D. Aldous and P. Diaconis, Hammersley's interacting particle process and longest increasing subsequences, Probab. Theory Related Fields, vol.103, pp.199-213, 1995.

A. L. Basdevant, L. Gerin, J. B. Gouéré, and A. Singh, From Hammersley's lines to Hammersley's trees, 2017.

B. Bollobás and P. Winkler, The longest chain among random points in Euclidean space, Proc. Amer. Math. Soc, vol.103, pp.347-353, 1988.

M. Bossy, O. Faugeras, and D. Talay, Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, J. Math. Neurosci, vol.5, p.p, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01098582

M. Cáceres, J. Carrillo, and B. Perthame, Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states, J. Math. Neurosci, vol.1, 2011.

J. Calder, S. Esedo¯-glu, and A. O. Hero, A Hamilton-Jacobi equation for the continuum limit of nondominated sorting, SIAM J. Math. Anal, vol.46, pp.603-638, 2014.

J. Carrillo, B. Perthame, D. Salort, and D. Smets, Qualitative properties of solutions for the noisy integrate and fire model in computational neuroscience, Nonlinearity, vol.28, pp.3365-3388, 2015.

E. Cator and P. Groeneboom, Hammersley's process with sources and sinks, Ann. Probab, vol.33, pp.879-903, 2005.

J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl, vol.127, pp.3870-3912, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01217407

J. Chevallier, M. Cáceres, M. Doumic, and P. Reynaud-bouret, Microscopic approach of a time elapsed neural model, Math. Models Methods Appl. Sci, vol.25, pp.2669-2719, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01159215

J. Chevallier, A. Duarte, E. Löcherbach, and G. Ost, Mean field limits for nonlinear spatially extended Hawkes processes with exponential memory kernels, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01489278

F. Delarue, J. Inglis, S. Rubenthaler, and E. Tanré, Global solvability of a networked integrate-and-fire model of McKean-Vlasov type, Ann. Appl. Probab, vol.25, pp.2096-2133, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00747565

F. Delarue, J. Inglis, S. Rubenthaler, and E. Tanré, Particle systems with a singular mean-field selfexcitation. Application to neuronal networks, Stochastic Process. Appl, vol.125, pp.2451-2492, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01001716

A. De-masi, A. Galves, E. Löcherbach, and E. Presutti, Hydrodynamic limit for interacting neurons, J. Stat. Phys, vol.158, pp.866-902, 2015.

J. D. Deuschel and O. Zeitouni, Limiting curves for i.i.d. records, Ann. Probab, vol.23, pp.852-878, 1995.

S. Ditlevsen and E. Löcherbach, Multi-class oscillating systems of interacting neurons, Stochastic Process. Appl, vol.127, pp.1840-1869, 2017.

N. Fournier and E. Löcherbach, On a toy model of interacting neurons, Ann. Inst. Henri Poincaré Probab. Stat, vol.52, pp.1844-1876, 2016.

T. Górski, R. Veltz, M. Galtier, H. Fragnaud, B. Telenczuk et al., Dendritic sodium spikes endow neurons with inverse firing rate response to correlated synaptic activity, J. Comput. Neurosci, vol.45, pp.223-234, 2018.

J. M. Hammersley, A few seedlings of research, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol.I, pp.345-394, 1970.

A. L. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, vol.117, pp.500-544, 1952.

J. Inglis and D. Talay, Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component, SIAM J. Math. Anal, vol.47, pp.3884-3916, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01069398

M. Kac, Foundations of kinetic theory, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol.III, pp.171-197, 1956.

E. R. Kandel, Principles of neural science, 2013.

C. Koch, Biophysics of Computation: Information Processing in Single Neurons, 2004.

B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math, vol.26, pp.206-222, 1977.

E. Luçon, Quenched limits and fluctuations of the empirical measure for plane rotators in random media Electron, J. Probab, vol.16, pp.792-829, 2011.

E. Luçon and W. Stannat, Mean field limit for disordered diffusions with singular interactions, Ann. Appl. Probab, vol.24, pp.1946-1993, 2014.

E. Luçon and W. Stannat, Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction, Ann. Appl. Probab, vol.26, pp.3840-3909, 2016.

H. P. Mckean, Propagation of chaos for a class of non-linear parabolic equations, Lecture series in differential equations, vol.7, pp.41-57, 1967.

S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Lecture Notes in Math, pp.42-95, 1627.

S. Ostojic, N. Brunel, and V. Hakim, Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities, J. Comput. Neurosci, vol.26, pp.369-392, 2009.

K. Pakdaman, M. Thieullen, and G. Wainrib, Fluid limit theorems for stochastic hybrid systems with application to neuron models, Adv. in Appl. Probab, vol.42, pp.761-794, 2010.
DOI : 10.1017/s0001867800050436
URL : https://hal.archives-ouvertes.fr/hal-00555398

A. Renart, N. Brunel, and X. Wang, Mean-field theory of irregularly spiking neuronal populations and working memory in recurrent cortical networks. Computational Neuroscience: A comprehensive approach, CRC Mathematical Biology and Medicine Series, pp.431-490, 2004.

M. Riedler, M. Thieullen, and G. Wainrib, Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models, Electron. J. Probab, vol.17, p.48, 2012.
DOI : 10.1214/ejp.v17-1946
URL : https://doi.org/10.1214/ejp.v17-1946

G. Stuart, N. Spruston, M. Häusser, and . Dendrites, , 2015.

A. Sznitman, InÉcoleIn´InÉcole d' ´ Eté de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math, vol.1464, pp.165-251, 1991.

S. M. Ulam, Monte Carlo calculations in problems of mathematical physics. Modern mathematics for the engineer: Second series, pp.261-281, 1961.

A. M. Versik and S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR, vol.233, pp.1024-1027, 1977.

I. Yakupov and M. Buzdalov, Incremental non-dominated sorting with O(N) insertion for the twodimensional case, IEEE Congress on Evolutionary Computation (CEC), pp.1853-1860, 2015.

N. Fournier, LPMA, UMR 7599, UPMC, vol.188

E. Tanré, route des Lucioles, Email address: Etienne.Tanre@inria.fr R. Veltz: Université Côte d'Azur, INRIA, 2004 route des Lucioles, vol.93, p.6902, 2004.