*This post continues the exploration of general theories of training relative to the elusive notion of recovery. (About 6.700 words, estimated reading time 31 minutes, dropping to 3.300 and 16 min. if you stick to the main text.)*

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# The Science & Bullshit of Recovery (2)

In Part 1, we saw how the bet on a theory of response to harmful stress to account for response to exercise had paid off.

Here’s a mathematical metaphor: the payoff was high in absolute value but it turned out to be negative. So in spite of its popularity with coaches and athletes, nobody really used Supercompensation Theory for research purposes and exercise science developed without unifying assumptions about response to training. But in the 1980s, a bunch of systems theorists decided that it was pretty much time to start over. The Fitness-Fatigue model was born.

This “system model” originate in the early efforts of holistic theoretical biology (“holistic” might sound suspect but really isn’t). That’s quite the upgrade from Selye’s rat-thing. Faithful to System Theory methodology, the model captures the response to exercise with just a handful of parameters. **Interestingly, neither “recovery” nor “recovery rate” are among the model’s parameters**. And that will also turn out to be an upgrade compared to Selye’s model.

Now, with all its groovy math and methodological issues, the subject is a trap for geeks. Predictably, I got trapped. I cut as much as I could from the early drafts and crammed what I could not cut in asides. My advice would be to **give the main text an interrupted first read and take a mental note of what you want to learn more about**, and then come back at it over a few days. And so, here’s the plan:

**The Fitness-Fatigue Model(s): A Tale of Two Glasses.**The first part presents two incarnations of the model: the vanilla one (with math) that does not fully do the job, and the revised one (less math) that does it but not really well.**Fitness-Fatigue vs. Supercompensation Theory: Active Recovery.**The second part pits the two models against one another to obtain a reconstruction of the notion recovery in both, and keep scores.

# The Fitness-Fatigue Model(s): A Tale of Two Glasses

The Fitness-Fatigue (FIFA) model postulates that repeated bouts of exercise have positive (fitness) and negative (fatigue) after-effects.

Fitness after-effects are beneficial to performance and fatigue after-effects detrimental to it. The FIFA model fits the data descriptively but has poor predictive power. That’s still better than Supercompensation Theory (ST) whose fit with data is poor both descriptively and predictively. Some applications map the FIFA model’s characteristic curve to findings in exercise physiology, wave out the mathematics behind the curve and use it as an interpretive device. What is lost in mathematical precision is recovered in explanatory power. Accordingly, I’ll split this section in two:

**The “mathematical” FIFA model (M-FIFA):**its parameters, assumptions, few strengths, many weaknesses, and why they matter;**The “interpretive” FIFA model (I-FIFA):**its many strengths, few weaknesses, and what would happen if the math caught up.

The M-FIFA and I-FIFA are not two different models but two ways of using the same underlying principles, one in mathematical English and the other in vernacular. I’ll use “FIFA model” *tout court* when referring to these principles irrespective of the variant of English they are expressed in.

## The mathematical FIFA (or: the half-empty glass)

Let’s start with what elementary math teacher should teach kids first but actually don’t: **mathematical English is a shorthand for vernacular English for talking about stuff that can be quantified**. Unlike vernacular English, mathematical English needs no translation in mathematical French, Chinese, or whatnot. But explaining what a mathematical English sentence means does require translation in vernacular English (French, Chinese, whatnot).

So we’d better be clear about what any given mathematical English sentence means in vernacular English in the first place before doing anything with it. Now, on occasion, **mathematical English sentences can be represented as curves** and curves as useful explanatory devices. The M-FIFA model exploits this by:

**expressing in mathematical English the impact of training* on performance***as time* goes by, under special assumptions* about training positive* and negative* after-effects;**representing the mathematical English sentence so obtained as a curve**that describes the evolution over time of a system (the athlete) according to its relevant characteristic (performance*).

In the above,** the *-marked terms are those that are mapped to hard numbers**, but we can understand the M-FIFA model through the vernacular English sentence that the model expresses in number, wave our hands at the math, flash the curve, and proceed. So, first with the sentence, which in mathematical English takes only a short line, but fill a full paragraph in vernacular (did I mention shorthand?):

**(FIFA)** Given (1) a base level of performance at time t and (2) the **sum of training impulses** between time t and time t’ (where t’ comes later than t), performance at any time t” such that t” comes later than t’ depends on: (3) how much the **positive after-effects** (fitness) of the sum of training impulses between t and t’ improve base performance level; (4) how much the **negative after-effects** (fatigue) of the sum of training impulses between t and t’ reduce base performance level; and: (5) the **time differential** between t’ and t”, because: (6) **effect (3) is smaller than effect (4) but lasts longer**.

Not worrying too much about time indexes, **there is a one-one map between numbered expressions in (FIFA) and the *-marked terms** in the bullet points, and so (by composition of the one-one maps) a one-one map between numbered expressions in (FIFA) and hard numbers. Setting the details aside (Putting numbers on FIFA) and waving a virtual ~~magic wand~~ chalk stick, the relations of those hard numbers look like the following curves on a virtual chalkboard.

Don’t fret over “main […] after-effect” and the implied meaning that there are others (I’ll come back to that in due time). Assuming that **the x-axis represents performance, the y-axis, time, and the orange block, the last training stimulus** (which is not equivalent to assuming that “no training at all” occurs after that point, I’ll come back to that too):

**the blue curve represents observable changes in performance**and (interestingly) matches the GAS/ST curve almost exactly;**the green and orange curves represent non-observable changes in performance**that ‘add up’ to yield the blue curve.

The values the green and orange curves represent are non-observables because they cannot be isolated experimentally. The main upshot is that **the FIFA model is not experimentally testable (falsifiable).** It’s not the first untestable model to be featured in this blog (nor is it the last one) and models like this may still be validated empirically [1]. I’ll leave the methodological implications of the above for an aside (FIFA Unobservables) because there are more pressing causes of concern.

One of those concerns is also raised by ST: **the M-FIFA model parameters have no explicit interpretation in the underlying physiology of the system** (the athlete). This does not preclude the M-FIFA to have good descriptive power but hinders its predictive power. This shortcoming can be partially addressed by backing down on mathematical modelling. Let’s see how.

## The Interpretive FIFA (or: the half-full glass)

One can **interpret empirical physiological data according to the FIFA model without engaging full mathematical modelling mode**. This interpretive approach turns a blind eye to the mathematical shortcomings of the M-FIFA model and rests on the claims that:

**if Supercompensation Theory can [do-x], the Fitness-Fatigue model can [do-x] better; and:****Even if Supercompensation Theory can’t [do-x], the Fitness-Fatigue model can.**

Of course, the variable [do-x] ranges over a restricted domain of natural language expression (in short: those that make (2) non-trivially true). With “impeach Donald Trump”, (1) holds but only trivially so and (2) does not hold at all. However, substituting [do-x] with “explain the delayed effect of training” or “explain why overtraining happens overnight” verifies both (1) and (2) non-trivially. I’ll assume implicitly that the domain is suitably restricted. An important caveat is that **relative to the M-FIFA ****(2) only holds in principle and not in fact**. Let’s consider an example right away.

Post-tetanic potentiation (PTP) is touted as critical supporting evidence for the FIFA model by Chiu et al. (2003): one of the motor manifestations of PTP is that a maximal voluntary contraction (MVC) immediately before an explosive effort may increase the rate of force development in that effort. **Substituting “explain the PTP effect of MVC” to [do-x] in (1) and (2) yields two true statements**: the first is trivially true, because the PTP-MVC effect is a complete mystery for ST (thus, the antecedent of (1) is false) but that’s what makes (2) non-trivially true.

The PTP-MVC after-effect has (in principle) a straightforward explanation in the FIFA model: the fatigue generated by an MVC dissipates fast enough for its fitness after-effect to be perceivable instants later. However, in practice, **the M-FIFA cannot deal with the PTP-MVC after-effect without collapsing into triviality** (see Trivializing Adjustments, A). A possible solution is to upgrade the model from 2 factors to 2*n* factors, that is, multiple pairs of performance modifiers and their associated decay rates.

In the PTP-MVC case, total workload for the explosive task and MVC would be, respectively, the main and secondary determinants for performance. A generalization of the FIFA model compounding multiple after-effects for multiple determinants would yield Fig. 2. below; but **absent mathematical modelling to support it Fig. 2 is a mere interpretive device** to account figuratively for compounded fitness and fatigue after-effects, which is what brings us into I-FIFA territory for good.

The PTP-MVC after-effect is, however, a double-edged sword. As argued by Chiu et. al. (2002) it demonstrates the existence of positive training after-effects, validating the FIFA model with a highly desirable link to physiology (something ST never properly achieved, cf Part 1, Punch two: the reduction of “preparedness”). But** the PTP-MVC after-effect also depends on training age and it’s an issue for both the M-FIFA and I-FIFA approaches.** The tentative explanation (that more athletic individuals have a different response to the training stimulus) is acceptable in the I-FIFA but the means to account for it in the M-FIFA may trivialize it (cf. Trivializing Adjustments, B).

Because of that, it does not do much good that the I-FIFA model can handle the PTP-MVC after-effect: **conceptual adjustments in the I-FIFA are only ok if the M-FIFA model is eventually revised to reflect them**. The case of the PTP-MVC after-effect shows that this revision may prove too a big a bullet to bite for the FIFA model, after all (further arguments in The Crude Math of FIFA).

# FIFA vs. ST: Active Recovery

The FIFA model has its shortcomings but they are nothing like those of the GAS/ST approach.

Here’s an ageist metaphor: if the FIFA’s problem is adolescence, then GAS/ST’s problem is senescence. ST is an old fart with delusions of relevance caused by people who use words like “recovery” and “supercompensation” out of respect but without really meaning it. FIFA is the ambitious teenager not yet sophisticated enough to take over the old fart but who’s attempting to anyway, and whose first order of business is not to sound like the old fart. The gist of the metaphor is that, at the end of the day, what ST and the FIFA model do with recovery is a matter of words. And since we are talking theories, matters of words are both of definition and of empirical correlate, so let’s proceed with the following plan:

**Beyond the curves.**“Recovery” is a primitive notion is GAS/ST without clear empirical correlate and no direct equivalent in the FIFA model.**Training to recover?**“Active recovery training” illustrates the divide between ST (where it has a dull interpretation) and the FIFA model (where it has a fascinating one).

## Beyond the curves

To put it in as few words as possible, the differences between ST and the FIFA model amount to this:

**ST is a curve**that: (1) modifies Selye’s GAS to account for the delayed effect of training based on the assumption that “training is stress”; and: (2) has only iffy empirical interpretations.**The FIFA model is a curve**that: (1) represents a function that

accounts for the delayed effect of training based on the assumption that training has positive and negative after-effects; and: (2) has some decent empirical interpretations.

“Recovery” is not always listed among the theoretical terms of GAS/ST but other terms are just synonyms: **recovery is “resistance” (GAS) and “restitution” (ST) by another name and is thus a primitive of the theory**. Let me abbreviating recovery-according-to-GAS/ST as (re-) for re◠*x* where “◠” is concatenation and x∈{covery, sistance, stitution, storation} (all appear in GAS/ST contexts).

The concept of (re-) may seem less problematic than the concept of supercompensation: it is reminiscent of the common sense concept of “rest” and the expression “getting one’s strength back” and it has known physiological correlates (although they do not supercompensate). However, **equating (re-) with rest is incorrect**: together with the Law of Adaptation, ST entails that some loads would provide no training stimulus, cause no significant depletion (of whatever is depleted) and therefore require no significant (re-) (see No Training Stimulus).

Now, flashing two summaries of ST and the FIFA model side by side shows that **the FIFA model re-interprets the Supercompensation curve** as the result of the composition of fitness and fatigue after-effects. It’s more conspicuous in Fig. 1 above, because Zatsiorsky & Kraemer skirt the tricky issue of scientific reduction with a superficial visual difference, but we’re doing Analytic Fitness™ here so let’s tackle it.

But the FIFA model does not feature any concept that immediately translates (re-). In particular, **the return to ‘baseline’ fatigue cannot alone translate (re-)** since fatigue after-effects are overcome by fitness after-effects before they dissipate entirely. The point is worth rephrasing: in the FIFA re-interpretation of the ST curve, the training effect mapped to supercompensation occurs before fatigue is fully dissipated. If you get that, you’ll get what comes next.

## Training to recover?

In ST, the interpretation of the vernacular English “active recovery training” as “active (re-)” is dull: **whatever workload that causes no significant depletion is “active recovery”**: it lets (re-) runs its course. Assuming furthermore that “active recovery training” aims at avoiding detraining, the workloads that fit the bill of “active recovery-*qua*-(re-) training” are Zatsiorsky & Kramer’s “retaining loads” (whatever the load is: there are options in the market that can fit any type of exercise).

Since it reinterprets ST, **the FIFA model supports retaining loads as active recovery measure** even without interpreting “recovery” as (re-). But it also supports a much more exciting interpretation. Before getting there, let’s re-interpret the theory-neutral caption of Fig. 1.1. in FIFA terms:

**a “detraining load”**causes negligible fatigue and fitness after-effects insufficient to maintain the initial level of performance.**a “retaining load”**causes little noticeable fatigue and fitness after-effects sufficient to maintain performance but insufficient to raise it.**a “stimulating load”**causes noticeable fatigue and fitness after-effects sufficient to raise performance when fatigue after-effects are overcome by fitness ones. [4]

But **what if stimulating loads could sometimes be used as “active recovery”?** The possibility may seem too good to be true but only if putting stock in ST (where a load is stimulating iff it causes depletion and supercompensation down the road (cf. No Training Stimulus). In fact, we have already encountered “stimulating loads” with a recovery effect with the PTP-MVC after-effect: a stimulating load whose fatigue after-effect is short-lived enough for the fitness after-effect to improve performance almost instantly.

“But,” I hear you thinking, “how does that amount to recovery?” Well, I’ll let you figure that one by yourself as a test of your understanding of the FIFA model (Hint: a 2*n* factor model is necessary). But if you’re ~~a lazy fuck~~ mentally exhausted by what you’ve read so far, the answer is below When PTP-MVC is “recovery”. But since this scenario is actually possible, it’s **a reason to adopt the FIFA model (under its I-FIFA guise) rather than ST** even though the model is incomplete.

# Conclusion: Recovery Training,

FIFA 1 : ST 0

The effect of MVC is enough to claim that exercise may help with recovery even under residual fatigue.

Of course, that’s because all we need to make true a possibility statement (featuring “may”) is one case. Then again, the PTP-MVC effect is transient and short-lived, and a more interesting question is whether the following scenario is possible at all :

**Day 1:**Athlete A performs a workout that predictably causes her performance at some task*t*to fall below baseline p* for at least 3 days.**Day 2:**A performs a different workout than on Day 1 that also causes her performance drop even further below p* on Day 2 than on Day 1, but whose fatigue after-effect will predicate dissipate before the Day 3 workout.**Day 3:**A tests her performance at task*t*, and in spite of the residual fatigue of Day 1, increases her performance compared to Day 1.

This scenario is simply the PTP-MVC story stretched over days instead of minutes. If established, **the existence of such a scenario would validate a concurrent training protocol** where different types of training performed on different days compound to improve performance under fatigue.

As it turns out, there is evidence that such scenarios exist. Chiu et al. (2003) propose **multiple examples of periodization based on compounding fitness after-effects** for FIFA-based periodization both short term (same-day multi-session, and weekly) and long term (monthly, yearly) based on the notion that ‘it is better to have high fitness with moderate fatigue, rather than moderate fitness and low fatigue” (p. 45).

I’ve been long enough already and this pertains to periodization which I’ll address in future posts. In the meantime, I’ll conclude that **the FIFA model turned recovery upside-down** and pushed it further away from the naive equation *recovery=rest* that a superficial reading of ST had let survive way past its time.

And as Jayne Cobb would say,