The game motoneurons play

We apply a Game Theoretical approach to analyze the competition between motoneurons (MNs) innervating a common muscle. A typical skeletal muscle consists of many thousands of fibers. At birth each muscle-fiber is innervated by several MNs, but during the first couple of weeks after birth, a competitive mechanism called “synapse elimination” abolishes all inputs but one, which we term “the winner at the muscle-fiber”. At birth, each MN innervates many muscle-fibers and therefore it engages in many competitions simultaneously, winning at some muscle-fibers and losing at others. The group of muscle-fibers that are eventually innervated by the same MN is called a “muscle unit”. In the adult system, MNs with successively higher activation-thresholds have successively larger muscle units. This is called “the size principle”. The size principle is well established experimentally, and is thought of as one of the most fundamental principles in the organization of motor-unit behavior. Therefore it is important to understand how the size principle evolves. In viewing the period of synapse elimination as a game in which MNs are competing to innervate a maximal number of muscle-fibers, the translation of the size principle is that less-active MNs (i.e., MNs with higher activation thresholds) win in more competitions than more-active MNs. However, different experiments that have selectively manipulated the activity of MNs, led to opposite views regarding whether more-active or less-active MNs have a competitive advantage in this process. This has been termed a paradox by several authors. Using strategic considerations we prove that less-active MNs indeed win the game (i.e., win in more competitions). This explains the emergence of the “size principle”. Using the same considerations, we resolve the paradox of seemingly contradictory experimental data described above. The main idea is as follows: On the basis of the experimental data, we conclude that more-active MNs are more involved in early competitions and lessactive MNs are more involved in later competitions. Thus the winner at early competitions is more likely to be more-active, and the winner at late competitions is more likely to be less-active. We prove that as the resource of a MN is limited, restricting the number of connections it can maintain, it is better to win in later competitions (as less-active MNs do), in order to win in more competitions. Thus the less-active MNs, which start winning only in later stages of the game, win in more competitions, and so the size principle emerges. Although there is no doubt that electrical activity plays a role in synapse elimination, researches disagree on its importance or centrality in the competition between MNs innervating the same muscle-fiber. Our proof shows that either way, (excluding only cases in which the competition is extremely biased in favor of moreactive MN), less-active MNs win the game. We believe that this work shows the necessity of using game theoretical approach, when analyzing a competitive process in the nervous system. First, it provides an appropriate framework for thinking about competition: Our game theoretical result; that the time of winning has a competitive value enabled us to explain the emergence of the size principle and to resolve the paradox of contradictory experimental data. Secondly, using our game theoretical approach may lead to conclusions about a competitive process, even when the mechanisms underlying the competition is not fully understood or is under debate, as happens in the competition at the neuromuscular junction. Our work explains the emergence of the size principle as a result of competition and thus suggests that the size principle may be adaptive (fine-tuned) to its environment. The evolutionary advantage of properties as the size principle, emerging as a consequence of competition rather than being genetically hardwired, is that it endows the system with plasticity (or adaptation capabilities), such that the outcome may be fine-tuned to fit the environment. In accordance with this idea, the model provides several new predictions regarding the degree of expression of the size principle in different muscles. The rules of the game we defined are based strictly on experimental results, thus providing testable new predictions. At the same time, these rules are easily generalized so as to be applicable to a much wider scope of situations. As explained, more-active MNs tend to win earlier and less-active MNs tend to win later. Thus a more-active MN can be viewed as “investing in early competitions”, and a less-active MN can be viewed as “investing in late competitions”. The general strategic conclusion which follows from this game-theoretic result, and may apply also to nonbiological scenarios, e.g., economical settings, is that when resources are limited (the limitation being as in our setting), one should invest in later competitions in order to win in more competitions. All the results in this work are proven mathematically, and are additionally illustrated by simulations using Matlab.