T. Morris-lecal-model and .. ,

A. For-bursting-immature-sacs and .. , , p.20

D. Deriving-sahp, 22 Saturated calmodulin production 22 Binding of calmodulin to SK terminals 23 Calcium concentration, 23 Calcium concentration dynamics. . . . . . . . . . . . . . . 23

, Full set of equations for the single SAC dynamics, p.25

R. Equations and . Multi-time-scale-analysis........, , p.25

F. Parameters, , p.25

U. , 25 Calibrating parameters from experiments, p.26

A. and .. , , p.26

P. , , p.26

E. Comparison-with, , p.26

.. Bifurcations and N. , 28 Slow-fast analysis 28 Dynamical changes with respect to parameters variations. 28 Dynamically driven bursting, 29 Robustness with respect to parameters variations. . . . . 33 CONTENTS Fast dynamics of V, p.111

.. Slow, , p.112

V. Medium-scale, , p.112

.. Equation-of-transport-for-ach, , p.112

. Ach-production-on-the-lower-branch........, , p.113

. Ach-production-on-the-upper-branch........, , p.113

T. , , p.113

?. Variable, , p.113

A. Piecewise-linear, , p.114

.. Laplacian-approximation, 114 Singular diffusion, p.114

C. , , p.115

D. , 115 Single neuron dynamics in the presence of a tunable sAHP and Ach currents, p.115

, Confronting our model to experimental recordings 119

, The role of potassium channels in waves characteristics, p.119

.. , Exploring the effect of the cholinergic transmission on the spatio-temporal patterns of stage II retinal waves, p.125

.. Objective-of-the-experiment, , p.125

P. , , p.125

.. Tissue-preparation, 125 Recordings of retinal waves with MEA of 256 electrodes . 125 Analysis of the MEA recordings, p.125

, The effect of hexametonium in early stage II retinal waves, p.126

, The effect of atropine in early stage II retinal waves, p.129

.. Power, , p.133

D. Conclusion, , p.134

, 7 Conclusions and Perspectives 137

.. Reflecting-on-possible-theoretical-extensions, 137 Synaptic coupling versus volume diffusion, p.137

.. , The role of variability in SACs on waves generation. . . . 138 Extending our model towards a generic dynamical system for retinal waves, p.139

, Can we use retinal waves to restore plasticity in pathological retinas, p.139

D. Karvouniari, L. Gil, O. Marre, S. Picaud, and B. , Cessac A biophysical model explains the spontaneous bursting behavior in the developing retina Nature Scientific Reports, under review, 2018.

D. Karvouniari, L. Gil, O. Marre, S. Picaud, and B. , Cessac Following stage II retinal waves with a biophysical model Bernstein Conference, p.2017

D. Karvouniari, L. Gil, O. Marre, S. Picaud, and B. , Cessac Mathematical and experimental studies of retinal waves ICMNS Conference, selected talk, 2016.

A. Ozaita, J. Petit-jacques, B. Volgy, C. S. Ho, H. Joho et al., A Unique Role for Kv3 Voltage-Gated Potassium Channels in Starburst Amacrine Cell Signaling in Mouse Retina, Journal of Neuroscience, vol.24, issue.33, pp.337335-7343, 2004.
DOI : 10.1523/JNEUROSCI.1275-04.2004

J. Touboul, O. Faugeras, and B. Rudy, The spikes trains probability distributions: A stochastic calculus approach, Journal of Physiology-Paris, vol.101, issue.1-3
DOI : 10.1016/j.jphysparis.2007.10.008

S. Fried and M. ,

T. Nch and F. Werblin, Mechanisms and circuitry underlying directional selectivity in the retina, Nature, 2002.

K. Tsumoto, H. Kitajima, T. Yoshinaga, K. Aihara, and H. Kawakami, Bifurcations in Morris???Lecar neuron model, Neurocomputing, vol.69, issue.4-6, pp.293-316, 2006.
DOI : 10.1016/j.neucom.2005.03.006

E. Marder and J. M. Goaillard, Variability, compensation and homeostasis in neuron and network function, Nature Reviews Neuroscience, vol.15, issue.7, pp.563-574, 2006.
DOI : 10.1016/j.neuint.2005.12.029

H. Xu, T. J. Burbridge, M. Ye, M. Chen, X. X. Ge et al., Retinal Wave Patterns Are Governed by Mutual Excitation among Starburst Amacrine Cells and Drive the Refinement and Maintenance of Visual Circuits, The Journal of Neuroscience, vol.36, issue.13, pp.36-3871, 2016.
DOI : 10.1523/JNEUROSCI.3549-15.2016

S. Firth, Retinal waves: mechanisms and function in visual system development Cell Calcium, 2004.

H. J. Abel, J. C. Lee, J. C. Callaway, and R. C. Foehring, Relationships between intracellular neocortical pyramidal neurons calcium and afterhyperpolarizations in neocortical pyramidal neurons, J Neurophysiol, 2004.

B. Lansdell, K. Ford, and J. Kutz, A Reaction-Diffusion Model of Cholinergic Retinal Waves, PLoS Computational Biology, vol.19, issue.12, pp.1-14, 2014.
DOI : 10.1371/journal.pcbi.1003953.s006

C. Morris and H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal, vol.35, issue.1, pp.193-213, 1981.
DOI : 10.1016/S0006-3495(81)84782-0

B. Cessac and . Blanchard-ph,

T. , Lyapunov exponents and transport in the Zhang model of Self-Organized Criticality, Phys. Rev. E, vol.64, p.16133, 2001.

J. Zhou and D. Zhao, Coordinated Transitions in Neurotransmitter Systems for the Initiation and Propagation of Spontaneous Retinal Waves, The Journal of Neuroscience, vol.20, issue.17, pp.6570-6577, 2000.
DOI : 10.1523/JNEUROSCI.20-17-06570.2000

M. Kardar, G. Parisi, and Y. , Zhang Dynamic Scaling of Growing Interfaces Phys, Rev. Lett, vol.56, issue.889 3, 1986.

P. Bak, C. Tang, and K. , noise, Physical Review Letters, vol.13, issue.4, p.381, 1987.
DOI : 10.1103/PhysRevB.13.556

O. Daniel and R. F. Cajueiro, Andrade Controlling self-organized criticality in sandpile models, Phys. Rev. E, vol.81, p.15102, 2010.

I. Vanessa, D. Pinto-thesis, . Homeobox, . Regulation, . Crx et al., , 2015.

M. Djilas, Pharmacologically induced wave-like activity in the adult retina IOVS, p.2012

M. Tauchi and R. H. Masland, The shape and arrangement of the cholinergic neurons in the rabbit retina Proc, 1984.

D. Goodman and R. Brette, The Brian simulator Front Neurosci, 2009.

S. De-lima, Full-length axon regeneration in the adult mouse optic nerve and partial recovery of simple visual behaviors, Proceedings of the National Academy of Sciences, vol.45, issue.12
DOI : 10.1167/iovs.04-0541

M. Sarter, V. Parikh, and W. Howe, Phasic acetylcholine release and the volume transmission hypothesis:time to move on, Nature Reviews. Neuroscience, 2009.

W. R. Taylor and R. G. , Smith The role of starburst amacrine cells in visual signal processing Vis Neurosci, 2012.

W. R. Taylor and R. Smith, Exp Eye Res, vol.33, issue.3, pp.315-347, 1981.

M. Mezard, G. Parisi, and M. A. , Virasoro Spin-glass theory and beyond World scientific Singapore, 1987.

, Starburst Amacrine Cells http://wiki.eyewire.org [29] Robert Clewley. Hybrid models and biological model reduction with pydstool, PLoS Computational Biology, 2012.

K. J. Ford, A. L. Felix, and M. B. Feller, Cellular Mechanisms Underlying Spatiotemporal Features of Cholinergic Retinal Waves The Journal of Neuroscience, pp.850-863, 2012.

E. Sernagor, S. J. Eglen, O. Donovan, and M. J. , t Differential Effects of Acetylcholine and Glutamate Blockade on the Spatiotemporal Dynamics of Retinal Waves, The Journal of Neuroscience, vol.20, issue.2, p.56, 2000.
DOI : 10.1523/JNEUROSCI.20-02-j0004.2000

J. Gjorgjieva, J. F. Evers, and S. J. , Eglen Homeostatic Activity-Dependent Tuning of Recurrent Networks for Robust Propagation of, Activity The Journal of Neuroscience, vol.20, p.56, 2000.

R. S. Mackay and C. , Tresser Transition to topological chaos for circle maps Physica D, 1986.

D. Butts, C. Feller, and . Shatz, DS Rokhsar Retinal waves are governed by collective network properties, Journal of Neuroscience, vol.1, pp.9-9

M. Iwata and S. , Shinichi Theoretical analysis for critical fluctuations of relaxation trajectory near a saddle-node bifurcation Phys, Rev. E, vol.82, p.11127, 2010.

M. B. Feller, D. A. Butts, H. L. Aaron, D. S. Rokhsar, and C. J. Shatz, Dynamic Processes Shape Spatiotemporal Properties of Retinal Waves, Neuron, vol.19, issue.2, pp.293-306, 1997.
DOI : 10.1016/S0896-6273(00)80940-X

K. J. Ford and M. B. Feller, Abstract, Visual Neuroscience, vol.10, issue.01, pp.61-71
DOI : 10.1016/j.neuron.2010.01.035

URL : https://hal.archives-ouvertes.fr/hal-01375359

B. Keith, . Godfrey, V. Nicholas, and . Swindale, Retinal wave behavior through activity-dependent refractory periods, PLoS Comput Biol, vol.3, issue.11, p.245, 2007.

A. Pikovsky and M. Scholarpedia, , 2007.

N. L. Lassignal and A. R. Martin, Effect of acetylcholine on postjunctional membrane permeability in eel electroplaque, The Journal of General Physiology, vol.70, issue.1, pp.23-36, 1977.
DOI : 10.1085/jgp.70.1.23

Z. Liu, J. Golowasch, E. Marder, and L. F. , Abbott A Model Neuron with Activity- Dependent Conductances Regulated by Multiple Calcium Sensors The Journal of Neuroscience, 1998.

W. Haq, Synaptic remodeling generates synchronous oscillations in the degenerated outer mouse retina Front Neural Circuits, 2014.

M. Graupner, A theory of Plasma Membrane calcium pump function and its consequences for presynaptic calcium dynamics, 2003.

M. Graupner, F. Erler, and M. Meyer-hermann, A Theory of Plasma Membrane Calcium Pump Stimulation and Activity, Journal of Biological Physics, vol.263, issue.2, pp.183-206, 2005.
DOI : 10.1042/bj3310763

J. Guckenheimer, . Ph, and . Holmes, Non linear oscillations, dynamical systems, and bifurcation of vector fields, 1983.

P. Bak, K. Chen, and C. Tang, A forest-fire model and some thoughts on turbulence, Physics Letters A, vol.147, issue.5-6, pp.297-300
DOI : 10.1016/0375-9601(90)90451-S

B. Drossel and F. Schwabl, Self-organized critical forest-fire model, Physical Review Letters, vol.40, issue.11, pp.1629-1632
DOI : 10.1103/PhysRevB.40.7425

V. I. Arnold, Supplementary chapters of the theory of ordinary differential equations, S Nauka, 1978.

E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, 2007.

K. Godfrey and S. Eglen, Theoretical models of spontaneous activity generation and propagation in the developing retina Molecular BioSystems, 2009.

J. Gjorgjieva and J. Stephen, Eglen Modeling developmental patterns of spontaneous activity, Curr Opin Neurobiol, 2011.

M. Kaneda, K. Ito, Y. Morishima, Y. Shigematsu, and Y. Shimoda, Characterization of Voltage-Gated Ionic Channels in Cholinergic Amacrine Cells in the Mouse Retina, Journal of Neurophysiology, vol.97, issue.6, 2007.
DOI : 10.1073/pnas.93.15.8057

A. Maccione, M. Hennig, M. , M. Gandolfo, O. Muthmann et al., Following the ontogeny of retinal waves: pan-retinal recordings of population dynamics in the neonatal mouse, The Journal of Physiology, vol.20, issue.7, pp.1545-1563, 2014.
DOI : 10.1113/jphysiol.2013.262840

M. Hennig, C. Adams, D. Willshaw, and E. Sernagor, Early-Stage Waves in the Retinal Network Emerge Close to a Critical State Transition between Local and Global Functional Connectivity, Journal of Neuroscience, vol.29, issue.4, 2009.
DOI : 10.1523/JNEUROSCI.4880-08.2009

E. Sernagor and N. Grzywacz, Spontaneous Activity in Developing Turtle Retinal Ganglion Cells: Pharmacological Studies, The Journal of Neuroscience, vol.19, issue.10, 1999.
DOI : 10.1523/JNEUROSCI.19-10-03874.1999

E. Sernagor and M. Hennig, Retinal Waves, 2012.
DOI : 10.1016/B978-0-12-397266-8.00151-4

M. H. Hennig and J. Grady, James van Coppenhagen, and Evelyne Sernagor Age-dependent Homeostatic Plasticity of GABAergic Signaling in Developing Retinal Networks Journal of Neuroscience, 2011.

A. Dhooge, W. Govaerts, and Y. A. Kuznetsov, MATCONT, ACM Transactions on Mathematical Software, vol.29, issue.2, pp.141-164, 2003.
DOI : 10.1145/779359.779362

A. L. Hodgkin and A. F. , A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, vol.117, issue.4, pp.500-544, 1952.
DOI : 10.1113/jphysiol.1952.sp004764

K. Yoshida, D. Watanabe, H. Ishikane, M. Tachibana, I. Pastan et al., A Key Role of Starburst Amacrine Cells in Originating Retinal Directional Selectivity and Optokinetic Eye Movement, Neuron, vol.30, issue.3, 2001.
DOI : 10.1016/S0896-6273(01)00316-6

R. Zwart and P. M. Henk, Vijverberg Potentiation and Inhibition of Neuronal Nicotinic Receptors by Atropine: Competitive and Noncompetitive Effects Molecular Pharmacology, 1997.
DOI : 10.1124/mol.52.5.886

URL : http://molpharm.aspetjournals.org/content/molpharm/52/5/886.full.pdf

J. Zheng, S. Lee, and Z. J. Zhou, A Developmental Switch in the Excitability and Function of the Starburst Network in the Mammalian Retina, Neuron, vol.44, issue.5, pp.851-864, 2004.
DOI : 10.1016/j.neuron.2004.11.015

J. Zheng, S. Lee, and Z. J. Zhou, A transient network of intrinsically bursting starburst cells underlies the generation of retinal waves, Nature Neuroscience, vol.94, issue.3, pp.363-371, 2006.
DOI : 10.1152/jn.00279.2005

A. M. Marritt, B. C. Cox, R. P. Yasuda, J. Mcintosh, X. Michael et al., Nicotinic Cholinergic Receptors in the Rat Retina: Simple and Mixed Heteromeric Subtypes Molecular Pharmacology, pp.1656-1668, 2005.

A. Katok, . Hasselblatt, and M. Boris, Leonardo Collaborateur Introduction to the modern theory of dynamical systems, 1995.

C. Park and J. E. Rubin, Cooperation of intrinsic bursting and calcium oscillations underlying activity patterns of model pre-B??tzinger complex neurons, Journal of Computational Neuroscience, vol.95, issue.4, 2013.
DOI : 10.1152/jn.01308.2005

A. Jung-min-han, V. Tanimura, and . Kirk, James Sneyd A mathematical model of calcium dynamics in HSY cells PLOS Comp Bio, p.2017

H. Choi, L. Zhang, M. S. Cembrowski, C. F. Sabottke, A. L. Markowitz et al., Hermann Riecke Intrinsic bursting of AII amacrine cells underlies oscillations in the rd1 mouse retina, Journal of Neurophysiology, 2014.

D. Warland, A. Huberman, and L. Chalupa, Dynamics of Spontaneous Activity in the Fetal Macaque Retina during Development of Retinogeniculate Pathways, Journal of Neuroscience, vol.26, issue.19, 2006.
DOI : 10.1523/JNEUROSCI.0328-06.2006

B. K. Stafford, A. Sher, A. M. Litke, and A. David, Feldheim, Spatial-Temporal Patterns of Retinal Waves Underlying Activity-Dependent Refinement of Retinofugal Projections Neuron, 2009.

J. Zhou, Direct Participation of Starburst Amacrine Cells in Spontaneous Rhythmic Activities in the Developing Mammalian Retina, The Journal of Neuroscience, vol.18, issue.11, pp.4155-4165, 1998.
DOI : 10.1523/JNEUROSCI.18-11-04155.1998

M. I. Fredlin and A. Wentzell, D Random Perturbations of Dynamical Systems Springer, Grundlehren der mathematischen Wissenschaften, 1998.

S. H. Strogatz, Nonlinear Dynamics and Chaos : with Applications to Physics, Biology, Chemistry, and Engineering. Boulder, CO, 2015.

H. Poincar?apoincar? and P. C. , Sur l'??quilibre d'une masse fluide anim??e d'un mouvement de rotation, Acta Mathematica, vol.7, issue.0, pp.259-380, 1885.
DOI : 10.1007/BF02402204