**3.1. Modeling the bioelectrical sources of muscle fibers**

## *3.1.1. The principles of bioelectricity*

The principle of the electromotive surface proposed by Helmholtz in 1853 provided the basis for the electrical potential theory of volume conductors. In addition, he introduced the concept of an electrical double layer into the theory of electricity and suggested its use for the solution of certain boundary problems in electrical potential theory. The concept of the double layer (or dipole) source, however, went unused for about 30 years until Wilson et al. (1933) demonstrated its appropriateness for modelling the excitation source of a single circular cylindrical fiber. After Wilson, it was not until 1974 that Plonsey definitively clarified the underlying electrostatic principles of the single and double layer sources and related them to the concepts of monopole and dipole, respectively.

Traditionally, two kinds of sources (generators) have been considered in the literature (Lorente de No, 1947; Plonsey, 1974), namely, the "monopole" and the "dipole". A monopole is a single (point) source or sink of current within a conducting medium. It is quite rare that problems in bioelectricity involve monopoles, since all bioelectric generators involve at least source and sink combinations (Plonsey, 1974).

### *3.1.2. A distributed presentation of the excitation source (excitation function)*

Because of its propagation along the fiber, an IAP does not merely exist as a function of time: it also spreads out along the fiber as a function of space. The length of the IAP profile along the fiber, *L*, is defined by the product of the IAP duration, *T*in, and the propagation velocity *v* [Fig. 1(b)]. In fact, the formation of an electrical field around a fiber depends on 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 as single-layer disks.

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

well established.

*3.1.1. The principles of bioelectricity* 

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.

In the last half of the 20th century EMG studies have directed attention to the calculation of single fiber action potentials (SFAPs) produced by excitable fibers and especially by fibers of finite length. The development of SFAP models was possible only after the principles of volume conductivity had been determined and the modelling of bioelectrical sources was

The principle of the electromotive surface proposed by Helmholtz in 1853 provided the basis for the electrical potential theory of volume conductors. In addition, he introduced the concept of an electrical double layer into the theory of electricity and suggested its use for the solution of certain boundary problems in electrical potential theory. The concept of the double layer (or dipole) source, however, went unused for about 30 years until Wilson et al. (1933) demonstrated its appropriateness for modelling the excitation source of a single circular cylindrical fiber. After Wilson, it was not until 1974 that Plonsey definitively clarified the underlying electrostatic principles of the single and double layer sources and

Traditionally, two kinds of sources (generators) have been considered in the literature (Lorente de No, 1947; Plonsey, 1974), namely, the "monopole" and the "dipole". A monopole is a single (point) source or sink of current within a conducting medium. It is quite rare that problems in bioelectricity involve monopoles, since all bioelectric generators

Because of its propagation along the fiber, an IAP does not merely exist as a function of time: it also spreads out along the fiber as a function of space. The length of the IAP profile along the fiber, *L*, is defined by the product of the IAP duration, *T*in, and the propagation velocity *v* [Fig. 1(b)]. In fact, the formation of an electrical field around a fiber depends on

**3. Modeling the electrical activity of the single muscle fiber** 

**3.1. Modeling the bioelectrical sources of muscle fibers** 

related them to the concepts of monopole and dipole, respectively.

involve at least source and sink combinations (Plonsey, 1974).

*3.1.2. A distributed presentation of the excitation source (excitation function)* 

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

From the above it follows that the calculation of the extracellular potential generated by a single excited fiber, Φe, can be reduced to the sum of the potentials produced by a sequence 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 expressed in terms of double layer disks is presented below.

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

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 derivative of the potential profile along the fiber ∂IAP(x)/∂x. Orientation of the dipoles is determined by the sign of ∂IAP(x)/∂x.

Let us consider a fiber element of infinitesimal length *dx* of Fig. 2(b) lying within the region occupied by the action potential. A current emerges from this differential fiber element into the extracellular region. If we assume that all the current is concentrated along the fiber axis, then this current can be expressed as⎯*p·dx*, where⎯*p* is the dipole current per unit length. Since this current emerges essentially from a point into an unbounded space, it behaves like a point source of current (dipole generator) that lies in an extensive conducting medium. Then, the contribution to the extracellular potential generated from this component can be expressed as

$$d\Phi\_{\varepsilon} = \frac{1}{4\pi\sigma\_{\varepsilon}} \cdot \frac{d\left(1/r\right)}{d\mathbf{x}} \cdot \overline{p}(\mathbf{x}, t)d\mathbf{x} \tag{2}$$

EMG Modeling 9

In (6) it can be seen that the term ∂(1/*r*)/∂x represents the scalar potential generated by a propagating dipole and the term ∂IAP/∂x corresponds to the distribution of moments of the

Calculation of the extracellular potential on the basis of monopoles can also be derived

Lorente de Nó (1947) was the first to obtain an expression for the potential of the external field of a nerve in a volume conductor as a function of an action potential. However, he did not provide a physical interpretation for the concepts of double and single layer sources. Nevertheless, his contribution to the understanding and modelling of the extracellular fields produced by an excitable fiber in a volume conductor was essential for subsequent

In 1968, Clark and Plonsey proposed a SFAP approximation based on formal solutions of Poisson's or Laplace's equation with the corresponding boundary conditions using a method of separation of variables. The solution was based on the Fourier transform technique and modified Bessel functions. The formal solution, however, gave no opportunity for transparent physical interpretations (Andreassen and Rosenfalck, 1981). In addition, the mathematical expressions describing SFAPs using the volume conductor theory were quite complex, computationally time consuming and, therefore, somewhat unsuitable for simulating motor unit potentials (MUPs). In response to these limitations, simplified models were presented. The dipole and tripole models approximated the transmembrane current by two and three point sources, respectively. These models, however, presented important shortcomings. The dipole model was not able to describe the effects of the excitation origin and extinction correctly (George, 1970; Boyd et al., 1978; Griep et al., 1978). The tripole models failed to correctly describe SFAPs close to fibers (Griep et al., 1978; Andreassen and

*3.2.2. Assumptions of the core-conductor model and the line-source model: the first* 

Clark and Plonsey (1966, 1968) and then Andreassen and Rosenfalck (1981), provided thorough compilations of the approaches that relate intracellular and extracellular potentials. Both works stressed the necessity of finding the conditions to simplify existing SFAP models. These conditions are summarized in the assumptions of the core-conductor:

a. Axial symmetry is assumed. That is ∂/∂*φ*=0 (where *φ* is the azimuth angle) so that, at most, all field quantities are functions of the cylindrical coordinates only. In fact, transmembrane currents as well as intra- and extra-cellular potentials are considered to

be functions only of *x* (i.e., the core-conductor is linear).

**3.2. Models of the extracellular potential generated by a single muscle fiber** 

following the steps outlined above, as shown in Plonsey and Barr (2000).

collection of dipoles that form the excitation function.

*3.2.1. The first SFAP models* 

researchers.

Rosenfalck, 1981).

*approach to convolutional models* 

where σe is the conductivity of the extracellular medium and *r* is the distance from excitation source to the recording point, P0 [see Fig. 3(e)]. If the element⎯*p dx* is located at the coordinate (x, y, z) and *P*0 is located at (x0, y0, z0) then

$$r = \left[ \left( \mathbf{x} - \mathbf{x}\_0 \right)^2 + \left( y - y\_0 \right)^2 + \left( z - z\_0 \right)^2 \right]^{1/2} \tag{3}$$

Normally, the coordinate origin is placed on the fiber axis, whereupon *y* = *z* = 0. The total field produced in the extracellular medium by the propagation of a single dipole along the fiber is found simply by integrating with respect to x (i.e., summing up the contributions to the potential from the propagation of this single dipole current element). The result is

$$\Phi\_{\varepsilon} \left( x\_0, y\_0, z\_0, t \right) = \int\_{x = -\infty}^{x = \infty} \frac{\overline{p}(\mathbf{x}, t)}{4\pi\sigma\_{\varepsilon} \cdot \left[ \left( \mathbf{x} - \mathbf{x}\_0 \right)^2 + y\_0^2 + z\_0^2 \right]^{3/2}} \, d\mathbf{x} \tag{4}$$

At this point, we should remember that the source of excitation is actually a "distributed source", which means that we do not have just one dipole but a sequence of dipoles distributed equidistantly along the IAP profile. The distribution of the dipole moments (strengths) along the fiber axis is determined by the function ∂IAP(x)/∂x. Thus, to be strictly correct, the actual source of excitation is a linear axial dipole source density

$$\overline{p} = -\pi a^2 \sigma\_i \cdot \frac{\partial IAP}{\partial x} \cdot \overline{a}\_{\mathcal{X}} \tag{5}$$

where σi is the intracellular conductivity and a is the fiber radius. If we now substitute (5) into (4), we obtain the dipole-based expression for the extracellular potential

$$\Phi\_{\varepsilon}(t) = -\frac{a^2 \cdot \sigma\_i}{4 \cdot \sigma\_{\varepsilon}} \cdot \int\_{-\infty}^{\infty} \frac{\partial IAP\{x, t\}}{\partial x} \cdot \overline{\mathfrak{a}}\_{\mathcal{X}} \cdot \frac{\partial}{\partial x} \left(\frac{1}{r(x)}\right) d\mathbf{x} \tag{6}$$

In (6) it can be seen that the term ∂(1/*r*)/∂x represents the scalar potential generated by a propagating dipole and the term ∂IAP/∂x corresponds to the distribution of moments of the collection of dipoles that form the excitation function.

Calculation of the extracellular potential on the basis of monopoles can also be derived following the steps outlined above, as shown in Plonsey and Barr (2000).
