Mathematics in Independent Component Analysis

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266 Chapter 20. Signal Processing 86(3):603-623, 2006 name meaning value num number of MUs 5 siz size of each MU, in number of fibres 20-30 (uniformly distr.) cfr central firing rate 20 Hz ED electrode inter-pole distance 2.54 mm CH number of generated channels 8 meanIPI mean inter-pulse interval 1/cfr · 1000 ms IPIstd Clamann formula for IPI deviation 9.1 · 10 −4 · meanIPI 2 + 4 SY radial conductivity 0.063 S/m SZ axial conductivity 0.328 S/m A anisotropy ratio SZ/SY = 5.2 SI intracellular conductivity 1.010 S/m D fibre diameter 50 · 10 −6 m CV conduction velocity 4.0 m/s FatHeight fat-skin layer thickness 3 mm MUterritory XY-plane radius of each MU 1.0 mm DistEndplate endplate-to-electrode distance 20 mm FibZsegment length of the endplate in the Z axis 0.4 mm CondVolCond conductivity of the volume conductor 0.08 S/m Table 1 Parameters used for artificial s-EMG generation. by the anisotropy factor A, calculated as the ratio between the radial conductivity (0.063 S/m) and the axial conductivity (0.328 S/m), see Tab. 1 for parameter details. Afterwards, the potential distribution can be calculated as if the volume conductor were isotropic. The channels composing the synthetic s-EMG contain the same MUAP without any time delay among themselves, hence producing a instantaneous mixture with the other source signals. We have generated 100 synthetic, eight-channel s-EMG signals of 1.5-second duration. Each s-EMG was composed of five MUs randomly located under a 3 mm fat and skin layer in the detection area of the electrode-array (up to ±2 mm of the sides of the electrode array and up to a depth of 6 mm). Each MU in turn consisted of 20 to 30 fibres (uniform distribution) of infinite length located randomly in a 1 mm circle in the XY plane. The conduction velocity of the fibres was 4.0 m/s. Note that an average biceps brachii MU is generally composed of at least 200-300 muscle fibres [23]; however, for the sake of computational speed, that number was reduced to one tenth, which 5
Chapter 20. Signal Processing 86(3):603-623, 2006 267 only resulted in the reduction of the amplitude of the final MUAPs. This does not have any influence in our case, as the algorithms are not affected by the general amplitude of the signals, only by their respective proportions, if at all. The simulated electrode-array used to generate the mixed recordings had the same dimensions and characteristics of the real one and was virtually located 20 mm apart from the endplate region, which had a thickness of 0.4 mm. X axis corresponded to the length of the electrode, Y axis to the depth, and Z axis to the fibres direction. The mean instantaneous firing rate of each MU was 20 firings/second (mean inter-pulse interval meanIPI of 50 ms), with a standard deviation calculated following Clamman’s model as 9.1 · 10 −4 · meanIPI 2 + 4 [21]. The s-EMG signals were generated as followed; each muscle fibre composing a firing MU generates an action potential following the moving tripole model ([24] and references therein); the final MUAP appearing in each channel is the summation of those action potentials after applying the anisotropy coefficient A and the fading produced by the media crossed by the signal ([19] and references therein). Note that the model generally holds for any skeletal muscle. However, some of the parameters (such as the fat layer and firing rate) were chosen following the average, normal values of the biceps brachii [25]. A single such data set is visualized in Fig. 2. The eight-dimensional s-EMG observations that have been generated from the five underlying MUAPTs are shown in Fig. 2(a). Their locations are depicted in Fig. 2(b) and the mixing matrix used to generate the synthetic data set (see Fig. 2(c)) is shown in Fig. 2(d). We will use several (100) of these signals with randomly chosen source locations in batch runs for statistical comparisons. 2.1.2 Real s-EMG signal acquisition Fig. 3 shows the experimental setting for the acquisition of real s-EMG recordings. The subject (after giving informed consent) sat on a chair and was strapped to it with nylon belts as shown in Fig. 3(a). The subject’s forearm was put in a cast fixed to the table, the shoulder angle being approximately 30 ◦ . To measure the torque exerted by the subject, a strain gauge was placed between the forearm cast and the table fixation. An electrode array was placed on the biceps short head muscle belly, out of the innervation zone, transverse to the direction of muscle fibers, after cleaning the skin with alcohol and Skin- Pure (Nihon Kohden Corp, Tokyo, Japan). The electrode array consists of 16 stainless steel poles of 1 mm in diameter, 3 mm in height, and 2.54 mm apart in both plane directions that are attached to a rigid acrylic plate, see Fig. 3(b). The poles of each bipolar electrode were placed at the shortest possible distance to obtain sharper MUAPs. 6
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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Mathematics in Independent Component Analysis

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