Mathematical Relationships between Spinal Motoneuron Properties

Abstract

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.

Estimation of the firing behaviour of a complete motoneuron pool by combining electromyography signal decomposition and realistic motoneuron modelling

“…From the previous findings, the continuous distribution of MN sizes S ( j ) (Figure 10A) is derived for the complete pool of N MNs. S ( j ) defines the electrophysiological properties (Caillet et al, 2021) of the MNs constituting the complete MN pool. The neural behaviour of the complete pool of N MNs is predicted in the 4 th step with a cohort of N S j -IP j -scaled LIF models and the application of the common synaptic drive I ( t ) (Figure 10).…”

“…A second limitation of the study is that, as discussed previously, the LIF parameters are tuned with typical cat data (Caillet et al 2021) while the experimental MN spike trains were obtained from humans. While the normalized mathematical relationships relating the MN electrophysiological parameters of the LIF model ( R, C,τ ,Δ V t h ) can be extrapolated between mammals (Caillet et al 2021) and thus to humans, no experimental data is yet available to scale these relationships to typical human data. A further limitation is that, in its current form, the method disregards the synaptic noise SN ( t ) specific to each MN and the inter-spike variability (ISV).…”

“…The multi-MU model in Figure 3 expands the classic multiscale approach of modelling a whole muscle as a single equivalent sarcomere or MU (Zajac Felix, 1989). Such multiscale models usually (1) cannot explicitly model the dynamics of MU recruitment, (2) disregard the continuous distribution of MU properties in the MU pool that typically affects the MN firing behaviour (Caillet et al, 2021), the calcium dynamics in the sarcolemma (Baylor & Hollingworth, 2003), the cross-bridge cycling dynamics (Rockenfeller & Günther, 2018), the force-velocity relationship (Alcazar et al, 2019) and the MU maximum isometric force distribution, and (3) cannot account for the passive properties of the intramuscular matrix of passive tissues (Herzog, 2019). Despite relevant modelling work in multi-scale modelling (Hatze, 1977; Hatze, 1980; Dorgan & O’Malley, 1997; Song et al, 2008; Dick et al, 2017; Hussein et al, 2021), multi-MU muscle models like the one proposed in Figure 3 provide a more convenient framework to answer the above limitations.…”

“…As I and C m typically vary over a 10-fold and 2.4-fold range respectively (Caillet et al, 2021), the FF predicted by the LIF is dominantly determined by the value of IP as the input current increasingly overcomes the MN current threshold: .…”

“…The capacitance parameter C in the LIF model is related to the MN size S as C = C m · S (Caillet et al, 2021). The specific capacitance C m , which dominantly affects the firing rate properties of the LIF model during the ramp of increasing I ( t ) over [ t tr 1 ; t tr 2 ], is accepted to be constant across a MN pool (Gentet et al, 2000; Caillet et al, 2021) with a reference value C m = 0.9 μF · cm 2 . However, C m cannot be directly measured in experiments and a wide range [0.8; 10.8] μF · cm 2 of C m values is found in the literature.…”