**2.1. Depolarization and repolarization of a muscle fiber membrane**

A muscle cell (fiber) is activated by electrical impulses coming from the motorneuron, which brings about the fiber's depolarization and the generation of a transmembrane voltage (normally defined as the extracellular electrical potential minus the intracellular one). Under normal conditions, the extracellular potential is practically zero, and so the transmembrane voltage can be considered to be practically the same as the intracellular action potential (IAP) (Burke, 1981; Plonsey and Barr, 2000). Membrane depolarization starts at the neuromuscular junction and extends along the muscle fiber towards both ends of the cell. As a result, two IAPs propagate without attenuation and with constant velocity *v* along the muscle fiber towards the tendons, where they extinguish (Lieber, 2010).

The plasma membrane of a muscle fiber has the well-studied property of actively maintaining a nearly constant potential difference between the intracellular and extracellular environment. This voltage is normally referred to as the resting potential and has a value of about -80 mV [Fig. 1(b)]. The negative sign indicates that the interior is more negative than the exterior (negative polarization). In Fig. 1(a) the polarization of the fiber membrane is represented by a number of layers of negative signs. When the fiber is at rest, the number of negative layers remains unchanged. After fiber activation, two potential profiles (one at each side of the neuromuscular junction) arise along the fiber membrane. These profiles are normally referred to as waves of excitation, excitation sources or simply IAPs [Fig. 1(a)]. In Fig. 1(a) it can be seen that, along the spatial profile of the IAP, the number of negative-signed layers changes progressively with axial distance (the *x*-axis in Fig. 1). This reflects the fact that the IAP profile has gradual depolarization and repolarization transitions, as shown in Fig. 1(b).

#### **2.2. The electric volume conduction**

Knowledge of muscle fiber excitability and IAP propagation would be clinically useless if the electromyographist had to penetrate muscle fibers to obtain information about membrane processes. Moreover, recording of the IAP *in situ* from human muscle fibers has not yet proved feasible in EMG studies (Ludin, 1973). All electromyography (EMG) techniques are based on the fact that local electrophysiological processes result in a detectable flow of the transmembrane current at a certain distance from the active sources (i.e., muscle fibers). This flow of current in the tissue (i.e., the volume conduction), allows EMG measurements to be made at a distance from the sources.

The so-called *Principle of Volume Conduction* can be considered as a three-dimensional version of Ohm's law, which establishes that an electric current *I*, flowing between two points connected through a resistance *R*, generates a potential difference *V* between these points: *V = I·R*. In the case of living tissue, the electrical impedance is the inverse of the electrical conductivity *σ*. So, the potential recorded at a point *P*0 (*x*0, *y*0, *z*0) within an infinite volume with uniform conductivity *σ*i produced by a current *I*s injected in the same volume at a point *P* (*x*, *y*, *z*) can be calculated as

$$V\_{P\_0} = \frac{1}{4\pi\sigma\_i} \frac{I\_s}{r\_i} \tag{1}$$

where *r*i is the shortest distance between the points *P*0 and *P*.

Computational Intelligence in Electromyography Analysis – 4 A Perspective on Current Applications and Future Challenges

physiological properties of the fibre and the shape of the potential.

**2. Modeling electrical conduction in skeletal muscle** 

**2.1. Depolarization and repolarization of a muscle fiber membrane** 

towards the tendons, where they extinguish (Lieber, 2010).

**2.2. The electric volume conduction** 

1(b).

potential, providing insight into the relationships between the anatomical and/or

Striated muscle is composed of a large number of striated muscle cells, also called muscle fibers. These elongated, cylindrical cells are arranged parallel to one another, and each one is surrounded by a plasma membrane called the sarcolemma. Muscle contraction is created via the repeated activation of several groups of muscle fibers, each of which is governed by a single motorneuron through its axon (Lieber, 2010). Figure 1(a) shows a portion of a muscle fiber that is attached, at the neuromuscular junction, to the terminal branch of its axon.

A muscle cell (fiber) is activated by electrical impulses coming from the motorneuron, which brings about the fiber's depolarization and the generation of a transmembrane voltage (normally defined as the extracellular electrical potential minus the intracellular one). Under normal conditions, the extracellular potential is practically zero, and so the transmembrane voltage can be considered to be practically the same as the intracellular action potential (IAP) (Burke, 1981; Plonsey and Barr, 2000). Membrane depolarization starts at the neuromuscular junction and extends along the muscle fiber towards both ends of the cell. As a result, two IAPs propagate without attenuation and with constant velocity *v* along the muscle fiber

The plasma membrane of a muscle fiber has the well-studied property of actively maintaining a nearly constant potential difference between the intracellular and extracellular environment. This voltage is normally referred to as the resting potential and has a value of about -80 mV [Fig. 1(b)]. The negative sign indicates that the interior is more negative than the exterior (negative polarization). In Fig. 1(a) the polarization of the fiber membrane is represented by a number of layers of negative signs. When the fiber is at rest, the number of negative layers remains unchanged. After fiber activation, two potential profiles (one at each side of the neuromuscular junction) arise along the fiber membrane. These profiles are normally referred to as waves of excitation, excitation sources or simply IAPs [Fig. 1(a)]. In Fig. 1(a) it can be seen that, along the spatial profile of the IAP, the number of negative-signed layers changes progressively with axial distance (the *x*-axis in Fig. 1). This reflects the fact that the IAP profile has gradual depolarization and repolarization transitions, as shown in Fig.

Knowledge of muscle fiber excitability and IAP propagation would be clinically useless if the electromyographist had to penetrate muscle fibers to obtain information about membrane processes. Moreover, recording of the IAP *in situ* from human muscle fibers has not yet proved feasible in EMG studies (Ludin, 1973). All electromyography (EMG) From inspection of (1), two main conclusions can be drawn. First, the potential recorded at a certain point is proportional to the strength of the current source, a feature highly desirable for electrodiagnostic medicine. Second, both *r*i and *σ*i are in the denominator of the equation (1). Thus, assuming a constant transmembrane current, the potential decreases with increasing radial distance and with increasing conductivity.

**Figure 1.** (a) Schematic representation of a portion of muscle fiber in which two excitation sources [IAP(x)] are propagating with velocity *v* from the neuromuscular junction (NMJ) to the fiber ends. The polarization of the fiber membrane is represented by several layers of negative signs. The number of negative-signed layers within the fiber region delimited by the intracellular action potential (IAP) changes gradually with axial distance, but it is constant within the regions where the fiber is at rest. The transmembrane ionic electric current, *I*m(z), is also indicated. (b) Spatial profile of the IAP with its depolarization and repolarization phases. L and *T*in are the spatial extension and temporal duration of the IAP, respectively.

Computational Intelligence in Electromyography Analysis – 6 A Perspective on Current Applications and Future Challenges

The principle of volume conduction is valid only as an intuitive approach to an understanding of the generation of an extracellular potential within a muscle. The simplicity of equation (1) hides important aspects that need to be clarified. First, the bioelectrical source cannot be described as a single injected current at a certain point*,* but it is rather a compound of multiple sources (see Section 3.1.2). Second, muscle fibers are of finite length, which implies that the assumption of an infinitely large volume conductor is never satisfied in practice. This will have important consequences: it will give rise to non-propagating components (see Section 2.2). Third, as muscle fibers can often be considered parallel to each other, conductivity of the muscle tissue in the longitudinal direction (*σ*x) is higher than in the transversal (*σ*r), i.e., the volume conductor is anisotropic with anisotropy ratio *K*an = *σ*x/*σ*r.

EMG Modeling 7

the spatial extension of the IAP along the fiber (Dimitrova and Dimitrov, 2006; Rodriguez et al., 2011). In addition, the spatial profile of the IAP is smooth [Fig. 1(a)], implying that the electric properties of the fiber membrane affected by the IAP change gradually with axial distance. Accordingly, a correct presentation of the excitation function consists of a sequence of cylinders (fiber portions) of equally-infinitesimal length *dx*, each cylinder containing a density of sources, as represented in Figs. 2(b) and (c). If these sources are considered dipoles, then each of these cylinders should be represented by a double-layer disk [Fig. 2(b)], whereas if the sources are regarded as monopoles, then cylinders should be modeled

**Figure 2.** (a) Representation of the IAP spatial profile and its first spatial derivative, ∂IAP/∂x. Schematic representations of the IAP as a sequence of double layer disks (b) (each disk comprising a density of

0 2 4 6 8 10 12 14 Space (mm)


IAP

<sup>100</sup> <sup>x</sup> (a)

IAP ∂ ∂

From the above it follows that the calculation of the extracellular potential generated by a

of double (or single) layer disks distributed along the IAP spatial course (Dimitrova and Dimitrov, 2006). The specific mathematical derivation by which the extracellular potential is

The dipole-based presentation of the source was first introduced by Wilson et al. (1933) and subsequently developed by Plonsey (1974). It is based on the hypothesis that the variation of the membrane electrical potential across an infinitesimal portion of the fiber membrane produces, in the extracellular medium, an electrical field that can be assumed to be equivalent to that produced by a lumped dipole (Wilson et al., 1933). So, the potentials produced by a double layer disk and a point dipole whose moment is proportional to the disk area are almost identical. This provides the basis for representing the portion of the fiber affected by the IAP as a sequence of dipoles distributed equidistantly along the IAP spatial profile (Dimitrova and Dimitrov, 2006; Rodriguez et al., 2011), as shown in Fig. 2(c). The strength of each of the dipoles (or dipole moment) is determined by the spatial

e, can be reduced to the sum of the potentials produced by a sequence

*x*

dipoles), and as a sequence of lumped (point) dipoles (c) lying along the axis of the fiber.

*dx*

as single-layer disks.

single excited fiber,

Φ

expressed in terms of double layer disks is presented below.

(b)

(c)

Voltage (mV)

*3.1.3. Calculation of the extracellular potential on the basis of dipoles* 
