Mathematical Relationships between Motoneuron Properties Derived by Empirical Data Analysis: Size Determines All Motoneuron Properties

Our understanding of the behaviour of motoneurons (MNs) in mammals partly relies on our knowledge of the relationships between MN membrane properties, such as MN size, resistance, rheobase, capacitance, time constant, axonal conduction velocity and afterhyperpolarization period. Based on scattered but converging evidence, current experimental studies and review papers qualitatively assumed that some of these MN properties are related. Here, we reprocessed the data from 27 experimental studies in cat and rat MN preparations to empirically demonstrate that all experimentally measured MN properties are associated to MN size. Moreover, we expanded this finding by deriving mathematical relationships between each pair of MN properties. These relationships were validated against independent experimental results not used to derive them. The obtained relationships support the classic description of a MN as a membrane equivalent electrical circuit and describe for the first time the association between MN size and MN membrane capacitance and time constant. The obtained relations indicate that motor units are recruited in order of increasing MN size, muscle unit size, MN rheobase, unit force recruitment thresholds and tetanic forces, but underlines that MN size and recruitment order may not be related to motor unit type.


Introduction
Our understanding of the behaviour of motoneurons (MNs) in mammals partly relies on the knowledge of the relationships between MN properties. The most relevant MN properties are reported in Table 1. Direct measurement of some of these properties has been performed in animal studies and yielded some significant correlations. For example, the size of MNs has been found to be strongly associated to the axonal conduction velocity (Cullheim, 1978;Kernell and Zwaagstra, 1981;Burke et al., 1982;Binder et al., 1996).
However, because of technical difficulties in measuring MN properties, measurements have been performed for a limited range of values and for only a few properties concurrently in each study. Our knowledge of MN properties and their associations is therefore limited to a series of datasets that provide a crude picture of MN physiological and biophysical features but not a detailed understanding of the working principles of MNs. For this reason, the statements reported in review papers on the association between MN properties rely on converging conclusions from independent studies and remain speculative (Henneman, 1981;Burke, 1981;Binder et al., 1996;Powers and Binder, 2001;Kernell, 2006;Heckman and Enoka, 2012). While qualitative associations, such as between MN size and resistance (Kernell and Zwaagstra, 1981;Burke et al., 1982), have led to extensions of the Henneman's size principle (Henneman, 1957;Wuerker et al., 1965;Henneman et al., 1965a;Henneman et al., 1965b;Henneman et al., 1974;Henneman, 1981;Henneman, 1985), other relationships, such as between MN size and membrane capacitance or time constant, are not known.
Due to the lack of mathematical descriptions of the relationships between MN properties and size, MN property profiles are typically built with scattered data from a relatively small number of experimental studies. Such generic MN profiles indicate reasonable orders of magnitude but result in a lack of interconsistency between the MN properties. This approach yields obvious limitations when the sets of physiological and MN-specific characteristics must be known, as in numerical models of a MN membrane equivalent electrical circuit (Negro et al., 2016;Teeter et al., 2018).
To tackle the above limitations in our understanding of MN characteristics, algebraic relationships must be derived between the main MN properties. This can be done either by extrapolating these relationships from existing data or by concurrently measuring all properties in single MNs. The latter approach is currently not possible because of technical constraints which make it impossible to date to experimentally measure a complete and reliable set of MN-specific values for a large sample of MNs. It has been possible in specialised experimental set-ups to measure a maximum of six MN properties concurrently in animal preparations, and for limited ranges of values (Gustafsson and Pinter, 1984a;Gustafsson and Pinter, 1984b). Conversely, these direct measurements are not feasible in humans in vivo, where only axonal conduction velocity can be estimated from indirect measurements (Freund et al., 1975;Dengler et al., 1988).
Here, we derived currently unknown mathematical associations between MN properties by digitizing, reprocessing, and merging the data from 27 available experimental studies in cat and rat preparations. In this way, we first demonstrate that the MN properties reported in Table 1 are all precisely predicted by MN size. Second, we derive mathematical relationships between any pair of the MN properties listed in Table 1. These empirical relationships were validated on new data from studies not used for their derivation and provide for the first time a mathematical framework for the association between any pair of MN properties. Finally, using additional correlations obtained between some MN and muscle unit (mU) properties, we discuss the empirical relationships obtained between MN properties in the context of the Henneman's size principle.  (Binder et al., 1996;Powers and Binder, 2001;Heckman and Enoka, 2012 (Gustafsson, B. and Pinter, 1984b;Zengel et al., 1985). ∆ ℎ is the constant amplitude of the membrane voltage depolarization threshold relative to resting state required to elicit an action potential. ℎ is the corresponding electrical current causing a membrane depolarization of ∆ ℎ . AHP is defined in most studies as the duration between the action potential onset and the time at which the MN membrane potential meets the resting state after being hyperpolarized. CV is the axonal conduction velocity of the elicited action potentials on the MN membrane.
is the size of the mU. As indicated in Table 2, the mU size is adequately described by measures of (1) the sum of the cross-sectional areas (CSAs) of the fibres composing the mU , (2) the mean fibre CSA , (3) the innervation ratio IR, i.e. the number of innervated fibres constituting the mU, and (4) the mU tetanic force . The muscle force at which a mU starts producing mU force is called mU force recruitment threshold ℎ .

Properties
Notation Unit

Methods
We analysed the results of the experimental studies in the literature that measured and provided direct comparisons between pairs of the MN properties reported in Table 1. From a first screening of the literature, most of the investigated pairs comprised either a direct measurement of MN size, noted as in this study, or another variable strongly associated to size, such as axonal conduction velocity ( ) or afterhyperpolarization period ( ). Accordingly, and consistent with the Henneman's size principle, we identify as the reference MN property with respect to which the relationship with the other MN properties in Table 1 is investigated. For convenience, in the following, the linear relationship between the properties and in the form = • with a constant gain, is noted as ∝ and reads ' is linearly related to '.

Definitions of MN Size , mU size and MU size
In the literature, the MN size is either a conceptual parameter or its definition varies among studies to be the measure of the MN head surface area ℎ (Burke et al., 1982), the dendritic surface area (Barrett and Crill, 1974), the soma diameter (Kernell and Zwaagstra, 1981), or the dendrites cross-sectional area (Kernell, 1966). To avoid confusion and to enable future inter-study comparisons in seeking relationships between and the MN properties reported in Table 1, we here provide a precise definition for . The membrane surface area ℎ of the head of the MN is an adequate geometrical definition of (Burke, 1981). ℎ was directly measured in a few animal studies as a spheric soma of diameter and surface connected to cylindric and branching dendrites of individual 1 st -order diameters and membrane surface (Kernell, 1966;Cullheim, 1978;Ulfhake and Kellerth, 1981;Kernell and Zwaagstra, 1981;Burke et al., 1982;Kernell and Zwaagstra, 1989). According to the direct measurements of performed in (Kernell, 1966;Barrett and Crill, 1974;Cullheim, 1978;Zwaagstra and Kernell, 1980;Kernell and Zwaagstra, 1981;Ulfhake and Kellerth, 1981;Burke et al., 1982), the total dendritic surface area is found to account for at least 85% of ℎ , so ℎ = + ≈ . Moreover, as ∝ according to (Ulfhake and Kellerth, 1981), and as the average is linearly correlated to (Ulfhake and Kellerth, 1981), we also obtain ∝ . As the axon diameter and are linearly correlated (Cullheim, 1978), overall we obtain that ℎ , and are linearly related, consistently with previous findings (Burke et al., 1982;Kernell and Zwaagstra, 1989): Consequently, the conceptual MN size is adequately and consistently described by the measurable and linearly inter-related MN head surface area ℎ , soma diameter , and axon diameter , as reported in Table 2. Therefore, the relationships between MN size and the other MN properties reported Table 1 can be obtained from experimental studies providing measures of ℎ , and , and not or for example.
Similarly, to enable future comparisons between MN and mU properties, we provide a precise definition of the mU size . The size of a mU ( ) can be geometrically defined as the sum of the cross-sectional areas CSAs of the innervated fibres composing the mU.
depends on the mU innervation ratio ( ) and on the mean CSA ( ) of the innervated fibres: was measured in a few studies on cat and rat muscles, either indirectly by histochemical fibre profiling (Burke and Tsairis, 1973;Dum and Kennedy, 1980;Burke, 1981), or directly by glycogen depletion, periodic acid Schiff (PAS) staining and fibre counting (Burke et al., 1982;Bodine et al., 1987;Chamberlain and Lewis, 1989;Totosy de Zepetnek, J E et al., 1992;Kanda and Hashizume, 1992;Rafuse et al., 1997). The mU tetanic force is however more commonly measured in animals. As the fibre mean specific force is considered constant among the mUs of one muscle in animals (Bodine et al., 1987;Lucas et al., 1987;Chamberlain and Lewis, 1989;Totosy de Zepetnek, J E et al., 1992;Enoka, 1995), the popular equation = • • (Burke, 1981;Enoka, 1995) returns a linear correlation ∝ • = in animals. Experimental results further provide the relationships ∝ (Bodine et al., 1987;Chamberlain and Lewis, 1989;Totosy de Zepetnek, J E et al., 1992;Kanda and Hashizume, 1992;Rafuse et al., 1997) and ∝ (Burke and Tsairis, 1973;Bodine et al., 1987;Totosy de Zepetnek, J E et al., 1992;Kanda and Hashizume, 1992). Consequently, , and are measurable, consistent, and valid measures of in animals, as summarized in Table 2: In the following, the MU size can refer to any of the size indices reported Table 2.

Relationships between MN properties
For convenience, in the following, the notation { ; } refers to the pair of properties and , to which a relationship can be defined in the form = ( ).

Digitized data and trendlines
Available studies on MN properties generally provide clouds of data points for pairs { ; } of concurrently measured MN properties through scatter graphs. These plots were digitized using the online tool WebPlotDigitizer (Ankit, 2020). To enable cross-study analysis, the coordinates of the digitized points were then normalized for each study and transformed as a percentage of the maximum property value measured in the same study. The normalized pairs of points for { ; } retrieved from different studies were then merged into a 'global' dataset dedicated to that property pair. A least square linear regression analysis was performed for the ln( ) − ln( ) transformation of each global dataset yielding ln( ) = • ln( ) + relationships which were converted into power relationships of the type = • , also noted as ∝ . Power fitting was chosen for flexibility and simplicity. The adequacy of these global power trendlines and the statistical significance of the correlations were assessed with the coefficient of determination 2 (squared value of the Pearson's correlation coefficient) and a threshold of 0.05 on the p-value of the regression analysis, respectively. To further assess the reliability of each derived trendline and the inter-study consistency, the equation of the global trendline was compared against the power relationships obtained from individual regression analyses performed for each paper constituting the dataset.

Size-dependent algebraic relationships
Once the ∝ relationships were obtained from trendline fitting for all pairs { ; } found in the literature, the MN properties in Table 1 were processed in a step-by-step manner in the order , , , ℎ , , to seek a power relationship ∝ between each of them and . For each investigated property and each fitted { ; } pair, two cases existed. If = , a size-dependent relationship ∝ was directly obtained. If ≠ , a statistically significant ∝ relationship was found between properties and , and if a power relationship ∝ had previously been derived for the pair { ; } , a consistent power relationship ∝ • , noted ∝ was mathematically derived for { ; } . With this dual approach, as many ∝ relationships as available { ; } pairs were obtained for the pair { ; } for each property . If the obtained -values were consistent, i.e. of the same sign and within an arbitrary 3-fold range, it was concluded that property was correlated to following an ∝ relationship. In this case, the -value was calculated as the average of the individual power values and rounded to the nearest integer. This new ∝ relationship, called 'final relationship' could be used in the derivation of the next-in-line property (in the second case), and the steps described above were repeated to seek a new ∝ final relationship for .

Scaling the normalized final relationships
The accuracy of the final ∝ relationships in reproducing existing data from the literature was first assessed against the typical fold range of the property . Minimum and maximum absolute values for the properties and were retrieved from the processed studies and from ten additional studies in the literature that, while not providing an analysis of the relations between MN properties (and therefore not being included in the derivation of the equations), reported ranges of experimental values for the analysed MN properties. An experimental ratio was calculated, as the average across studies of the ratios of minimum and maximum values measured for . An experimental ratio was similarly obtained for the property .
was then compared to a third theoretical ratio = ( ) | | , with taken from the final relationship ∝ , as previously derived. If was in the range [0.75 ; 1.25], the global ∝ relationship was considered an accurate method to predict the physiological fold range of MN property from the physiological fold range of .
Then, the intercept of the normalized final = • relationships was scaled using the average across studies of the minimum and maximum absolute values for and . Using the additional ten studies is adequate for scaling and does not affect the quality of the previously fitted trendlines if there is consistency in the property fold ranges between the fitted data and the additional set of experimental data. In this respect, the ratio was calculated as restricted to the fitted studies, and it was assessed  Table 1.

Relationships between MN and mU properties
To assess whether the empirical relationships ∝ between MN properties and MN size derived in this study were in accordance with the Henneman's size principle of MU recruitment, we identified a set of twelve experimental studies that concurrently measured a MN property and a muscle unit (mU) property for the same MU. The data obtained for the pairs { ; } were fitted with power trendlines, as previously described for MN properties, yielding ∝ relationships. Using both the definition of and the ∝ final relationships derived previously, the ∝ relationships were then mathematically transformed into ∝ relationships. If all -values were of the same sign, it was concluded that mU and MN sizes were correlated.

Results
We identified 27 experimental studies on cats and rats that report direct comparisons and processable experimental data for the 15 pairs of MN properties and the 5 pairs of one MN and one mU property represented in the bubble diagram of Figure 1(A).  Table 3
Lastly, these size-dependent relationships were used to mathematically derive algebraic relationships between any of the MN properties , , , ℎ , , and populating and ℎ = 2.9 • 10 −3 • ℎ 0.5 , the relationship = 10 −2 ℎ = ∆ ℎ ℎ was obtained. All constants and relationships are given in SI base units. The specific capacitance and the membrane voltage threshold ∆ ℎ were found to be constant among MNs, a property discussed in the Discussion section.  The mathematical relationships in Table 6 are reliable in explaining all the existing data retrieved from the literature. First, they remain consistent (Table 7) with the power relationships that were experimentally derived in the studies listed in Table 3 for the pairs { ℎ ; }, { ; }, { ; ℎ }, { ; } and { ; ℎ }. In Table 7, experimental and empirical -values show a strong match for the 5 pairs of MN properties, demonstrating that the empirical relationships derived in this study are expedient in predicting the inter-relationships between MN properties. Relationship -value (fitted relationships) -value (mathematical relationships) The mathematical relationships in Table 6 are moreover strongly consistent with further experimental measurements that were not included in the data processing used for deriving our relationships and displayed in Figure 2. The relationships ℎ ∝ 1 2 , 1 ∝ , ∝ and ∝ 1 2 (Table 6) are, when combined, perfectly consistent with the relationship ℎ ∝ 1 experimentally observed in Gustafsson and Pinter (1985) and 1 • ∝ measured in Kernell and Zwaagstra (1981). No correlation between ∆ ℎ • ℎ and is reported by the empirical relationships in Table 6, consistent with measurements performed in Gustafsson, B. and Pinter (1984b), substantiating that the dynamics of MN recruitment dominantly rely on , ℎ and ∆ ℎ (Heckman and Enoka, 2012). The stronger-than-linear inverse relationship ∝ 1 2 is consistent with the phenomenological conclusions from Kernell and Zwaagstra (1981). Similarly, the retrieved ∝ 1 relationship is consistent with the modelling conclusions from Barrett and Crill (1974), who reported a significant ∝ • relationship and a weak but significant ∝ 1 relationship. The indirect conclusion on a positive correlation between and in Gustafsson and Pinter (1984b) is finally consistent with ∝ (Table 6). However, interestingly, the relationship ∝ 1 contradicts the speculations in Kernell and Zwaagstra (1981) and Gustafsson and Pinter (1984b) that is the dominant factor influencing the distribution of values rather than .
This study also predicts the correlations between MN properties that were either never reported in past review studies, such as the positive − relationship, or never concurrently measured in the literature, as displayed in Figure 1(B). Such unknown relationships were indirectly extracted from the combination of known relationships (Table 3) and typical ranges of values obtained from the literature for these properties. For example, ℎ ∝ 2 was predicted from the following combinations of known and validated relationships: { ℎ ∝ −1 ; ∝ −2 }, or { ℎ ∝ −2 ; ∝ }. Due to the prior validation of the relationships in Table 6, these findings are reliable as indirectly consistent with the literature data processed in this study and provide new insights on the size-dependency of the MN recruitment mechanisms.

Relationships between MN and mU properties
As shown in Figure 1(A), five pairs of one MN and one mU property were investigated in twelve studies in the literature in cats and rats, and none in the past 30 years, as remarked by Heckman and Enoka (2012). One study on the rat gastrocnemius muscle (Kanda and Hashizume, 1992) indicated no correlation between and . However, after removing from the dataset 2 outliers that fell outside two standard deviations of the mean data, a statistically significant correlation ( < 0.05) between and was successfully fitted with a power trendline ( 2 = 0.43). Eight studies, dominantly focusing on the cat soleus and medial gastrocnemius muscles, found a strong correlation between and , while one study showed a significant correlation for the pair { ; } in both the cat tibialis anterior and extensor digitorum longus muscles. Finally, one study (Burke et al., 1982) on the cat soleus, medial and lateral gastrocnemius muscles inferred a statistically significant correlation for { ; }. As and are reliable indices of , by using the MN relationships in Table 3, four ∝ relationships were obtained between mU and MN properties (Table 8) McPhedran et al., 1965;Wuerker et al., 1965;Appelberg and Emonet-Dénand, 1967;Proske and Waite, 1974;Bagust, 1974;Jami and Petit, 1975;Stephens and Stuart, 1975;Burke et al., 1982;Emonet-Dénand et al., 1988) 4.3 ∝ −1.3 0.27 6 • 10 −5 (Dum and Kennedy, 1980) 2.6 2.0 0.21 0.02 (Burke et al., 1982) 2.0

Discussion
We processed the data from previous experimental studies to extract mathematical relationships between several MN properties and MN size. This allowed us to demonstrate that all investigated MN properties are predicted by MN size and that properties at the level of individual MNs are interrelated. We established mathematical relations linking all the pairs of MN properties (Table 6). These relationships were validated with respect to the ranges of the predicted MN properties against a set of typical literature range values (Table 5), against directly fitted experimental data (Table 7) and against other results available in the literature.
The findings are consistent with considerations from previous papers and literature reviews, either drawn from direct but isolated measurements or speculated. For example, Binder et al. (1996) concluded that the duration was inversely related to MN size based on measures of a single experimental study. Conversely, the size-dependencies of and ℎ were qualitatively assumed by Powers and Binder (2001) from the equations obtained from a variant of the Rall's model of MN membrane equivalent electrical circuit, and not from experimental data. The negative correlation between and was predicted by Binder et al. (1996) from the findings of Henneman's studies on the correlation between extracellular spike amplitude and (Henneman, 1957;Henneman et al., 1965a). The fact that defines the MN rheobase and thus dictates the size order of MN recruitment was similarly justified in Binder et al. (1996) and Powers and Binder (2001) from a combination of Henneman's findings and Rall's model equations. Moreover, all previous papers used the debated association between MU type and , mainly obtained from measures in the cat gastrocnemius muscle (Fleshman et al., 1981;Burke et al., 1982;Zengel et al., 1985;Bakels and Kernell, 1993), to indirectly speculate on correlations between and (Heckman and Enoka, 2012), (Powers and Binder, 2001;Heckman and Enoka, 2012), (Binder et al., 1996;Powers and Binder, 2001) or ℎ (Binder et al., 1996). Finally, we could find a single paper that inferred a positive correlation between and (Heckman and Enoka, 2012) but without reference to direct experimental results.

Relevance for MN modelling
The empirical equations in Table 6 support the common approach of modelling the MN membrane behaviour with an equivalent resistance-capacitance electrical circuit as variants of the Rall's cable model (Rall, 1957;Rall, 1959;Rall, 1960). The relationship between and reported in Table 6 validates the definition = • , as well as it emphasizes that the specific capacitance per unit area is constant among the MN pool, and indirectly yields, from typical ranges of and in the literature, the relation = 1.8 • 10 −2 • −2 , which is highly consistent with the average ranges of values reported in (Lux and Pollen, 1966;Albuquerque and Thesleff, 1968;Barrett and Crill, 1974;Adrian and Hodgkin, 1975;Sukhorukov et al., 1993;Major et al., 1994;Solsona et al., 1998;Thurbon et al., 1998;Gentet et al., 2000). Similarly, the empirical relationship ℎ = 10 −2 (Table 6) yields ∆ ℎ = 10 , consistently with (Brock et al., 1952;Eccles et al., 1958), despite uncertainties in the value of the membrane resting potential (Heckman and Enoka, 2012). This supports the conclusions that the relative voltage threshold ∆ ℎ is constant within the MN pool (Coombs et al., 1955;Gustafsson and Pinter, 1984a;Gustafsson and Pinter, 1984b;Powers and Binder, 2001), and that Ohm's law is followed in MNs (Glenn and Dement, 1981;Spruston and Johnston, 1992;Kernell, 2006). Finally, the findings ∝ 2 , ∝ 1 and ∝ in Table 6 numerically validate the classic empirical relationship = = (Gustafsson and Pinter, 1984a;Zengel et al., 1985). From the typical ranges of values obtained from the literature for { ; } or { ; }, the latter relationship enforces ∈ [0.16; 0.62]Ω • 2 , which is consistent with the ranges of values speculated in (Albuquerque and Thesleff, 1968;Barrett and Crill, 1974;Burke et al., 1982;Gustafsson and Pinter, 1984a;Kernell and Zwaagstra, 1989) (Table  5). Table 8 reports statistically significant power relationships ∝ of positive -values between MN and mU indices of size. These results substantiate the concept that and are positively correlated in a MU pool and that large MNs innervate large mUs (Henneman, 1981;Heckman and Enoka, 2012), a statement that has never been demonstrated from the concurrent direct measurement of and . Besides, considering that ℎ ∝ 2 (Table 6), and that the mU force recruitment threshold ℎ is positively correlated to (Heckman and Enoka, 2012) and thus to (Table 2), larger MUs have both larger current and force recruitment thresholds ℎ and ℎ than relatively smaller MUs, which are thus recruited first, consistently with the Henneman's size principle of MU recruitment (Henneman, 1957;Wuerker et al., 1965;Henneman et al., 1965a;Henneman et al., 1965b;Henneman et al., 1974;Henneman, 1981;Henneman, 1985). The terminologies 'small MU', 'low-force MU' and 'low-threshold MU' are thus equivalent. Henneman's size principle thus entirely relies on the amplitude of the MN membrane resistance ∝ −2 , as inferred in (Binder et al., 1996;Powers and Binder, 2001;Heckman and Enoka, 2012). Finally, the relationships ∝ 0.6 ∝ −1 ∝ −1 (Table 6) suggest that high-threshold MUs rely on relatively faster MN dynamics, which might partially explain why large MNs can attain relatively larger firing rates than low-thresholds MNs.
It has been repeatedly attempted to extend Henneman's size principle and the correlations between the MU properties in Table 1 to the concept of 'MU type' (Burke and Ten Bruggencate, 1971;Burke, 1981;Bakels and Kernell, 1993;Powers and Binder, 2001). While a significant association between 'MU type' and indices of MU size has been observed in some animal (Fleshman et al., 1981;Burke et al., 1982;Zengel et al., 1985) and a few human (Milner-Brown et al., 1973;Stephens and Usherwood, 1977;Garnett et al., 1979;Andreassen and Arendt-Nielsen, 1987) studies, it has however not been observed in other animal studies ((Bigland-Ritchie et al., 1998) for a review) and in the majority of human investigations (Sica and McComas, 1971;Goldberg and Derfler, 1977;Yemm, 1977;Young and Mayer, 1982;Thomas et al., 1990;Nordstrom and Miles, 1990;Elek et al., 1992;Macefield et al., 1996;Cutsem et al., 1997;Mateika et al., 1998;Fuglevand et al., 1999;Keen and Fuglevand, 2004). Moreover, the reliability of these results is weakened by the strong limitations of the typical MU type identification protocols. Sag experiments are irrelevant in humans (Buchthal and Schmalbruch, 1970;Thomas et al., 1991;Bakels and Kernell, 1993;Macefield et al., 1996;Bigland-Ritchie et al., 1998;Fuglevand et al., 1999), and lack consistency with other identification methods (Nordstrom and Miles, 1990). MU type identification by twitch contraction time measurements is limited by the strong sources of inaccuracy involved in the transcutaneous stimulation, intramuscular microstimulation, intraneural stimulation, and spike-triggered averaging techniques (Taylor et al., 2002;Keen and Fuglevand, 2004;McNulty and Macefield, 2005;Negro et al., 2014;Dideriksen and Negro, 2018). Finally, as muscle fibres show a continuous distribution of contractile properties among the MU pool, some MUs fail to be categorized in discrete MU types in some animal studies by histochemical approaches (Reinking et al., 1975;Totosy de Zepetnek, J E et al., 1992). Owing to these conflicting results and technical limitations, MU type may not be related to MN size and the basis for MU recruitment during voluntary contractions (McNulty and Macefield, 2005;Duchateau and Enoka, 2011).

Limitations
The mathematical relationships derived Table 6 and the conclusions drawn in the Discussion are constrained by some limitations.
A first limitation is due to the limited experimental data available in the literature, as also discussed by Heckman and Enoka (2012). Some pairs of MN properties were investigated in only one study, such as { ; } or { ; }, preventing inter-study comparisons. There have been no studies in the past 30 years on direct measures of these properties. Moreover, measurements obtained from different species (cat, rat) and different muscles were merged into unique datasets, implicitly assuming similar distributions of MN properties within the MN pool of different muscles and species.
A second limitation is related to the methods chosen for processing the retrieved data. The measurements were reproduced from a digitization of scatter plots, which may have determined small inaccuracies. All datasets were besides normalized to the highest measured value retrieved in each study; this approach is only valid if the experimental studies identified the same largest MN relatively to the MN populations under investigation, which cannot be verified.

Conclusion
This study provides the first empirical and algebraical proof that the MN size precisely determines all other MN properties ( , , , ℎ , and ). The derived mathematical relationships between any of these MN properties and/or are provided in Table 6. They accurately describe the experimental data available in the literature and provide for the first time a method for building virtual MN profiles of inter-consistent MN-specific properties.