*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.)*

# 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.

**Putting Numbers on FIFA.** Transforming (1)-(6) into hard numbers that can be operated on is not very difficult but there are a few constraints and some limitations. But first, let’s look at the list which introduces the common mathematical English names for these parameters in brackets after their vernacular English names.

**Baseline performance (p*)**. In FIFA studies, baseline performance is typically defined by some quantifiable athletic performance (fastest 100-m run, longest hammer throw, powerlifting/weightlifting total) and is tested at t, that is, before application of the first training impulse (t’).**Sum of training impulses (∑**The training impulse at time i (^{t”-1}_{i=t’}w_{i}).**w**) is typically expressed as a total workload (number of meters run at full speed, number of hammer throws, total weight moved in a session). Notice that time is “discretized” and that every point in time between t and t” corresponds to one day of training (or rest). The total workload is the sum of all workloads for days from t to t”-1, the day before the final performance test performed on day t”._{i}**Positive after-effect (k**The positive after-effect or fitness effect is represented by a positive real number that multiplies the sum of training impulses (more about that below)._{1}).**Negative after-effect (k**The negative after-effect or fatigue effect is represented by a negative real number that also multiplies the sum of training impulses (more about that too below)._{2}).**The time differential between t’ and t”.**This one needs no explicit expression since it is implicitly accounted for in how (2) and (6) are obtained.**Decay rates of after-effects (τ**In (FIFA) (6) handles both comparative magnitudes and decay rates for training after-effects. Since k_{1}and τ_{2})._{1}and k_{2}already give us magnitudes, all that is left are decay rates. The decay rates modify the result of multiplying the training impulse by k_{1}(τ_{1}) and k_{2}(τ_{2}) reducing the impact of the multipliers as time passes after the last training impulse, with τ_{2}translating a faster decay than τ_{1}.

There are some minutiae in the relations between the above parameters. For instance, k_{1} being positive and k_{2} negative, it is always the case that k_{1}>k_{2} , so the hypothesis that negative after-effects exceed in magnitude positive after-effects translates in absolute values, namely that |k_{2}|>|k_{1}|. Other than that, there’s not much left to say. I won’t introduce the actual formula yielding the curve of Fig. 1, because the mathematical ~~tricks~~ operations used to translate how the parameters work together would be a distraction, and a hybrid between vernacular English and mathematical English suffices, namely: the performance p(t”) at t” is a sum of the following terms:

- the base performance p*,
- the positive effect of the total workload as perceived at t” (the sum of total workload, multiplied by k
_{1}and modified by τ_{1}, the latter depending on how far in time t” is from the last fitness-inducing training impulse), - the negative effect of the total workload as perceived at t” (the sum of total workload, multiplied by k2 and modified by τ2 the latter depending on how far in time t” is from the last fatigue-inducing training impulse).

Clearly, the closest in time one is to t (at which p* was measured before training impulse was applied), the more pronounced the fatigue and the less pronounced the fitness effects, resulting in loss of performance compared to p*. Conversely, as one gets farther from the last training effect, fatigue decays faster, and fitness effects become perceivabl (although are less pronounced than earlier, where they were still masked by fatigue).

**FIFA Unobservables.** Changes represented by the green and orange curve are not observable because it is not possible to control experimentally the effect of fatigue and fitness. In order to do so, one would need to be able to train a bunch of people with the same program for a while, then let them rest, and from that point divide them into three groups:

**a control group,**whose performance would be tested, and expected to follow the blue curve of Fig. 1**a “fatigue” group**, in which fitness after-effects would be canceled, and whose performance, when tested, would be expected to follow the orange curve;**a “fitness” group**, in which the fatigue after-effects would be canceled and whose performance, when tested, would be expected to follow the blue curve.

While not impossible in principle, canceling the fitness or fatigue effects is beyond the reach of exercises science of today. Even if it became possible in practice, for instance, by canceling chemically the effects of fatigue, the theory would still be the reference for evaluating how much fatigue would have to be canceled, leading to underdetermination problems (see SBL (I): Science Basics (1); the same would go *mutatis mutandis* for fitness). Subsequently, the only way to obtain the green and orange curve are mathematical manipulations of the blue curve under the assumption that the model is broadly correct which in practice entails that the values of parameters k_{1}, k_{2}, τ_{1} and τ_{2} are estimated in order to fit the data, under the constraints that |k_{2}|>|k_{1}| and that τ_{2} translates a faster decay than τ_{1}. When such parameters are found, the model is validated, but not tested in such a way that it could be confirmed or falsified.

## 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).

**Trivializing adjustments.**The M-FIFA model has a generality problem, because parameters of the model are often adjusted to match the data in case studies (see this 1994 study on a hammer thrower) and case studies are how scientists bypass pesky demands for generality and statistical significance **[3]**. But they are adjusted in accordance with the assumptions of the model to fit observations, and that’s ok, as long as it is understood that the model will eventually be revised to derive these values deductively. Incidentally, that’s also why Ptolemaic epicycles where good science after all. Then again, Ptolemaic astronomy was eventually superseded by Copernican astronomy after the revision had failed to happen. Below is an argument to the effect that failure threatens the M-FIFA model upon attempting to revise it to account for the PTP-MVC after-effect within the current mathematical framerwork of the model.

**A. MVC after-effect.** Assume the standard M-FIFA model with its standard 5-uple of parameters (p*,k_{1}, k_{2}, τ_{1}, τ_{2}), its standard variables t,t’,t”, for (discretized) time and total workload ∑w_{i}. Assume now two seasoned athletes, A and B, who have so accrued the same box-jump workload (abbreviated w_{t-1,A}=w_{t-1,B}) after having demonstrated the same base box–jump performance (abbreviated p*_{A}=p*_{B}). Let us assume that the performance of both athletes is tested at t, and that right before t, A has performed an MVC, and B hasn’t. Assume that scientist S gains access to the data set for A’s training and performance p_{t,A}, but neither the data set for B’s training nor B’s performance p_{t,B}. If S infers values for k_{1}, k_{2}, τ_{1}, τ_{2} under the assumptions of the M-FIFA, that match the data set for A’s training and performance p_{t,A} and then tries to predict the performance p_{t,B} of B at t, her prediction for B’s performance will be too high. Conversely, if she accesses the data set for B’s training and performance p_{t,B} first, and goes through the same steps, her prediction for B’s performance will be too low. The only way is to index k_{1}, k_{2}, τ_{1}, τ_{2} to A and B and set k_{1,A}>k_{1,B}, k_{2,A}=k_{2,B} (because the fatigue cost of MVC is too short, and thus negligible), as well as τ_{1,A}≠τ_{1,B} to reflect that A’s performance will degrade slower due to A’s increased fitness (the direction of the inequality is irrelevant, but to fix intuitions, decay rates can be specified as values that are ‘chipped out’ over time, in which case τ_{1,A}>τ_{1,B}), and τ_{1,A}=τ_{1,B} to reflect that there is no specific fatigue associated with the MVC. This, however, is tantamount to introducing implicitly another 4-uple k’_{1}, k’_{2}, τ’_{1}, τ’_{2} and compounding theses parameters with k_{1}, k_{2}, τ_{1}, τ_{2} for A, but not for B. Thus, the model is now explicitly 2-factor but implicitly 4-factor wich is still a minor injury.

**B. MVC and training age.** Let’s add now insult to injury, and assume that a new data set about a beginning (but gifted) box-jump athlete C becomes available, and to simplify utterly, that C initial performance matches A’s and B’s; has followed the same box-jump regimen as A and B; and has performed an MVC before the test on day t. Assume that scientist S accesses all the data available about C, minus C’s training age and actual test result. Based on the data, S predicts that p_{t,C}=p_{t,A} because she expects that (k_{1,C}, k_{2,C}, τ_{1,C}, τ_{2,C})≡(k_{1,A}, k_{2,A}, τ_{1,A}, τ_{2,A}). However, by our assumptions, p_{t,A}>p_{t,C}. Adjusting k_{1,C}, k_{2,C}, τ_{1,C}, τ_{2,C} is not equivalent to introduce a 3rd pair of factors but tantamount to introducing a modifier that affects the 2nd pair of factors. If we call the 4-uple (k_{1,I}, k_{2,I}, τ_{1,I}, τ_{2,I}) “first-order” parameters, then the new modifier is a “second-order” parameter modifying the first-order ones. In the worst case scenario, every new individual added to the population {A,B,C} comes with a new implicit pair of factors, or a new second-order parameter and the only generality left is that for any I, the relations between k_{1,I} and k_{2,I} and between τ_{1,I} and τ_{2,I} are characterized by inequalities. It cannot even be assumed that for all I, k_{1,I}<k_{2,I} because if enough first- and second-order parameters are integrated that have the same effect as PTP-MVC, the compound effect may make it look like fatigue after-effects are of lower magnitude than fitness after-effects. Clearly, trying to accommodate effects such as PTP-MVC incurs a risk of collapsing the M-FIFA model into triviality. On the other hand, aiming for a 2*n*-factor model with second-order parameters would collapse the methodology of the FIFA approach into that of the analysis of Complex Adaptive Sytemssee below The Crude Math fo FIFA (an aside to an aside, how geek is that?).

**The Crude Math of FIFA.** System theory originates in the efforts of biologist Karl Ludvig von Bertalanffy to model mathematically whole systems rather than their parts. Bertalanffy proposed in the 1930s a simple mathematical model of the growth of an animal given only two parameters (the animal’s age, and a growth coefficient). About 30 years later, Bertalanffy generalized the methodology to arbitrary systems under the moniker of *General System Theory* and published a book with that title in 1968. General System Theory had a raw deal you can read about in Wikipedia, being for a time considered pseudoscience by a bunch of influential people.
This criticism is not totally undeserved but is so by association only: many over the years have claimed to develop “system models” without having a clear idea of what a system model is (not unlike people who talk about “blockchain” without any idea of what a blockchain is). Nevertheless, Bertalanffy’s ideas spread among applied mathematicians, and System Theory reached a peak in popularity around the 1980s. The hallmark of system models is the appeal to few parameters and possibly sophisticated functions (although Bertalanffy’s function is not that sophisticated) but not all systems can actually be modelled that way. And so General System Theory’s ideas found a new incarnation in Chaos Theory and the theory of Complex Adaptive Systems. Nobody complained about pseudoscience when they did.

So in the 1990s, mathematicians expanded the mathematical foundations of System Theory to better capture systems that are sensitive to small perturbations in initial conditions, in what would become known as Chaos Theory. Thanks to the efforts of overenthusiastic and undereducated publicists (including the screenwriters of the otherwise excellent Jurassic Park [1996]) everybody would come to believe that it was about butterfly flapping their wings in China and causing hurricanes in Texas, while it’s actually about how much time it would take for the perturbation introduced in the Earth climate system to cause a hurricane in Texas and the description of the evolution of the Earth climate system in between the butterflying and the hurricaning [2].

Not all dynamic systems are chaotic though, and during the 2000s chaotic systems began to lose the popularity contest to Complex Adaptive Systems (CAS). CAS are collections of individuals entities that self-organize in response to changes in their environment and/or interaction with other entities. The collection of individuals can be human or non-human animals (anthills are popular examples) but also self-organizing parts of single biological systems, like the human’s brain (and central nervous system) and the human’s immune system. Given that an athlete’s performance depends on neurological adaptations and modulation of the immune response (the latter for sure with endurance athletes and possibly for strength athletes as well), it is no surprise that the simple math of the FIFA model fails miserably at predicting performance: the M-FIFA is way too crude to factor in self-organization of an athlete’s neural and immune sub-systems. Too bad that it is precisely the kind of effect that the I-FIFA is betting on to validate the model.Indeed, the tentative explanation of the PTP effect of MVC on power development summed up in Chiu et al. (2003) makes clear (2nd paragraph, p. 44) rests on long-term physiological adaptations, some of which are neurological. Theses neurological adaptations entail that the same training impulse would not have the same k_{1}, k_{2}, τ_{1} and τ_{2} depending on training age. But these adaptations are also the result of the auto-organization of the athlete’s nervous system to respond to repeated training impulses. Thus, the I-FIFA is actually betting on the M-FIFA to become a full-vanilla CAS-model but it’s tantamount to a bet against the M-FIFA. It is indeed equivalent to advocating a sophisticated type functional model that would take as input physiological responses to types of exercise and would output predictions about future performance. And such functional models are, in facts, alternatives to the system-model approach of the M-FIFA (cf. Taha & Thomas (2003), “Systems Modelling of the Relationship Between Training and Performance“, *Sports Medicine*, **33**(14), pp. 1061-1073). Notice in passing that this proposal comes full circle with Bertalanffy’s first “system model” as the size of animals is actually a physiological parameter, although it depends in turn of underlying physiological processes that should ultimately be described by a full CAS-model.

# 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.

Zatsiorsky & Kraemer (2006) Science & Practice of Strength Training, Second Edition, p. 10

[Click to enlarge]Zatsiorsky & Kraemer, Science & Practice of Strength Training, Second Edition, p. 13

[Click to enlarge]

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.

**No training stimulus.** The absence of stimulus does not entail the absence of training, provided that the training is *not* a stimulus (for instance, if it is below what the athlete is accommodated to). Both ST studies and M-FIFA studies often include periods of “reduced training” where performance is tested (data points for the ST/FIFA post-training curve) but the workload is kept below what would induce significant drops of performance. The ST study (Hartman et al, 2008) discussed in Part I, Punch two: the reduction of “preparedness” included several weeks of “active rest training” to get the hormonal profile back to baseline (and hopefully, to supercompensate). The M-FIFA study discussed in footnote [3] also included reduced training to dissipate fatigue.

Assuming the FIFA model rather than ST (more on the reasons to do so in the next section) determining how much training does not cause significant fitness or fatigue after-effects is rather straightforward. Since the magnitude of the fatigue after-effect is assumed to exceed that of the fitness after’effect, one can use the fatigue after-effect as a guide. Then, finding the workload to which an athlete is accommodated is an iterative trial-and-error process: (1) test performance on day *d* and then apply workload *w* of ; and (2) on day *d*+1, check if performance has dropped. If it has not, (3a) keep applying *w*. If it has, (3b) rest long enough for performance to go back to *d*-level, then repeat (1)-(2) with a lower workload. Alternatively, (3a*) repeat (1)-(2) with incrementally higher workloads until (3b) must be applied, to find the highest possible dose the athlete is accommodated to. It is not guaranteed by the model that a workload to which the athlete is accommodated fatigue-wise does not produce a fitness after-effect. However, since the magnitude of the fitness after-effect is assumed to be lower, and the fitness after-effect is masked by fatigue, the absence of a drop in performance can be taken to indicate that there should be no observable positive after-effect (the full reasoning is by *reductio*, and left to the reader).

The process just described above works best with training stimuli that have relatively long fatigue after-effects (a day or more). With a stimulus like MVC, the decay rate of fatigue would require trials so close to one another that the total workload of performance attempts (rather than the isometric contraction) would end up causing significant fatigue after-effects. In practice, this is of no consequence, since there would be little point in finding which SMVC (sub-maximal voluntary contraction) does not cause fatigue nor improvements in the rate of force production, because nobody ever cared about submaximal isometrics.

## 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.

**When PTP-MVC is “recovery”.** Assume a seasoned box jumper B with a base performance p*=h on day 1. On day 2, B trains with a workload above her retaining load. On day 3, her performance p_{3} is tested first thing in the workout and measured as p_{3.0}=h-i, indicating that the fatigue after-effect of the training of day 2 is not yet overcome by the fitness after-effect of the training of day 2. Then B performs a MVC and her performance is tested again and measured as p_{3.1}=h-j. Ex hypothesis, B is a seasoned box jumper; therefore, we can safely assume that i>j. From there, the reasoning is by cases with 3 cases to consider:

**j>0 and p*>p**the fitness after-effect of the MVC is not enough to offset fatigue after-effect of day 2_{3.1}:**j=0 and p*=p**the fitness after-effect of the MVC is enough to offset fatigue after-effect of day 2, but not to raise performance above baseline._{3.1}:**j<0 and p*<p**the fitness after-effect of the MVC is enough not only to offset the fatigue after-effect of day 2, but also to raise performance above baseline._{3.1}:

In **case 1**, B has partially recovered her level of baseline performance. In **case 2**, B has fully recovered her level of baseline performance. And in **case 3**, B has not only recovered her level of baseline performance, but improved upon it. Hence, in all three cases, there is an X such that **“B has recovered X”** is a true statement. Furthermore, since ex hypothesis the fatigue after-effect of the training of day 2 is not yet overcome by the fitness after-effect of the training of day 2 recovery so construed happens under residual fatigue and the notion of recovery just constructed has no counterpart in ST.

# 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.

it is better to have high fitness with moderate fatigue, rather than moderate fitness and low fatigue

Chiu et al. (2003:45)

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,

### Notes

**[1]** The Cholowicki-McGill model of the *in vivo* spine discussed in ~~Trick~~ Train Your CNS, Get Stronger can be validated empirically, but cannot actually be tested (falsified) because it incorporates hypotheses about unobservables. More generally, the naive view that a theory must be falsifiable is not tenable in practice, since the failure of a prediction can always be blamed on auxiliary assumptions, instruments, etc., a problem known as underdetermination (cf. SBL (I): Science Basics (1)). Consequently, the lifespan of a theory does not depend on whether its predictions are falsified or not, but of how many hits it can take before the scientific community decides that it should be shelved (cf. SBL (II): Science Basics (2) which incidentally took ST as an example of falsified-but-still-not-shelved theory).

**[2]** I’ve read a lot of pop-sci on Chaos Theory, but this blog post by Geoff Boeing (which I got from the footnote section of the Wikipedia page) is the shortest and clearest explanation I’ve seen of what “butterfly effect” can refer to. It’s probably derivative on other works, as popsci is, but at least it emphasizes the right thing, namely that today’s Texans need worry about today’s Chinese butterflies because they will be long dead before any flapping of Chinese butterfly wings affects Texas climate.

**[3]** The case study (Busso, Candau & Lacour (1994), “Fatigue and fitness modelled from the effects of training on performance“, *European Journal of Applied Physiology*, 69, pp. 50-54) actually boast statistically significant correlations between fitness and fatigue after-effects and performance, even with a sample-of-one study. That is because the data set is actually the set of performances of their single hammer thrower, and because the goal of the study was to adjudicate between two types of functions for the estimation of fitness and fatigue after-effects. Accordingly, what was shown was not that the model predicts the performance of hammer throwers in general, but that one way to cash (FIFA) mathematically correlated better than another way to cash (FIFA) mathematically with the performance of a single hammer thrower. Furthermore, the fitness and fatigue after-effects being unobservables (see FIFA Unobservables) this study can only count as a validation of the model, not as an attempt-at-falsification-that-went-well-after-all.

**[4]** The first two bullet points should also be completed with “when fatigue after-effects are overcome by fitness ones”. But the precision ~~is utterly pedantic ~~ can be omitted without loss of generality since the fatigue effects are negligible in the first case, and accommodated to in the second.