Biological Cybernetics

, Volume 111, Issue 5–6, pp 421–438 | Cite as

Fuzzy neuronal model of motor control inspired by cerebellar pathways to online and gradually learn inverse biomechanical functions in the presence of delay

  • Armin Salimi-Badr
  • Mohammad Mehdi EbadzadehEmail author
  • Christian Darlot
Original Article


Contrary to forward biomechanical functions, which are deterministic, inverse biomechanical functions are generally not. Calculating an inverse biomechanical function is an ill-posed problem, which has no unique solution for a manipulator with several degrees of freedom. Studies of the command and control of biological movements suggest that the cerebellum takes part in the computation of approximate inverse functions, and this ability can control fast movements by predicting the consequence of current motor command. Limb movements toward a goal are defined as fast if they last less than the total duration of the processing and transmission delays in the motor and sensory pathways. Because of these delays, fast movements cannot be continuously controlled in a closed loop by use of sensory signals. Thus, fast movements must be controlled by some open loop controller, of which cerebellar pathways constitute an important part. This article presents a system-level fuzzy neuronal motor control circuit, inspired by the cerebellar pathways. The cerebellar cortex (CC) is assumed to embed internal models of the biomechanical functions of the limb segments. Such neural models are able to predict the consequences of motor commands and issue predictive signals encoding movement variables, which are sent to the controller via internal feedback loops. Differences between desired and expected values of variables of movements are calculated in the deep cerebellar nuclei (DCN). After motor learning, the whole circuit can approximate the inverse function of the biomechanical function of a limb and acts as a controller. In this research, internal models of direct biomechanical functions are learned and embedded in the connectivity of the cerebellar pathways. Two fuzzy neural networks represent the two parts of the cerebellum, and an online gradual learning drives the acquisition of the internal models in CC and the controlling rules in DCN. As during real learning, exercise and repetition increase skill and speed. The learning procedure is started by a simple and slow movement, controlled in the presence of delays by a simple closed loop controller comparable to the spinal reflexes. The speed of the movements is then increased gradually, and output error signals are used to compute teaching signals and drive learning. Repetition of movements at each speed level allows to properly set the two neural networks, and progressively learn the movement. Finally, conditions of stability of the proposed model as an inverter are identified. Next, the control of a single segment arm, moved by two muscles, is simulated. After proper setting by motor learning, the circuit is able to reject perturbations.


Cerebellum Online gradual learning Approximate inverse function Fast movements Forward model LTD/LTP Deep cerebellar nuclei (DCN) Reflex pathways 


  1. Albus JS (1975a) Data storage in the cerebellar model articulation controller (CMAC). J Dyn Syst Meas Control 97:228–233CrossRefGoogle Scholar
  2. Albus JS (1975b) A new approach to manipulator control: the cerebellar model articulation controller (CMAC). J Dyn Syst Meas Control 97:220–227CrossRefGoogle Scholar
  3. Armano S, Rossi P, Taglietti V, D’Angelo E (2000) Long-term potentiation of intrinsic excitability at the mossy fiber-granule cell synapse of rat cerebellum. J Neurosci 20(14):5208–5216PubMedGoogle Scholar
  4. Asadi-Eydivand M, Ebadzadeh M, Solati-Hashjin M, Darlot C, Osman N (2015) Cerebellum-inspired neural network solution of the inverse kinematics problem. Biol Cybern 109:561–574CrossRefPubMedPubMedCentralGoogle Scholar
  5. Barto AG, Fagg AH, Sitkoff N, Houk JC (1999) A cerebellar model of timing and prediction in the control of reaching. Neural Comput 11:565–594CrossRefPubMedGoogle Scholar
  6. Bostan AC, Dum RP, Strick PL (2013) Cerebellar networks with the cerebral cortex and basal ganglia. Trends Cogn Sci 17(4):241–254. doi: 10.1016/j.tics.2013.03.003 CrossRefPubMedPubMedCentralGoogle Scholar
  7. Chapeau-Blondeau P, Chauvet G (1991) A neural network model of the cerebellar cortex performing dynamic associations. Biol Cybern 65:267–279CrossRefPubMedGoogle Scholar
  8. D’Angelo E, Solinas S, Mapelli J, Gandolfi D, Mapelli L, Prestori F (2013) The cerebellar Golgi cell and spatiotemporal organization of granular layer activity. Front Neural Circuits 7(93):1–21Google Scholar
  9. Darban ZZ, Ebadzadeh M (2012) Anatomical model of VOR using fuzzy neural network. Proc Eng 41:561–566CrossRefGoogle Scholar
  10. Darlot C (1993) The Cerebellum as a predictor of neural messages—I. The stable estimator hypothesis. Neuroscience 56:617–646CrossRefPubMedGoogle Scholar
  11. Darlot C, Zupan L, Etard O, Denise P, Maruani A (1996) Computation of inverse dynamics for the control of movements. Biol Cybern 75:173–186CrossRefPubMedGoogle Scholar
  12. Denise P, Darlot C (1993) The cerebellum as a predictor of neural messages—II. Role in motor control and motion sickness. Neuroscience 56(3):647–655CrossRefPubMedGoogle Scholar
  13. Droulez J, Darlot C (1990) The geometric and dynamic implications of the coherence constraints in three-dimensional sensorimotor inter-actions. In: Jeannerod M (ed) Attention and performance XIII. Lawrence Erlbaum, Hillsdale, pp 495–526Google Scholar
  14. Ebadzadeh M, Darlot C (2003) Cerebellar learning of bio-mechanical functions of extra-ocular muscles: modeling by artificial neural networks. Neuroscience 122:941–966CrossRefPubMedGoogle Scholar
  15. Ebadzadeh M, Salimi-Badr A (2015) CFNN: correlated fuzzy neural network. Neurocomputing 148:430–444CrossRefGoogle Scholar
  16. Ebadzadeh M, Salimi-Badr A (2017) IC-FNN: a novel fuzzy neural network with interpretable intuitive and correlated-contours fuzzy rules for function approximation. IEEE Trans Fuzzy Syst. doi: 10.1109/TFUZZ.2017.2718497 Google Scholar
  17. Ebadzadeh M, Tondu B, Darlot C (2005) Computation of inverse functions in a model of cerebellar and reflex pathways allows to control a mobile mechanical segment. Neuroscience 133:29–49CrossRefPubMedGoogle Scholar
  18. Eccles JC, Ito M, Szentágothai J (1967) The cerebellum as a neuronal machine. Springer, New YorkCrossRefGoogle Scholar
  19. Eskiizmirliler S, Forestier N, Tondu B, Darlot C (2002) A model of the cerebellar pathways applied to the control of a single-joint robot arm actuated by McKibben artificial muscles. Biol Cybern 86:379–394CrossRefPubMedGoogle Scholar
  20. Forestier N (1999) Modélisation du contrôle moteur cérébelleux par réseaux de neurones formels. Thèse de doctorat. Ecole Nationale Supérieure des Télécommunications. ENST 99 E 009Google Scholar
  21. Fujita M (1982) Adaptive filter model of the cerebellum. Biol Cybern 45:195–206CrossRefPubMedGoogle Scholar
  22. Garrido J, Luque NR, D’Angelo E, Ros E (2013) Distributed cerebellar plasticity implements adaptable gain control in a manipulation task: a closed-loop robotic simulation. Front Neural Circuits 7(159):1–20Google Scholar
  23. Gentili RJ, Papaxanthis C, Ebadzadeh M, Eskiizmirliler S, Ouanezar S, Darlot C (2009) Integration of gravitational torques in cerebellar pathways allows for the dynamic inverse computation of vertical pointing movements of a robot arm. PloS ONE 4:e5176CrossRefPubMedPubMedCentralGoogle Scholar
  24. Han H, Qiao J (2010) A self-organizing fuzzy neural network based on a growing and pruning algorithm. IEEE Trans Fuzzy Syst 18:1129–1143CrossRefGoogle Scholar
  25. Hirano T (2013) Long-term depression and other synaptic plasticity in the cerebellum. Proc Jpn Acad Ser B Phys Biol Sci 89(4):183–195CrossRefPubMedPubMedCentralGoogle Scholar
  26. Houk JC, Buckingham JT, Barto AG (1996) Models of the cerebellum and motor control. Behav Brain Sci 19:368–383Google Scholar
  27. Huang G, Saratchandran P, Sundararajan N (2005) A generalized growing and pruning RBF (GGAP-RBF) neural network for function approximation. IEEE Trans Neural Netw 16:57–67CrossRefPubMedGoogle Scholar
  28. Ito M (1986) Long-term depression as a memory process in the cerebellum. Neurosci Res 3:531–539CrossRefPubMedGoogle Scholar
  29. Ito M (2006) Cerebellar circuitry as a neuronal machine. Prog Neurobiol 78:272–303CrossRefPubMedGoogle Scholar
  30. Ito M, Kano M (1982) Long-lasting depression of parallel fiber-Purkinje cell transmission induced by conjunctive stimulation of parallel fibers and climbing fibers in the cerebellar cortex. Neurosci Lett 33:253–258CrossRefPubMedGoogle Scholar
  31. Jaberi J, Gambrell K, Tiwana P, Madden C, Finn R (2013) Long-term clinical outcome analysis of poly-methyl methacrylate cranio-plasty for large skull defects. J Oral Maxillofac Surg 71:e81–e88CrossRefPubMedGoogle Scholar
  32. Jaeger D (2013) Cerebellar nuclei and cerebellar learning. Handbook of the cerebellum and cerebellar disorders. Springer, BerlinGoogle Scholar
  33. Kandel E, Schwartz J (2013) Principles of neural science, 5th edn. McGraw-Hill Education, New YorkGoogle Scholar
  34. Kawato M, Gomi H (1992) A computational model of four regions of the cerebellum based on feedback-error learning. Biol Cybern 68:95–103CrossRefPubMedGoogle Scholar
  35. Kawato M, Furukawa K, Suzuki R (1987) A hierarchical neural-network model for control and learning of voluntary movement. Biol Cybern 57:169–185CrossRefPubMedGoogle Scholar
  36. Khayat O, Ebadzadeh M, Shahdoosti H, Rajaei R, Khajehnasiri I (2009) A novel hybrid algorithm for creating self-organizing fuzzy neural networks. Neurocomputing 73:517–524CrossRefGoogle Scholar
  37. Koene A, Erkelens C (2002) Cause of kinematic differences during centrifugal and centripetal saccades. Vis Res 42:1797–1808CrossRefPubMedGoogle Scholar
  38. Kosko B (1994) Fuzzy systems as universal approximators. IEEE Trans Comput 43:1329–1333CrossRefGoogle Scholar
  39. Luque NR, Garrido JA, Carrillo RR, D’Angelo E, Ros E (2014) Fast convergence of learning requires plasticity between inferior olive and deep cerebellar nuclei in a manipulation task: a closed-loop robotic simulation. Front Comput Neurosci 8(97):1–16Google Scholar
  40. Malek H, Ebadzadeh MM, Rahmati M (2012) Three new fuzzy neural networks learning algorithms based on clustering, training error and genetic algorithm. Appl Intell 37:280–289CrossRefGoogle Scholar
  41. Mapelli J, D’Angelo E (2005) The spatial organization of long-term synaptic plasticity at the input stage of cerebellum. J Neurosci 25:1285–1296Google Scholar
  42. Marr D (1969) A theory of cerebellar cortex. J Physiol 202:437–470CrossRefPubMedPubMedCentralGoogle Scholar
  43. Miall R (1998) The cerebellum, predictive control andmotor coordination. Sens Guid Mov 218:272–290Google Scholar
  44. Miall R, Wolpert DM (1996) Forward models for physiological motor control. Neural Netw 9:1265–1279CrossRefPubMedGoogle Scholar
  45. Miall R, Weir D, Wolpert D, Stein J (1993) Is the cerebellum a Smith predictor? J Mot Behav 25:203–216CrossRefPubMedGoogle Scholar
  46. Nieus T, Sola E, Mapelli J, Saftenku E, Rossi P, D’Angelo E (2006) LTP regulates burst initiation and frequency at mossy fiber-granule cell synapses of rat cerebellum: experimental observations and theoretical predictions. J Neurophysiol 95:686–699CrossRefPubMedGoogle Scholar
  47. Ouanezar S, Jean F, Tondu B, Maier M, Darlot C, Eskiizmirliler S (2011) Biologically inspired sensory motor control of a 2-link robotic arm actuated by McKibben muscles. In: Proceedings of the IEEE international conference on robotics and automation. Shanghai International Conference Center, ShanghaiGoogle Scholar
  48. Passino M, Yurkovich S (1998) Fuzzy control. Addison-Wesley, ReadingGoogle Scholar
  49. Riahi-Madvar H, Ayyoubzadeh S, Khadangi E, Ebadzadeh M (2009) An expert system for predicting longitudinal dispersion coefficient in natural streams by using ANFIS. Expert Syst Appl 36:8589–8596CrossRefGoogle Scholar
  50. Roggeri L, Rivieccio B, Rossi P, D’Angelo E (2008) Tactile stimulation evokes long-term synaptic plasticity in the granular layer of cerebellum. J Neurosci 28:6354–6359CrossRefPubMedGoogle Scholar
  51. Rubio JJ (2009) SOFMLS: online self organizing fuzzy modified least squares network. IEEE Trans Fuzzy Syst 17:1296–1309CrossRefGoogle Scholar
  52. Schweighofer N, Lang EJ, Kawato M (2013) Role of the olivo-cerebellar complex in motor learning and control. Front Neural Circuits 7:94. doi: 10.3389/fncir.2013.00094 CrossRefPubMedPubMedCentralGoogle Scholar
  53. Shadmehr R (2009) Computational approaches to motor control. Encyclopedia of Neuroscience Oxford, vol 3. Academic Press, New York, pp 9–17Google Scholar
  54. Shadmehr R, Krakauer J (2008) A computational neuroanatomy for motor control. Exp Brain Res 185:359–381CrossRefPubMedPubMedCentralGoogle Scholar
  55. Uttley A (1979) Information transmission in the nervous system. Academic Press, New YorkGoogle Scholar
  56. Uusisaari M, Knopfel T (2011) Functional classification of neurons in the mouse lateral cerebellar nuclei. Cerebellum 10:637–646CrossRefPubMedGoogle Scholar
  57. Wang N, Er MJ, Meng X (2009) A fast and accurate online self-organizing scheme for parsimonious fuzzy neural networks. Neurocomputing 72:3818–3829CrossRefGoogle Scholar
  58. Wolpert DM, Miall RC, Kawato M (1998) Internal models in the cerebellum. Trends Cogn Sci 2:338–347CrossRefPubMedGoogle Scholar
  59. Xu WL, Zhang NY, Zeng K (2000) A comparative study on sufficient conditions for Takagi-Sugeno fuzzy systems as universal approximators. IEEE Trans Fuzzy Syst 8(5):773–780Google Scholar
  60. Ying H (1998) General SISO Takagi-Sugeno fuzzy systems with linear rule consequent are universal approximators. IEEE Trans Fuzzy Syst 6(4):582–587CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Computer Engineering and Information TechnologyAmirkabir University of TechnologyTehranIran
  2. 2.INSERM U1093, Laboratoire de Cognition, Action et Plasticité Sensorimotrice, UFR STAPSUniversité de BourgogneDijonFrance

Personalised recommendations