Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach


Electromyography (EMG) can be used to study the behaviour of the motor neurons and thus provides insights into the physiology of the central nervous system. However, due to the high complexity of neuromuscular control, EMG signals are challenging to interpret. While the exact knowledge of the excitation patterns of a specific muscle within an in vivo experimental setting remains elusive, simulations allow to systematically investigate EMG signals in a controlled environment. Within this context, simulations can provide virtual EMG data, which, for example, can be used to validate and optimise signal analysis methods that aim to estimate the relationship between EMG signals and the output of motor neuron pools. However, since existing methods, which are employed to compute EMG signals, exhibit deficiencies with respect to the physical model itself as well as with respect to numerical aspects, we propose a novel homogenised continuum model that closely resolves the electro-physiological behaviour of skeletal muscle tissue. The proposed model is based on an extension of the well-established bidomain model and includes a biophysically detailed description of the electrical activity within the tissue, which is due to the depolarisation of the muscle fibre membranes. In contrast to all other published EMG models, which assume that the electrical potential field for each muscle fibre can be calculated independently, the proposed model assumes that the electrical potential in the muscle fibres is coupled to the electrical potential in the extracellular space. We show that the newly proposed model is able to simulate realistic EMG signals and demonstrate the potential to employ the predicted virtual EMG signal in order to evaluate the goodness of automated decomposition algorithms.

This is a preview of subscription content, log in to check access.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. Adrian RH, Chandler WK, Hodgkin AL (1970) Voltage clamp experiments in striated muscle fibres. J Physiol 208(3):607–644

  2. Blijham PJ, Ter Laak HJ, Schelhaas HJ, Van Engelen B, Stegeman DF, Zwarts MJ (2006) Relation between muscle fiber conduction velocity and fiber size in neuromuscular disorders. J Appl Physiol 100(6):1837–1841

  3. Bradley CP, Emamy N, Ertl T, Göddeke D, Hessenthaler A, Klotz T, Krämer A, Krone M, Maier B, Mehl M, Rau T, Röhrle O (2018) Enabling detailed, biophysics-based skeletal muscle models on hpc systems. Front Physiol 9:816.

  4. Bryant SH (1969) Cable properties of external intercostal muscle fibres from myotonic and nonmyotonic goats. J Physiol 204:539–550.

  5. Buist ML, Poh YC (2010) An extended bidomain framework incorporating multiple cell types. Biophys J 99(1):13–18.

  6. Cannon S, Brown R, Corey D (1993) Theoretical reconstruction of myotonia and paralysis caused by incomplete inactivation of sodium channels. Biophys J 65(1):270–288

  7. Carriou V, Boudaoud S, Laforet J (2018) Speedup computation of hd-semg signals using a motor unit-specific electrical source model. Med Biol Eng Comput 56(8):1459–1473.

  8. Clayton R, Bernus O, Cherry E, Dierckx H, Fenton FH, Mirabella L, Panfilov AV, Sachse FB, Seemann G, Zhang H (2011) Models of cardiac tissue electrophysiology: progress, challenges and open questions. Prog Biophys Mol Biol 104(1–3):22–48

  9. Corrias A, Pathmanathan P, Gavaghan DJ, Buist ML (2012) Modelling tissue electrophysiology with multiple cell types: applications of the extended bidomain framework. Integr Biol 4(2):192–201

  10. Del Vecchio A, Negro F, Felici F, Farina D (2017) Distribution of muscle fibre conduction velocity for representative samples of motor units in the full recruitment range of the tibialis anterior muscle. Acta Physiol 222(2):e12,930.

  11. Dimitrova NA, Dimitrov AG, Dimitrov GV (1999) Calculation of extracellular potentials produced by an inclined muscle fibre at a rectangular plate electrode. Med Eng Phys 21:583–588.

  12. Epstein BR, Foster KR (1983) Anisotropy in the dielectric properties of skeletal muscle. Med Biol Eng Comput 21(1):51.

  13. Farina D, Merletti R (2001) A novel approach for precise simulation of the EMG signal detected by surface electrodes. IEEE Trans Biomed Eng 48:637–646.

  14. Farina D, Merletti R (2004) Methods for estimating muscle fibre conduction velocity from surface electromyographic signals. Med Biol Eng Comput 42(4):432–445

  15. Farina D, Mesin L, Martina S (2004) Advances in surface electromyographic signal simulation with analytical and numerical descriptions of the volume conductor. Med Biol Eng Comput 42:467–476.

  16. Farina D, Cescon C, Negro F, Enoka RM (2008) Amplitude cancellation of motor-unit action potentials in the surface electromyogram can be estimated with spike-triggered averaging. J Neurophysiol 100(1):431–440

  17. FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466

  18. Fuglevand AJ, Winter DA, Patla AE, Stashuk D (1992) Detection of motor unit action potentials with surface electrodes: influence of electrode size and spacing. Biol Cybern 67(2):143–153

  19. Fuglevand AJ, Winter DA, Patla AE (1993) Models of recruitment and rate coding organization in motor unit pools. J Neurophys 70(6):2470–2488

  20. Gabriel S, Lau RW, Gabriel C (1996) The dielectric properties of biological tissues: Ii. measurements in the frequency range 10 hz to 20 ghz. Phys Med Biol 41(11):2251

  21. Geers MGD, Kouznetsova VG, Matouš K, Yvonnet J (2017) Homogenization methods and multiscale modeling: nonlinear problems. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics second edition.

  22. Gielen FLH, Wallinga-de Jonge W, Boon KL (1984) Electrical conductivity of skeletal muscle tissue: experimental results from different musclesin vivo. Med Biol Eng Comput 22(6):569–577.

  23. Gizzi L, Lenti M, Felici F, Filligoi G (2008) Muscle fibers conduction velocity in cycling: a cross correlation-based application for dynamic exercise. In: IET Conference proceedings

  24. Griffiths DJ (2013) Introduction to electrodynamics, 4th edn. Pearson, Boston

  25. Hakansson C (1956) Conduction velocity and amplitude of the action potential as related to circumference in the isolated fibre of frog muscle. Acta Physiol Scand 37(1):14–34

  26. Heidlauf T, Röhrle O (2013) Modeling the chemoelectromechanical behavior of skeletal muscle using the parallel open-source software library open CMISS. Comput Math Methods Med 2013:1–14.

  27. Henneman E, Somjen G, Carpenter DO (1965) Functional significance of cell size in spinal motoneurons. J Neurophysiol 28(3):560–580

  28. Henriquez CS (1993) Simulating the electrical behavior of cardiac tissue using the bidomain model. Crit Rev Biomed Eng 21(1):1–77

  29. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11:357–372

  30. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544.

  31. Holobar A, Minetto MA, Botter A, Negro F, Farina D (2010) Experimental analysis of accuracy in the identification of motor unit spike trains from high-density surface emg. IEEE Trans Neural Syst Rehabil Eng 18(3):221–229

  32. Huang Q, Eason JC, Claydon FJ (1999) Membrane polarization induced in the myocardium by defibrillation fields: an idealized 3-d finite element bidomain/monodomain torso model. IEEE Trans Biomed Eng 46(1):26–34

  33. Kandel ER, Schwartz JH, Jessell TM et al (2000) Principles of neural science, vol 4. McGraw-Hill, New York

  34. Keener J, Sneyd J (2009) Mathematical physiology II: cellular physiology, vol 2, 2nd edn. Springer, Berlin

  35. Kim JH, Trew ML, Pullan AJ, Röhrle O (2012) Simulating a dual-array electrode configuration to investigate the influence of skeletal muscle fatigue following functional electrical stimulation. Comput Biol Med 42(9):915–924.

  36. Lloyd CM, Halstead MD, Nielsen PF (2004) Cellml: its future, present and past. Prog Biophys Mol Biol 85(2):433–450. (Modelling cellular and tissue function)

  37. Lowery MM, Stoykov NS, Taflove A, Kuiken TA (2002) A multiple-layer finite-element model of the surface EMG signal. IEEE Trans Biomed Eng 49(5):446–454.

  38. Lowery MM, Stoykov NS, Dewald PA, Kuiken TA (2004) Volume conduction in an anatomically based surface EMG model. IEEE Trans Biomed Eng 51:2138–2147.

  39. MacIntosh RB, Gardiner FP, McComas JA (2006) Skeletal muscle: form and function, 2nd edn. Human Kinetics, New York

  40. Maffiuletti NA, Minetto MA, Farina D, Bottinelli R (2011) Electrical stimulation for neuromuscular testing and training: state-of-the art and unresolved issues. Eur J Appl Physiol 111:2391.

  41. MATLAB (2016) Version (R2016a). Natick, Massachusetts: The MathWorks Inc

  42. Merletti R, Parker PA (2004) Electromyography: physiology, engineering, and non-invasive applications, vol 11. Wiley, London

  43. Merletti R, Farina D, Gazzoni M, Schieroni MP (2002) Effect of age on muscle functions investigated with surface electromyography. Muscle Nerve 25(1):65–76

  44. Mesin L (2005) Analytical generation model of surface electromyogram for multi-layer volume conductors. Model Med Biol VI WIT 8:95–110.

  45. Mesin L (2013) Volume conductor models in surface electromyography: computational techniques. Comput Biol Med 43(7):942–952.

  46. Mesin L, Joubert M, Hanekom T, Merletti R, Farina D (2006) A finite element model for describing the effect of muscle shortening on surface EMG. IEEE Trans Biomed Eng 53:600–693.

  47. Miller WT, Geselowitz DB (1978) Simulation studies of the electrocardiogram. I. The normal heart. Circ Res 43(2):301–315.

  48. Mordhorst M, Heidlauf T, Röhrle O (2015) Predicting electromyographic signals under realistic conditions using a multiscale chemo-electro-mechanical finite element model. Interface Focus 5(2):1–11.

  49. Mordhorst M, Strecker T, Wirtz D, Heidlauf T, Röhrle O (2017) POD-DEIM reduction of computational EMG models. J Comput Sci 19:86–96.

  50. Negro F, Farina D (2011) Decorrelation of cortical inputs and motoneuron output. J Neurophysiol 106(5):2688–2697

  51. Nielsen BF, Ruud TS, Lines GT, Tveito A (2007) Optimal monodomain approximations of the bidomain equations. Appl Math Comput 184(2):276–290

  52. Oudeman J, Mazzoli V, Marra MA, Nicolay K, Maas M, Verdonschot N, Sprengers AM, Nederveen AJ, Strijkers GJ, Froeling M (2016) A novel diffusion-tensor MRI approach for skeletal muscle fascicle length measurements. Physiol Rep 4(24):e13012.

  53. Pullan AJ, Buist ML, Cheng LK (2005) Mathematically modelling the electrical activity of the heart: from cell to body surface and back again. World Scientific, Singapore.

  54. Qu Z, Garfinkel A (1999) An advanced algorithm for solving partial differential equation in cardiac conduction. IEEE Trans Biomed Eng 46(9):1166–1168

  55. Ramasamy E, Avci O, Dorow B, Chong SY, Gizzi L, Steidle G, Schick F, Röhrle O (2018) An efficient modelling-simulation-analysis workflow to investigate stump-socket interaction using patient-specific, three-dimensional, continuum-mechanical, finite element residual limb models. Front Bioeng Biotechnol 6:126

  56. Röhrle O, Davidson JB, Pullan AJ (2012) A physiologically based, multi-scale model of skeletal muscle structure and function. Front Physiol 3:1–14

  57. Röhrle O, Yavuz U, Klotz T, Negro F, Heidlauf T (2019) Multiscale modelling of the neuromuscular system: coupling neurophysiology and skeletal muscle mechanics. Wiley, Berlin

  58. Rush S, Abildskov J, McFee R (1963) Resistivity of body tissues at low frequencies. Circ Res 12(1):40–50

  59. Saad Y, Schultz MH (1986) Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J Sci Stat Comput 7(3):856–869.

  60. Sbriccoli P, Sacchetti M, Felici F, Gizzi L, Lenti M, Scotto A, De Vito G (2009) Non-invasive assessment of muscle fiber conduction velocity during an incremental maximal cycling test. J Electromyogr Kinesiol 19(6):e380–e386

  61. Shampine LF, Reichelt MW (1997) The matlab ode suite. SIAM J Sci Comput 18(1):1–22

  62. Shorten PR, O’Callaghan P, Davidson JB, Soboleva TK (2007) A mathematical model of fatigue in skeletal muscle force contraction. J Muscle Res Cell Motil 28(6):293–313

  63. Sinha U, Yao L (2002) In vivo diffusion tensor imaging of human calf muscle. J Mag Reson Imaging 15(1):87–95.

  64. Sundnes J, Lines GT, Tveito A (2005) An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso. Math Biosci 194:233–248

  65. Sundnes J, Nielsen BF, Mardal K, Cai X, Lines GT, Tveito A (2006) On the computational complexity of the bidomain and the monodomain models of electrophysiology. Ann Biomed Eng 34(7):1088–1097.

  66. Tung L (1978) A bi-domain model for describing ischemic myocardial dc potentials. PhD thesis, Massachusetts Institute of Technology

  67. Whiteley JP (2006) An efficient numerical technique for the solution of the monodomain and bidomain equations. IEEE Trans Biomed Eng 53(11):2139–2147

Download references


This research was funded by the Baden-Württemberg Stiftung as part of the DiHu project of the High Performance Computing II program and by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2075 – 390740016.

Author information

Correspondence to Thomas Klotz.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Klotz, T., Gizzi, L., Yavuz, U.Ş. et al. Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach. Biomech Model Mechanobiol 19, 335–349 (2020).

Download citation


  • EMG
  • Biophysical
  • Bidomain
  • Multi-scale
  • Decomposition algorithms
  • Motor units