Mathematical relationships between spinal motoneuron properties
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Evaluation Summary:
This study describes the correlations between different membrane properties and the size of the soma of spinal alpha-motoneurons (MNs) using data from 40 experimental in vivo studies. The authors have distilled decades of research on motoneuron properties into a set of mathematical relationships that can guide both experimentalists and modelers interested in developing realistic models of populations of motoneurons. The key result is a complete table of the empirical relationships between the anatomical and physiological properties of MNs. Overall, the dataset approach is interesting, although a detailed analysis of the variability within and between datasets is urgently needed. In addition, a simpler framing of the paper could make the main message easier to grasp.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 and Reviewer #3 agreed to share their name with the authors.)
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Abstract
Our understanding of the behaviour of spinal alpha-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 duration. We reprocessed the data from 40 experimental studies in adult cat, rat, and mouse MN preparations to empirically derive a set of quantitative mathematical relationships between these MN electrophysiological and anatomical properties. This validated mathematical framework, which supports past findings that the MN membrane properties are all related to each other and clarifies the nature of their associations, is besides consistent with the Henneman’s size principle and Rall’s cable theory. The derived mathematical relationships provide a convenient tool for neuroscientists and experimenters to complete experimental datasets, explore the relationships between pairs of MN properties never concurrently observed in previous experiments, or investigate inter-mammalian-species variations in MN membrane properties. Using this mathematical framework, modellers can build profiles of inter-consistent MN-specific properties to scale pools of MN models, with consequences on the accuracy and the interpretability of the simulations.
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Evaluation Summary:
This study describes the correlations between different membrane properties and the size of the soma of spinal alpha-motoneurons (MNs) using data from 40 experimental in vivo studies. The authors have distilled decades of research on motoneuron properties into a set of mathematical relationships that can guide both experimentalists and modelers interested in developing realistic models of populations of motoneurons. The key result is a complete table of the empirical relationships between the anatomical and physiological properties of MNs. Overall, the dataset approach is interesting, although a detailed analysis of the variability within and between datasets is urgently needed. In addition, a simpler framing of the paper could make the main message easier to grasp.
(This preprint has been reviewed by eLife. We …
Evaluation Summary:
This study describes the correlations between different membrane properties and the size of the soma of spinal alpha-motoneurons (MNs) using data from 40 experimental in vivo studies. The authors have distilled decades of research on motoneuron properties into a set of mathematical relationships that can guide both experimentalists and modelers interested in developing realistic models of populations of motoneurons. The key result is a complete table of the empirical relationships between the anatomical and physiological properties of MNs. Overall, the dataset approach is interesting, although a detailed analysis of the variability within and between datasets is urgently needed. In addition, a simpler framing of the paper could make the main message easier to grasp.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 and Reviewer #3 agreed to share their name with the authors.)
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Reviewer #1 (Public Review):
This manuscript focuses on establishing the parametric relations between critical membrane and morphological properties of spinal alpha-motoneurons (MNs), using data from 40 experimental in vivo studies, most of them in the cat. Importantly, the authors digitalized the data from original papers, then created a global dataset for each property pair using normalized values, and lastly used a step-by step inference approach to create final datasets, which allowed to computed correlations between all pairs of parameters, even for those with no direct experimental data. In addition, a validation method was successfully performed using crossvalidation with 70% of data for training and 30% for testing. The authors also performed an extrapolation to the data in rats and mice, using the scaled relationships obtained …
Reviewer #1 (Public Review):
This manuscript focuses on establishing the parametric relations between critical membrane and morphological properties of spinal alpha-motoneurons (MNs), using data from 40 experimental in vivo studies, most of them in the cat. Importantly, the authors digitalized the data from original papers, then created a global dataset for each property pair using normalized values, and lastly used a step-by step inference approach to create final datasets, which allowed to computed correlations between all pairs of parameters, even for those with no direct experimental data. In addition, a validation method was successfully performed using crossvalidation with 70% of data for training and 30% for testing. The authors also performed an extrapolation to the data in rats and mice, using the scaled relationships obtained from cat anatomical and physiological information. Finally, the relationships between MN and mU properties were determined using the same methodology. Overall, the analytical approach is technically sound, and the results are potentially useful for neurophysiologists and modelling scientists. Indeed, Table 6 is a great summary of the pair-wise correlations for all nine anatomofunctional MN parameters, which can be utilized by people from different scientific communities. However, a simpler framing of the paper could make the main message easier to grasp. The abstract and introduction should be briefer, using a language that considers a broad audience. The results section should be distilled to the main findings, decreasing the number of main tables, and sending most of the detailed computations to the supplementary material section. I also recommend highlighting the procedure used to extract data from published experiments and the normalization and validation scheme to compute final correlation between parameters. This approach can be used to crease datasets in a wide variety of disciplines.
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Reviewer #2 (Public Review):
Spinal motoneurons were the first neurons in the mammalian CNS to be studied via intracellular recording and in the seventy years that have elapsed since then a number of studies have examined the potential relationship between motoneuron morphology, intrinsic properties and excitability. Elwood Henneman (Henneman, Science 1957) proposed that motoneurons are activated in order of increasing size and much of the experimental work that followed has been directed toward understanding the mechanism of the "size principle". Caillet and colleagues have done the field a valuable service by collating and normalizing data from 18 experimental studies in cats in which at least two morphometric or electrophysiological parameters were measured to determine the best power-relation fits between estimates of size (based on …
Reviewer #2 (Public Review):
Spinal motoneurons were the first neurons in the mammalian CNS to be studied via intracellular recording and in the seventy years that have elapsed since then a number of studies have examined the potential relationship between motoneuron morphology, intrinsic properties and excitability. Elwood Henneman (Henneman, Science 1957) proposed that motoneurons are activated in order of increasing size and much of the experimental work that followed has been directed toward understanding the mechanism of the "size principle". Caillet and colleagues have done the field a valuable service by collating and normalizing data from 18 experimental studies in cats in which at least two morphometric or electrophysiological parameters were measured to determine the best power-relation fits between estimates of size (based on cell soma diameter) and all other measures of cell properties such as axonal conduction velocity, input resistance and rheobase. They then invert these size-related relationships to predict relationships between all other parameter pairs. Finally, they show that with some exceptions, these relationships can be applied to intrinsic properties measured in other species (rats and mice).
The authors discuss the potential inaccuracies of estimates of cell size at great length, but do not consider other sources of error in any detail. For example, measurement of membrane time constant can be difficult due to the contribution of hyperpolarization-activated conductances to the measured voltage decay (cf. Fleshman et al., J Neurophysiol., 1988). Estimates of specific membrane properties are even more problematic - specific membrane resistivity (Rm) can be estimated from the membrane time constant assuming that specific capacitance is constant (e.g., Gustafsson and Pinter, J Physiol,1984) or it can be estimated from an equivalent cable model of the motoneuron, given input resistance, estimated electrotonic length and cell surface area (e.g., Ulfhake and Kellerth, Brain Res, 1984). Finally, adding further complexity, estimates of Rm based on completely reconstructed motoneurons suggest that it may vary over the surface of the motoneuron (Fleshman et al., J Neurophyiol, 1988; Clements and Redman, J Physiol, 1989). This is important because it has been argued that variations in Rm are as important as variations in cell size in determining motoneuron excitability (Gustafsson and Pinter, Trends in Neurosci., 1985).
The authors could focus more on the range of variation of different variables and its implications for mechanisms. The ranges of variation (ratio of the minimum to maximum values) of size-related variables (soma diameter, cell surface area) are around 3, whereas input resistance and rheobase show much larger ranges of variation. This suggests that size alone can not explain the observed variation in relative excitability. In cases where size, input resistance, specific membrane resistivity and rheobase have all been measured it might be possible to use some combination of the other variables to predict rheobase.
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Reviewer #3 (Public Review):
Caillet et al. performed a regression analysis to estimate mathematical functions relating electrophysiological and morphological properties of spinal motor neurons, as well as motor neuron and muscle unit properties. Power functions relating to pairs of properties were parameterized using data from cat motor neurons with a subsequent validation using rodent data. The study's main conclusion is that the estimated relations extend Henneman's size principle, providing "unknown" associations between motor unit properties.
Despite the great effort of the authors in reconciling a large amount of data from several studies, my primary concern is that the proposed method neglect data variability and their ensuing distribution. It is well known that several electrophysiological properties present a multimodal …
Reviewer #3 (Public Review):
Caillet et al. performed a regression analysis to estimate mathematical functions relating electrophysiological and morphological properties of spinal motor neurons, as well as motor neuron and muscle unit properties. Power functions relating to pairs of properties were parameterized using data from cat motor neurons with a subsequent validation using rodent data. The study's main conclusion is that the estimated relations extend Henneman's size principle, providing "unknown" associations between motor unit properties.
Despite the great effort of the authors in reconciling a large amount of data from several studies, my primary concern is that the proposed method neglect data variability and their ensuing distribution. It is well known that several electrophysiological properties present a multimodal distribution with overlapping values for motor neurons of different sizes (see Figure 7 in Zengel et al. 1985, for instance). Additionally, the authors disregard the well-defined biophysical relations, for example, the relation between membrane time constant and the product of membrane resistance and capacitance (tau = R . C). We do not have any experimental evidence to hypothesize the motor neuron will behave differently from a resistive-capacitive system (at least in subthreshold potentials); on the contrary, several experimental results are showing the RC behavior of the motoneuronal membrane, along with different computational models (conceptualized as an RC network) that closely resemble the electrophysiological behaviors of individual neurons.
The enthusiastic tone adopted in the paper regarding the novelty and potential relevance for future studies seems unsupported. For instance, in p. 4, l. 66-67, the authors state that the estimated relationships "can accelerate future research in the behavioural of individual MNs". However, the relations obtained between electrophysiological and morphological properties of motor neurons did not consider important aspects of motor neuron physiology, namely the active properties yielded by ionic channels and a distributed synaptic integration along the dendritic tree of the motor neuron. Properties such as f-I gain, hysteresis, EPSP and IPSP amplitude, electrotonic length, and others are lacking. If the authors want to make such assertions, they should provide how the empirical static relations among (mostly) passive properties would be translated to a functional context.
Another problematic aspect of the authors' arguments is that they did not consider previous evidence from the literature as valid quantitative relations among variables. Eccles et al. (J. Physiol. 142: 275-91, 1958) have shown a clear linear relation (strong correlation indeed) between AHP and axon conduction velocity. The provided graph (with a regression) in the referred paper is not "speculative", despite the fact the authors did not give the slope and intercept of the curve (can be easily estimated). Also, Powers and Binder (2001) present clear evidence that electrophysiological data from cat motor neurons can be adjusted by the theoretical functions from Rall's cable theory (see Figure 4 in the referred review paper). Considering Rall's approach, it is not surprising that all parameters used in the present study are related. Nonetheless, the unexpected point here is the unjustified choice of power functions to fit the relations. How "flexibility and simplicity" would affect the interpretations and results? Why not use functions more adherent to the theoretical relations expected from biophysical studies?
Another aspect not fully explained in the manuscript (and related to my first point) is how data scarcity in some dataset areas would influence the proposed method. There are regions with a limited number of data points, while other areas are very dense. R-squared values (I am unsure if they are valid for non-linear regressions) are not high enough (< 0.70), and the validation results show normalized mean errors as high as 400%. Moreover, the authors did not discuss the possible bias of using several datasets from the same research group (or lab). Methodological bias can also be a confounding factor in the analysis. Finally, in the inter-species analysis, the authors did use data from studies with mutated animals (SOD-1 rodents, for example). It is not clear if the data included in the analysis were from wild-type animals or all animals in the study dataset. Particularly for the Huh et al. (2021) dataset, data from animals of different ages can also influence the analysis (see Highlander et al. 2020).
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