S. Billakota, O. Devinsky, and K. Kim, Why we urgently need improved epilepsy therapies 694 for adult patients, Neuropharmacology, p.107855, 2019.

M. Avoli, M. De-curtis, V. Gnatkovsky, J. Gotman, R. Köhling et al., Specific 697 imbalance of excitatory/inhibitory signaling establishes seizure onset pattern in temporal 698 lobe epilepsy, J Neurophysiol, vol.115, issue.6, pp.3229-3266, 2016.

K. T. Kahle, A. R. Khanna, J. Duan, K. J. Staley, E. Delpire et al., The KCC2 Cotransporter 700 and Human Epilepsy: Getting Excited About Inhibition, The Neuroscientist, vol.701, issue.6, pp.555-62, 2016.

F. E. Dudek, K. J. Staley, M. Avoli, M. A. Rogawski, R. W. Olsen et al., The Time Course and Circuit Mechanisms of Acquired 703 Epileptogenesis, Noebels JL

, Jasper's Basic Mechanisms of the Epilepsies, National Center for Biotechnology Information (US, p.705, 2012.

H. Morris, I. Najm, and P. Kahane, 707 Textbook of Epilepsy Surgery. London: Informa Healthcare, pp.230-237, 2008.

R. S. Fisher and A. L. Velasco, Electrical brain stimulation for epilepsy, Nat Rev Neurol, vol.10, issue.5, pp.261-70, 2014709-05.

M. Sprengers, K. Vonck, E. Carrette, A. G. Marson, and P. Boon, Deep brain and cortical 711 stimulation for epilepsy, Cochrane Database Syst Rev, issue.7, p.2017, 2017.

R. P. Lesser, H. W. Lee, W. Webber, B. Prince, N. E. Crone et al., Short-term 715 variations in response distribution to cortical stimulation, Brain, vol.131, issue.6, pp.1528-1567, 2008.

N. G. Laxpati, W. S. Kasoff, and R. E. Gross, Deep Brain Stimulation for the Treatment of 718 Epilepsy: Circuits, Targets, and Trials, Neurotherapeutics, vol.11, issue.3, pp.508-534, 2014.

C. N. Heck, D. King-stephens, A. D. Massey, D. R. Nair, B. C. Jobst et al., Two-721 year seizure reduction in adults with medically intractable partial onset epilepsy treated 722 with responsive neurostimulation: Final results of the RNS System Pivotal trial, Epilepsia, vol.723, issue.3, pp.432-473, 2014.

S. Lee, D. D. Spencer, and S. S. Spencer, Intracranial EEG Seizure-Onset Patterns in 725 Neocortical Epilepsy, Epilepsia, vol.41, issue.3, pp.297-307, 2000.

F. Bartolomei, F. Wendling, J. Régis, M. Gavaret, M. Guye et al., Pre-ictal 727 synchronicity in limbic networks of mesial temporal lobe epilepsy, Epilepsy Res, vol.728, issue.1, pp.89-104, 2004.

A. L. Velascol, C. L. Wilson, T. L. Babb, E. Jr, and J. , Functional and Anatomic Correlates of 730 Two Frequently Observed Temporal Lobe Seizure-Onset Patterns, Neural Plasticity, vol.731, issue.1-2, pp.49-63, 2000.

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J 734 Differ Equ, vol.31, issue.1, pp.53-98, 1979.

C. Jones, L. Arnold, and C. Jones, Geometric singular perturbation theory, p.736

K. Mischaikow, G. Raugel, and R. Johnson, Dynamical Systems: Lectures Given at the 737 2nd Session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini 738

I. Terme, Lecture 739 Notes in Mathematics). Available from, pp.44-118, 1994.

F. Wendling, F. Bartolomei, J. J. Bellanger, and P. Chauvel, Epileptic fast activity can be 741 explained by a model of impaired GABAergic dendritic inhibition, Eur J Neurosci, vol.742, issue.9, pp.1499-508, 2002.

C. Kuehn, Multiple Time Scale Dynamics, Applied Mathematical Sciences, issue.744, 2015.

M. Krupa and P. Szmolyan, Extending Geometric Singular Perturbation Theory to 747 Nonhyperbolic Points---Fold and Canard Points in Two Dimensions, SIAM J Math Anal, vol.748, issue.2, pp.286-314, 2001.

J. Rubin and M. Wechselberger, Giant squid-hidden canard: the 3D geometry of the Hodgkin-750 Huxley model, Biol Cybern, vol.97, issue.1, pp.5-32, 2007.

M. Desroches and K. V. Spike, Adding in a Canonical Three-Time-Scale Model: Superslow 752 Explosion and Folded-Saddle Canards, SIAM J Appl Dyn Syst, vol.17, issue.3, pp.1989-2017, 2018.

P. Nan, Y. Wang, V. Kirk, and J. E. Rubin, Understanding and Distinguishing Three-Time-Scale 755 Oscillations: Case Study in a Coupled Morris--Lecar System, SIAM J Appl Dyn Syst, vol.756, issue.3, pp.1518-57, 2015.

J. Talairach, J. Bancaud, G. Szikla, A. Bonis, S. Geier et al., New approach to the 758 neurosurgery of epilepsy. Stereotaxic methodology and therapeutic results. 1. Introduction 759 and history, Neurochirurgie, vol.20, 1974.

E. J. Doedel, A. Champneys, T. F. Fairgrieve, A. B. Yu, A. P. Kuznetsov et al.,

, Auto-07p: Continuation and bifurcation software for ordinary differential equations, p.762, 2007.

B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems, Society for 764 Industrial and Applied Mathematics; 2002. 312 p. (Software, Environments and Tools)

,

B. Molaee-ardekani, P. Benquet, F. Bartolomei, and F. Wendling, Computational modeling of 767 high-frequency oscillations at the onset of neocortical partial seizures: From 'altered 768 structure' to 'dysfunction, NeuroImage, vol.52, issue.3, pp.1109-1131, 2010.

F. Bartolomei, P. Chauvel, and F. Wendling, Epileptogenicity of brain structures in human 771 temporal lobe epilepsy: a quantified study from intracerebral EEG, Brain, vol.772, issue.7, pp.1818-1848, 2008.

R. G. Andrzejak, O. David, V. Gnatkovsky, F. Wendling, F. Bartolomei et al.,

, Localization of Epileptogenic Zone on Pre-surgical Intracranial EEG Recordings: Toward 775 a Validation of Quantitative Signal Analysis Approaches, Brain Topogr, vol.28, issue.6, pp.832-776, 2015.

S. Lagarde, S. Buzori, A. Trebuchon, R. Carron, D. Scavarda et al., The repertoire 778 of seizure onset patterns in human focal epilepsies: Determinants and prognostic values

, Epilepsia, vol.60, issue.1, pp.85-95, 2019.

F. Wendling, P. Benquet, F. Bartolomei, and V. Jirsa, Computational models of epileptiform 781 activity, J Neurosci Methods, vol.260, pp.233-51, 2016.

E. M. Izhikevich, Neural excitability, spiking and bursting, Int J Bifurc Chaos, vol.783, issue.06, pp.1171-266, 2000.

P. Suffczynski, S. Kalitzin, F. L. Da-silva, J. Parra, D. Velis et al., Active paradigms of 785 seizure anticipation: Computer model evidence for necessity of stimulation, Phys Rev E, vol.786, issue.5, 2008.

F. Da-silva, W. Blanes, S. N. Kalitzin, J. Parra, P. Suffczynski et al., Dynamical 789 diseases of brain systems: different routes to epileptic seizures, IEEE Trans Biomed Eng, vol.790, issue.5, pp.540-548, 2003.

F. Silva, W. Blanes, S. N. Kalitzin, J. Parra, P. Suffczynski et al., , p.792

, Dynamical Diseases of Brain Systems: Basic Models of the Transition Between Normal 793 and Epileptic Activity, Epilepsia, vol.44, issue.s12, pp.72-83, 2003.

P. Perucca, F. Dubeau, and J. Gotman, Intracranial electroencephalographic seizure-onset 796 patterns: effect of underlying pathology, Brain J Neurol, vol.137, pp.183-96, 2014.

S. Lagarde, F. Bonini, A. Mcgonigal, P. Chauvel, M. Gavaret et al., Seizure-799 onset patterns in focal cortical dysplasia and neurodevelopmental tumors: Relationship 800 with surgical prognosis and neuropathologic subtypes, Epilepsia, vol.57, issue.9, pp.1426-1461, 2016.

G. Huberfeld, L. Menendez-de-la-prida, J. Pallud, I. Cohen, L. Van-quyen et al., Glutamatergic pre-ictal discharges emerge at the transition to seizure in human 804 epilepsy, Nat Neurosci, vol.14, issue.5, pp.627-661, 2011.

M. Avoli and M. De-curtis, GABAergic synchronization in the limbic system and its role in the 806 generation of epileptiform activity, Prog Neurobiol, vol.95, issue.2, pp.104-136, 2011.

B. Lasztóczi, G. Nyitrai, L. Héja, and J. Kardos, Synchronization of GABAergic Inputs to CA3

, Pyramidal Cells Precedes Seizure-Like Event Onset in Juvenile Rat Hippocampal Slices

, J Neurophysiol, vol.102, issue.4, pp.2538-53, 2009.

S. F. Muldoon, F. Pasqualetti, S. Gu, M. Cieslak, S. T. Grafton et al., Stimulation-812 Based Control of Dynamic Brain Networks, PLoS Comput Biol, vol.12, issue.9, 2016.

C. Chiang, T. P. Ladas, L. E. Gonzalez-reyes, and D. M. Durand, Seizure Suppression by High 816 Frequency Optogenetic Stimulation Using In Vitro and In Vivo Animal Models of 817 Epilepsy, Brain Stimulat, vol.7, issue.6, pp.890-899, 2014.

T. P. Ladas, C. Chiang, L. E. Gonzalez-reyes, T. Nowak, and D. M. Durand, Seizure reduction 819 through interneuron-mediated entrainment using low frequency optical stimulation, Exp 820 Neurol, vol.269, pp.120-152, 2015.

S. Rashid, G. Pho, M. Czigler, M. A. Werz, D. M. Durand et al., Low frequency stimulation of 822 ventral hippocampal commissures reduces seizures in a rat model of chronic temporal 823 lobe epilepsy, Epilepsia, vol.53, issue.1, pp.5765-90, 2012.

T. Kano, Y. Inaba, D. 'antuono, M. Biagini, G. Levésque et al., Blockade of in vitro 828 ictogenesis by low-frequency stimulation coincides with increased epileptiform response 829 latency, J Neurophysiol, vol.114, issue.1, pp.21-29, 2015.

B. Svejgaard, M. Andreasen, and S. Nedergaard, Role of GABAB receptors in proepileptic and 831 antiepileptic effects of an applied electric field in rat hippocampus in vitro, Brain Res, vol.832, pp.157-62, 2019.

N. H. Couturier and D. M. Durand, Corpus callosum low-frequency stimulation suppresses 834 seizures in an acute rat model of focal cortical seizures, Epilepsia, vol.59, issue.12, 2018.

N. H. Couturier and D. M. Durand, Comparison of fiber tract low frequency stimulation to focal 837 and ANT stimulation in an acute rat model of focal cortical seizures, Brain Stimulat, vol.838, issue.2, pp.499-506, 2019.

X. Li, P. Somogyi, A. Ylinen, and G. Buzsáki, The hippocampal CA3 network: An in vivo 841 intracellular labeling study, J Comp Neurol, vol.339, issue.2, pp.181-208, 1994.

A. Sik, M. Penttonen, A. Ylinen, and G. Buzsaki, Hippocampal CA1 interneurons: an in vivo 844 intracellular labeling study, J Neurosci, vol.15, issue.10, pp.6651-65, 1995.

F. Assaf and Y. Schiller, The antiepileptic and ictogenic effects of optogenetic 847 neurostimulation of PV-expressing interneurons, J Neurophysiol, vol.116, issue.4, pp.1694-704, 2016.

M. Lévesque, L. Chen, G. Etter, Z. Shiri, S. Wang et al., Paradoxical effects 850 of optogenetic stimulation in mesial temporal lobe epilepsy, Ann Neurol, vol.851, issue.5, pp.714-742, 2019.

E. Krook-magnuson, The Devil's in the Details: How to Harness Inhibition for Seizure 853 Control, Epilepsy Curr, vol.20, issue.2, pp.99-101, 2020.

N. Zangiabadi, L. D. Ladino, F. Sina, J. P. Orozco-hernández, A. Carter et al.,

, Deep Brain Stimulation and Drug-Resistant Epilepsy: A Review of the Literature. Front 856 Neurol, vol.10, p.601, 2019.

J. Yamamoto, A. Ikeda, M. Kinoshita, R. Matsumoto, T. Satow et al., Low-859 frequency electric cortical stimulation decreases interictal and ictal activity in human 860 epilepsy, Seizure, vol.15, issue.7, pp.520-527, 2006.

Y. Schiller and Y. Bankirer, Cellular Mechanisms Underlying Antiepileptic Effects of Low-862 and High-Frequency Electrical Stimulation in Acute Epilepsy in Neocortical Brain Slices 863 In Vitro, J Neurophysiol, vol.97, issue.3, pp.1887-902, 2007.

C. Chiang, C. Lin, M. Ju, and D. M. Durand, High frequency stimulation can suppress 865 globally seizures induced by 4-AP in the rat hippocampus: An acute in vivo study, Brain 866 Stimulat, vol.6, issue.2, pp.180-189, 2013.

A. L. Velasco, M. Velasco, F. Velasco, D. Menes, F. Gordon et al., Subacute and 868 Chronic Electrical Stimulation of the Hippocampus on Intractable Temporal Lobe 869 Seizures: Preliminary Report, Arch Med Res, vol.31, issue.3, pp.64-67, 2000.

N. Sah and S. K. Sikdar, Transition in subicular burst firing neurons from epileptiform activity 872 to suppressed state by feedforward inhibition, Eur J Neurosci, vol.38, issue.4, pp.2542-56, 2013.

Z. Xu, Y. Wang, B. Chen, C. Xu, X. Wu et al., Entorhinal Principal Neurons 875 Mediate Brain-stimulation Treatments for Epilepsy, EBioMedicine, vol.14, pp.148-60, 2016.

E. Krook-magnuson, C. Armstrong, M. Oijala, and I. Soltesz, On-demand optogenetic control of 878 spontaneous seizures in temporal lobe epilepsy, Nat Commun, vol.4, p.1376, 2013.

B. J. Bacak, T. Kim, J. C. Smith, J. E. Rubin, and I. A. Rybak, Mixed-mode oscillations and 881 population bursting in the pre-Bötzinger complex, eLife, vol.5, p.13403, 2016.

J. Tabak, J. Rinzel, and M. J. O'donovan, The Role of Activity-Dependent Network Depression 885 in the Expression and Self-Regulation of Spontaneous Activity in the Developing Spinal 886 Cord, J Neurosci, vol.21, issue.22, pp.8966-78, 2001.

J. Tabak, W. Senn, M. J. O'donovan, and J. Rinzel, Modeling of Spontaneous Activity in 888 Developing Spinal Cord Using Activity-Dependent Depression in an Excitatory Network

, J Neurosci, vol.20, issue.8, pp.3041-56, 2000.

C. S. Nunemaker, R. Bertram, A. Sherman, K. Tsaneva-atanasova, C. R. Daniel et al.,

, Glucose Modulates [Ca2+]i Oscillations in Pancreatic Islets via Ionic and Glycolytic 892 Mechanisms, Biophys J, vol.91, issue.6, pp.2082-96, 2006.

C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural 894, Sciences. Society for Industrial and Applied Mathematics, 1988.

W. Duan, K. Lee, A. E. Herbison, and J. Sneyd, A mathematical model of adult GnRH neurons in 897 mouse brain and its bifurcation analysis, J Theor Biol, vol.276, issue.1, pp.22-34, 2011.

J. E. Rubin, J. A. Hayes, J. L. Mendenhall, and C. Negro, Calcium-activated nonspecific cation 900 current and synaptic depression promote network-dependent burst oscillations, Proc Natl, vol.901

A. Sci, , vol.106, pp.2939-2983, 2009.

R. Fitzhugh, Impulses and Physiological States in Theoretical Models of Nerve 903 Membrane, Biophys J, vol.1, issue.6, pp.86902-86908, 1961.

R. Bertram, J. Previte, A. Sherman, T. A. Kinard, and L. S. Satin, The Phantom Burster Model for 905 Pancreatic ?-Cells, Biophys J, vol.79, issue.6, pp.76525-76533, 2000.

H. G. Rotstein, D. D. Pervouchine, C. D. Acker, M. J. Gillies, J. A. White et al., Slow 908 and Fast Inhibition and an H-Current Interact to Create a Theta Rhythm in a Model of 909 CA1 Interneuron Network, J Neurophysiol, vol.94, issue.2, pp.1509-1527, 2005.

E. Harvey, V. Kirk, M. Wechselberger, and J. Sneyd, Multiple Timescales, Mixed Mode 912 Oscillations and Canards in Models of Intracellular Calcium Dynamics, J Nonlinear Sci, vol.913, issue.5, p.639, 2011.

M. Krupa, N. Popovi?, N. Kopell, and H. G. Rotstein, Mixed-mode oscillations in a three time-915 scale model for the dopaminergic neuron, Chaos Interdiscip J Nonlinear Sci, vol.916, issue.1, p.15106, 2008.

H. G. Rotstein, T. Oppermann, J. A. White, and N. Kopell, The dynamic structure underlying 918 subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal 919 cortex stellate cells, J Comput Neurosci, vol.21, issue.3, pp.271-92, 2006.

H. G. Rotstein, M. Wechselberger, and N. Kopell, Canard Induced Mixed-Mode Oscillations in a 922

, Medial Entorhinal Cortex Layer II Stellate Cell Model, SIAM J Appl Dyn Syst, vol.923, issue.4, pp.1582-611, 2008.

P. De-maesschalck and M. Desroches, Numerical Continuation Techniques for Planar Slow-925 Fast Systems, SIAM J Appl Dyn Syst, vol.12, issue.3, pp.1159-80, 2013.

P. De-maesschalck and M. Wechselberger, Neural Excitability and Singular Bifurcations, J 927 Math Neurosci JMN, vol.5, issue.1, p.16, 2015.

J. Mitry, M. Mccarthy, N. Kopell, and M. Wechselberger, Excitable Neurons, Firing Threshold 929 Manifolds and Canards, J Math Neurosci, vol.3, issue.1, p.12, 2013.

R. Bertram and J. E. Rubin, Multi-timescale systems and fast-slow analysis, Math Biosci, vol.931, pp.105-126, 2017.

K. Ersöz, E. Desroches, M. Guillamon, A. Rinzel, J. Tabak et al., Canard-induced complex 933 oscillations in an excitatory network, J Math Biol, vol.80, pp.2075-2107, 2020.

J. E. Rubin and D. Terman, High Frequency Stimulation of the Subthalamic Nucleus Eliminates 936 Pathological Thalamic Rhythmicity in a Computational Model, J Comput Neurosci, vol.937, issue.3, pp.211-246, 2004.

D. Wilson and J. Moehlis, A Hamilton-Jacobi-Bellman approach for termination of seizure-939 like bursting, J Comput Neurosci, vol.37, issue.2, pp.345-55, 2014.

J. Hebbink, S. A. Van-gils, and H. Meijer, On analysis of inputs triggering large nonlinear 941 neural responses Slow-fast dynamics in the Wendling neural mass model, Commun 942 Nonlinear Sci Numer Simul, vol.83, p.105103, 2020.

A. Weigenand, M. S. Costa, H. Ngo, J. C. Claussen, and T. Martinetz, Characterization of K-944 Complexes and Slow Wave Activity in a Neural Mass Model, PLOS Comput Biol, vol.10, issue.11, p.1003923, 201494513.

M. Desroches, O. Fougeras, and M. Krupa, Slow-Fast Transitions to Seizure States in the
URL : https://hal.archives-ouvertes.fr/hal-01404623

. Wendling-chauvel, Neural Mass Model, Opera Med Physiol, vol.2, issue.3-4, pp.228-234, 2016.

T. Vo, R. Bertram, and M. Wechselberger, Multiple Geometric Viewpoints of Mixed Mode 950 Dynamics Associated with Pseudo-plateau Bursting, SIAM J Appl Dyn Syst, vol.951, issue.2, pp.789-830, 2013.

S. Bensaid, J. Modolo, I. Merlet, F. Wendling, and P. Benquet, COALIA: A Computational ratio of / is kept constant. (a) Pulse width is 0.5 ms

, Pulse width is 50 ms