**1. Introduction**

The aim of this chapter is to describe the approaches used for modelling electromyographic (EMG) signals as well as the principles of electrical conduction within the muscle. Sections are organized into a progressive, step-by-step EMG modeling of structures of increasing complexity. First, the basis of the electrical conduction that allows for the propagation of the EMG signals within the muscle is presented. Second, the models used for describing the electrical activity generated by a single fibre described. The third section is devoted to modeling the organization of the motor unit and the generation of motor unit potentials. Based on models of the architectural organization of motor units and their activation and firing mechanisms, the last section focuses on modeling the electrical activity of a complete muscle as recorded at the surface.

A mathematical model of a system describes the relations between a number of physical variables involved in the system. A mathematical model is a set of equations that can be implemented on a computer to study and to simulate the behaviour (response) of the system under specific conditions. EMG models presented in this chapter are structure based or structural, which means that they describe elements of the real biological structure and characterize them in a reductional way in order to represent the system's elements, behaviours or mechanism that are of importance. In the EMG models outlined here, the input variables or parameters are those that describe the anatomical, physiological, and functional properties of the biological structure under study (single fibre, motor unit, or entire muscle), whereas the output parameters are typically the extracellular generated potentials and/or specific quantitative measurements of these potentials.

Models of EMG activity are useful to address the "forward problem", that is, how specific mechanism and phenomena influence the generated potentials, as well as the "inverse problem", that is, how the extracellular potentials provide information about the underlying mechanism and phenomena. Accordingly, a desirable feature of an EMG model is that it allows studying the effect of the model's (input) parameters on the waveform of the

© 2012 Rodriguez-Falces et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Rodriguez-Falces et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

potential, providing insight into the relationships between the anatomical and/or physiological properties of the fibre and the shape of the potential.

EMG Modeling 5

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

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

> 1 4

*<sup>I</sup> <sup>V</sup>* πσ

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

**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.

*s*

*<sup>r</sup>* <sup>=</sup> (1)

*i i*

0

*P*

EMG measurements to be made at a distance from the sources.

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

increasing radial distance and with increasing conductivity.

at a point *P* (*x*, *y*, *z*) can be calculated as
