Perspectives on the modeling of the neuromusculoskeletal system to investigate the influence of neurodegenerative diseases on sensorimotor control
Leonardo Abdala Elias, Débora Elisa da Costa Matoso, Renato Naville Watanabe, André Fabio Kohn
Abstract
Introduction: The understanding of the neurophysiological mechanisms underlying movement control can be much furthered using computational models of the neuromusculoskeletal system. Biologically based multi-scale neuromusculoskeletal models have a great potential to provide new theories and explanations related to mechanisms behind muscle force generation at the molecular, cellular, synaptic, and systems levels. Albeit some efforts have been made to investigate how neurodegenerative diseases alter the dynamics of individual elements of the neuromuscular system, such diseases have not been analyzed from a systems viewpoint using multi-scale models.
Overview and Perspectives: This perspective article synthesizes what has been done in terms of multi-scale neuromuscular development and points to a few directions where such models could be extended so that they can be useful in the future to discover early predictors of neurodegenerative diseases, as well as to propose new quantitative clinical neurophysiology approaches to follow the course of improvements associated with different therapies (drugs or others).
Concluding Remarks: Therefore, this article will present how existing biologically based multi-scale models of the neuromusculoskeletal system could be expanded and adapted for clinical applications. It will point to mechanisms operating at different levels that would be relevant to be considered during model development, along with implications for interpreting experimental results from neurological patients.
Keywords
References
Allen JM, Elbasiouny SM. The effects of model composition design choices on high-fidelity simulations of motoneuron recruitment and firing behaviors. J Neural Eng. 2017; 15(3):036024. http://dx.doi.org/10.1088/1741-2552/aa9db5. PMid:29182156.
Amendola J, Durand J. Morphological differences between wild-type and transgenic superoxide dismutase 1 lumbar motoneurons in postnatal mice. J Comp Neurol. 2008; 511(3):329-41. http://dx.doi.org/10.1002/cne.21818. PMid:18803237.
Arnold EM, Ward SR, Lieber RL, Delp SL. A model of the lower limb for analysis of human movement. Ann Biomed Eng. 2010; 38(2):269-79. http://dx.doi.org/10.1007/s10439-009-9852-5. PMid:19957039.
Boe SG, Stashuk DW, Doherty TJ. Motor unit number estimates and quantitative motor unit analysis in healthy subjects and patients with amyotrophic lateral sclerosis. Muscle Nerve. 2007; 36(1):62-70. http://dx.doi.org/10.1002/mus.20784. PMid:17455264.
Bostock H, Baker M, Reid G. Changes in excitability of human motor axons underlying post-ischaemic fasciculations: evidence for two stable states. J Physiol. 1991; 441(1):537-57. http://dx.doi.org/10.1113/jphysiol.1991.sp018766. PMid:1667800.
Bouhy D, Timmerman V. Animal models and therapeutic prospects for Charcot-Marie-Tooth disease. Ann Neurol. 2013; 74(3):391-6. http://dx.doi.org/10.1002/ana.23987. PMid:23913540.
Brennan KM, Bai Y, Shy ME. Demyelinating CMT–what’s known, what’s new and what’s in store? Neurosci Lett. 2015; 596:14-26. http://dx.doi.org/10.1016/j.neulet.2015.01.059. PMid:25625223.
Bromberg MB. Motor neuron disease in adults. New York: Oxford Univeristy Press; 2015.
Brown IE, Scott SH, Loeb GE. Mechanics of feline soleus. 2. Design and validation of a mathematical model. J Muscle Res Cell Motil. 1996; 17(2):221-33. http://dx.doi.org/10.1007/BF00124244. PMid:8793724.
Burke RE, Levine DN, Zajac FE, Tsairis P, Engel WK. Mammalian motor units: physiological-histochemical correlation in three types in cat gastrocnemius. Science. 1971;174(4010):709-12. PMID: 4107849.
Chang Q, Martin LJ. Voltage-gated calcium channels are abnormal in cultured spinal motoneurons in the G93A-SOD1 transgenic mouse model of ALS. Neurobiol Dis. 2016; 93:78-95. http://dx.doi.org/10.1016/j.nbd.2016.04.009. PMid:27151771.
Chaud VM, Elias LA, Watanabe RN, Kohn AF. A simulation study of the effects of activation-dependent muscle stiffness on proprioceptive feedback and short-latency reflex. In: Proceedings of the 4th IEEE RAS/EMBS International Conference Biomedical Robotics and Biomechatronics; 2012; Rome. USA: IEEE; 2012. p. 133-8.
Cisi RRL, Kohn AF. Simulation system of spinal cord motor nuclei and associated nerves and muscles, in a web-based architecture. J Comput Neurosci. 2008; 25(3):520-42. http://dx.doi.org/10.1007/s10827-008-0092-8. PMid:18506610.
Cullheim S, Fleshman JW, Glenn LL, Burke RE. Three-Dimensional architecture of dendritic trees in type-identified alpha-motoneurons. J Comp Neurol. 1987; 255(1):82-96. http://dx.doi.org/10.1002/cne.902550107. PMid:3819011.
De Luca CJ, Hostage EC. Relationship Between Firing Rate and Recruitment Threshold of Motoneurons in Voluntary Isometric Contractions. J Neurophysiol. 2010; 104(2):1034-46. http://dx.doi.org/10.1152/jn.01018.2009. PMid:20554838.
Delestrée N, Manuel M, Iglesias C, Elbasiouny SM, Heckman CJ, Zytnicki D. Adult spinal motoneurones are not hyperexcitable in a mouse model of inherited amyotrophic lateral sclerosis. J Physiol. 2014; 592(7):1687-703. http://dx.doi.org/10.1113/jphysiol.2013.265843. PMid:24445319.
Destexhe A, Mainen ZF, Sejnowski TJ. An efficient method for computing synaptic conductances based on a kinetic-model of receptor-binding. Neural Comput. 1994; 6(1):14-8. http://dx.doi.org/10.1162/neco.1994.6.1.14.
Durandau G, Farina D, Sartori M. Robust real-time musculoskeletal modeling driven by electromyograms. IEEE Trans Biomed Eng. 2018; 65(3):556-64. PMid:28504931.
Eccles JC, Eccles RM, Lundberg A. Synaptic actions on motoneurones in relation to the two components of the group i muscle afferent volley. J Physiol. 1957; 136(3):527-46. http://dx.doi.org/10.1113/jphysiol.1957.sp005778. PMid:13429518.
Eisen A, Entezari-Taher M, Stewart H. Cortical projections to spinal motoneurons: changes with aging and amyotrophic lateral sclerosis. Neurology. 1996; 46(5):1396-1396. http://dx.doi.org/10.1212/WNL.46.5.1396. PMid:8628489.
Eisen A, Swash M. Clinical neurophysiology of ALS. Clin Neurophysiol. 2001; 112(12):2190-201. http://dx.doi.org/10.1016/S1388-2457(01)00692-7. PMid:11738189.
ElBasiouny SM, Amendola J, Durand J, Heckman CJ. Evidence from computer simulations for alterations in the membrane biophysical properties and dendritic processing of synaptic inputs in mutant superoxide dismutase-1 motoneurons. J Neurosci. 2010; 30(16):5544-58. http://dx.doi.org/10.1523/JNEUROSCI.0434-10.2010. PMid:20410108.
Elias LA, Chaud VM, Kohn AF. Models of passive and active dendrite motoneuron pools and their differences in muscle force control. J Comput Neurosci. 2012; 33(3):515-31. http://dx.doi.org/10.1007/s10827-012-0398-4. PMid:22562305.
Elias LA, Kohn AF. Individual and collective properties of computationally efficient motoneuron models of types S and F with active dendrites. Neurocomputing. 2013; 99:521-33. http://dx.doi.org/10.1016/j.neucom.2012.06.038.
Elias LA, Watanabe RN, Kohn AF. Spinal mechanisms may provide a combination of intermittent and continuous control of human posture: predictions from a biologically based neuromusculoskeletal model. PLOS Comput Biol. 2014; 10(11):e1003944. http://dx.doi.org/10.1371/journal.pcbi.1003944. PMid:25393548.
Eliasmith C, Stewart TC, Choo X, Bekolay T, DeWolf T, Tang Y, Rasmussen D. A large-scale model of the functioning brain. Science. 2012;30(611):338:1202-5. http://dx.doi.org/10.1126/science.1225266.
Farina D, Negro F, Dideriksen JL. The effective neural drive to muscles is the common synaptic input to motor neurons. J Physiol. 2014; 592(16):3427-41. http://dx.doi.org/10.1113/jphysiol.2014.273581. PMid:24860172.
Fisher KM, Zaaimi B, Williams TL, Baker SN, Baker MR. Beta-band intermuscular coherence: a novel biomarker of upper motor neuron dysfunction in motor neuron disease. Brain. 2012; 135(Pt 9):2849-64. http://dx.doi.org/10.1093/brain/aws150. PMid:22734124.
Fuglevand AJ, Winter DA, Patla AE. Models of recruitment and rate coding organization in motor-unit pools. J Neurophysiol. 1993; 70(6):2470-88. http://dx.doi.org/10.1152/jn.1993.70.6.2470. PMid:8120594.
Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol. 1952; 117(4):500-44. http://dx.doi.org/10.1113/jphysiol.1952.sp004764. PMid:12991237.
Howells J, Trevillion L, Bostock H, Burke D. The voltage dependence of I h in human myelinated axons. J Physiol. 2012; 590(7):1625-40. http://dx.doi.org/10.1113/jphysiol.2011.225573. PMid:22310314.
Jalaleddini K, Nagamori A, Laine CM, Golkar MA, Kearney RE, Valero-Cuevas FJ. Physiological tremor increases when skeletal muscle is shortened: implications for fusimotor control. J Physiol. 2017a; 595(24):7331-46. http://dx.doi.org/10.1113/JP274899. PMid:29023731.
Jalaleddini K, Niu C, Chakravarthi Raja S, Sohn WJ, Loeb G, Sanger T, Valero-Cuevas F. Neuromorphic meets neuromechanics, Part II: The role of fusimotor drive. J Neural Eng. 2017b; 14(2):158-62. PMid:28094764.
Jankowska E, Hammar I. Spinal interneurones; how can studies in animals contribute to the understanding of spinal interneuronal systems in man? Brain Res Brain Res Rev. 2002; 40(1-3):19-28. http://dx.doi.org/10.1016/S0165-0173(02)00185-6. PMid:12589903.
Jankowska E. Interneuronal relay in spinal pathways from proprioceptors. Prog Neurobiol. 1992; 38(4):335-78. http://dx.doi.org/10.1016/0301-0082(92)90024-9. PMid:1315446.
Jauregui-Renaud K. Postural balance and peripheral neuropathy. In Souayah N (Ed.). Peripheral neuropathy - a new insight into the mechanism, evaluation and management of a complex disorder. Rijeka: InTech; 2013. p. 125-46. http://dx.doi.org/10.5772/55344.
Kanai K, Kuwabara S, Misawa S, Tamura N, Ogawara K, Nakata M, Sawai S, Hattori T, Bostock H. Altered axonal excitability properties in amyotrophic lateral sclerosis: impaired potassium channel function related to disease stage. Brain. 2006; 129(Pt 4):953-62. http://dx.doi.org/10.1093/brain/awl024. PMid:16467388.
Kristeva R, Patino L, Omlor W. Beta-range cortical motor spectral power and corticomuscular coherence as a mechanism for effective corticospinal interaction during steady-state motor output. Neuroimage. 2007; 36(3):785-92. http://dx.doi.org/10.1016/j.neuroimage.2007.03.025. PMid:17493837.
Kumaravelu K, Brocker DT, Grill WM. A biophysical model of the cortex-basal ganglia-thalamus network in the 6-OHDA lesioned rat model of Parkinson’s disease. J Comput Neurosci. 2016; 40(2):207-29. http://dx.doi.org/10.1007/s10827-016-0593-9. PMid:26867734.
Kuo JJ, Siddique T, Fu R, Heckman CJ. Increased persistent Na + current and its effect on excitability in motoneurones cultured from mutant SOD1 mice. J Physiol. 2005; 563(Pt 3):843-54. http://dx.doi.org/10.1113/jphysiol.2004.074138. PMid:15649979.
Kuwabara S, Yuki N. Axonal Guillain-Barré syndrome: concepts and controversies. Lancet Neurol. 2013; 12(12):1180-8. http://dx.doi.org/10.1016/S1474-4422(13)70215-1. PMid:24229616.
Laine CM, Nagamori A, Valero-Cuevas FJ. The dynamics of voluntary force production in afferented muscle influence involuntary tremor. Front Comput Neurosci. 2016; 10:1-14. http://dx.doi.org/10.3389/fncom.2016.00086. PMid:27594832.
Le Masson G, Przedborski S, Abbott LF. A computational model of motor neuron degeneration. Neuron. 2014; 83(4):975-88. http://dx.doi.org/10.1016/j.neuron.2014.07.001. PMid:25088365.
Lin CC, Crago PE. Neural and mechanical contributions to the stretch reflex: a model synthesis. Ann Biomed Eng. 2002; 30(1):54-67. http://dx.doi.org/10.1114/1.1432692. PMid:11874142.
Lin MT, Beal MFF. Mitochondrial dysfunction and oxidative stress in neurodegenerative diseases. Nature. 2006; 443(7113):787-95. http://dx.doi.org/10.1038/nature05292. PMid:17051205.
Loeb G, Davoodi R. Musculoskeletal mechanics and modeling. Scholarpedia. 2016; 11(11):12389. http://dx.doi.org/10.4249/scholarpedia.12389.
Loeb GE, Tsianos GA. Major remaining gaps in models of sensorimotor systems. Front Comput Neurosci. 2015; 9:1-11. http://dx.doi.org/10.3389/fncom.2015.00070.
Magalhães FH, Elias LA, Silva CR, Lima FF, Toledo DR, Kohn AF. D1 and D2 inhibitions of the soleus H-Reflex are differentially modulated during plantarflexion force and position tasks. PLoS One. 2015; 10(11):e0143862. http://dx.doi.org/10.1371/journal.pone.0143862. PMid:26599909.
Manuel M, Zytnicki D. Alpha, beta and gamma motoneurons: functional diversity in the motor system’s final pathway. J Integr Neurosci. 2011; 10(3):243-76. http://dx.doi.org/10.1142/S0219635211002786. PMid:21960303.
Martin LJ, Chang Q. Inhibitory synaptic regulation of motoneurons: a new target of disease mechanisms in amyotrophic lateral sclerosis. Mol Neurobiol. 2012; 45(1):30-42. http://dx.doi.org/10.1007/s12035-011-8217-x. PMid:22072396.
Mathey EK, Park SB, Hughes RAC, Pollard JD, Armati PJ, Barnett MH, Taylor BV, Dyck PJB, Kiernan MC, Lin CS-Y. Chronic inflammatory demyelinating polyradiculoneuropathy: from pathology to phenotype. J Neurol Neurosurg Psychiatry. 2015; 86(9):973-85. http://dx.doi.org/10.1136/jnnp-2014-309697. PMid:25677463.
McIntyre CC, Richardson AG, Grill WM. Modeling the excitability of mammalian nerve fibers: influence of afterpotentials on the recovery cycle. J Neurophysiol. 2002; 87(2):995-1006. http://dx.doi.org/10.1152/jn.00353.2001. PMid:11826063.
Mileusnic MP, Brown IE, Lan N, Loeb GE. Mathematical models of proprioceptors. I. Control and transduction in the muscle spindle. J Neurophysiol. 2006; 96(4):1772-88. http://dx.doi.org/10.1152/jn.00868.2005. PMid:16672301.
Mileusnic MP, Loeb GE. Mathematical models of proprioceptors. II. Structure and function of the Golgi tendon organ. J Neurophysiol. 2006; 96(4):1789-802. http://dx.doi.org/10.1152/jn.00869.2005. PMid:16672300.
Nagamori A, Laine CM, Valero-Cuevas FJ. Cardinal features of involuntary force variability can arise from the closed-loop control of viscoelastic afferented muscles. PLOS Comput Biol. 2018; 14(1):e1005884. http://dx.doi.org/10.1371/journal.pcbi.1005884. PMid:29309405.
Nakajima T, Tazoe T, Sakamoto M, Endoh T, Shibuya S, Elias LA, Mezzarane RA, Komiyama T, Ohki Y. Reassessment of non-monosynaptic excitation from the motor cortex to motoneurons in single motor units of the human biceps brachii. Front Hum Neurosci. 2017; 11:19. http://dx.doi.org/10.3389/fnhum.2017.00019. PMid:28194103.
Nardone A, Grasso M, Schieppati M. Balance control in peripheral neuropathy: are patients equally unstable under static and dynamic conditions? Gait Posture. 2006; 23(3):364-73. http://dx.doi.org/10.1016/j.gaitpost.2005.04.002. PMid:15896962.
Negro F, Farina D. Decorrelation of cortical inputs and motoneuron output. J Neurophysiol. 2011; 106(5):2688-97. http://dx.doi.org/10.1152/jn.00336.2011. PMid:21795617.
Negro F, Yavuz UŞ, Farina D. The human motor neuron pools receive a dominant slow-varying common synaptic input. J Physiol. 2016; 594(19):5491-505. http://dx.doi.org/10.1113/JP271748. PMid:27151459.
Nijssen J, Comley LH, Hedlund E. Motor neuron vulnerability and resistance in amyotrophic lateral sclerosis. Acta Neuropathol. 2017; 133(6):863-85. http://dx.doi.org/10.1007/s00401-017-1708-8. PMid:28409282.
Niu CM, Jalaleddini K, Sohn WJ, Rocamora J, Sanger T, Valero-Cuevas F. Neuromorphic meets neuromechanics PART I: the methodology and implementation. J Neural Eng. 2017; 14(2):025001. http://dx.doi.org/10.1088/1741-2552/aa593c. PMid:28084217.
Passmore E, Lai A, Sangeux M, Schache AG, Pandy MG. Application of ultrasound imaging to subject-specific modelling of the human musculoskeletal system. Meccanica. 2016
Pierrot-Deseilligny E, Burke D. The circuitry of the human spinal cord: spinal and corticospinal mechanisms of movement. Cambridge: Cambridge University Press; 2012. http://dx.doi.org/10.1017/CBO9781139026727.
Piotrkiewicz M, Hausmanowa-Petrusewicz I. Motoneuron afterhyperpolarisation duration in amyotrophic lateral sclerosis. J Physiol. 2011; 589(Pt 11):2745-54. http://dx.doi.org/10.1113/jphysiol.2011.204891. PMid:21486815.
Porter R, Lemon R. Corticospinal function and voluntary movement. Oxford: Oxford University Press; 1995. http://dx.doi.org/10.1093/acprof:oso/9780198523758.001.0001.
Priori A, Bossi B, Ardolino G, Bertolasi L, Carpo M, Nobile-Orazio E, Barbieri S. Pathophysiological heterogeneity of conduction blocks in multifocal motor neuropathy. Brain. 2005; 128(Pt 7):1642-8. http://dx.doi.org/10.1093/brain/awh513. PMid:15888541.
Prochazka A, Gorassini M. Models of ensemble firing of muscle spindle afferents recorded during normal locomotion in cats. J Physiol. 1998; 507(Pt 1):277-91. http://dx.doi.org/10.1111/j.1469-7793.1998.277bu.x. PMid:9490851.
Raikova RT, Aladjov HT. Hierarchical genetic algorithm versus static optimization—investigation of elbow flexion and extension movements. J Biomech. 2002; 35(8):1123-35. http://dx.doi.org/10.1016/S0021-9290(02)00031-3. PMid:12126671.
Raphael G, Tsianos GA, Loeb GE. Spinal-like regulator facilitates control of a two-degree-of-freedom wrist. J Neurosci. 2010; 30(28):9431-44. http://dx.doi.org/10.1523/JNEUROSCI.5537-09.2010. PMid:20631172.
Raynor EM, Shefner JM. Recurrent inhibition is decreased in patients with amyotrophic lateral sclerosis. Neurology. 1994; 44(11):2148-53. http://dx.doi.org/10.1212/WNL.44.11.2148. PMid:7969975.
Revill AL, Fuglevand AJ. Effects of persistent inward currents, accommodation, and adaptation on motor unit behavior: a simulation study. J Neurophysiol. 2011; 106(3):1467-79. http://dx.doi.org/10.1152/jn.00419.2011. PMid:21697447.
Schmied A, Pouget J, Vedel J. Electromechanical coupling and synchronous firing of single wrist extensor motor units in sporadic amyotrophic lateral sclerosis. Clin Neurophysiol. 1999; 110(5):960-74. http://dx.doi.org/10.1016/S1388-2457(99)00032-2. PMid:10400212.
Schuurmans J, de Vlugt E, Schouten AC, Meskers CGM, de Groot JH, van der Helm FCT. The monosynaptic Ia afferent pathway can largely explain the stretch duration effect of the long latency M2 response. Exp Brain Res. 2009; 193(4):491-500. http://dx.doi.org/10.1007/s00221-008-1647-7. PMid:19048240.
Scott SH, Brown IE, Loeb GE. Mechanics of feline soleus. 1. Effect of fascicle length and velocity on force output. J Muscle Res Cell Motil. 1996; 17(2):207-19. http://dx.doi.org/10.1007/BF00124243. PMid:8793723.
Stephanova DI, Daskalova M. Electrotonic potentials in simulated chronic inflammatory demyelinating polyneuropathy at 20°C–42°C. J Integr Neurosci. 2015; 14(2):235-52. http://dx.doi.org/10.1142/S0219635215500119. PMid:25916252.
Stephanova DI, Dimitrov B. Computational neuroscience: simulated demyelinating neuropathies and neuronopathies. Boca Raton, FL: CRC Press; 2013. http://dx.doi.org/10.1201/b14589.
Taylor JP, Brown RH Jr, Cleveland DW. Decoding ALS: from genes to mechanism. Nature. 2016; 539(7628):197-206. http://dx.doi.org/10.1038/nature20413. PMid:27830784.
Tsianos GA, Goodner J, Loeb GE. Useful properties of spinal circuits for learning and performing planar reaches. J Neural Eng. 2014; 11(5):56006. http://dx.doi.org/10.1088/1741-2560/11/5/056006.
Tsianos GA, Loeb GE. Muscle and limb mechanics. Compr Physiol. 2017;7(2):429-62. https://doi.org/10.1002/cphy.c160009.
Tsianos GA, Rustin C, Loeb GE. Mammalian muscle model for predicting force and energetics during physiological behaviors. IEEE Trans Neural Syst Rehabil Eng. 2012; 20(2):117-33. http://dx.doi.org/10.1109/TNSRE.2011.2162851. PMid:21859633.
Ushiyama J, Masakado Y, Fujiwara T, Tsuji T, Hase K, Kimura A, Liu M, Ushiba J. Contraction level-related modulation of corticomuscular coherence differs between the tibialis anterior and soleus muscles in humans. J Appl Physiol. 2012; 112(8):1258-67. http://dx.doi.org/10.1152/japplphysiol.01291.2011. PMid:22302959.
Valero-Cuevas FJ, Hoffmann H, Kurse MU, Kutch JJ, Theodorou EA. Computational models for neuromuscular function. IEEE Rev Biomed Eng. 2009; 2:110-35. http://dx.doi.org/10.1109/RBME.2009.2034981. PMid:21687779.
Vucic S, Kiernan MC. Pathophysiology of neurodegeneration in familial amyotrophic lateral sclerosis. Curr Mol Med. 2009; 9(3):255-72. http://dx.doi.org/10.2174/156652409787847173. PMid:19355908.
Watanabe RN, Kohn F. Fast oscillatory commands from the motor cortex can be decoded by the spinal cord for force control. J Neurosci. 2015; 35(40):13687-97. http://dx.doi.org/10.1523/JNEUROSCI.1950-15.2015. PMid:26446221.
Watanabe RN, Magalhães FH, Elias LA, Chaud VM, Mello EM, Kohn AF. Influences of premotoneuronal command statistics on the scaling of motor output variability during isometric plantar flexion. J Neurophysiol. 2013; 110(11):2592-606. http://dx.doi.org/10.1152/jn.00073.2013. PMid:24027105.
Williams ER, Baker SN. Renshaw cell recurrent inhibition improves physiological tremor by reducing corticomuscular coupling at 10 Hz. J Neurosci. 2009a; 29(20):6616-24. http://dx.doi.org/10.1523/JNEUROSCI.0272-09.2009. PMid:19458232.
Williams ER, Baker SN. Circuits generating corticomuscular coherence investigated using a biophysically based computational model. I. Descending Systems. J Neurophysiol. 2009b; 101(1):31-41. http://dx.doi.org/10.1152/jn.90362.2008. PMid:19019981.
Yavuz SU, Mrachacz-Kersting N, Sebik O, Berna Ünver M, Farina D, Türker KS. Human stretch reflex pathways reexamined. J Neurophysiol. 2014; 111(3):602-12. http://dx.doi.org/10.1152/jn.00295.2013. PMid:24225537.
Zahalak GI. A distribution-moment approximation for kinetic theories of muscular-contraction. Math Biosci. 1981; 55(1-2):89-114. http://dx.doi.org/10.1016/0025-5564(81)90014-6.
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