*3.2.3. Towards an easy formulation of the SFAP convolutional model*

After the key contributions of Clark and Plonsey and then Andreassen and Rosenfalck, great effort was directed towards including in SFAP models the effects of the finite size (Gootzen et al., 1991), inhomogeneneity (Dimitrova and Dimitrov, 2001), and frequency dependency (Albers et al., 1988) of the volume conductor. All of these authors were more interested in improving the accuracy of the SFAP model at the expense of reducing the physical transparency and the time efficiency of their approximations. However, these latter two considerations were precisely two features that scientists in other areas of EMG research and practice were increasingly demanding from SFAP models.

The key step towards an easy and intuitive presentation of the SFAP convolutional model was made by Nandedkar and Stalberg in 1983. The authors substituted the complex weight function used by Andreassen and Rosenfalck (that involved Bessel functions) for the simpler expression of a potential generated by a point current source (i.e., a monopole) that propagates along the fiber axis from the neuromuscular junction towards the tendons.

Although the Nandedkar and Stalberg model was found computationally more efficient and more accurate than the dipole and tripole models, it still lacked a suitable approach for dealing with the excitation onset and extinction. To overcome this problem, the excitation wave should be represented with two stacks of double-layer disks distributed equidistantly along the fiber axis. Following this idea, Dimitrov and Dimitrova (1998) replaced the stacks of distributed current dipoles by dipoles lumped along the axis of the fiber (line source model). The weight function was computed as the potential produced by two lumped current dipoles propagating in opposite directions from the endplate toward the fiber ends.

Let us assume that a unit dipole originates at time zero at the neuromuscular junction and propagates along a fiber with a constant velocity *v*. The potential produced by this unit dipole at the electrode is *IR*(t) [Fig. 3(a)]. However, the source of excitation is actually a sequence of *n*

dipoles distributed equidistantly along the IAP profile, whose strengths (moments) are given by the function ∂IAP(x,t)/∂t. The rst dipole originates at time zero, and each subsequent dipole at an interval *∆t*. Let the amplitude of these dipoles be *a*1, *a*2,…, *a*n [Fig. 3(b)]. The first dipole originates at time zero, propagates towards the tendons, generating a potential *a*1·*IR*(t). The second dipole originates at time *∆t*, propagates towards the tendons, generating a potential *a*2·*IR*(t-∆t) [Fig. 3(c)]. Hence the total potential recorded by the electrode is

Computational Intelligence in Electromyography Analysis – 10 A Perspective on Current Applications and Future Challenges

normally justified.

medium surrounding the active fiber is neglected.

*3.2.3. Towards an easy formulation of the SFAP convolutional model* 

practice were increasingly demanding from SFAP models.

b. For a fiber in an extracellular medium of considerable extent, it is assumed that the resistance of the extracellular medium is practically 0, and so the influence of the

c. The excitation source is assumed to be distributed along the axis of the fiber. Since in general fiber radius is many times smaller than fiber length, this approximation is

On the basis of the volume conductor theory, Plonsey (1974) was the first to show that a SFAP can be expressed as a convolution of an excitation source and a weight function. An important step towards the simplification of this model was made by Andreassen and Rosenfalck (1981) who assumed that the transmembrane current was distributed and concentrated along the axis of the fiber (a line-source model). In addition, the authors theorized that the error caused by such a source simplification would be less than 5% provided the radial distance was 5 fiber radii or greater from the fiber axis. However, both the approach of Clark and Plonsey and that of Andreassen and Rosenfalck were still far from being simple and intuitive as they included an intricate mathematical formulation.

After the key contributions of Clark and Plonsey and then Andreassen and Rosenfalck, great effort was directed towards including in SFAP models the effects of the finite size (Gootzen et al., 1991), inhomogeneneity (Dimitrova and Dimitrov, 2001), and frequency dependency (Albers et al., 1988) of the volume conductor. All of these authors were more interested in improving the accuracy of the SFAP model at the expense of reducing the physical transparency and the time efficiency of their approximations. However, these latter two considerations were precisely two features that scientists in other areas of EMG research and

The key step towards an easy and intuitive presentation of the SFAP convolutional model was made by Nandedkar and Stalberg in 1983. The authors substituted the complex weight function used by Andreassen and Rosenfalck (that involved Bessel functions) for the simpler expression of a potential generated by a point current source (i.e., a monopole) that propagates along the fiber axis from the neuromuscular junction towards the tendons.

Although the Nandedkar and Stalberg model was found computationally more efficient and more accurate than the dipole and tripole models, it still lacked a suitable approach for dealing with the excitation onset and extinction. To overcome this problem, the excitation wave should be represented with two stacks of double-layer disks distributed equidistantly along the fiber axis. Following this idea, Dimitrov and Dimitrova (1998) replaced the stacks of distributed current dipoles by dipoles lumped along the axis of the fiber (line source model). The weight function was computed as the potential produced by two lumped current dipoles propagating in opposite directions from the endplate toward the fiber ends. Let us assume that a unit dipole originates at time zero at the neuromuscular junction and propagates along a fiber with a constant velocity *v*. The potential produced by this unit dipole at the electrode is *IR*(t) [Fig. 3(a)]. However, the source of excitation is actually a sequence of *n*

$$SFAP(t) = a\_1IR\left(t\right) + a\_2IR\left(t - \Delta t\right) + \dots + a\_nIR\left(t - (n-1)\Delta t\right) \tag{7}$$

As *∆t* tends to zero and *n* tends to infinity, the total potential produced at the point *P*0 by the propagating excitation function represents the convolution of d*IAP*(x,t)/dt and *IR*(t):

$$SFAP(t) = \frac{dIAP(t)}{dt} \* IR(t) \tag{8}$$

**Figure 3.** Presentation of the potential generated by a single fiber (SFAP) (d) as the output of a linear timeshift invariant system whose impulse response *IR* is the potential produced by a moving unit dipole (a) and whose input signal is a function of distributed dipoles whose strengths (moments) are determined by d*IAP*(x,t)/dt (b). In (c) the contribution of each dipole to the total SFAP is shown separately. (e) Schematic representation of a muscle fiber innervated by the axon of a motorneuron.

Thus, an active fiber can be considered as a linear timeshift-invariant system for the generation of an extracellular potential. Specifically, the SFAP is the output of the system with the impulse response *IR*(t) [Fig. 3(d)] and input signal d*IAP*(x,t)/dt. Since the activation of a skeletal muscle fiber gives rise to two oppositely-directed propagating IAPs, *IR*(t) is the sum of potentials generated at the observation point by two dipoles moving in opposite directions from the neuromuscular junction to the fiber ends, where they disappear (McGill et al., 2001; Dimitrov and Dimitrova, 1998). This sum of potentials can be described as

Computational Intelligence in Electromyography Analysis –

12 A Perspective on Current Applications and Future Challenges

$$SFAP(t) = K \cdot \int\_0^t \frac{dIAP(\tau)}{d\tau} \cdot \frac{1}{v} \cdot \left[IR\_1(t-\tau) + IR\_2(t-\tau)\right] \cdot d\tau = K \cdot \frac{1}{v} \cdot \frac{dIAP(t)}{dt} \* IR(t) \tag{9}$$

EMG Modeling 13

**Figure 4.** A schematic representation of the motor unit cross-section showing the MUFs of two motor units and delineating their MUTs which may overlap and may occupy only a fraction of the muscle

muscle cross section motor unit territory muscle fibers

By delineating the boundary enclosing the MUFs, we obtain a picture of the MUT. MUTs have been shown to be distributed over a localized region of a muscle's volume, with an approximately elliptical shape in cross-section [4] (see Fig. 4). That is, in terms of the muscle cross-section, only a fraction of its area is occupied by each motor unit (Bodine et al, 1988). This spatial feature was confirmed in humans using multilead-electrode-EMG, and using scanning-EMG (Stålberg and Antoni, 1980), with typical cross-sectional corridors between 5 and 10 mm long. Research based on glycogen-depletion techniques demonstrates that motor unit fibers of different motor units are intermingled, hence motor unit territories overlap

Defining the MUFD as the ratio of the MUFN to the MUTA, it is found that the ranges for MUFD are independent of the MUFN, while MUTA is strongly correlated with MUFN (Kanda and Hashizume, 1992). In reconstructions of the three-dimensional structure of motor units, MUFD is almost constant within a single motor unit when comparing the values obtained over the different cross-sections of the unit along the longitudinal axis of the

Another important feature related to motor unit fibers is their spatial distribution within the motor unit territory. Quantitative statistical studies in glycogen-depleted motor units by means of Monte Carlo simulation analysis (Bodine et al, 1988), suggest that MUFs are not homogeneously distributed: there are alternating regions of high and low MUF density within the MUT. High-density MUF areas are not usually located in the center of the MUT, as had been previously assumed. The various studies indicate that MUFs are arranged in small subclusters separated by holes with relatively few fibers, and researchers have suggested that if any clustering is present, it should be related to the axonal branching pattern established during development. The scanning EMG has confirmed the presence of silent areas within motor units of human muscles (Stålberg and Antoni, 1980), which most

cross-section.

muscle.

within the muscle cross-section.

*4.1.2. Spatial distribution of motor unit fibers* 

where \* means convolution and *IR* = *IR*1 + *IR*2, 0 ≤ *t* ≤ *t*max. Constant *K* is equal to (*a*2·*σ*i)/(4·*σ*e). The expressions for *IR*1 and *IR*2 and can be found in Dimitrov and Dimitrova (1998).

#### **3.3. Models of electrodes for recording single-fibre potentials**

The advent of single fiber electromyography (SFEMG) allowed investigators to analyze the shape peculiarities of the SFAP. The routine single fibre (SF) electrode consists of a montage of a 25-μm recording port (leading-off surface) referenced to the cannula (Ekstedt, 1964; Stålberg and Trontelj, 1979). When modelling SF electrodes, the leading-off surface is normally considered as a detection point, whereas the cannula is simulated through a plate electrode (Stalberg and Trontelj, 1992). As established by Ekstedt (1964), the conditions for recording an SFAP in a voluntarily activated muscle using an SF electrode are: (1) that the fibre is close to the electrode and thus gives rise to a potential with a short peak-to-peak interval (rise-time, RT), and (2) that any other fibres in the same motor unit that have coincident action potentials are sufficiently remote from the electrode that their contribution to the recorded signal is small.

#### **4.1. Anatomy and physiology of the motor unit**

#### *4.1.1. Motoneuron, motor unit fibers, and motor unit territory*

The motor unit is the entity that serves as a functional building block for the production of force and movement, both in reflex and voluntary contractions. Sherrington defined the motor unit as the set comprising a single motoneuron axon and the many muscle fibers to which that axon runs and which are hence innervated by it. The motor unit fibers (MUFs) are the different muscle fibers innervated by a single motoneuron. The numbers of muscle fibers supplied by a single motor neuron, through branching of its axons, is often referred to as the motor unit size or innervation ratio, and we will refer to this number as the motor unit fiber number (MUFN). In addition, we can consider the extent of the muscle crosssection that these fibers occupy, which is the motor unit territory (MUT), and the spatial distribution of these fibers within the motor unit territory area (MUTA). Subsequently, the motor unit can be characterization in terms of the motor unit fiber density (MUFD), measured as the number of MUFs per unit of area of its territory.

The MUFN varies greatly from one muscle to another and from one motor unit to another within the same muscle. The glycogen-depletion technique allows identification of the MUFs of a single motor unit within a muscle cross-section. Studies using this technique have demonstrated the large range of variation in MUFN (Burke and Tsairis, 1973), which extends from just a few fibers per motor unit in some muscles to thousands in others. Studies also show great variation within a single muscle: up to 100-fold from the smallest to the largest motor units.

**Figure 4.** A schematic representation of the motor unit cross-section showing the MUFs of two motor units and delineating their MUTs which may overlap and may occupy only a fraction of the muscle cross-section.

By delineating the boundary enclosing the MUFs, we obtain a picture of the MUT. MUTs have been shown to be distributed over a localized region of a muscle's volume, with an approximately elliptical shape in cross-section [4] (see Fig. 4). That is, in terms of the muscle cross-section, only a fraction of its area is occupied by each motor unit (Bodine et al, 1988). This spatial feature was confirmed in humans using multilead-electrode-EMG, and using scanning-EMG (Stålberg and Antoni, 1980), with typical cross-sectional corridors between 5 and 10 mm long. Research based on glycogen-depletion techniques demonstrates that motor unit fibers of different motor units are intermingled, hence motor unit territories overlap within the muscle cross-section.

Defining the MUFD as the ratio of the MUFN to the MUTA, it is found that the ranges for MUFD are independent of the MUFN, while MUTA is strongly correlated with MUFN (Kanda and Hashizume, 1992). In reconstructions of the three-dimensional structure of motor units, MUFD is almost constant within a single motor unit when comparing the values obtained over the different cross-sections of the unit along the longitudinal axis of the muscle.

#### *4.1.2. Spatial distribution of motor unit fibers*

Computational Intelligence in Electromyography Analysis – 12 A Perspective on Current Applications and Future Challenges

to the recorded signal is small.

the largest motor units.

τ

τ

**4.1. Anatomy and physiology of the motor unit** 

*4.1.1. Motoneuron, motor unit fibers, and motor unit territory* 

measured as the number of MUFs per unit of area of its territory.

() () ( ) 1 2 <sup>0</sup>

 ττ

= ⋅ ⋅⋅ − + − ⋅ = ⋅⋅ (9)

*d v v dt*

() 1 1 () ( ) \* *<sup>t</sup> dIAP dIAP t SFAP t K IR t IR t d K IR t*

where \* means convolution and *IR* = *IR*1 + *IR*2, 0 ≤ *t* ≤ *t*max. Constant *K* is equal to (*a*2·*σ*i)/(4·*σ*e).

The advent of single fiber electromyography (SFEMG) allowed investigators to analyze the shape peculiarities of the SFAP. The routine single fibre (SF) electrode consists of a montage of a 25-μm recording port (leading-off surface) referenced to the cannula (Ekstedt, 1964; Stålberg and Trontelj, 1979). When modelling SF electrodes, the leading-off surface is normally considered as a detection point, whereas the cannula is simulated through a plate electrode (Stalberg and Trontelj, 1992). As established by Ekstedt (1964), the conditions for recording an SFAP in a voluntarily activated muscle using an SF electrode are: (1) that the fibre is close to the electrode and thus gives rise to a potential with a short peak-to-peak interval (rise-time, RT), and (2) that any other fibres in the same motor unit that have coincident action potentials are sufficiently remote from the electrode that their contribution

The motor unit is the entity that serves as a functional building block for the production of force and movement, both in reflex and voluntary contractions. Sherrington defined the motor unit as the set comprising a single motoneuron axon and the many muscle fibers to which that axon runs and which are hence innervated by it. The motor unit fibers (MUFs) are the different muscle fibers innervated by a single motoneuron. The numbers of muscle fibers supplied by a single motor neuron, through branching of its axons, is often referred to as the motor unit size or innervation ratio, and we will refer to this number as the motor unit fiber number (MUFN). In addition, we can consider the extent of the muscle crosssection that these fibers occupy, which is the motor unit territory (MUT), and the spatial distribution of these fibers within the motor unit territory area (MUTA). Subsequently, the motor unit can be characterization in terms of the motor unit fiber density (MUFD),

The MUFN varies greatly from one muscle to another and from one motor unit to another within the same muscle. The glycogen-depletion technique allows identification of the MUFs of a single motor unit within a muscle cross-section. Studies using this technique have demonstrated the large range of variation in MUFN (Burke and Tsairis, 1973), which extends from just a few fibers per motor unit in some muscles to thousands in others. Studies also show great variation within a single muscle: up to 100-fold from the smallest to

τ

The expressions for *IR*1 and *IR*2 and can be found in Dimitrov and Dimitrova (1998).

**3.3. Models of electrodes for recording single-fibre potentials** 

Another important feature related to motor unit fibers is their spatial distribution within the motor unit territory. Quantitative statistical studies in glycogen-depleted motor units by means of Monte Carlo simulation analysis (Bodine et al, 1988), suggest that MUFs are not homogeneously distributed: there are alternating regions of high and low MUF density within the MUT. High-density MUF areas are not usually located in the center of the MUT, as had been previously assumed. The various studies indicate that MUFs are arranged in small subclusters separated by holes with relatively few fibers, and researchers have suggested that if any clustering is present, it should be related to the axonal branching pattern established during development. The scanning EMG has confirmed the presence of silent areas within motor units of human muscles (Stålberg and Antoni, 1980), which most likely correspond to the holes observed in glycogen-depletion studies. Hence these results seem to agree with the long-range distribution findings in non-human vertebrates by the glycogen-depletion technique.

EMG Modeling 15

muscle fiber diameters and positions

**Figure 5.** Representation of a simulated motor unit including the most relevant anatomical and

The first motor unit model (Griep et al, 1978) was proposed in order to study the properties of the motor unit potential (MUP) by means of computer simulation. Other researchers, (Miller-Larsson, 1980; Gath and Stålberg, 1982; Hilton-Brown et al, 1985), were concerned with the modeling of fiber density and spatial distribution of muscle fibers of a single motor unit, in order to determine the influence of architectural changes produced by neuromuscular disease on clinical electrophysiological recordings. Other single motor unit models have been developed more recently. Other studies (Nandedkar et al, 1988; Stålberg and Karlsson, 2001) proposed a muscle model to study the correlation between anatomical parameters and MUP signals by means of simulations, allowing the study of the MUP variations under different

muscle fiber lengths motor unit radius

concentric needle orientation and position

muscle fiber conduction velocity

Two main approaches are used to model the spatial distribution of MUFs: random location and random selection from a muscle fiber grid. In the first approach, a number of MUFs are placed following a given spatial distribution, usually normal or uniform. In the second approach, a predefined grid of evenly distributed muscle fibers is created, and a number of

**4.3. Models of the physiological parameters affecting the temporal dispersion of** 

In all the above-mentioned models, muscle fiber conduction velocity, delay of initiation of depolarization, and longitudinal position of the motor end-plate are modeled as random variables from statistical distributions. It is important to note that, as individual motor units are being modeled, simulated distributions refer to the MUFs and not to all the fibers of the

Muscle fiber diameter is modeled as a normally distributed variable with mean and standard deviation fixed to match corresponding values observed in real counterparts of the simulated muscle. Muscle fiber conduction velocity (MFCV) is modeled as a normally distributed variable, obtained from a linear transformation of the muscle fiber diameter. The

physiological factor to reproduce the architecture and physiology of the motor unit.

motor end-plate position

axonal delay

pathological conditions such as denervation and re-innervation.

**a MUAP** 

muscle bundle.

them are selected to be innervated by the motor unit under simulation.
