A simulation model of the surface EMG signal for analysis of muscle activity during the gait cycle

This work describes a model able to synthetize the surface EMG (electromyography) signal acquired from tibialis anterior and gastrocnemious medialis muscles during walking of asymptomatic adult subjects. The model assumes a muscle structure where the volume conductor is represented by multiple layers of anisotropic media. This model originates from analysis of the single fiber action potential characterized by the conduction velocity. The surface EMG of voluntary contraction is calculated by gathering motor unit action potentials estimated by the summation of all activities of muscle fibers assumed to have a uniformly parallel distribution. The parameters related to the gait cycle, such as onset and cessation timings of muscle activation, amplitude of muscle contraction, periods and sequences of motor units' recruitment, are included in the model presented. In addition, the relative positions of the electrodes during gait can also be specified in order to adapt the simulation to the different acquisition settings.     

Power spectra of single infinite fibre extracellular potentials recorded by a bipolar electrode

The power spectra of bipolarly recorded extracellular action potentials (EAPs) generated by an infinite, homogeneous, excitable fibre in an infinite, resistive, isotropic and homogeneous volume conductor were theoretically analysed. The changes in the power spectrum of EAP, which occurred as a result of alterations in the propagation velocity v and duration T_in of the intracellular action potential IAP, were analytically determined for bipolar parallel and radial electrodes with a small interpole distance. It was found that the sensitivity of the spectral characteristics to alteration in v, T_in and/or the IAP asymmetry substantially depends on the fibre-electrode distance; information on the IAP fast changes, that seems to be lost in unipolar recording as a result of the filtering effect of the fibre-electrode distance, can be restored. The orientation of the recording electrode need not be taken into account when a qualitative analysis is carried out, but when a quantitative analysis has to be performed, then the electrode orientation has a significant influence. A method is suggested for determination of the fibres' orientation by means of the spectrum of EAPs recorded bipolarly. The selectivity of the bipolar electrodes is analysed.     

Power spectra of extracellular potentials generated by an infinite,homogeneous excitable fibre

The power spectra of the extracellular potentials (EPs) generated under activation of an infinite, homogeneous excitable fibre immersed into an infinite, resistive, isotropic and homogeneous volume conductor are theoretically analysed. The changes in the power spectrum related to the changes in the propagation velocity v, amplitudes V_m and duration T_in of the intracellular action potential (IAP) are analytically determined. It is found that in the ultra-low-frequency region the EP spectral power follows the course of alteration in the square of the modified Bessel function of the second kind and order zero multiplied by the fourth power of the frequency, and the T_in can be assessed by the deviation of the EP power spectrum from this function. It is shown why the sensitivity of the spectral characteristics depends substantially on the radial distance y₀ from the activated fibre to the point of observation; why the total spectral amplitude depends directly on the IAP wavelength but the total spectral power depends on the IAP wavelength as well as on its duration and propagation velocity; and why the EPs are not proportional to the IAP second spatial derivative even in close proximity to the fibre.     

Simulation of single muscle fibre action potentials

Using the volume conductor model, a single muscle fibre action potential can be expressed as a convolution of the transmembrane current and a weighting function. By simplifying the weighting function, the line source model is derived. We have developed similar expressions to compute the single muscle fibre action potential using simple models and physical considerations without any mathematical complexity. The relationship between the conduction velocity and amplitude is analysed and it is concluded that, for a given fibre, the amplitude is inversely proportional to the conduction velocity. This agrees with the experimental data reported in the literature. The relation between amplitude and fibre diameter is studied. The amplitude increases with diameter owing to the increase in membrane current, but it is counteracted by the increase in conduction velocity. Because of the opposing effects, a point of inflection in the amplitude/diameter relationship is observed. Squareroot, square and linear dependencies of conduction velocity on fibre diameter were used. The difference in the peak-to-peak amplitude with these relationships is small and a linear relation between amplitude and fibre diameter seems reasonable irrespective of the conduction velocity/fibre diameter relationship. The effect of lumping the transmembrane current at the axis of the fibre is discussed. This approximation results in an underestimation of the peak-to-peak amplitude near the surface, and the error is close to 50% for very large fibres. This error is accounted for in the model. The recording distance at which the peak-to-peak amplitude of the signal is 100 μV is found to be 400 μm in the model. This is in good agreement with values obtained from a single-fibre electrode recording. The model is computationally fast and precise. It can easily be used with few modifications to simulate single fibre action potentials recorded using different electrodes.     

Application of the boundary element method to the solution of anisotropic electromagnetic problems

The paper discusses the application of the boundary element method to the computation of the electric potential and magnetic field generated by bioelectric sources in an anisotropic inhomogeneous volume conductor, using a proper coordinate transformation. It is shown that the co-ordinate transformation generally not only affects the conductivity and geometry of the volume conductor under consideration, but also the current source term and the continuity relation on the interfaces bounding regions of different conductivity. To illustrate these results, the electric potentials in an anisotropic finite length cylinder and in an anisotropic volume conductor of irregular (torso) shape, computed by the boundary element method, are compared with the results obtained by the analytical solution and the finite element method, respectively.     

Radial changes of extracellular potential amplitude and integral characteristics and the inverse problem in electroneurography

The possibility of solving the inverse problem in electroneurography, i.e. of estimating the main parameters specifying the activated fibre's functional state, using the amplitude and integral characteristics of the surface potentials generated by infinite homogeneous fibres, has been analysed. An analytical expression has been found for the amplitude of the negative phase A_nph of the single fibre extracellular action potential (SFEAP) as a function of the wavelength b, the fibre-electrode distance y and a scale factor A_o proportional to the intracellular action potential amplitude V_m, to the square of the fibre radius a and to the ratio of the axoplasm conductivity τ_a and volume conductor conductivity τ_e. For a large fibre-electrode distance, typical of surface recordings, an analytical expression of the integral of the negative phase I_nph of the SFEAP as a function of A_o, b, y and the propagation velocity v was also found. Simple methods are proposed for estimating v, the location of the electrical centre of the activated fibres' territory and the product of the number of activated fibres N, duration T_in of the intracellular action, potential and of the factor A_o. The estimation errors due to the temporal and spatial dispersion of the activated fibres were analysed as a function of the fibre-electrode distance and the territory shape.     

Radial decline of the extracellular action potential

A modified line source model presented earlier has been used to study the decline of the extracellular single muscle fibre action potential. The muscle tissue is modelled as a low-pass filter. The transfer function of the filter declines more slowly than a first order low-pass filter at low frequencies, but much faster at high frequencies. The cutoff frequency of the filter increases when the anisotropy of the muscle decreases. It also increases proportionally with the propagation velocity of the action potential. The decline of different frequency components obtained from the modified line source volume conductor and a filter model derived from experimental measurements are compared and their differences explained. The modified line source model was found to be identical to the volume conductor model in terms of results and at the same time conceptually simple for applications.     

柴胡桂枝汤对离体蟾蜍坐骨神经动作电位的影响

目的:研究柴胡桂枝汤对蟾蜍离体坐骨神经干动作电位传导阻滞作用。方法:观察柴胡桂枝汤对蟾蜍离体坐骨神经复合动作电位的振幅和传导速度的影响。结果:柴胡桂枝汤作用0min、2min、4min、6min、8min、10min后均使坐骨神经干复合动作电位的振幅变小(P0.01),传导速度变慢(P0.01)并最终使坐骨神经动作电位消失。结论:柴胡桂枝汤能阻滞神经动作电位的传导。     

蛇床子提取液对离体蟾蜍坐骨神经动作电位的影响

目的:研究蛇床子提取液对蟾蜍离体坐骨神经动作电位传导阻滞作用。方法:观察3种浓度蛇床子提取液(1g·ml-1、0.5g·ml-1、0.2g·ml-1)对蟾蜍离体坐骨神经复合动作电位的振幅和传导速度的影响。结果:3种浓度的蛇床子提取液均可使坐骨神经复合动作电位的振幅变小(P0.01),传导速度变慢(P0.01)并最终使坐骨神经动作电位消失。结论:蛇床子提取液能阻滞神经动作电位的传导。     

A model for compound action potentials and currents in a nerve bundle I: The forward calculation

We describe a model for the Compound Action Currents (CACs) and Compound Action Potentials (CAPs) produced by a peripheral nerve bundlein vitro. The Single Fiber Action Currents (SFACs) and the extracellular Single Fiber Action Potentials (SFAPs) are calculated using a generalized volume conduction model. Frequency-dependent conductivities, variations in the intracellular action potentials with recording temperature and axon conduction velocity, and the effects of axonal myelination are incorporated into the volume conduction calculation. We demonstrate how the propagation distance and the recording radius affect the simulated Compound Action Signals (CASs) of various nerve bundles. We also demonstrate how the frequency-dependent and-independent conductivities affect the CASs simulated by our model. For this simulation, some of the parameters for the nerve bundles and Conduction Velocity Distributions (CVDs) were obtained from the literature. In accompanying papers, we use the simulated CASs to investigate the effects of variations in the model parameters on the CVDs predicted by our inverse model.     

Simulation of propagation along an isolated skeletal muscle fiber in an isotropic volume conductor

This paper describes a model of the frog skeletal muscle fiber that includes the effects of the transverse tubular system (T system) on propagation. Uniform propagation on an isolated fiber suspended in Ringer’s solution or in air is simulated by placing the cylindrical fiber model in a concentric three-dimensional isotropic volume conductor. The current through the T system outlets at the sarcolemmal surface is comparable in magnitude to the sarcolemmal current density, but is of opposite polarity. When it is added to the sarcolemmal current, the resulting triphasic waveform has a 100% increase in the leading positive peak, a 50% reduction in the negative peak, and more than 60% reduction in the trailing positive peak. As a result the tubular output current causes a reduction in the conduction, velocity, a decrease in the maximum rate of rise of the action potential, and an important modification of the extracellular potential. Compared to an isolated fiber in a large volume of Ringer’s solution, uniform propagation within a 2-μm-thick volume conductor annulus is slowed down from 1.92 to 0.72 m/s, and the extracellular potential is increased from 1 to 108 mV peak to peak, in agreement with published experimental measurements.     

Extracellular potentials and currents of a single active fiber in a restricted volume conductor

Based on mathematical expressions governing the electric field, the extracellular potentials generated by a single active fiber in a restricted circular cylindrical volume conductor are evaluated. This paper examines the effect of the extent of the volume conductor, with radius b, on the extracellular potentials at different field points. For values of b less than 1.5 times the fiber radius, the extracellular potentials in the volume conductor are always the core conductor potentials, independent of the shape and amplitude of the transmembrane potential. For b greater than a critical radius (a value that depends on the transmembrane potential waveform), the extracellular potentials at and near the membrane are the same as if the volume conductor were unbounded. Near the boundary with the insulator, the amplitude of the extracellular potentials is equal to the core conductor amplitude, although the potentials are much broader than the core conductor potential.     

Influence of motor unit properties on the size of the simulated evoked surface EMG potential

The purpose of the study was to quantify the influence of selected motor unit properties on the simulated amplitude and area of evoked muscle potentials detected at the skin surface. The study was restricted to a motor unit population simulating a hand muscle whose potentials were recorded on the skin over the muscle. Peak-to-peak amplitude and area of the evoked potential were calculated from the summed motor unit potentials and compared across conditions that simulated variation in different motor unit properties. The simulations involved varying the number of activated motor units, muscle fiber conduction velocities, axonal conduction velocities, neuronal activation times, the shape of the intracellular action potential, and recording configurations commonly used over hand muscles. The results obtained for the default condition simulated in this study indicated that ~7% of the motor unit potentials were responsible for 50% of the size of the evoked potential. Variation in the amplitude and area of the evoked muscle potential was directly related to the number of active motor units only when the stimulus activated motor units randomly, and not when activation was based on a parameter such as motor unit size. Independent adjustments in motor unit properties had variable effects on the size of the evoked muscle potential, including when the stimulus activated only a subpopulation of motor units. These results provide reference information that can be used to assist in the interpretation of experimentally observed changes in the size of evoked muscle potentials.     

Conduction velocity is inversely related to action potential threshold in rat motoneuron axons

Intra-axonal recordings were performed in ventral roots of rats in vitro to study the conduction velocity and firing threshold properties of motoneuron axons. Mean values ± SD were 30.5±5.6 m/s for conduction velocity and 11.6±4.5 mV for the depolarization from the resting potential required to reach firing threshold (threshold depolarization). Conduction velocity varied inversely and significantly with threshold depolarization (P=0.0002 by linear regression). This relationship was evident even after accounting for variation in conduction velocity associated with action potential amplitude, injected current amplitude, or body weight. Conduction velocity also varied inversely with the time to action potential onset during just-threshold current pulse injection. These data suggest that the time course of depolarization leading to action potential initiation contributes to the speed of conduction in motoneuron axons. Electronic Publication     

Direct detection of a single evoked action potential with MRS in Lumbricus terrestris

Functional MRI (fMRI) measures neural activity indirectly by detecting the signal change associated with the hemodynamic response following brain activation. In order to alleviate the temporal and spatial specificity problems associated with fMRI, a number of attempts have been made to detect neural magnetic fields (NMFs) with MRI directly, but have thus far provided conflicting results. In this study, we used MR to detect axonal NMFs in the median giant fiber of the earthworm, Lumbricus terrestris, by examining the free induction decay (FID) with a sampling interval of 0.32 ms. The earthworm nerve cords were isolated from the vasculature and stimulated at the threshold of action potential generation. FIDs were acquired shortly after the stimulation, and simultaneous field potential recordings identified the presence or absence of single evoked action potentials. FIDs acquired when the stimulus did not evoke an action potential were summed as background. The phase of the background-subtracted FID exhibited a systematic change, with a peak phase difference of (-1.2 ± 0.3) × 10(-5) radians occurring at a time corresponding to the timing of the action potential. In addition, we calculated the possible changes in the FID magnitude and phase caused by a simulated action potential using a volume conductor model. The measured phase difference matched the theoretical prediction well in both amplitude and temporal characteristics. This study provides the first evidence for the direct detection of a magnetic field from an evoked action potential using MR.     

Interactions between adjacent fibers in a cardiac muscle bundle

A strand of cardiac muscle was modeled as a small bundle of individual fibers surrounded by a large volume conductor. The bundle is a uniform assembly of small identical cylindrical fibers, arranged as a series of concentric layers, and its behavior is examined in the presence (coupled bundle) or absence (uncoupled bundle) of transverse resistive coupling between adjacent fibers. Individual fibers are continuous cables of excitable membrane, with circumferential segmentation into 12 equal patches to make the membrane potential changes dependent upon the local interstitial potential. The minimum spacing (d) between adjacent fibers is used to modify the interstitial microstructural organization and the intracellular volume fraction (f _i). Whend is small enough (d<0.01 μm),f _i remains unchanged at its maximum of about 90%, the interstitial potential is large, the transveerse interstitial resistance is high, and the proximity effect arising from the close juxtaposition of adjacent fibers is important. A surface fiber of the uncoupled bundle exhibits little sensitivity to changes in the intestitial microstructure, owing to the dominant influence of the external volume conductor, whereas the central fiber shows a large decrease in velocity, substantial waveshape modifications, and a large increase in interstitial potential asd is reduced. In the coupled bundle, all fibers adopt the same velocity during uniformo propagation owing to the strong transverse resistive coupling; whend is reduced in the range ofd<0.01 μm, the velocity and interstitial potential changes are less pronounced than in the uncoupled bundle. Whend is large enough (d>0.01 μm), the bundle behavior (coupled and uncoupled) approaches that obtained with a bidomain formulation.     

Propagation on a central fiber surrounded by inactive fibers in a multifibered bundle model

We studied uniform propagation on a central active fiber surrounded by inactive fibers in a multifibered bundle model lying in a large volume conductor. The behavior of a fully active bundle is considered in a companion paper. The bundle is formed by concentric layers of small cylindrical fibers (radius 5 μm), with a uniform minimum distance (d) between any two adjacent fibers, to yield a bundle radius of about 72μm. Individual vidual fibers are identical continuous cables of excitable membrane based on a modified Beeler-Reuter model. The intracellular volume fraction (f _i) increases to a maximum of about 90% asd is reduced and remains unchanged ford<0.01 μm. In the range ofd<0.01 μm, the central fiber is effectively shielded from external effects by the first concentric layer of inactive fibers, and a large capacitive load current flows across the surrounding inactive membranes. In addition, the fiber proximity produces a circumferentially nonuniform, current density (proximity effect) that is equivalent to an increased average longitudinal interstitial resistance. The conduction velocity is reduced asd becomes smaller in the range ofd<0.1 μm, the interstitial potential becomes larger, and both the maximum rate of rise and time constant of the foot of the upstroke are increased. On the other hand, ford>0.1 μm, there are negligible changes in the shape of the upstroke, and the, behavior of the central fiber is close to that of a uniform cable in a restricted volume conductor. Ford larger than about 1.2 μm, the active fiber environment is close to an unbounded isotropic volume conductor.     

Simulation of the Interaction Between Muscle Fiber Conduction Velocity and Instantaneous Firing Rate

In this study, the relationships between the early and late afterpotentials and velocity and amplitude recovery functions (VRF and ARF) in skeletal muscle were examined using model simulation. A mathematical model of the muscle fiber action potential, that incorporated a tubular slow potassium conductance, was developed and used to simulate muscle fiber action potentials at a range of interpulse intervals. The slow potassium conductance produced an afterhyperpolarization which resulted in supernormal action potential conduction velocity and amplitude for interpulse intervals >7 ms. Increasing the number of conditioning stimuli caused a further increase in conduction velocity and amplitude, and an additional phase of supernormality, with a peak at approximately 100 ms. Positive correlations between instantaneous firing rate and both conduction velocity and amplitude were also observed during simulation of repetitive stimulation of the muscle fiber. The relationships were eliminated when the slow potassium conductance channel was removed from the model. The results suggest that an afterhyperpolarization, possibly due to a slow tubular potassium conductance, could cause the VRF and ARF observed in muscle. They additionally suggest that the positive correlations between instantaneous firing rate, conduction velocity, and amplitude are directly related to the VRF and ARF.     

Non-invasive conductivity technique to detect changes in haematocrit:in vitro validation

An on-line haematocrit measurement in extracorporeal circuits might be useful under some clinical circumstances (e.g. haemodialysis or cardiac surgery). As no such measurement exists, a device has been developed that makes it possible to detect haematocrit (Ht) continuously without a loss of blood. It is a multi-frequency system for the detection of electrical conductivities. The aim of this study was to investigate whether this device can measure Ht alterations properly. Ht alterations were induced by adding pure mannitol and 20% mannitol to fresh human blood. Furthermore, the effect of both mannitol substances on the intracellular ion content, intracellular conductivity and Ht were investigated. Alternations in Ht were established by the addition of 1000, 800, 600, 400, 200 and 0 mg of pure mannitol to 10 ml of fresh human blood, and 3.0, 2.5, 2.0, 2.0, 1.5, 1.0, 0.5 and 0 ml of 20% mannitol to fresh human blood until a total volume of 10 ml was achieved. Although their effects were significantly different, pure mannitol and 20% mannitol both caused a reduction in mean cellular volume, and thus in Ht. A highly significant correlation was found between Ht and intracellular conductivity (r=0.90, p<0.001). In addition to these effects, addition of pure mannitol and 20% mannitol had different effects on the intracellular ion content. Pure mannitol caused an increase in intracellular ion content due to a transcellular ion shift, whereas 20% mannitol induced a decrease. From this study, it can be concluded that the multi-frequency conductivity method observes changes in Ht (and intracellular fluid volume) in an accurate manner. Changes in intracellular ion content of erythrocytes depend on the sort of mannitol substance that is added. The intracellular ion concentration can be calculated from measured intracellular conductivity and Ht. The total number of intracellular ions can be derived from intracellular conductivity and the number of erythrocytes.