Losing the battle but winning the war: game theoretic analysis of the competition between motoneurons innervating a skeletal muscle

Abstract

The fibers in a skeletal muscle are divided into groups called "muscle units" whereby each muscle unit is innervated by a single neuron. It was found that neurons with low activation thresholds have smaller muscle units than neurons with higher activation thresholds. This results in a fixed recruitment order of muscle units, from smallest to largest, called the "size principle." It is thought that the size principle results from a competitive process-taking place after birth-between the neurons innervating the muscle. The underlying mechanism of the competition was not understood. Moreover, the results in the majority of experiments that manipulated the activity during the competition period seemed to contradict the size principle. Experiments at the isolated muscle fibers showed that the competition is governed by a Hebbian-like rule, whereby neurons with low activation thresholds have a competitive advantage at any single muscle fiber. Thus neurons with low activation thresholds are expected to have larger muscle units in contradiction to what is seen empirically. This state of affairs was termed "paradoxical." In the present study we developed a new game theoretic framework to analyze such competitive biological processes. In this game, neurons are the players competing to innervate a maximal number of muscle fibers. We showed that in order to innervate more muscle fibers, it is advantageous to win (as the neurons with higher activation thresholds do) later competitions. This both explains the size principle and resolves the seemingly paradoxical experimental data. Our model establishes that the competition at each muscle fiber may indeed be Hebbian and that the size principle still emerges from these competitions as an overall property of the system. Thus, the less active neurons "lose the battle but win the war." Our model provides experimentally testable predictions. The new game-theoretic approach may be applied to competitions in other biological systems.

Keywords: Hebbian rule; game theory; neuromuscular junction; size principle; synapse elimination.

Figure 1

Schematics depicting the development of motor-muscle innervation. The motoneurons (MNs) innervating the muscle are clustered into a nucleus within the ventral spinal cord (top). Two (green and yellow) MNs and a group of muscle fibers are shown. (A) At birth each muscle fiber is innervated by several MNs. (B) When synapse elimination ends, each muscle fiber is innervated by a single MN but each MN innervates a set of several muscle fibers called a muscle unit. Two muscle units are shown—the green muscle fibers and the yellow muscle fibers, each innervated by a corresponding MN.

Figure 2

An example of a simulation of the initial conditions. Top. The activity level of four MNs is depicted (numbers in circles): the two most active MNs constitute the M-group whereas the two less active MNs are the L-group. Below, three muscle fibers are presented (thick lines) together with their innervation pattern, selected at random. For each muscle fiber, the activity level of the corresponding MNs is summed up and the muscle fibers are sequenced according to a decreasing level of activity: 0.9 > 0.73 > 0.62.

Figure 3

The emergence of the size principle. Even when the game is strongly biased in favor of the more active MNs, the less active MNs still win the game. (A) Two examples of prior winning functions of the M-group at a single muscle fiber are shown: a biased function (solid line) and an unbiased function (dashed line). The x-axis denotes the fraction of M-group connections at a muscle fiber and the y-axis represents the winning probability of the M-group at that muscle fiber. (B) Simulations of the biased and unbiased games implementing the biased (solid line) and the unbiased (dashed line) prior winning functions depicted in A. These prior winning probabilities were updated along the game. The x-axis is the relative stage of the game and the y-axis shows the normalized difference in the number of victories throughout the game. As seen, at the end of both games, the L-group wins over the M-group as the lines are below the zero line.

Figure 4

Resolving the paradox in the blocking experiments results. Simulations of the short blocking procedure of Callaway et al. (, ; black solid line) and the long blocking procedure of Ribchester and Taxt (1983; broken line), against control (gray solid line). Depicted is the normalized difference in the number of victories between the blocked and unblocked groups. Activity is blocked in the middle of the game. In the imitation of Ribchester and Taxt (broken line), the blocked group loses whereas in the imitation of Callaway's experiments, the blocked group wins significantly more than control as the black solid line is higher than the gray solid line at x = 1 (p-value < 10⁻²⁷).