Properties of Relaxed Trajectories for a Class of Nonlinear Evolution Equations on a Banach Space, SIAM Journal on Control and Optimization, vol.21, issue.6, pp.953-967, 1983. ,
DOI : 10.1137/0321058
Optimal control of systems governed by a class of nonlinear evolution equations in a reflexive Banach space, Journal of Optimization Theory and Applications, vol.61, issue.1, pp.57-81, 1978. ,
DOI : 10.1007/BF00933255
The emergence of the deterministic Hodgkin-Huxley equations as a limit from the underlying stochastic ion-channel mechanism, Ann. Appl. Probab, vol.18, pp.1279-1325, 2008. ,
Neural substrates of awakening probed with optogentic control of hypocretin neurons, Nature, vol.450, pp.420-424, 2007. ,
A general denseness result for relaxed control theory, Bull. Austral . Math. Soc, vol.30, pp.463-475, 1984. ,
Towards a geometric theory in the time minimal control of chemical batch reactors, SIAM J. Contr. Opt, vol.33, pp.1279-1311, 1998. ,
Numerical methods for the exit time of a Piecewise Deterministic Markov Process, Adv. in Appl. Probab, vol.44, issue.1, pp.196-225, 2012. ,
Convergence of probability measures, 1968. ,
??-convergence and optimal control problems, Journal of Optimization Theory and Applications, vol.110, issue.3, pp.385-407, 1982. ,
DOI : 10.1007/BF00935345
On the Phase Reduction and Response Dynamics of Neural Oscillator Populations, Neural Computation, vol.14, issue.4, pp.673-715, 2004. ,
DOI : 10.1007/BF02339491
Optogenetics and the future of neuroscience, Nature Neuroscience, vol.18, pp.1200-1201, 2015. ,
Two open states with progressive proton selectivities in the branched channelrhodopsin-2 photocycle, Biophysical Journal, vol.98, pp.753-761, 2010. ,
MDP algorithms for portfolio optimization problems in??pure jump markets, Finance and Stochastics, vol.62, issue.4, pp.591-611, 2009. ,
DOI : 10.1007/s00780-009-0093-0
An exact stochastic hybrid model of excitable membranes including spatio-temporal evolution, J. Math. Biol, vol.63, issue.6, pp.1051-1093, 2011. ,
Markov decision processes with applications to finance, 2011. ,
Stochastic optimal control: the discrete-time case, 1978. ,
Millisecond-timescale, genetically targeted optical contro of neural activity, Nature Neuroscience, vol.8, issue.9, pp.1263-1268, 2005. ,
Stability and Ergodicity of Piecewise Deterministic Markov Processes, SIAM Journal on Control and Optimization, vol.47, issue.2, pp.1053-1077, 2008. ,
DOI : 10.1137/060670109
URL : https://hal.archives-ouvertes.fr/hal-00268162
Optimal stopping with continuous control of piecewise deterministic Markov processes, Stochastics An International Journal of Probability and Stochastic Processes, vol.70, issue.1, pp.41-73, 2000. ,
DOI : 10.1080/17442500008834245
Spontaneous action potentials due to channel fluctuations, Biophysical Journal, vol.71, issue.6, pp.3013-3021, 1996. ,
DOI : 10.1016/S0006-3495(96)79494-8
URL : http://doi.org/10.1016/s0006-3495(96)79494-8
Illuminating Parkinson's therapy with optogenetics, Nature Biotechnology, vol.78, issue.2, pp.149-150, 2015. ,
DOI : 10.1073/pnas.0700293104
URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4339091
Piecewise-Deterministic Markov Processes: a general class of non-diffusion stochastic models, J. R. Statist. Soc, vol.46, issue.3, pp.353-388, 1984. ,
Markov models and optimization, Chapman and Hall, Dei11] K. Deisseroth, Optogenetics, pp.26-29, 1993. ,
The Morris???Lecar neuron model embeds a leaky integrate-and-fire model, Journal of Mathematical Biology, vol.14, issue.6, pp.239-259, 2013. ,
DOI : 10.1007/s00285-012-0552-7
URL : http://arxiv.org/abs/1108.0073
A Markovian analysis of additive-increase multiplicative-decrease algorithms, Advances in Applied Probability, vol.III, issue.01, pp.85-111, 2002. ,
DOI : 10.1145/122431.122434
Linear operators. part i: General theory Numerical methods for simulation and optimization of piecewise deterministic markov processes, 1977. ,
One-parameter semigroups for linear evolution equations, Semigroup Forum, vol.63, issue.2, 2000. ,
DOI : 10.1007/s002330010042
Theoretical principles underlying optical stimulation of a channelrhodopsin-2 positive pyramidal neuron, Journal of Neurophysiology, vol.107, issue.12, pp.3235-3245, 2012. ,
DOI : 10.1152/jn.00501.2011
The Cauchy problem, 1983. ,
Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophysical Journal, vol.1, issue.6, pp.445-466, 1961. ,
DOI : 10.1016/S0006-3495(61)86902-6
Principle of optimal control theory [Gen13] A. Genadot, A multiscale study of stochastic spatially-extended conductancebased models for excitable systems, 1987. ,
Algebraic invariance conditions in the study of approximate (null-)controllability of Markov switch processes, Mathematics of Control, Signals, and Systems, vol.16, issue.3, pp.551-578, 2015. ,
DOI : 10.1007/s00498-015-0146-1
URL : https://hal.archives-ouvertes.fr/hal-01061543
Optogenetics vision restoration using rhodopsin for enhanced sensitivity, Molecular Therapy, vol.23, issue.10, pp.1562-1571, 2015. ,
Quantization, IEEE Transactions on Information Theory, vol.44, issue.6, pp.2325-2383, 1998. ,
DOI : 10.1109/18.720541
Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions, Advances in Applied Probability, vol.20, issue.03, pp.749-773, 2012. ,
DOI : 10.1007/BF01053972
URL : https://hal.archives-ouvertes.fr/hal-00662268
Multiple Photocycles of Channelrhodopsin, Biophysical Journal, vol.89, issue.6, pp.3911-3918, 2005. ,
DOI : 10.1529/biophysj.105.069716
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
Multiple Photocycles of Channelrhodopsin, Biophysical Journal, vol.89, issue.6, pp.3911-3918, 2005. ,
DOI : 10.1529/biophysj.105.069716
URL : http://doi.org/10.1529/biophysj.105.069716
Multivariate point processes: predictable projections, Radon- Nikodym derivatives, representation of martingales, Z. Wahrsag. Verw. Gebiete, vol.34, pp.235-253, 1975. ,
Point process theory and applications: marked point processes and piecewise deterministic processes, 2006. ,
Recherche quantitatives sur l'excitation électrique des nerfs traitée comme une polarization, J. Physiol. Pathol. Gen, vol.9, pp.620-635, 1907. ,
Stochastic optimal control of single neuron spike trains, J. Neural. Eng, p.11, 2014. ,
Control and Synchronization of Neuron Ensembles, IEEE Transactions on Automatic Control, vol.58, issue.8, 1919. ,
DOI : 10.1109/TAC.2013.2250112
Voltage oscillations in the barnacle giant muscle fiber, Biophysical J, vol.35, pp.193-213, 1981. ,
Potential Utility of Optogenetics in the Study of Depression, Biological Psychiatry, vol.71, issue.12, pp.1068-1074, 2012. ,
DOI : 10.1016/j.biopsych.2011.12.026
Geometric optimal control: theory, methods, examples ,
Optimal control theory for infinite dimensional systems, 1995. ,
DOI : 10.1007/978-1-4612-4260-4
General Pontryagin-type stochastic maximum principle and backward stochastic evolution equations in infinite dimensions, 2014. ,
An introduction to ?-convergence [MYM77] A. Matsuno-Yagi and Y. Mukohata, Two possible roles of bacteriorhodopsin; a comparative study of strains of halobacterium halobium differing in pigmentation, Biochem. Biophys. Res. Commun, pp.78-237, 1977. ,
Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de sobolev, Ann. Inst. H. Poincaré Prob. Statist, vol.20, pp.329-348, 1984. ,
Channelrhodopsin-1: A Light-Gated Proton Channel in Green Algae, Science, vol.296, issue.5577, pp.2395-2398, 2002. ,
DOI : 10.1126/science.1072068
An Active Pulse Transmission Line Simulating Nerve Axon, Proceedings of the IRE, vol.50, issue.10, pp.2061-2070, 1962. ,
DOI : 10.1109/JRPROC.1962.288235
Computational models of optogenetic tools for controlling neural circuits with light, 2013 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), pp.5934-5937, 2013. ,
DOI : 10.1109/EMBC.2013.6610903
Single input optimal control for globally coupled neuron networks, Journal of Neural Engineering, vol.8, issue.6, pp.3911-3918, 2011. ,
DOI : 10.1088/1741-2560/8/6/065008
Rhodopsin-like Protein from the Purple Membrane of Halobacterium halobium, Nature New Biology, vol.233, issue.39, pp.149-152, 1971. ,
DOI : 10.1038/newbio233149a0
Properties of the relaxed trajectories of evolution equations and optimal control, SIAM J. Control Optim, vol.27, issue.2, pp.267-288, 1989. ,
Optogenetics and Epilepsy: Past, Present and Future, Epilepsy Currents, vol.15, issue.1, pp.34-38, 2015. ,
DOI : 10.5698/1535-7597-15.1.34
URL : http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4320957
The adjoint semi-group, Pacific J. Math, vol.5, issue.2, pp.269-283, 1955. ,
Optimal quantization methods and applications to numerical problems in finance, Handbook of computational and numerical methods in finance, pp.253-297, 2004. ,
Fluid limit theorems for stochastic hybrid systems and applications to neuron models, Adv ,
DOI : 10.1239/aap/1282924062
URL : http://arxiv.org/abs/1001.2474
Engram cells retain memory under retrograde amnesia, Science, vol.348, issue.6238, pp.1007-1013, 2015. ,
Limit theorems for infinitedimensional Piecewise Deterministic Markov Processes. Applications to stochastic excitable membrane models, Electron, J. Probab, vol.17, issue.55, pp.1-48, 2012. ,
The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales, Chaos: An Interdisciplinary Journal of Nonlinear Science, vol.49, issue.1, p.15105, 2008. ,
DOI : 10.1023/A:1008871803040
Archaeal-type rhodopsins in Chlamydomonas: model structure and intracellular localization, Biochemical and Biophysical Research Communications, vol.301, issue.3, pp.711-717, 2003. ,
DOI : 10.1016/S0006-291X(02)03079-6
Comparison of Coding Capabilities of Type I and Type II Neurons, Journal of Computational Neuroscience, vol.16, issue.3, pp.299-313, 2004. ,
DOI : 10.1023/B:JCNS.0000025690.02886.93
Two rhodopsins mediate phototaxis to low- and high-intensity light in Chlamydomonas reinhardtii, Proc. Natl. Acad. Sci. USA 99, pp.8689-8694, 2002. ,
DOI : 10.1016/S1350-9462(00)00002-1
Optimal control based seizure abatement using patient derived connectivity, Frontiers in Neuroscience, vol.9, issue.202, 2015. ,
De l'aléa dans les neurones : une analyse probabiliste multiéchelle, 2010. ,
Computational optogenetics: A novel continuum framework for the photoelectrochemistry of living systems, Journal of the Mechanics and Physics of Solids, vol.60, issue.6 ,
DOI : 10.1016/j.jmps.2012.02.004
Necessary conditions for minimum in relaxed variational problems, J. Math. Anal. Appl, vol.4, pp.129-145, 1962. ,
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, vol.10, issue.1, pp.25-27, 2006. ,
DOI : 10.1007/s10107-004-0559-y
Noise from voltage-gated ion channels may influence neuronal dynamics in the entorhinal cortex, Journal of neurophysiology, vol.80, issue.1, pp.262-269, 1998. ,
Computational Optogenetics: epirically-derived voltage-and light-sensitive Channelrhodopsin-2 model, PLoS Comput Biol, vol.9, issue.9, pp.1-19, 2013. ,