Biological Cybernetics

, Volume 110, Issue 1, pp 55–71 | Cite as

Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia

  • Ihab Haidar
  • William Pasillas-LépineEmail author
  • Antoine Chaillet
  • Elena Panteley
  • Stéphane Palfi
  • Suhan Senova
Original Article


This paper develops a new closed-loop firing rate regulation strategy for a population of neurons in the subthalamic nucleus, derived using a model-based analysis of the basal ganglia. The system is described using a firing rate model, in order to analyse the generation of beta-band oscillations. On this system, a proportional regulation of the firing rate reduces the gain of the subthalamo-pallidal loop in the parkinsonian case, thus impeding pathological oscillation generation. A filter with a well-chosen frequency is added to this proportional scheme, in order to avoid a potential instability of the feedback loop due to actuation and measurement delays. Our main result is a set of conditions on the parameters of the stimulation strategy that guarantee both its stability and a prescribed delay margin. A discussion on the applicability of the proposed method and a complete set of mathematical proofs is included.


Neural oscillations Firing rate models Time-delay systems Basal ganglia Parkinson’s disease Deep brain stimulation Closed-loop stimulation 



This work was financially supported by the European Commission through the FP7 NoE HYCON2 and by the region Ile-de-France through the Neurosynch project (RTRA Digiteo). The work of the fourth author was supported by the Government of the Russian Federation (grant 074-U01).


  1. Agarwal R, Sarma SV (2010) Restoring the basal ganglia in Parkinson’s disease to normal via multi-input phase-shifted deep brain stimulation. In: IEEE/EMBS conference on neural engineering (NER)Google Scholar
  2. Aström KJ, Murray RM (2010) Feedback systems: an introduction for scientists and engineers. Princeton University Press, PrincetonGoogle Scholar
  3. Benabid AL, Pollak P, Gervason C, Hoffmann D, Gao DM, Hommel M, Perret JE, de Rougemont J (1991) Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus. Lancet 337:403–406CrossRefPubMedGoogle Scholar
  4. Bergman H, Wichmann T, Delong MR (1990) Reversal of experimental parkinsonism by lesions of the subthalamic nucleus. Science 249:1436–1438CrossRefPubMedGoogle Scholar
  5. Boyden ES (2011) A history of optogenetics: the development of tools for controlling brain circuits with light. F1000 Biol Rep 3:1–12. Art ID 11Google Scholar
  6. Bragin A, Wilson CL, Staba RJ, Reddick M, Fried I, Engel J (2002) Interictal high-frequency oscillations (80–50 Hz) in the human epileptic brain: entorhinal cortex. Ann Neurol 52(4):407–415CrossRefPubMedGoogle Scholar
  7. Carron R, Chaillet A, Filipchuk A, Pasillas-Lépine W, Hammond C (2013) Closing the loop of deep brain stimulation. Front Syst Neurosci 7:1–18. Art ID 112Google Scholar
  8. Carron R, Filipchuk A, Nardou R, Singh A, Michel FJ, Humphries MD, Hammond C (2014) Early hypersynchrony in juvenile pink1(-)/(-) motor cortex is rescued by antidromic stimulation. Front Syst Neurosci 8:1–12. Art ID 95Google Scholar
  9. Coombes S, Laing C (2009) Delays in activity-based neural networks. Philos Trans R Soc A Math Phys Eng Sci 367(1891):1117–1129CrossRefGoogle Scholar
  10. Curtain RF, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, volume 21 of texts in applied mathematics, 1st edn. Springer, New YorkCrossRefGoogle Scholar
  11. Davie CA (2008) A review of Parkinson’s disease. Br Med Bull 86(1):109–127CrossRefPubMedGoogle Scholar
  12. Dayan P, Abbott LF (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. The MIT Press, CambridgeGoogle Scholar
  13. Dovzhenok A, Park C, Worth RM, Rubchinsky LL (2013) Failure of delayed feedback deep brain stimulation for intermittent pathological synchronization in Parkinson’s disease. PLoS ONE 8(3):e58264PubMedCentralCrossRefPubMedGoogle Scholar
  14. Doyle J, Francis B, Tannenbaum A (1990) Feedback control theory. Macmillan Publishing Co., LondonGoogle Scholar
  15. Doyle JC, Glover K, Khargonekar PP, Francis B (1989) State-space solutions to standard \(h_2\) and \(h_\infty \) control problems. IEEE Trans Autom Control 34(8):831–847CrossRefGoogle Scholar
  16. Dunn EM, Lowery MM (2013) Simulation of PID control schemes for closed-loop deep brain stimulation. In: IEEE/EMBS conference on neural engineering (NER), San Diego (California)Google Scholar
  17. Feng X, Greenwald B, Rabitz H, Shea-Brown E, Kosut R (2007) Toward closed-loop optimization of deep brain stimulation for parkinson’s disease: concepts and lessons from a computational model. J Neural Eng 4:L14CrossRefPubMedGoogle Scholar
  18. Franci A, Chaillet A, Pasillas-Lépine W (2011) Existence and robustness of phase-locking in coupled Kuramoto oscillators under mean-field feedback. Automatica 47(6):1193–1202CrossRefGoogle Scholar
  19. Fridman E, Shaked U (2002) An improved stabilization method for linear time-delay systems. IEEE Trans Autom Control 47(11):1931–1937CrossRefGoogle Scholar
  20. Gahinet P, Nemirovskii A, Laub AJ, Chilali M (1994) The LMI control toolbox. In: Proceedings of the IEEE conference on decision and control, pp 2038–2041Google Scholar
  21. Gonzalez-Burgos G, Lewis DA (2008) GABA neurons and the mechanisms of network oscillations: implications for understanding cortical dysfunction in schizophrenia. Schizophr Bull 34(5):944–961PubMedCentralCrossRefPubMedGoogle Scholar
  22. Gradinaru V, Mogri M, Thompson KR, Henderson JM, Deisseroth K (2009) Optical deconstruction of parkinsonian neural circuitry. Science 324(5925):354–359CrossRefPubMedGoogle Scholar
  23. Gray CM, König P, Engel AK, Singer W (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338(6213):334–337CrossRefPubMedGoogle Scholar
  24. Haidar I, Pasillas-Lépine W, Panteley E, Chaillet A, Palfi S, Senova S (2014) Analysis of delay-induced basal ganglia oscillations: the role of external excitatory nuclei. Int J Control 87(9):1936–1956CrossRefGoogle Scholar
  25. Hammond C, Bergman H, Brown P (2007) Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends Neurosci 30(7):357–364 INMED/TINS special issue–Physiogenic and pathogenic oscillations: the beauty and the beastCrossRefPubMedGoogle Scholar
  26. Han X, Boyden ES (2007) Multiple-color optical activation, silencing, and desynchronization of neural activity, with single-spike temporal resolution. PLoS ONE 2(3):299CrossRefGoogle Scholar
  27. Han X, Qian X, Stern P, Chuong AS, Boyden ES (2009) Informational lesions: optical perturbation of spike timing and neural synchrony via microbial opsin gene fusions. Front Mol Neurosci 2:1–9. Art ID 12Google Scholar
  28. Han X, Chow BY, Zhou H, Klapoetke NC, Chuong A, Rajimehr R, Yang A, Baratta MV, Winkle J, Desimone R et al (2011) A high-light sensitivity optical neural silencer: development and application to optogenetic control of non-human primate cortex. Front Syst Neurosci 5:18PubMedCentralCrossRefPubMedGoogle Scholar
  29. Hauptmann C, Popovych O, Tass PA (2005a) Delayed feedback control of synchronization in locally coupled neuronal networks. Neurocomputing 65:759–767CrossRefGoogle Scholar
  30. Hauptmann C, Popovych O, Tass PA (2005b) Effectively desynchronizing deep brain stimulation based on a coordinated delayed feedback stimulation via several sites: a computational study. Biol Cybern 93(6):463–470CrossRefPubMedGoogle Scholar
  31. He Y, Wang QG, Lin C, Wu M (2007) Delay-range-dependent stability for systems with time-varying delay. Automatica 43(2):371–376CrossRefGoogle Scholar
  32. Jenkinson N, Brown P (2011) New insights into the relationship between dopamine, beta oscillations and motor function. Trends Neurosci 34(12):611–618CrossRefPubMedGoogle Scholar
  33. Kumar R, Lozano AM, Sime E, Lang AE (2003) Long-term follow-up of thalamic deep brain stimulation for essential and parkinsonian tremor. Neurology 61(11):1601–1604CrossRefPubMedGoogle Scholar
  34. Leblois A, Boraud T, Meissner W, Bergman H, Hansel D (2006) Competition between feedback loops underlies normal and pathological dynamics in the basal ganglia. J Neurosci 26(13):3567–3583CrossRefPubMedGoogle Scholar
  35. Legatt AD, Arezzo J, Vaughan HG Jr (1980) Averaged multiple unit activity as an estimate of phasic changes in local neuronal activity: effects of volume-conducted potentials. J Neurosci Methods 2(2):203–217CrossRefPubMedGoogle Scholar
  36. Limousin P, Pollak P, Benazzouz A, Hoffmann D, Le Bas J-F, Perret J-E, Benabid A-L, Broussolle E (1995) Effect on parkinsonian signs and symptoms of bilateral subthalamic nucleus stimulation. Lancet 345(8942):91–95CrossRefPubMedGoogle Scholar
  37. Little S, Pogosyan A, Kuhn AA, Brown P (2012) Beta band stability over time correlates with Parkinsonian rigidity and bradykinesia. Exp Neurol 236(2):383–388PubMedCentralCrossRefPubMedGoogle Scholar
  38. Liu J, Oweiss KG, Khalil HK (2010) Feedback control of the spatiotemporal firing patterns of neural microcircuits. In: IEEE conference on decision and controlGoogle Scholar
  39. Liu J, Khalil HK, Oweiss KG (2011) Model-based spatiotemporal analysis and control of a network of spiking basal ganglia neurons. In: IEEE/EMBS conference on neural engineering (NER)Google Scholar
  40. Lysyansky B, Popovych OV, Tass PA (2011) Desynchronizing anti-resonance effect of m: n on-off coordinated reset stimulation. J Neural Eng 8:036019CrossRefPubMedGoogle Scholar
  41. McIntyre C, Savasta M, Kerkerian-Le Goff L, Vitek JL (2004a) Uncovering the mechanism(s) of action of deep brain stimulation: activation, inhibition, or both. Clin Neurophysiol 115(6):1239–1248CrossRefPubMedGoogle Scholar
  42. McIntyre CC, Grill WM, Sherman DL, Thakor NV (2004b) Cellular effects of deep brain stimulation: model-based analysis of activation and inhibition. J Neurophysiol 91(4):1457–1469CrossRefPubMedGoogle Scholar
  43. Michmizos KP, Sakas D, Nikita KS (2012) Prediction of the timing and the rhythm of the parkinsonian subthalamic nucleus neural spikes using the local field potentials. IEEE Trans Inf Technol Biomed 16(2):190–197CrossRefPubMedGoogle Scholar
  44. Middleton RH, Miller DE (2007) On the achievable delay margin using LTI control for unstable plants. IEEE Trans Autom Control 52(7):1194–1207CrossRefGoogle Scholar
  45. Mitzdorf U (1987) Properties of the evoked potential generators: current source-density analysis of visually evoked potentials in the cat cortex. Int J Neurosci 33(1–2):33CrossRefPubMedGoogle Scholar
  46. Modolo J, Henry J, Beuter A (2008) Dynamics of the subthalamo-pallidal complex in parkinson’s disease during deep brain stimulation. J Biol Phys 34(3–4):251–266CrossRefPubMedGoogle Scholar
  47. Montaseri G, Yazdanpanah MJ, Pikovsky A, Rosenblum M (2013) Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback. Chaos: an interdisciplinary. J Nonlinear Sci 23(3):033122Google Scholar
  48. Nambu A (2011) Somatotopic organization of the primate basal ganglia. Front Neuroanat 5:1–9. Art ID 26Google Scholar
  49. Nevado-Holgado AL, Terry JR, Bogacz R (2010) Conditions for the generation of beta oscillations in the subthalamic nucleus–globus pallidus network. J Neurosci 30(37):12340–12352CrossRefGoogle Scholar
  50. Nini A, Feingold A, Slovin H, Bergman H (1995) Neurons in the globus pallidus do not show correlated activity in the normal monkey, but phase-locked oscillations appear in the MPTP model of parkinsonism. J Neurophysiol 74(4):1800–1805PubMedGoogle Scholar
  51. Omel’chenko OE, Hauptmann C, Maistrenko YL, Tass PA (2008) Collective dynamics of globally coupled phase oscillators under multisite delayed feedback stimulation. Phys D 237(3):365–384CrossRefGoogle Scholar
  52. Park C, Worth RM, Rubchinsky LL (2010) Fine temporal structure of beta oscillations synchronization in subthalamic nucleus in Parkinson’s disease. J Neurophysiol 103(5):2707–2716PubMedCentralCrossRefPubMedGoogle Scholar
  53. Pascual A, Modolo J, Beuter A (2006) Is a computational model useful to understand the effect of deep brain stimulation in Parkinson’s disease? J Integr Neurosci 5(04):541–559CrossRefPubMedGoogle Scholar
  54. Pasillas-Lépine W (2013) Delay-induced oscillations in Wilson and Cowan’s model: an analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects. Biol Cybern 107(3):289–308CrossRefPubMedGoogle Scholar
  55. Pavlides A, Hogan J, Bogacz R (2012) Improved conditions for the generation of beta oscillations in the subthalamic nucleus–globus pallidus network. Eur J Neurosci 36(2):2229–2239CrossRefPubMedGoogle Scholar
  56. Plenz D, Kital ST (1999) A basal ganglia pacemaker formed by the subthalamic nucleus and external globus pallidus. Nature 400(6745):677–682CrossRefPubMedGoogle Scholar
  57. Pogosyan A, Yoshida F, Chen CC, Martinez-Torres I, Foltynie T, Limousin P, Zrinzo L, Hariz MI, Brown P (2010) Parkinsonian impairment correlates with spatially extensive subthalamic oscillatory synchronization. Neuroscience 171(1):245–257CrossRefPubMedGoogle Scholar
  58. Pyragas K, Popovych OV, Tass PA (2007) Controlling synchrony in oscillatory networks with a separate stimulation-registration setup. Europhys Lett 80(4):40002-p1–40002-p6Google Scholar
  59. Rosin B, Slovik M, Mitelman R, Rivlin-Etzion M, Haber SN, Israel Z, Vaadia E, Bergman H (2011) Closed-loop deep brain stimulation is superior in ameliorating parkinsonism. Neuron 72(2):370–384CrossRefPubMedGoogle Scholar
  60. Rubin JE, Terman D (2004) High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. J Comput Neurosci 16(3):211–235CrossRefPubMedGoogle Scholar
  61. Santaniello S, Fiengo G, Glielmo L, Grill WM (2011) Closed-loop control of deep brain stimulation: a simulation study. IEEE Trans Neural Syst Rehabil Eng 19(1):15–24CrossRefPubMedGoogle Scholar
  62. Schiff SJ (2010) Towards model-based control of Parkinson’s disease. Philos Trans R Soc A Math Phys Eng Sci 368(1918):2269–2308Google Scholar
  63. Seuret A, Gouaisbaut F, Fridman E (2013) Stability of systems with fast-varying delay using improved Wirtinger’s inequality. In: IEEE conference on decision and control (CDC), pp 946–951, Florence (Italy)Google Scholar
  64. Sharott A, Magill PJ, Harnack D, Kupsch A, Meissner W, Brown P (2005) Dopamine depletion increases the power and coherence of \(\beta \)-oscillations in the cerebral cortex and subthalamic nucleus of the awake rat. Eur J Neurosci 21(5):1413–1422CrossRefPubMedGoogle Scholar
  65. Sipahi R, Niculescu SI, Abdallah CT, Michiels W, Gu K (2011) Stability and stabilization of systems with time delay. IEEE Control Syst Mag 31(1):38–65CrossRefGoogle Scholar
  66. Strafella AP, Vanderwerf Y, Sadikot AF (2004) Transcranial magnetic stimulation of the human motor cortex influences the neuronal activity of subthalamic nucleus. Eur J Neurosci 20(8):2245–2249CrossRefPubMedGoogle Scholar
  67. Teleńczuk B, Destexhe A (2014) Local field potential, relationship to unit activity. In: Jaeger D, Jung R (eds) Encyclopedia of computational neuroscience. Springer, New York, pp 1–6CrossRefGoogle Scholar
  68. Tufail Y, Matyushov A, Baldwin N, Tauchmann ML, Georges J, Yoshihiro A, Tillery SIH, Tyler WJ (2010) Transcranial pulsed ultrasound stimulates intact brain circuits. Neuron 66(5):681–694CrossRefPubMedGoogle Scholar
  69. Tukhlina N, Rosenblum M, Pikovsky A, Kurths J (2007) Feedback suppression of neural synchrony by vanishing stimulation. Phys Rev E 75(1):011918CrossRefGoogle Scholar
  70. Tye KM, Deisseroth K (2012) Optogenetic investigation of neural circuits underlying brain disease in animal models. Nat Rev Neurosci 13(4):251–266CrossRefPubMedGoogle Scholar
  71. Wagenaar DA, Madhavan R, Pine J, Potter SM (2005) Controlling bursting in cortical cultures with closed-loop multi-electrode stimulation. J Neurosci 25(3):680–688PubMedCentralCrossRefPubMedGoogle Scholar
  72. Zaidel A, Spivak A, Grieb B, Bergman H, Israel Z (2010) Subthalamic span of oscillations predicts deep brain stimulation efficacy for patients with Parkinson’s disease. Brain 133(7):2007–2021CrossRefPubMedGoogle Scholar
  73. Zhang F, Wang L-P, Brauner M, Liewald JF, Kay K, Watzke N, Wood PG, Bamberg E, Nagel G, Gottschalk A et al (2007) Multimodal fast optical interrogation of neural circuitry. Nature 446(7136):633–639CrossRefPubMedGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ihab Haidar
    • 1
  • William Pasillas-Lépine
    • 1
    Email author
  • Antoine Chaillet
    • 1
  • Elena Panteley
    • 1
    • 5
  • Stéphane Palfi
    • 2
    • 3
    • 4
  • Suhan Senova
    • 2
    • 3
    • 4
  1. 1.Laboratoire des signaux et systèmesCNRS – CentraleSupélec – Univ. Paris SudGif-sur-YvetteFrance
  2. 2.AP-HP, Hôpital H. Mondor, Service de NeurochirurgieCréteilFrance
  3. 3.IMRB, Inserm, U955, Equipe 14CréteilFrance
  4. 4.Faculté de médecineUniversité Paris EstCréteilFrance
  5. 5.ITMO UniversitySaint PetersburgRussia

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