. Rauch, and less invasive motor-cortex stimulation, 2004.

R. L. Albin, A. B. Young, and J. B. Penney, The functional anatomy of basal ganglia disorders, Trends in neurosciences, vol.12, issue.10, pp.366-375, 1989.

, Allen Human Brain Atlas, 2010.

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological cybernetics, vol.27, issue.2, pp.77-87, 1977.

O. J. Andy, Thalamic stimulation for control of movement disorders, Stereotactic and Functional Neurosurgery, vol.46, issue.1-4, pp.107-111, 1983.

D. Angeli, A lyapunov approach to incremental stability properties, IEEE Transactions on Automatic Control, vol.47, issue.3, pp.410-421, 2002.

M. Arlotti, S. Marceglia, G. Foffani, J. Volkmann, A. M. Lozano et al., Eight-hours adaptive deep brain stimulation in patients with Parkinson disease, Neurology, vol.90, issue.11, pp.971-976, 2018.

M. Atmaca, Drug-induced impulse control disorders: a review, Current clinical pharmacology, vol.9, issue.1, pp.70-74, 2014.

I. Bar-gad, G. Heimer, Y. Ritov, and H. Bergman, Functional correlations between neighboring neurons in the primate globus pallidus are weak or nonexistent, Journal of Neuroscience, vol.23, issue.10, pp.4012-4016, 2003.

A. L. Benabid, S. Chabardes, J. Mitrofanis, and P. Pollak, Deep brain stimulation of the subthalamic nucleus for the treatment of Parkinson's disease, The Lancet Neurology, vol.8, issue.1, pp.67-81, 2009.

A. L. Benabid, P. Pollak, D. Hoffmann, C. Gervason, M. Hommel et al., Long-term suppression of tremor by chronic stimulation of the ventral intermediate thalamic nucleus, The Lancet, vol.337, issue.8738, pp.403-406, 1991.

A. L. Benabid, P. Pollak, A. Louveau, S. Henry, and J. De-rougemont, Combined (thalamotomy and stimulation) stereotactic surgery of the VIM thalamic nucleus for bilateral Parkinson disease, Stereotactic and Functional Neurosurgery, vol.50, issue.1-6, pp.344-346, 1987.

M. Beudel, S. Little, A. Pogosyan, K. Ashkan, T. Foltynie et al., Tremor reduction by deep brain stimulation is associated with gamma power suppression in Parkinson's disease. Neuromodulation: Technology at the Neural Interface, vol.18, pp.349-354, 2015.

C. Beurrier, Y. Ben-ari, and C. Hammond, Preservation of the direct and indirect pathways in an in vitro preparation of the mouse basal ganglia, Neuroscience, vol.140, issue.1, pp.77-86, 2006.
URL : https://hal.archives-ouvertes.fr/inserm-00484325

A. Beuter, J. Lefaucheur, and J. Modolo, Closed-loop cortical neuromodulation in Parkinson's disease: An alternative to deep brain stimulation, Clinical Neurophysiology, vol.125, issue.5, pp.874-885, 2014.

E. Bezard and S. Przedborski, A tale on animal models of Parkinson's disease, Movement Disorders, vol.26, issue.6, pp.993-1002, 2011.

A. Blenkinsop, S. Anderson, and K. Gurney, Frequency and function in the basal ganglia: the origins of beta and gamma band activity, The Journal of physiology, vol.595, issue.13, pp.4525-4548, 2017.

M. Boldrini, C. A. Fulmore, A. N. Tartt, L. R. Simeon, I. Pavlova et al., Human hippocampal neurogenesis persists throughout aging, Cell Stem Cell, vol.22, issue.4, pp.589-599, 2018.

E. Bossy-wetzel, R. Schwarzenbacher, and S. A. Lipton, Molecular pathways to neurodegeneration, Nature medicine, vol.10, issue.7s, p.2, 2004.

P. C. Bressloff, Spatiotemporal dynamics of continuum neural fields, vol.45, p.33001, 2011.

J. Brittain and P. Brown, Oscillations and the basal ganglia: motor control and beyond, Neuroimage, vol.85, pp.637-647, 2014.

P. Brown, A. Oliviero, P. Mazzone, A. Insola, P. Tonali et al., Dopamine dependency of oscillations between subthalamic nucleus and pallidum in Parkinson's disease, Journal of Neuroscience, vol.21, issue.3, pp.1033-1038, 2001.

J. Bugaysen, I. Bar-gad, and A. Korngreen, Continuous modulation of action potential firing by a unitary gabaergic connection in the globus pallidus in vitro, Journal of Neuroscience, vol.33, issue.31, pp.12805-12809, 2013.

R. Carron, A. Chaillet, A. Filipchuk, W. Pasillas-lepine, and C. Hammond, Closing the loop of deep brain stimulation, Frontiers in systems neuroscience, vol.7, pp.1-18, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01264998

T. Y. Chai and T. Zhang, A new model reference robust adaptive controller in the presence of unmodeled dynamics and bounded disturbances, Automatica, vol.30, issue.5, pp.865-869, 1994.

A. Chaillet, G. I. Detorakis, S. Palfi, and S. Senova, Robust stabilization of delayed neural fields with partial measurement and actuation, Automatica, vol.83, pp.262-274, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01522308

A. Chaillet, J. Or-lowski, P. , and P. , A relaxed Lyapunov-Krasovskii condition for global exponential stability of Lipschitz time-delay systems, Proc. IEEE Conf. on Decision and Control, 2019.

A. Chaillet and P. Pepe, Integral input-to-state stability of delay systems based on Lyapunov-Krasovskii functionals with point-wise dissipation rate, Proc. IEEE Conf. on Decision and Control, 2018.

A. Chaillet, P. Pepe, P. Mason, and Y. Chitour, Is a point-wise dissipation rate enough to show ISS for time-delay systems?, IFAC World Congress, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01493650

C. W. Christine, K. S. Bankiewicz, A. D. Van-laar, R. M. Richardson, B. Ravina et al., Magnetic resonance imaging-guided phase 1 trial of putaminal AADC gene therapy for Parkinson's disease, Annals of neurology, vol.85, issue.5, pp.704-714, 2019.

S. R. Cole, R. Van-der-meij, E. J. Peterson, C. De-hemptinne, P. A. Starr et al., Nonsinusoidal beta oscillations reflect cortical pathophysiology in Parkinson's disease, Journal of Neuroscience, vol.37, issue.18, pp.4830-4840, 2017.

S. Coombes, P. Beim-graben, R. Potthast, W. , and J. , Neural fields: theory and applications, 2014.

P. Dayan and L. F. Abbott, Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, 2000.

A. Destexhe and T. J. Sejnowski, The Wilson-Cowan model, 36 years later, Biological Cybernetics, vol.101, issue.1, pp.1-2, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00444899

G. I. Detorakis and A. Chaillet, Incremental stability of spatiotemporal delayed dynamics and application to neural fields, Proc. IEEE Conf. on Decision and Control, pp.5937-5942, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01813326

G. I. Detorakis, A. Chaillet, S. Palfi, and S. Senova, Closed-loop stimulation of a delayed neural fields model of parkinsonian STN-GPe network: a theoretical and computational study, Frontiers in Neuroscience, issue.237, p.9, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01251739

M. Di-volo, A. Romagnoni, C. Capone, and A. Destexhe, Biologically realistic mean-field models of conductance-based networks of spiking neurons with adaptation, Neural computation, vol.31, issue.4, pp.653-680, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02363362

R. D. Driver, Existence and stability of solutions of a delay-differential system, Arch. Rational Mech. Anal, vol.10, issue.1, pp.401-426, 1962.

X. Drouot, S. Oshino, B. Jarraya, L. Besret, H. Kishima et al., Functional recovery in a primate model of Parkinson's disease following motor cortex stimulation, Neuron, vol.44, issue.5, pp.769-778, 2004.

E. M. Dunn and M. M. Lowery, Simulation of PID control schemes for closed-loop deep brain stimulation, 6th International IEEE/EMBS Conference on Neural Engineering (NER), pp.1182-1185, 2013.

E. M. Dunn and M. M. Lowery, A model of the cortico-basal ganglia network and local field potential during deep brain stimulation, 7th International IEEE/EMBS Conference on Neural Engineering (NER), pp.848-851, 2015.

H. Ehringer and O. Hornykiewicz, Verteilung Von Noradrenalin Und Dopamin (3-Hydroxytyramin) Im Gehirn Des Menschen Und Ihr Verhalten Bei Erkrankungen Des Extrapyramidalen Systems, Klinische Wochenschrift, vol.38, issue.24, pp.1236-1239, 1960.

R. Eitan, H. Bergman, and Z. Israel, Closed-loop deep brain stimulation for Parkinson's disease, Surgery for Parkinson's Disease, pp.131-149, 2019.

A. Ernst, K. Alkass, S. Bernard, M. Salehpour, S. Perl et al., Neurogenesis in the striatum of the adult human brain, vol.156, pp.1072-1083, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00952021

A. Eusebio, W. Thevathasan, L. Doyle-gaynor, A. Pogosyan, E. Bye et al., Deep brain stimulation can suppress pathological synchronisation in parkinsonian patients, Journal of neurology, vol.82, issue.5, pp.569-573, 2011.

G. Faye and O. Faugeras, Some theoretical and numerical results for delayed neural field equations, Physica D: Nonlinear Phenomena, vol.239, issue.9, pp.561-578, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00847433

F. Forni and R. Sepulchre, A differential Lyapunov framework for contraction analysis, IEEE Transactions on Automatic Control, vol.59, issue.3, pp.614-628, 2013.

A. L. Fradkov, I. V. Miroshnik, and V. O. Nikiforov, Nonlinear and adaptive control of complex systems, 1999.

A. M. Graybiel, The basal ganglia, Current biology, vol.10, issue.14, pp.509-511, 2000.

W. Hahn, Stability of motion, 1967.

I. Haidar, P. Mason, and M. Sigalotti, Converse Lyapunov-Krasovskii theorems for uncertain retarded differential equations, Automatica, vol.62, pp.263-273, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00924252

I. Haidar, W. Pasillas-lépine, A. Chaillet, E. Panteley, S. Palfi et al., , 2016.

, Closed-loop firing rate regulation of two interacting excitatory and inhibitory neural populations of the basal ganglia, Biological Cybernetics, vol.110, issue.1, pp.55-71

J. K. Hale and S. M. Verduyn-lunel, Introduction to functional differential equations, Applied Mathematical Sciences, vol.99, 1993.

C. Hammond, H. Bergman, and P. Brown, Pathological synchronization in Parkinson's disease: networks, models and treatments, Trends in Neurosciences, vol.30, issue.7, pp.357-364, 2007.

X. Han and E. S. Boyden, Multiple-color optical activation, silencing, and desynchronization of neural activity, with single-spike temporal resolution, PLoS ONE, vol.2, issue.3, p.299, 2007.

A. B. Holt and T. I. Netoff, Origins and suppression of oscillations in a computational model of Parkinson's disease, Journal of Computational Neuroscience, vol.37, issue.3, pp.505-521, 2014.

L. Hsu and R. Costa, Bursting phenomena in continuous-time adaptive systems with a ?-modification, IEEE Transactions on Automatic Control, vol.32, issue.1, pp.84-86, 1987.

W. D. Hutchison, J. O. Dostrovsky, J. R. Walters, R. Courtemanche, T. Boraud et al., Neuronal oscillations in the basal ganglia and movement disorders: evidence from whole animal and human recordings, Journal of Neuroscience, vol.24, issue.42, pp.9240-9243, 2004.

A. Ichikawa, Equivalence of L p stability and exponential stability for a class of nonlinear semigroups. Nonlinear analysis, theory, methods, and applications, vol.8, pp.805-815, 1984.

A. Ilchmann and E. P. Ryan, Universal ?-tracking for non-linearly-perturbed systems in the presence of noise, Automatica, vol.30, pp.337-346, 1994.

P. Ioannou and B. Fidan, Adaptive control tutorial. Advances in Design and Control, 2006.

P. Ioannou and K. Tsakalis, A robust direct adaptive controller, IEEE Transactions on Automatic Control, vol.31, issue.11, pp.1033-1043, 1986.

P. A. Ioannou and P. V. Kokotovic, Instability analysis and improvement of robustness of adaptive control, Automatica, vol.20, issue.5, pp.583-594, 1984.

H. Ito, P. Pepe, and Z. P. Jiang, A small-gain condition for iISS of interconnected retarded systems based on Lyapunov-Krasovskii functionals, Automatica, vol.46, issue.10, pp.1646-1656, 2010.

J. Jankovic, Parkinson's disease: clinical features and diagnosis, neurosurgery & psychiatry, vol.79, issue.4, pp.368-376, 2008.

H. Jin, A. Kanthasamy, V. Anantharam, and A. G. Kanthasamy, Biomarkers of Parkinson's disease, Biomarkers in Toxicology, pp.895-909, 2019.

G. Kang and M. M. Lowery, Interaction of oscillations, and their suppression via deep brain stimulation, in a model of the cortico-basal ganglia network, IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.21, issue.2, pp.244-253, 2013.

H. G. Kankanamalage, Y. Lin, W. , and Y. , On Lyapunov-Krasovskii characterizations of input-to-output stability, Proceedings of the IFAC World Congress, pp.1-6, 2017.

I. Karafyllis and Z. P. Jiang, Stability and stabilization of nonlinear systems, Communications and Control Engineering Series, 2011.

I. Karafyllis, P. Pepe, and Z. P. Jiang, Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations, European Journal of Control, vol.14, issue.6, pp.516-536, 2008.

I. Karafyllis, P. Pepe, and Z. P. Jiang, Input-to-Output Stability for systems described by retarded functional differential equations, European Journal of Control, vol.14, issue.6, pp.539-555, 2008.

S. Katzner, I. Nauhaus, A. Benucci, V. Bonin, D. L. Ringach et al., Local origin of field potentials in visual cortex, Neuron, vol.61, issue.1, pp.35-41, 2009.

H. K. Khalil, Nonlinear systems, 2002.

N. N. Krasovskii, Stability of motion, 1963.

A. A. Kühn, A. Kupsch, G. H. Schneider, and P. Brown, Reduction in subthalamic 8-35 Hz oscillatory activity correlates with clinical improvement in Parkinson's disease, European Journal of Neuroscience, vol.23, issue.7, pp.1956-1960, 2006.

A. A. Kühn, D. Williams, A. Kupsch, P. Limousin, M. Hariz et al., Event-related beta desynchronization in human subthalamic nucleus correlates with motor performance, Brain, vol.127, issue.4, pp.735-746, 2004.

A. Kumar, S. Cardanobile, S. Rotter, A. , and A. , The role of inhibition in generating and controlling Parkinson's disease oscillations in the basal ganglia, Frontiers in Systems Neuroscience, p.5, 2011.

V. Lakshmikantham and X. Liu, Stability analysis in terms of two measures, 1993.

P. Lamberti, S. Armenise, V. Castaldo, M. De-mari, G. Iliceto et al., Freezing gait in Parkinson's disease, European neurology, vol.38, issue.4, pp.297-301, 1997.

J. L. Lanciego, N. Luquin, and J. A. Obeso, Functional neuroanatomy of the basal ganglia. Cold Spring Harbor Perspectives in Medicine, vol.2, p.9621, 2012.

D. P. Lindorff and R. L. Carroll, Survey of adaptive control using Liapunov design, International Journal of Control, vol.18, issue.5, pp.897-914, 1973.

S. Little, M. Beudel, L. Zrinzo, T. Foltynie, P. Limousin et al., Bilateral adaptive deep brain stimulation is effective in Parkinson's disease, J Neurol Neurosurg Psychiatry, vol.87, issue.7, pp.717-721, 2016.

S. Little and P. Brown, The functional role of beta oscillations in Parkinson's disease. Parkinsonism and Related Disorders, vol.20, 2014.

S. Little, A. Pogosyan, S. Neal, B. Zavala, L. Zrinzo et al., , 2013.

, Adaptive deep brain stimulation in advanced Parkinson disease, Annals of neurology, vol.74, issue.3, pp.449-457

S. Little, A. Pogosyan, S. Neal, L. Zrinzo, M. Hariz et al., Controlling Parkinson's disease with adaptive deep brain stimulation, Journal of Visualized Experiments, issue.89, p.51403, 2014.

C. Liu, J. Wang, H. , L. Fietkiewicz, C. Loparo et al., Modeling and analysis of beta oscillations in the basal ganglia, IEEE Transactions on Neural Networks and Learning Systems, vol.29, issue.5, pp.1864-1875, 2018.

W. Lohmiller and J. E. Slotine, On contraction analysis for non-linear systems, Automatica, vol.34, issue.6, pp.683-696, 1998.

A. M. Lyapunov, Probléme général de la stabilité du mouvement. (French translation of a Russian paper dated 1892). Les Annales de la Faculté des, Sciences de Toulouse, vol.2, issue.9, pp.27-247, 1907.

C. D. Mackinnon, R. M. Webb, P. Silberstein, S. Tisch, P. Asselman et al., Stimulation through electrodes implanted near the subthalamic nucleus activates projections to motor areas of cerebral cortex in patients with Parkinson's disease, European Journal of Neuroscience, vol.21, issue.5, pp.1394-1402, 2005.

P. J. Magill, J. P. Bolam, and M. D. Bevan, Dopamine regulates the impact of the cerebral cortex on the subthalamic nucleus-globus pallidus network, Neuroscience, vol.106, issue.2, pp.313-330, 2001.

E. Marani, T. Heida, E. A. Lakke, and K. G. Usunoff, The subthalamic nucleus: Part I: Development, cytology, topography and connections, vol.198, 2008.

I. Mareels and J. W. Polderman, Adaptive Systems, 1996.

J. L. Massera, Contributions to stability theory, pp.182-206, 1956.

F. Mazenc, H. Ito, P. , and P. , Construction of Lyapunov functionals for coupled differential and continuous time difference equations, Proc. IEEE Conf. on Decision and Control, pp.2245-2250, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01088235

M. M. Mccarthy, C. Moore-kochlacs, X. Gu, E. S. Boyden, X. Han et al., Striatal origin of the pathologic beta oscillations in Parkinson's disease, Proceedings of the National Academy of Sciences of the United States of America, vol.108, pp.11620-11625, 2011.

C. C. Mcintyre, S. Mori, D. L. Sherman, N. V. Thakor, and J. L. Vitek, Electric field and stimulating influence generated by deep brain stimulation of the subthalamic nucleus, Clinical Neurophysiology, vol.115, issue.3, pp.589-595, 2004.

C. C. Mcintyre, M. Savasta, . Kerkerian-le, L. Goff, and J. L. Vitek, Uncovering the mechanism(s) of action of deep brain stimulation: activation, inhibition, or both, Clinical Neurophysiology, vol.115, issue.6, pp.1239-1248, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00306742

A. Mironchenko and F. Wirth, Global converse Lyapunov theorems for infinitedimensional systems, Proceedings of 10th IFAC Symposium on Nonlinear Control Systems, 2016.

T. J. Moore, J. Glenmullen, and D. R. Mattison, Reports of pathological gambling, hypersexuality, and compulsive shopping associated with dopamine receptor agonist drugs, JAMA internal medicine, vol.174, issue.12, pp.1930-1933, 2014.

K. S. Narendra and A. M. Annaswamy, A new adaptive law for robust adaptation without persistent excitation, IEEE Transactions on Automatic Control, vol.32, issue.2, pp.134-145, 1987.

K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems, 2005.

K. S. Narendra, S. S. Tripathi, G. Lueders, and P. Kudva, Adaptive control using Lyapunov's direct method, 1971.

N. Holgado, A. J. Terry, J. R. Bogacz, and R. , Conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network, Journal of Neuroscience, issue.37, p.30, 2010.

S. I. Niculescu, Delay effects on stability: A robust control approach, Lecture Notes in Control and Information Sciences, vol.269, 2001.
URL : https://hal.archives-ouvertes.fr/hal-02183198

J. A. Obeso, C. W. Olanow, and J. G. Nutt, Levodopa motor complications in Parkinson's disease, Trends in Neurosciences, vol.23, issue.10, pp.2-7, 2000.

M. S. Okun, Deep-brain stimulation for Parkinson's disease, New England Journal of Medicine, vol.367, issue.16, pp.1529-1538, 2012.

J. Or-lowski, A. Chaillet, and M. Sigalotti, Counterexample to a Lyapunov condition for uniform asymptotic partial stability, IEEE Control Systems Letters, vol.4, issue.2, pp.397-401, 2019.

J. Or-lowski, A. Chaillet, M. Sigalotti, and A. Destexhe, Adaptive scheme for pathological oscillations disruption in a delayed neuronal population model, Proc. IEEE Conf. on Decision and Control, 2018.

R. Ortega and Y. Tang, Robustness of adaptive controllers -a survey, Automatica, vol.25, issue.5, pp.651-677, 1989.

A. S. Oziraner and V. V. Rumiantsev, The method of Liapunov functions in the stability problem for motion with respect to a part of the variables, Prikl. Mat. Meh, vol.36, pp.364-384, 1972.

S. Palfi, J. M. Gurruchaga, H. Lepetit, K. Howard, G. S. Ralph et al., Long-term follow-up of a phase I/II study of ProSavin, a lentiviral vector gene therapy for Parkinson's disease, Human Gene Therapy Clinical Development, vol.29, issue.3, pp.148-155, 2018.

M. Parastarfeizabadi and A. Z. Kouzani, Advances in closed-loop deep brain stimulation devices, Journal of neuroengineering and rehabilitation, vol.14, issue.1, p.79, 2017.

J. Parkinson, An Essay on the Shaking Palsy, 1817.

W. Pasillas-lépine, Delay-induced oscillations in Wilson and Cowan's model: An analysis of the subthalamo-pallidal feedback loop in healthy and parkinsonian subjects, Biological Cybernetics, vol.107, issue.3, pp.289-308, 2013.

A. Pavlides, S. John-hogan, and R. Bogacz, Improved conditions for the generation of beta oscillations in the subthalamic nucleus-globus pallidus network, European Journal of Neuroscience, vol.36, issue.2, pp.2229-2239, 2012.

A. Pavlov, N. Van-de-wouw, and H. Nijmeijer, Frequency response functions for nonlinear convergent systems, IEEE Transactions on Automatic Control, vol.52, issue.6, pp.1159-1165, 2007.

A. V. Pavlov, N. Van-de-wouw, and H. Nijmeijer, Uniform output regulation of nonlinear systems: a convergent dynamics approach, Systems & Control: Foundations & Applications, 2006.

P. Pepe, On Liapunov-Krasovskii functionals under Carathéodory conditions, Automatica, vol.43, issue.4, pp.701-706, 2007.

P. Pepe, The problem of the absolute continuity for Lyapunov-Krasovskii functionals, IEEE Transactions on Automatic Control, vol.52, issue.5, pp.953-957, 2007.

P. Pepe and Z. P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems & Control Letters, vol.55, issue.12, pp.1006-1014, 2006.

P. Pepe and I. Karafyllis, Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form, International Journal of Control, vol.86, issue.2, pp.232-243, 2013.

J. S. Perlmutter and J. W. Mink, Deep brain stimulation, Annu. Rev. Neurosci, vol.29, pp.229-257, 2006.

B. Peterson and K. Narendra, Bounded error adaptive control, IEEE Transactions on Automatic Control, vol.27, issue.6, pp.1161-1168, 1982.

M. Picillo and A. Fasano, Recent advances in essential tremor: Surgical treatment, Parkinsonism & Related Disorders, vol.22, pp.171-175, 2016.

D. Piña-fuentes, S. Little, M. Oterdoom, S. Neal, A. Pogosyan et al., Adaptive DBS in a Parkinson's patient with chronically implanted DBS: a proof of principle, Movement disorders: official journal of the Movement Disorder Society, vol.32, issue.8, p.1253, 2017.

D. Plenz and S. T. Kital, A basal ganglia pacemaker formed by the subthalamic nucleus and external globus pallidus, Nature, vol.400, issue.6745, pp.677-682, 1999.

R. B. Postuma, D. Berg, M. Stern, W. Poewe, C. W. Olanow et al., MDS clinical diagnostic criteria for Parkinson's disease, Movement Disorders, vol.30, issue.12, pp.1591-1601, 2015.

T. Pringsheim, N. Jette, A. Frolkis, and T. D. Steeves, The prevalence of Parkinson's disease: A systematic review and meta-analysis, Movement disorders, vol.29, issue.13, pp.1583-1590, 2014.

A. Priori, G. Foffani, A. Pesenti, F. Tamma, A. M. Bianchi et al., Rhythm-specific pharmacological modulation of subthalamic activity in Parkinson's disease, Experimental Neurology, vol.189, issue.2, pp.369-379, 2004.

K. Pyragas, V. Pyragas, I. Z. Kiss, and J. L. Hudson, Adaptive control of unknown unstable steady states of dynamical systems, Plasmas, Fluids, and Related Interdisciplinary Topics, vol.70, issue.2, p.12, 2004.

F. Rauch, K. Schwabe, and J. K. Krauss, Effect of deep brain stimulation in the pedunculopontine nucleus on motor function in the rat 6-hydroxydopamine Parkinson model, Behavioural Brain Research, vol.210, issue.1, pp.46-53, 2010.

M. Rosa, M. Arlotti, G. Ardolino, F. Cogiamanian, S. Marceglia et al., Adaptive deep brain stimulation in a freely moving Parkinsonian patient, Movement Disorders, vol.30, issue.7, pp.1003-1005, 2015.

B. Rosin, M. Slovik, R. Mitelman, M. Rivlin-etzion, S. N. Haber et al., Closed-loop deep brain stimulation is superior in ameliorating parkinsonism, Neuron, vol.72, issue.2, pp.370-384, 2011.

J. E. Rubin and D. Terman, High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model, Journal of computational neuroscience, vol.16, issue.3, pp.211-235, 2004.

B. S. Rüffer, N. Van-de-wouw, and M. Mueller, Convergent systems vs. incremental stability, Systems & Control Letters, vol.62, issue.3, pp.277-285, 2013.

A. R. Sadek, P. J. Magill, and J. P. Bolam, A single-cell analysis of intrinsic connectivity in the rat globus pallidus, Journal of Neuroscience, vol.27, issue.24, pp.6352-6362, 2007.

G. Sadoc, G. Le-masson, B. Foutry, Y. Le-franc, Z. Piwkowska et al., Re-creating in vivo-like activity and investigating the signal transfer capabilities of neurons: Dynamic-clamp applications using real-time neuron, Dynamic-Clamp, pp.287-320, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00378069

T. H. Sanders and D. Jaeger, Optogenetic stimulation of cortico-subthalamic projections is sufficient to ameliorate bradykinesia in 6-ohda lesioned mice, Neurobiology of disease, vol.95, pp.225-237, 2016.

S. Santaniello, G. Fiengo, L. Glielmo, and W. M. Grill, Closed-loop control of deep brain stimulation: a simulation study, IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol.19, issue.1, pp.15-24, 2010.

S. Santaniello, M. M. Mccarthy, E. B. Montgomery, J. T. Gale, N. Kopell et al., Therapeutic mechanisms of high-frequency stimulation in Parkinson's disease and neural restoration via loop-based reinforcement, Proceedings of the National Academy of Sciences, vol.112, issue.6, pp.586-595, 2015.

F. J. Santos, R. M. Costa, and F. Tecuapetla, Stimulation on demand: Closing the loop on deep brain stimulation, vol.72, pp.197-198, 2011.

S. A. Shah, G. Tinkhauser, C. C. Chen, S. Little, and P. Brown, Parkinsonian tremor detection from subthalamic nucleus local field potentials for closed-loop deep brain stimulation, Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, pp.2320-2324, 2018.

P. Silberstein, A. Pogosyan, A. A. Kühn, G. Hotton, S. Tisch et al., Cortico-cortical coupling in Parkinson's disease and its modulation by therapy, Brain, vol.128, issue.6, pp.1277-1291, 2005.

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Transactions on Automatic Control, vol.34, issue.4, pp.435-443, 1989.

E. D. Sontag, Input to state stability: Basic concepts and results, Lecture Notes in Mathematics, pp.163-220, 2008.

E. D. Sontag and Y. Wang, Notions of input-to-output stability, Systems & Control Letters, vol.38, pp.235-248, 1999.

E. D. Sontag and Y. Wang, Lyapunov characterizations of input to output stability, SIAM Journal on Control and Optimization, vol.39, pp.226-249, 2001.

S. F. Sorrells, M. F. Paredes, A. Cebrian-silla, K. Sandoval, D. Qi et al., Human hippocampal neurogenesis drops sharply in children to undetectable levels in adults, Nature, vol.555, issue.7696, p.377, 2018.

K. L. Spalding, O. Bergmann, K. Alkass, S. Bernard, M. Salehpour et al., Dynamics of hippocampal neurogenesis in adult humans, Cell, vol.153, issue.6, pp.1219-1227, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00839513

P. Tass, D. Smirnov, A. Karavaev, U. Barnikol, T. Barnikol et al., The causal relationship between subcortical local field potential oscillations and parkinsonian resting tremor, Journal of Neural Engineering, vol.7, issue.1, p.16009, 2010.

A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Transactions on Automatic Control, vol.43, issue.7, pp.960-964, 1998.

A. R. Teel and L. Praly, A smooth Lyapunov function from a class-K L estimate involving two positive semidefinite functions, ESAIM Control Optim. Calc. Var, vol.5, pp.313-367, 2000.

D. Terman, J. E. Rubin, A. C. Yew, and C. J. Wilson, Activity patterns in a model for the subthalamopallidal network of the basal ganglia, Journal of Neuroscience, vol.22, issue.7, pp.2963-2976, 2002.

E. Tolosa, G. Wenning, and W. Poewe, The diagnosis of Parkinson's disease, The Lancet Neurology, vol.5, issue.1, pp.75-86, 2006.

S. J. Van-albada, T. , G. R. Drysdale, P. M. , R. et al., Mean-field modeling of the basal ganglia-thalamocortical system. II, Journal of Theoretical Biology, vol.257, issue.4, pp.664-688, 2009.

V. Vedam-mai, C. Rodgers, A. Gureck, M. Vincent, G. Ippolito et al., Deep brain stimulation associated gliosis: A post-mortem study, Parkinsonism & related disorders, vol.54, pp.51-55, 2018.

A. Velisar, J. Syrkin-nikolau, Z. Blumenfeld, M. H. Trager, M. F. Afzal et al., Dual threshold neural closed loop deep brain stimulation in Parkinson disease patients, 2019.

R. Veltz and O. Faugeras, Stability of the stationary solutions of neural field equations with propagation delays, The Journal of Mathematical Neuroscience, vol.1, issue.1, p.1, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00784425

V. I. Vorotnikov, Partial stability and control, 1998.

V. I. Vorotnikov, Partial stability and control: The state-of-the-art and development prospects. Automation and Remote Control, vol.66, pp.511-561, 2005.

H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophysical Journal, vol.12, issue.1, pp.1-24, 1972.

H. R. Wilson and J. D. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, vol.13, pp.55-80, 1973.

N. Yamawaki, I. M. Stanford, S. D. Hall, and G. L. Woodhall, Pharmacologically induced and stimulus evoked rhythmic neuronal oscillatory activity in the primary motor cortex in vitro, Neuroscience, vol.151, issue.2, pp.386-395, 2008.

N. Yeganefar, P. Pepe, and M. Dambrine, Input-to-State Stability of time-delay systems: a link with exponential stability, IEEE Trans. Automatic Control, vol.53, issue.6, pp.1526-1531, 2008.