Derivation of cable parameters for a reduced model that retains asymmetric voltage attenuation of reconstructed spinal motor neuron dendrites

A geometric approach to quantifying the neuromodulatory effects of persistent inward currents on individual motor unit discharge patterns

Objective.All motor commands flow through motoneurons, which entrain control of their innervated muscle fibers, forming a motor unit (MU). Owing to the high fidelity of action potentials within MUs, their discharge profiles detail the organization of ionotropic excitatory/inhibitory as well as metabotropic neuromodulatory commands to motoneurons. Neuromodulatory inputs (e.g. norepinephrine, serotonin) enhance motoneuron excitability and facilitate persistent inward currents (PICs). PICs introduce quantifiable properties in MU discharge profiles by augmenting depolarizing currents upon activation (i.e. PIC amplification) and facilitating discharge at lower levels of excitatory input than required for recruitment (i.e. PIC prolongation).Approach. Here, we introduce a novel geometric approach to estimate neuromodulatory and inhibitory contributions to MU discharge by exploiting discharge non-linearities introduced by PIC amplification during time-varying linear tasks. In specific, we quantify the deviation from linear discharge ('brace height') and the rate of change in discharge (i.e. acceleration slope, attenuation slope, angle). We further characterize these metrics on a simulated motoneuron pool with known excitatory, inhibitory, and neuromodulatory inputs and on human MUs (number of MUs; Tibialis Anterior: 1448, Medial Gastrocnemius: 2100, Soleus: 1062, First Dorsal Interosseus: 2296).Main results. In the simulated motor pool, we found brace height and attenuation slope to consistently indicate changes in neuromodulation and the pattern of inhibition (excitation-inhibition coupling), respectively, whereas the paired MU analysis (ΔF) was dependent on both neuromodulation and inhibition pattern. Furthermore, we provide estimates of these metrics in human MUs and show comparable variability in ΔFand brace height measures for MUs matched across multiple trials.Significance. Spanning both datasets, we found brace height quantification to provide an intuitive method for achieving graded estimates of neuromodulatory and inhibitory drive to individual MUs. This complements common techniques and provides an avenue for decoupling changes in the level of neuromodulatory and pattern of inhibitory motor commands.

The Cellular Basis for the Generation of Firing Patterns in Human Motor Units

Motor units, which comprise a motoneuron and the set of muscle fibers it innervates, are the fundamental neuromuscular transducers for all motor commands. The one to one relationship between a motoneuron and its innervated muscle fibers allow motoneuron firing patterns to be readily measured in humans. In this chapter, we summarize the current understanding of the cellular basis for the generation of firing patterns in human motor units. We provide a brief review of landmark insights from classic studies and then proceed to consider the features of motor unit firing patterns that are most likely to be sensitive estimators of motoneuron inputs and properties. In addition, we discuss recent advances in technology for recording human motor unit firing patterns and highly realistic computer simulations of motoneurons. The final section presents our recent efforts to use the power of supercomputers for implementation of the motoneuron models, with a goal of achieving a true "reverse engineering" approach that maximizes the insights from motor unit firing patterns into the synaptic structure of motor commands.

Linking Motoneuron PIC Location to Motor Function in Closed-Loop Motor Unit System Including Afferent Feedback: A Computational Investigation

The goal of this study is to investigate how the activation location of persistent inward current (PIC) over motoneuron dendrites is linked to motor output in the closed-loop motor unit. Here, a physiologically realistic model of a motor unit including afferent inputs from muscle spindles was comprehensively analyzed under intracellular stimulation at the soma and synaptic inputs over the dendrites during isometric contractions over a full physiological range of muscle lengths. The motor output of the motor unit model was operationally assessed by evaluating the rate of force development, the degree of force potentiation and the capability of self-sustaining force production. Simulations of the model motor unit demonstrated a tendency for a faster rate of force development, a greater degree of force potentiation, and greater capacity for self-sustaining force production under both somatic and dendritic stimulation of the motoneuron as the PIC channels were positioned farther from the soma along the path of motoneuron dendrites. Interestingly, these effects of PIC activation location on force generation significantly differed among different states of muscle length. The rate of force development and the degree of force potentiation were systematically modulated by the variation of PIC channel location for shorter-than-optimal muscles but not for optimal and longer-than-optimal muscles. Similarly, the warm-up behavior of the motor unit depended on the interplay between PIC channel location and muscle length variation. These results suggest that the location of PIC activation over motoneuron dendrites may be distinctively reflected in the motor performance during shortening muscle contractions.

PyMUS: Python-Based Simulation Software for Virtual Experiments on Motor Unit System

We constructed a physiologically plausible computationally efficient model of a motor unit and developed simulation software that allows for integrative investigations of the input-output processing in the motor unit system. The model motor unit was first built by coupling the motoneuron model and muscle unit model to a simplified axon model. To build the motoneuron model, we used a recently reported two-compartment modeling approach that accurately captures the key cell-type-related electrical properties under both passive conditions (somatic input resistance, membrane time constant, and signal attenuation properties between the soma and the dendrites) and active conditions (rheobase current and afterhyperpolarization duration at the soma and plateau behavior at the dendrites). To construct the muscle unit, we used a recently developed muscle modeling approach that reflects the experimentally identified dependencies of muscle activation dynamics on isometric, isokinetic and dynamic variation in muscle length over a full range of stimulation frequencies. Then, we designed the simulation software based on the object-oriented programing paradigm and developed the software using open-source Python language to be fully operational using graphical user interfaces. Using the developed software, separate simulations could be performed for a single motoneuron, muscle unit and motor unit under a wide range of experimental input protocols, and a hierarchical analysis could be performed from a single channel to the entire system behavior. Our model motor unit and simulation software may represent efficient tools not only for researchers studying the neural control of force production from a cellular perspective but also for instructors and students in motor physiology classroom settings.

Impact of the localization of dendritic calcium persistent inward current on the input-output properties of spinal motoneuron pool: a computational study

The goal of this study is to investigate how the dendritic Ca-PIC location influences nonlinear input-output properties and depends on the type of motoneurons across the motoneuron pool. A model motoneuron pool consisting of 10 motoneurons was constructed using a recently developed two-compartment modeling approach that reflected key cell type-associated properties experimentally identified. The dendritic excitability and firing output depended systematically on both the PIC location and the motoneuron type. The PIC onset and offset in the current-voltage (I-V) relationship tended to occur at more hyperpolarized voltages as the path length to the PIC channels from the soma increased and as the cell type shifted from high- to low-threshold motoneurons. At the same time, the firing acceleration and frequency hysteresis in the frequency-current (F-I) relationship became faster and larger, respectively. However, the PIC onset-offset hysteresis increased as the path length and the recruitment threshold increased. Furthermore, the gain of frequency-current function before full PIC activation was larger for PIC channels located over distal dendritic regions in low- compared with high-threshold motoneurons. When compared with previously published experimental observations, the modeling concurred when Ca-PIC channels were placed closer to the soma in high- than low-threshold motoneurons in the model motoneuron pool. All of these results suggest that the negative relationship of Ca-PIC location and cell recruitment threshold may underlie the systematic variation in I-V and F-I transformation across the motoneuron pool.NEW & NOTEWORTHY How does the dendritic location of calcium persistent inward current (Ca-PIC) influence dendritic excitability and firing behavior across the spinal motoneuron pool? This issue was investigated developing a model motoneuron pool that reflected key motoneuron type-specific properties experimentally identified. The simulation results point out the negative relationship between the distance of Ca-PIC source from the soma and cell recruitment threshold as a basis underlying the systematic variation in input-output properties of motoneurons over the motoneuron pool.

Synaptic control of the shape of the motoneuron pool input-output function

Although motoneurons have often been considered to be fairly linear transducers of synaptic input, recent evidence suggests that strong persistent inward currents (PICs) in motoneurons allow neuromodulatory and inhibitory synaptic inputs to induce large nonlinearities in the relation between the level of excitatory input and motor output. To try to estimate the possible extent of this nonlinearity, we developed a pool of model motoneurons designed to replicate the characteristics of motoneuron input-output properties measured in medial gastrocnemius motoneurons in the decerebrate cat with voltage-clamp and current-clamp techniques. We drove the model pool with a range of synaptic inputs consisting of various mixtures of excitation, inhibition, and neuromodulation. We then looked at the relation between excitatory drive and total pool output. Our results revealed that the PICs not only enhance gain but also induce a strong nonlinearity in the relation between the average firing rate of the motoneuron pool and the level of excitatory input. The relation between the total simulated force output and input was somewhat more linear because of higher force outputs in later-recruited units. We also found that the nonlinearity can be increased by increasing neuromodulatory input and/or balanced inhibitory input and minimized by a reciprocal, push-pull pattern of inhibition. We consider the possibility that a flexible input-output function may allow motor output to be tuned to match the widely varying demands of the normal motor repertoire.NEW & NOTEWORTHY Motoneuron activity is generally considered to reflect the level of excitatory drive. However, the activation of voltage-dependent intrinsic conductances can distort the relation between excitatory drive and the total output of a pool of motoneurons. Using a pool of realistic motoneuron models, we show that pool output can be a highly nonlinear function of synaptic input but linearity can be achieved through adjusting the time course of excitatory and inhibitory synaptic inputs.

The tight relationship between asymmetric signaling and locational excitability in motoneuron dendrites

Spinal motoneurons possess large, highly branching dendritic structures that contain thousands of synaptic contacts and various voltage-gated ion channels (VGICs). Research has indicated that dendritic arborization and cable properties provide the basis for foundational dendritic processing, which is characterized by direction-dependent signal propagation and location-dependent channel activation in dendritic arbors. Due to these arbors' complex structure, signals attenuate differentially depending on whether propagation occurs from the soma to the dendrite or in the opposite direction. In addition, current thresholds for the activation of dendritic ion channels differ depending on the location of these channels within dendrites. However, whether and how these foundational properties for dendritic signaling and excitability are related in motoneurons remains unclear. Based on our analyses of anatomically reconstructed motoneurons and novel reduced models, we propose that 1) directional signal propagation is similar across spinal motoneurons, regardless of cell type-specific structures; 2) reduced models that retain dendritic signaling asymmetry can accurately replicate anatomical dendritic excitability in both passive and active modes; and 3) asymmetric signal propagation and locational dendritic excitability are closely related, irrespective of motoneurons' arbor structures.

Development of modified cable models to simulate accurate neuronal active behaviors

In large network and single three-dimensional (3-D) neuron simulations, high computing speed dictates using reduced cable models to simulate neuronal firing behaviors. However, these models are unwarranted under active conditions and lack accurate representation of dendritic active conductances that greatly shape neuronal firing. Here, realistic 3-D (R3D) models (which contain full anatomical details of dendrites) of spinal motoneurons were systematically compared with their reduced single unbranched cable (SUC, which reduces the dendrites to a single electrically equivalent cable) counterpart under passive and active conditions. The SUC models matched the R3D model's passive properties but failed to match key active properties, especially active behaviors originating from dendrites. For instance, persistent inward currents (PIC) hysteresis, frequency-current (FI) relationship secondary range slope, firing hysteresis, plateau potential partial deactivation, staircase currents, synaptic current transfer ratio, and regional FI relationships were not accurately reproduced by the SUC models. The dendritic morphology oversimplification and lack of dendritic active conductances spatial segregation in the SUC models caused significant underestimation of those behaviors. Next, SUC models were modified by adding key branching features in an attempt to restore their active behaviors. The addition of primary dendritic branching only partially restored some active behaviors, whereas the addition of secondary dendritic branching restored most behaviors. Importantly, the proposed modified models successfully replicated the active properties without sacrificing model simplicity, making them attractive candidates for running R3D single neuron and network simulations with accurate firing behaviors. The present results indicate that using reduced models to examine PIC behaviors in spinal motoneurons is unwarranted.

Neuromodulation impact on nonlinear firing behavior of a reduced model motoneuron with the active dendrite

Neuromodulatory inputs from brainstem systems modulate the normal function of spinal motoneurons by altering the activation properties of persistent inward currents (PICs) in their dendrites. However, the effect of the PIC on firing outputs also depends on its location in the dendritic tree. To investigate the interaction between PIC neuromodulation and PIC location dependence, we used a two-compartment model that was biologically realistic in that it retains directional and frequency-dependent electrical coupling between the soma and the dendrites, as seen in multi-compartment models based on full anatomical reconstructions of motoneurons. Our two-compartment approach allowed us to systematically vary the coupling parameters between the soma and the dendrite to accurately reproduce the effect of location of the dendritic PIC on the generation of nonlinear (hysteretic) motoneuron firing patterns. Our results show that as a single parameter value for PIC activation was either increased or decreased by 20% from its default value, the solution space of the coupling parameter values for nonlinear firing outputs was drastically reduced by approximately 80%. As a result, the model tended to fire only in a linear mode at the majority of dendritic PIC sites. The same results were obtained when all parameters for the PIC activation simultaneously changed only by approximately ±10%. Our results suggest the democratization effect of neuromodulation: the neuromodulation by the brainstem systems may play a role in switching the motoneurons with PICs at different dendritic locations to a similar mode of firing by reducing the effect of the dendritic location of PICs on the firing behavior.

Asymmetry in signal propagation between the soma and dendrites plays a key role in determining dendritic excitability in motoneurons

It is widely recognized that propagation of electrophysiological signals between the soma and dendrites of neurons differs depending on direction, i.e. it is asymmetric. How this asymmetry influences the activation of voltage-gated dendritic channels, and consequent neuronal behavior, remains unclear. Based on the analysis of asymmetry in several types of motoneurons, we extended our previous methodology for reducing a fully reconstructed motoneuron model to a two-compartment representation that preserved asymmetric signal propagation. The reduced models accurately replicated the dendritic excitability and the dynamics of the anatomical model involving a persistent inward current (PIC) dispersed over the dendrites. The relationship between asymmetric signal propagation and dendritic excitability was investigated using the reduced models while varying the asymmetry in signal propagation between the soma and the dendrite with PIC density constant. We found that increases in signal attenuation from soma to dendrites increased the activation threshold of a PIC (hypo-excitability), whereas increases in signal attenuation from dendrites to soma decreased the activation threshold of a PIC (hyper-excitability). These effects were so strong that reversing the asymmetry in the soma-to-dendrite vs. dendrite-to-soma attenuation, reversed the correlation between PIC threshold and distance of this current source from the soma. We propose the tight relation of the asymmetric signal propagation to the input resistance in the dendrites as a mechanism underlying the influence of the asymmetric signal propagation on the dendritic excitability. All these results emphasize the importance of maintaining the physiological asymmetry in dendritic signaling not only for normal function of the cells but also for biophysically realistic simulations of dendritic excitability.

Statistical computer model analysis of the reciprocal and recurrent inhibitions of the Ia-EPSP in α-motoneurons

We simulate the inhibition of Ia-glutamatergic excitatory postsynaptic potential (EPSP) by preceding it with glycinergic recurrent (REN) and reciprocal (REC) inhibitory postsynaptic potentials (IPSPs). The inhibition is evaluated in the presence of voltage-dependent conductances of sodium, delayed rectifier potassium, and slow potassium in five α-motoneurons (MNs). We distribute the channels along the neuronal dendrites using, alternatively, a density function of exponential rise (ER), exponential decay (ED), or a step function (ST). We examine the change in EPSP amplitude, the rate of rise (RR), and the time integral (TI) due to inhibition. The results yield six major conclusions. First, the EPSP peak and the kinetics depending on the time interval are either amplified or depressed by the REC and REN shunting inhibitions. Second, the mean EPSP peak, its TI, and RR inhibition of ST, ER, and ED distributions turn out to be similar for analogous ranges of G. Third, for identical G, the large variations in the parameters' values can be attributed to the sodium conductance step (g(Na_step)) and the active dendritic area. We find that small g(Na_step) on a few dendrites maintains the EPSP peak, its TI, and RR inhibition similar to the passive state, but high g(Na_step) on many dendrites decrease the inhibition and sometimes generates even an excitatory effect. Fourth, the MN's input resistance does not alter the efficacy of EPSP inhibition. Fifth, the REC and REN inhibitions slightly change the EPSP peak and its RR. However, EPSP TI is depressed by the REN inhibition more than the REC inhibition. Finally, only an inhibitory effect shows up during the EPSP TI inhibition, while there are both inhibitory and excitatory impacts on the EPSP peak and its RR.

A simple method for characterizing passive and active neuronal properties: application to striatal neurons

The study of active and passive neuronal dynamics usually relies on a sophisticated array of electrophysiological, staining and pharmacological techniques. We describe here a simple complementary method that recovers many findings of these more complex methods but relies only on a basic patch-clamp recording approach. Somatic short and long current pulses were applied in vitro to striatal medium spiny (MS) and fast spiking (FS) neurons from juvenile rats. The passive dynamics were quantified by fitting two-compartment models to the short current pulse data. Lumped conductances for the active dynamics were then found by compensating this fitted passive dynamics within the current-voltage relationship from the long current pulse data. These estimated passive and active properties were consistent with previous more complex estimations of the neuron properties, supporting the approach. Relationships within the MS and FS neuron types were also evident, including a graduation of MS neuron properties consistent with recent findings about D1 and D2 dopamine receptor expression. Application of the method to simulated neuron data supported the hypothesis that it gives reasonable estimates of membrane properties and gross morphology. Therefore detailed information about the biophysics can be gained from this simple approach, which is useful for both classification of neuron type and biophysical modelling. Furthermore, because these methods rely upon no manipulations to the cell other than patch clamping, they are ideally suited to in vivo electrophysiology.

Asymmetric electrotonic coupling between the soma and dendrites alters the bistable firing behaviour of reduced models

The goal of the study was to investigate the influence of asymmetric coupling, between the soma and dendrites, on the nonlinear dynamic behaviour of a two-compartment model. We used a recently published method for generating reduced two-compartment models that retain the asymmetric coupling of anatomically reconstructed motor neurons. The passive input-output relationship of the asymmetrically coupled model was analytically compared to the symmetrically coupled case. Predictions based on the analytic comparison were tested using numerical simulations. The simulations evaluated the nonlinear dynamics of the models as a function of coupling parameters. Analytical results showed that the input resistance at the dendrite of the asymmetric model was directly related to the degree of coupling asymmetry. In contrast, a comparable symmetric model had identical input resistances at both the soma and dendrite regardless of coupling strength. These findings lead to predictions that variations in dendritic excitability, subsequent to changes in input resistance, might change the current threshold and onset timing of the plateau potential generated in the dendrite. Since the plateau potential underlies bistable firing, these results further predicted that asymmetric coupling might alter nonlinear (i.e. bistable) firing patterns. The numerical simulations supported analytical predictions, showing that the fully bistable firing pattern of the asymmetric model depended on the degree of coupling asymmetry and its correlated dendritic excitability. The physiological property of asymmetric coupling plays an important role in generating and stabilizing the bistability of motor neurons by interacting with the excitability of dendritic branches.

Analysis of impulse adaptation in motoneurons

Animal locomotion results from muscle contraction and relaxation cycles that are generated within the central nervous system and then are relayed to the periphery by motoneurons. Thus, motoneuron function is an essential element for understanding control of animal locomotion. This paper presents motoneuron input-output relationships, including impulse adaptation, in the medicinal leech. We found that although frequency-current graphs generated by passing 1-s current pulses in neuron somata were non-linear, peak and steady-state graphs of frequency against membrane potential were linear, with slopes of 5.2 and 2.9 Hz/mV, respectively. Systems analysis of impulse frequency adaptation revealed a static threshold nonlinearity at -43 mV (impulse threshold) and a single time constant (tau = 88 ms). This simple model accurately predicts motoneuron impulse frequency when tested by intracellular injection of sinusoidal current. We investigated electrical coupling within motoneurons by modeling these as three-compartment structures. This model, combined with the membrane potential-impulse frequency relationship, accurately predicted motoneuron impulse frequency from intracellular records of soma potentials obtained during fictive swimming. A corollary result was that the product of soma-to-neurite and neurite-to-soma coupling coefficients in leech motoneurons is large, 0.85, implying that the soma and neurite are electrically compact.