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

#### *3.2.1. The first SFAP models*

Computational Intelligence in Electromyography Analysis – 8 A Perspective on Current Applications and Future Challenges

coordinate (x, y, z) and *P*0 is located at (x0, y0, z0) then

( )

Φ =

*e*

determined by the sign of ∂IAP(x)/∂x.

where σ

where σ

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

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

<sup>1</sup> ( ) 1 / ( ,) <sup>4</sup> *<sup>e</sup>*

source to the recording point, P0 [see Fig. 3(e)]. If the element⎯*p dx* is located at the

( )( )( )

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

<sup>000</sup> 3/2 <sup>2</sup> 2 2

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

*pxt xyzt dx*

the potential from the propagation of this single dipole current element). The result is

πσ

2 *i*

( ,) ,,, 4

*<sup>x</sup> <sup>e</sup>*

*x*

=∞

=−∞

correct, the actual source of excitation is a linear axial dipole source density

*p a*

into (4), we obtain the dipole-based expression for the extracellular potential

( ) ( )

*i*

σ

*e*

σ

4

*e*

π σ

*a IAP x t*

Φ =− ⋅ ⋅ ⋅

+∞

−∞

*d r d p x t dx*

e is the conductivity of the extracellular medium and *r* is the distance from excitation

1/2 2 22

( )

*IAP*

*x*

i is the intracellular conductivity and a is the fiber radius. If we now substitute (5)

<sup>2</sup> , 1

⋅ ∂ ∂

*t dx*

*ax*

*x x r x ax*

*xx y z*

 ⋅ − ++ 

Φ= ⋅ ⋅ (2)

0 00 *r xx yy zz* = − +− +− (3)

0 00

(4)

<sup>∂</sup> <sup>⋅</sup> <sup>∂</sup> =− ⋅ (5)

( )

⋅ ∂∂ (6)

*e*

πσ*dx* 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 researchers.

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 Rosenfalck, 1981).
