Delay-induced oscillations in Wilson and Cowan's model: An analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects

Biological Cybernetics

Volumn 107, Issue 3, 2013, Pages 289-308

  Pasillas Lépine, William ^a  

^a UNIV PARIS SUD   (France)

Author keywords

Basal ganglia; Delays; Firing rate models; Neural oscillations; Parkinson's disease; Wilson and Cowan's model

Indexed keywords

BASAL GANGLIA; DELAYS; LOCAL ASYMPTOTIC STABILITY; NEURAL OSCILLATIONS; OSCILLATORY BEHAVIORS; PARKINSON'S DISEASE; SPONTANEOUS OSCILLATIONS; SUFFICIENT CONDITIONS;

ASYMPTOTIC STABILITY; NEURAL NETWORKS;

NEURODEGENERATIVE DISEASES;

PARKINSONIA;

ARTICLE; BIOLOGICAL MODEL; BIOLOGICAL RHYTHM; FEEDBACK SYSTEM; GLOBUS PALLIDUS; HUMAN; NERVE CELL; NONLINEAR SYSTEM; PARKINSON DISEASE; PATHOLOGY; PHYSIOLOGY; SUBTHALAMIC NUCLEUS;

BIOLOGICAL CLOCKS; FEEDBACK, PHYSIOLOGICAL; GLOBUS PALLIDUS; HUMANS; MODELS, NEUROLOGICAL; NEURONS; NONLINEAR DYNAMICS; PARKINSON DISEASE; SUBTHALAMIC NUCLEUS;

EID: 84878563435     PISSN: 03401200     EISSN: 14320770     Source Type: Journal    
DOI: 10.1007/s00422-013-0549-3     Document Type: Article

References (37)

1. Asl F, Ulsoy A (2003) Analysis of a system of linear delay differential equations. ASME J Dyn Syst Meas Cont 125:215-223
2. Bhatia N, Szegö G (1970) Stability theory of dynamical systems, vol 161. Springer, Berlin
3. Boraud T, Brown P, Goldberg J, Graybiel A, Magill P (2005) Oscillations in the basal ganglia: the good, the bad, and the unexpected. In: et al JB (ed) The basal ganglia VIII, Advances in Behavioral Biology, vol 56, Springer, Berlin, pp 1-24
4. Coombes S, Laing C (2009) Delays in activity-based neural networks. Philos Trans R Soc A: Math, Phys Eng Sci 367(1891):1117-1129
5. Curtain R, Zwart H (1995) An introduction to infinite-dimensional linear systems theory, vol 21. Springer, Berlin
6. Curtu R, Ermentrout B (2001) Oscillations in a refractory neural net. J Math Biol 43(1):81-100 (Pubitemid 33772352)
7. Dayan P, Abbott LF (2001) Theoretical neuroscience: computational and mathematical modelling of neural systems. MIT Press, Cambridge
8. Desoer CA, Vidyasagar M (1975) Feedback systems: input-output properties. Academic Press, New York
9. Doyle J, Francis B, Tannenbaum A (1992) Feedback control theory. Macmillan, New York
10. Engelborghs K, Luzyanina T, Roose D (2002) Numerical bifurcation analysis of delay differential equations using dde-biftool. ACM Trans Math Software (TOMS) 28(1):1-21 (Pubitemid 135686769)
11. Ermentrout G, Terman D (2010) Mathematical foundations of neuroscience, vol 35. Springer, Berlin
12. Ermentrout GB, Cowan JD (1979a) A mathematical theory of visual hallucination patterns. Biol Cybern 34(3):137-150 (Pubitemid 10246235)
13. Ermentrout GB, Cowan JD (1979b) Temporal oscillations in neuronal nets. J Math Biol 7(3):265-280 (Pubitemid 9255718)
14. Gillies A, Willshaw D, Li Z (2002) Subthalamic-pallidal interactions are critical in determining normal and abnormal functioning of the basal ganglia. Proc R Soc London Ser B: Biol Sci 269(1491):545-551 (Pubitemid 34303988)
15. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer, Berlin
16. Hale J (1977) Theory of functional differential equations, vol 3. Springer, New York.
17. Khalil HK (2002) Nonlinear systems. Prentice Hall, Upper Saddle River
18. Kumar A, Cardanobile S, Rotter S, Aertsen A (2011) The role of inhibition in generating and controlling parkinsons disease oscillations in the basal ganglia. Front Syst Neurosci 5:86
19. 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):3567-3583
20. Ledoux E, Brunel N (2011) Dynamics of networks of excitatory and inhibitory neurons in response to time-dependent inputs. Front Comput Neurosci 5:25
21. Li Z, Hopfield JJ (1989) Modeling the olfactory bulb and its neural oscillatory processings. Biol Cybern 61(5):379-392 (Pubitemid 19225842)
22. Marsden J, McCracken M (1976) The Hopf bifurcation and its applications, vol 19. Springe, Berlin
23. Middleton R, Miller D (2007) On the achievable delay margin using lti control for unstable plants. IEEE Trans Autom Control 52(7): 1194-1207 (Pubitemid 47098932)
24. Monteiro LHA, Bussab MA (2002) Analytical results on a Wilson-Cowan neuronal network modified model. J Theor Biol 219(1): 83-91
25. Nambu A, Llinas R (1994) Electrophysiology of globus pallidus neurons in vitro. J Neurophysiol 72(3):1127-1139 (Pubitemid 24294830)
26. Nevado-Holgado AJ, Terry JR, Bogacz R (2010) Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network. J Neurosci 30(37):12340-12352
27. Nyquist H (1932) Regeneration theory. Bell Syst Techn J 11(1):126-147
28. Ogata K (2001) Modern control engineering. Prentice Hall, New York
29. Pavlides A, Hogan S, Bogacz R (2012) Improved conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network. Eur J Neurosci 36(2):2229-2239
30. Pinto DJ, Brumberg JC, Simons DJ, Ermentrout GB, Traub R (1996) A quantitative population model of whisker barrels: re-examining the Wilson-Cowan equations. J Comput Neurosc 3(3):247-264 (Pubitemid 26273238)
31. Plenz D, Kitai S (1999) A basal ganglia pacemaker formed by the subthalamic nucleus and external globus pallidus. Nature 400(6745):677-682, in the published version of this paper the second author's name is 'Kital' instead of 'Kitai'
32. Schuster H, Wagner P (1990) A model for neuronal oscillations in the visual cortex. Biol Cybern 64(1):77-82
33. Seung HS, Richardson TJ, Lagarias JC, Hopfield JJ (1997) Minimax and Hamiltonian dynamics of excitatory-inhibitory networks. Proceedings of advances in neural information processing systems, Denver
34. Sipahi R, Niculescu S, Abdallah C, Michiels W, Gu K (2011) Stability and stabilization of systems with time delay. IEEE Control Syst Mag 31(1):38-65
35. Wei J, Ruan S (1999) Stability and bifurcation in a neural network model with two delays. Phys D: Nonlinear Phenom 130(3-4):255-272 (Pubitemid 129587672)
36. Wilson C, Weyrick A, Terman D, Hallworth N, Bevan M (2004) A model of reverse spike frequency adaptation and repetitive firing of subthalamic nucleus neurons. J Neurophysiol 91(5):1963-1980 (Pubitemid 38500916)
37. Wilson HR, Cowan JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12(1):1-24