*5.2.3. MUF innervation: scatter, random, and weighted models*

In the literature, some models explicitly state the algorithm used to model the innervation process. In these models, prior to innervation, a square or hexagonal grid of muscle fibers is created within the muscle cross-section. Then, for each of the muscle fibers, an innervating motor unit is selected from the set of units whose territories cover the fiber position. This assignment can be done in a completely random manner (Stashuk, 1993) or by weighting the innervation probabilities. The idea behind such assignments is to model non-homogeneous distributions of muscle fibers within the motor unit territory (Cohen et al, 1987; Hamilton-Wright and Stashuk, 2005; Navallas, 2010), or to control the number of innervated fibers (Hamilton-Wright and Stashuk, 2005). A further refineed algorithm (Navallas, 2010) uses an analytical expression of the probability distributions of the outcomes (MUFN, MUFD, and MUTA) to determine the innervation probabilities in order to satisfy certain target distributions.

Other approaches simply state that muscle fibers of a given motor unit are scattered within the motor unit territory (Duchêne and Hogrel, 2000; Stålberg and Karlsson, 2001; Farina et al, 2002; Dimitrov et al, 2008; Keenan et al 2006) with no prior grid of muscle fibers within the muscle cross-section. Often, papers describing scatter models do not provide the algorithms used for the placement of the motor unit fibers within the motor unit territory. The reader might guess that one of two possible approaches have been used: we can assume that the exact number of motor unit fibers is assigned to each motor unit, or that the number of motor unit fibers assigned to each motor unit is calculated to satisfy the expected MUFD. Note that, as long as MUTA remains unchanged, both approaches would be equivalent. In both cases, the final motor unit fiber positions can be obtained at random from a uniform distribution over the motor unit territory. This approach should imply the use of a particular mechanism to avoid collisions between muscle fibers, solving the so called "fiber packing problem".

All of these models are adequate as long as optimized-augmented MUT placement has been carried out previously (Navallas et al, 2009b). However, only the inverse-model for weighted innervation ensures that exactly the intended MUFN, MUFD, and MUTA distributions will be obtained for the simulated muscle (Navallas et al, 2010).

#### *5.2.4. Motor unit temporal dispersion parameters*

If whole-muscle distribution characteristics are used in modeling the distributions of muscle fiber conduction velocities and motor end-plate locations of the motor unit, the result is usually overly complex MUPs. Muscle fiber diameters can be modeled to follow a normal distribution within individual motor units but with an increasing mean and standard deviation as a function of the motor unit index (Hamilton-Wright and Stashuk, 2005). The rationale behind this approach lies in the differences in diameters of fibers of different types. With this approach, the variability in muscle fiber conduction velocities within individual motor units is narrower, leading to more accurate levels of MUP complexity, while the overall distribution for the whole muscle follows a wider and almost normal distribution, as expected.

Generally, the approach to modeling end-plate locations is the same as that used for individual motor units, hence drawing the locations from normal or uniform distributions as described in section 3.3. Thus, end-plate locations are assumed to be uncorrelated between different motor units. However, more complex approaches (Navallas and Stålberg, 2009) can also be used to ensure further realism in the degree of complexity of the simulated MUPs, although end-plate locations are always independent from one motor unit to another.

Axonal delays and the mean neuromuscular transmission delay, as in the case of single motor units, are usually neglected and left unmodeled. However, in more accurate models (Hamilton-Wright and Stashuk, 2005), the neuromuscular junction transmission delay variability is accommodated. This model of "jitter" values, in which individual transmission delay variability can be modeled using the actual values found in real single-fiber EMG recordings, provides simulated jitter values that are highly correlated with those observed in reality.
