*3.2.2. Assumptions of the core-conductor model and the line-source model: the first approach to convolutional models*

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).

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

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 medium surrounding the active fiber is neglected.

EMG Modeling 11

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

As *∆t* tends to zero and *n* tends to infinity, the total potential produced at the point *P*0 by the

**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

( ) () \* ( ) *dIAP t SFAP t IR t*

() ( ) ( ) ( ) 1 2 ( ) ... <sup>1</sup> *<sup>n</sup> SFAP t a IR t a IR t t a IR t n t* = + −Δ + + − − Δ (7)

*dt* <sup>=</sup> (8)

potential *a*2·*IR*(t-∆t) [Fig. 3(c)]. Hence the total potential recorded by the electrode is

propagating excitation function represents the convolution of d*IAP*(x,t)/dt and *IR*(t):

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 normally justified.

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.
