal cycle.

There are two formulations for the LoA: one in vernacular English, and one in the universal language of mathematics (probability theory, to be precise). It will surprise no-one that I prefer the second, but here’s the first just in case you don’t share my preferences:

Every adaptive system converges to a state in which all kind of stimulation ceases.

Martin H, De Lope & Maravall (2009)[2]

Equivalently, an adaptive system is adapted2 to a stimulus when the odds that the system would change state in response to the stimulus are no greater than the odds that the system would undertake a similar change without the stimulus.

A simple illustration of the dynamics of adaptive systems converging to a state in which all kind of stimulation ceases can be illustrated with Conway’s Game of Life. In one interpretation of the game, dots represent ‘organisms’ that have two states (‘alive’ or ‘dead’) and react to stimuli (what happens in adjacent cells). Each organism in the game converges where a state where all stimulation ceases: it dies (disappears) or moves minimally (in the best case, coming to a dead stop, as with what happens at 0m11s of the video below).

Moving ‘minimally’. In the Game of Life, the response to the environment is to move, or stay put. The ‘state where all stimulation ceases’ should thus be the state where a dot stands still. However, that does not always happen, and that’s why I identified this state as ‘moving minimally’. The reason is that dots are imperfect representations of complex living organisms: they do not have finite lifespans (and therefore do not have ‘reproductive fitness’) and they cannot alter their own behavior. Subsequently, the dots may end up moving in infinite cycles rather than going into a dead stop. And sometimes, ‘minimally’ is the same as ‘maximally’ because the initial configuration does not converge to a stable solution. Then again, if the dots are programmed to have finite lifespans, they eventually converge to a state where all stimulation ceases, albeit trivially so (they all die eventually).

The math that underlies both definitions (and the programming of the game of life) is unpacked in the next section. It does not come biology but from an artificial intelligence, and presents a generalization of the notion of biological adaptation2 that applies to training without the need for an analogy. An example will suffice to show that the application is immediate..

Exercise-wise, the LoA translates as follows: assume that Jane Doe is an athlete training for some sport, say rowing, to fix intuitions; assume furthermore that Jane follows a training regimen that involves (unsurprisingly) a lot of rowing and some strength training. We’ll say that Jane Doe is adapted2 to her training regimen when:

  1. Jane’s rowing performance plateaus. Jane’s body does not need to become stronger or more endurant to sustain the training regimen, and so her performance ceases to improve.
  2. Jane’s risk of training-related injury or illness drops to a minimum. Jane’s body can sustain the training regimen without increased risk of breakdown either local (injury) or systemic (illness).

I don’t think that anyone would object if I said that (1) is ‘bad’ and (2) is ‘good’ and that the whole thing about adaptation is a compromise between the two. Specifically, rowing is a power-endurance sport that requires both strength, speed, and stamina, and puts the back under loads in a compromised position (flexed spine). Furthermore, like any endurance sport, high workloads depress the immune system.

Rowing puts Jane, or any rower for that matter, at a risk of injury and illness as long as she’s not adapted to it, but the risk is the price to pay for improving her performance as a rower. Most people who cross-train for general fitness face the same dilemma as Jane and should want to maintain a compromise between improving their performance in some benchmark fitness tests and lowering their risk of injury and illness.

Doing the Math I: The Law

The mathematical formulation of the Law of Adaptation is a beautiful thing, but maybe less so if you’re not into mathematics.

Before you can dive into the maths of adaptation we must look at shallower mathematical waters, namely the mathematical definition of what a stimulus is. I’ll assume here that my reader is more-or-less familiar with the notion of probability, but there will really be no number-crunching, as the point of those maths is to make the definitions as unambiguous and general as possible.

Let me first introduce the notation S⟶S’ to denote the transition of some system S from a state S to a state S’. With this, we can express:

  • Pt(S⟶S’) the probability that S transitions at time t from S to S’ on its own (for instance, as the result of its internal activity); and:
  • Pt(S⟶S’|E) is the probability that S transitions at time t from S to S’ given the event E.

We will say that E is a stimulus for S (at time t) if and only if S and E validate the following inequality:

(1) Pt(S⟶S’|E) > Pt(S⟶S’) > 0

The following example will illustrate the notion of ‘stimulus’ relative to training.

Example 1. Let S be Jane Doe; S, her current state characterized by her current rowing performance; S’, a state characterized by an improved rowing performance; and let E be a challenging but manageable training session. Assuming that E is specific enough to rowing, completing E makes it more likely that Jane will transition from S to S’. The probability that S (Jane) would transition on her own from S to S’ must be non-zero: it could result from hormonal changes caused by earlier training sessions. But E is a stimulus for Jane relative to her performance if it makes it more likely that Jane’s performance will improve if she completes it than if she ‘rides’ the effect of her earlier training sessions.

We are now equipped to express mathematically the Law of Adaptation. (1) defines mathematically a stimulus E for a system S, so all we have to manage is to find a mathematical expression for “converges to a state in which all kind of stimulation ceases”. The mathematical formulation is relativized to a given stimulus E, and goes as follow:

(2) (lim t→∞) Pt(S⟶S’|E) = Pt(S⟶S’)

In (2), “(lim t→∞)” is just a fancy way to say that we are considering an ideal time (with no end). Given that understanding, (2) reads as this: as time goes by, the probability that S would transition from S to S’ as a result of E becomes increasingly closer to the probability that S would transitions from S to S’ on its own. Or equivalently: as time goes by, the response of S to E is blunted to the point of being non-existent.

Doing the Math (II): Consequences of the Law

The first general training principle singled out by Zatsiorsky and Kraemer (Z&S) is the so-called Principle of Accommodation

Usually, ‘principle’ (which means ‘stuff that comes first’) refers to a fundamental hypothesis (here, of training theory) that is not the consequence of another underlying hypothesis. Still, Z&S hint that the Principle of Accommodation is backed by the biological process of adaptation (adaptation2) but do not explain the link. So let’s do just that.

First, let’s assume that human beings are adaptive systems relative to training stimuli. If we do so, accommodation becomes a special case of the LoA, when the system S is a human trainee, and when S and S’ are states characterized by poorer and better performance, respectively. I used this assumption in the examples above.

So, insofar as accommodation is a consequence of the LoA, it is not a ‘principle’ in the narrow sense of ‘hypothesis not backed by another’. But it’s still among the stuff that comes first when you think about how you should train.

From the Principle of Accommodation, Z&S justify their second general principle, the Principle of Overload, which states that workload must increase over time to avoid accommodation. Having defined adaptation in a precise way allows for considering Overload as the flip side of a two-sided coin that would materialize the trade-off relation between accommodation and risk of injury and illness where the latter are the typical consequences of overtraining. We now have the math to formalize the trade-off relation sketched informally.

Example 2. Let S be Jane Doe again; and S, S’, and E be as in Example 1, namely states characterized by her current rowing performance, an improved rowing performance, and a typical challenging but manageable workout). Over time, repeating E will become easier for Jane and less likely to make her transition from S to S’. Let now S” be a state where Jane’s rowing performance is the same as in S, but where Jane has contracted a respiratory tract infection (RTI). Assuming that E is specific to rowing and includes some cardiovascular endurance work, and given that cardiovascular endurance work blunts immune response, we also have:

(3) Pt(S⟶S”|E) > Pt(S⟶S”) > 0

(4) (lim t→∞) Pt(S⟶S”|E) = Pt(S⟶S”)

Jane is more likely to contract an RTI following the performance of E (3) but this probability also diminishes over time (4). Hence, Jane’s accommodation to E (2) may be ‘bad’ performance-wise, but it’s ‘good’ health-wise. Applying the Principle of Overload means substituting E’ to E, where E’ is a workout more challenging than E, such that both of the following hold:

(5) Pt(S⟶S’|E’) > Pt(S⟶S’|E)

(6) Pt(S⟶S”|E’) > Pt(S⟶S”|E)

The example of Jane easily generalizes. Below is the sketch of one such generalization, with others left to the reader.

Generalizing Jane’s example. Jane’s example generalizes to other activities than rowing and other risks than RTI. For instance, you can take S to be John Doe, a powerlifter, S (S’) a state characterized by John’s current (improved) 1RM at the Squat, Bench Press, and (or) Deadlift; E, one of John’s typical workouts; E’, a workout with a higher workload than E; and S”, a state characterized by the same performance as in S, but where John suffers from some training-related injury. Then, (1)-(6) express, respectively, that E is specific to John’s performance (1); that John would adapt to E (2); that E also increases John’s risk of injury (3), but that accommodation to E decreases that risk, too (4); and that John could expect to improve his performance with E’ more than with E, but that E’ puts him at a higher risk of injury. I leave other substitutions to the reader’s imagination.

Wrapping Up

Adaptation, properly construed, is key to understanding how training should be planned.

Ideally, the training of anyone, whether for athletic and competitive purposes or for ‘general health and fitness’ could be laid down according to the Law of Adaptation. Assuming that the trainee is characterized as a system S, this would imply the following steps:

  1. Identify the current state of performance to be improved (represented by S).
  2. Identify the desired state of improved performance (represented by S’) .
  3. Identify a variety of training stimuli that would make the transition S⟶S’ more likely to happen (represented by E, E’, etc.) and have at least a guesstimate of the likelihood of the transition(represented by the probability distribution Pt)
  4. Identify the risk associated with the training stimuli, that is, the non-desirable states that S can transition into conditional on E, E’, etc. (represented by non-desirable states like S” in (4) and (6)).

Laid down that way, training is closer to risk management than is usually realized. As long as the current state S falls short of the desired performance, any training stimulus that can bring about the desired state S’ incurs an increased risk of bringing about the non-desirable state S”.

In practice, risk management is often thought of as a balance between ‘training’ and ‘pre-habilitation’. While there is some wisdom to that, there is an alternative way to think about it: once S” has been identified, ‘pre-habilitation’ can be thought of as accommodation to harmful stimuli. In fact, tests of ‘functional’ strength-endurance (as for instance those presented in McGill’s Low Back Disorders) are precisely meant to test whether an individual is resilient enough to spinal load to avoid the consequence of cumulative trauma from everyday tasks.

Let’s wrap things up for today with one last remark: training for competitive athletes and for the general public obey the same principles. There are minimal levels of cardiovascular fitness and strengths necessary to avert risks of premature death associated with sitting (discussed here for cardio, here for strength). The diagram below represents the current hypothesis about these levels.[3]

These levels would characterize the desirable state S’ for the general population. Consequently, health policies should be much more informed by training theory than they currently are, if the goal of improving the general public’s health is to be taken seriously.


[1]^ Martín H., J.A., de Lope, J. & Maravall, D (2009), Adaptation, anticipation and rationality in natural and artificial systems: computational paradigms mimicking nature, Natural Computing, 8: 757, p.2.

[2]^ Martín H, de Lope & Maravall, (2009), p.8.

[3]^ From Strasser, Barbara, and Martin Burtscher. “Survival of the fittest: VO2max, a key predictor of longevity?.” Front Biosci Landmark Ed 23 (2018): 1505-16.

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