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A mathematical model of neuromuscular adaptation to resistance training and its application in a computer simulation of accommodating loads

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Abstract

A large corpus of data obtained by means of empirical study of neuromuscular adaptation is currently of limited use to athletes and their coaches. One of the reasons lies in the unclear direct practical utility of many individual trials. This paper introduces a mathematical model of adaptation to resistance training, which derives its elements from physiological fundamentals on the one side, and empirical findings on the other. The key element of the proposed model is what is here termed the athlete’s capability profile. This is a generalization of length and velocity dependent force production characteristics of individual muscles, to an exercise with arbitrary biomechanics. The capability profile, a two-dimensional function over the capability plane, plays the central role in the proposed model of the training-adaptation feedback loop. Together with a dynamic model of resistance the capability profile is used in the model’s predictive stage when exercise performance is simulated using a numerical approximation of differential equations of motion. Simulation results are used to infer the adaptational stimulus, which manifests itself through a fed back modification of the capability profile. It is shown how empirical evidence of exercise specificity can be formulated mathematically and integrated in this framework. A detailed description of the proposed model is followed by examples of its application—new insights into the effects of accommodating loading for powerlifting are demonstrated. This is followed by a discussion of the limitations of the proposed model and an overview of avenues for future work.

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Notes

  1. For nonlinearly elastic bands the equation can be modified according to its force profile, e.g. using empirical data obtained as in (Shoepe et al. 2010).

  2. Note that regardless of the lift considered, x = 0 is taken to correspond to the bottom-most position in the lift. This allows for clearer formulations of relevant equations and does not affect any of the underlying physics. However, it should be kept in mind that this means that when the weight of the fixed load is referred to, the initial pull of the accommodating loading is implicitly added to that of the actual fixed load (usually a barbell and plates).

  3. Throughout the paper the same notation will be adopted to refer to the maximal load which can be used for a set of multiple repetitions. Thus nRM signifies the greatest load which can be used for n consecutive repetitions

  4. Note that this does not mean that the effects of fatigue on performance are small but rather its differential effects due to adaptation to high intensity strength training.

  5. Practical consequences of this choice on training program design and the detection of an athlete’s weaknesses are discussed in “General discussion”.

  6. For a discussion on the interpretation and nature of this limit see “General discussion”.

  7. Note that an athlete with a mid-range weakness in terms of force production can nonetheless have an end of the ROM weakness in terms of actual performance (or, in other words, that an athlete experiences failure near the top of a lift with a marginally supramaximal load), as is indeed the case in this experiment. This seemingly paradoxical fact can be explained by observing that high force production in the early stages of a lift can supply sufficient momentum to the load to overcome transient dips in force production.

  8. Note that the term “constant resistance” (or “constant load”) is used to refer to non-progressive training, i.e. one in which loading parameters are unchanged over time. “Fixed resistance” (or “fixed load”), on the other hand, implies the absence of any accommodating resistance.

  9. Note that l was used instead of x to denote elevation as a means of emphasizing that unlike x and \(\dot{x}\), l and v are independent integration variables. Thus \(\dot{l}\) is not equal to v—indeed, neither l nor v are functions of time.

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Acknowledgments

The author would like to thank Trinity College Cambridge for their kind support of this research. My gratitude also goes to the anonymous reviewers whose suggestions were invaluable in improving the clarity of presentation.

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  1. Trinity College, Trinity Street, Cambridge, CB2 1TQ, UK

    Ognjen Arandjelović

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Correspondence to Ognjen Arandjelović.

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Communicated by Fausto Baldissera.

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Arandjelović, O. A mathematical model of neuromuscular adaptation to resistance training and its application in a computer simulation of accommodating loads. Eur J Appl Physiol 110, 523–538 (2010). https://doi.org/10.1007/s00421-010-1526-3

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