Mathematical relationships between spinal motoneuron properties

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 period. 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, to explore relationships between pairs of MN properties never concurrently observed in previous experiments, or to investigate inter-mammalian-species variations in MN membrane properties. Using this mathematical framework, modelers 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.

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

“…Some realistic aspects of the MN neurophysiology are also maintained. For example, the range of MN surface areas [0.15; 0.36]mm 2 and input resistances [0.5; 4.1]MO obtained from the S(j) distribution falls into the classic range of MN properties for cats [21,42], which is consistent with the distribution of rheobase values used in Step (2) that was obtained from cat data. The distribution of MN input resistance in the MN pool, obtained from the calibrated sizes S(j), defines the individual MN rheobase thresholds and predicts that 80% of the human TA MU pool is recruited at 35%MVC (Fig 10C ), which is consistent with previous findings in the literature [20,64].…”

“…where S is the MN surface area, I th the MN current threshold for recruitment, IR the MU innervation ratio defining the MU size, F th is the MU force recruitment threshold, and f max iso is the MU maximum isometric force. (4) The MN-specific electrophysiological properties are mathematically defined by the MN size S [42]. This extends the Henneman's size principle to:…”

“…It is worth noting that the experimental I th were obtained by injecting current pulses directly into the soma of identified MNs, thus bypassing the dendritic activity and most of the aforementioned PIC-generated nonlinear mechanisms [40]. A Δ I = 9.1-fold range in MN rheobase in [3.9; 35.5]nA was taken [21,42], while a larger Δ I is also consistent with the literature ( [42], Table 4) and larger values of I th can be expected for humans [21].…”

“…However, these phenomenological models cannot account for the MN-specific nonlinear mechanisms that dominate the MN pool behaviour [20,35,40,41]. MN leaky integrate-and-fire (LIF) models, the parameters of which can be defined by MN membrane electrophysiological properties for which mathematical relationships are available [42], are an acceptable trade-off between Hodgkin-Huxley-type and Fuglevand-type MN models with intermediate computational cost and complexity, and accurate descriptions of the MN macroscopic discharge behaviour [43] without detailing the MN's underlying neurophysiology [44]. While repeatedly used for the modelling and the investigation of individual MN neural dynamics [45][46][47], MN LIF models are however not commonly used for the description of MN pools.…”

“…The model of the MN pool was built upon a cohort of N single-compartment LIF models of MNs. The LIF parameters are derived from the available HDEMG data, are MN-specific, and account for the inter-relations existing between mammalian MN properties [ 42 ]. The distribution of N MN input resistances hence obtained defines the recruitment dynamics of the MN pool.…”