Mathematical Neuroscience Team

Principal Investigator: Jonathan TOUBOUL, CR Inria

The mathematical neuroscience laboratory is interested in the theoretical understanding of the functioning of brain and neuron through the analysis of mathematical models. We focus on the modeling of brain and neuron functions and on the mathematical analysis of such systems. A central questions we address in our lab is the emergence of macroscopic behaviors from the interaction of noisy, unreliable cells.

In order to address this central question, our lab particularly focuses on the nonlinear and stochastic properties of the brain activity, at different scales. At the microscopic scale, individual neurons are adequately described by nonlinear or stochastic processes. We investigate the properties of individual cells described through mathematical models, such as for instance hybrid dynamical systems coupling continuous and discrete dynamics.


Illustration 1 : Excitability type (top), and spike dynamics (bottom), illustrating bursting and chaotic firing in the adaptive exponential neuron model.

We analyze the link between this microscopic scale and large-scale neuronal networks through asymptotic regimes characterized by infinite numbers of neurons. These limit give access to the behavior of finite-populations or spatially extended systems. Obtaining these limit characterizations is one of the aspects of our research, together with the analysis of finite-size effects, i.e. qualitative differences between networks with a large but finite number of neurons and the asymptotic regimes.

Illustration 2 : Qualitative transitions in the activity of spatially extended neuronal networks of firing-rate neurons with 2 layers, as noise is increased.

These mathematical approaches then are applied to current neuroscience questions, such as the identification of parameters related to epileptic behaviors or the presence of self-organized criticality in neuronal networks.
Illustration 3 : An epileptic seizure in Jansen and Rit model, with simulation of the injection of convulsant drug.

Selected Publications

- Touboul, J. & Destexhe, A., (2017), Power-law statistics and universal scaling in the absence of criticality. Phys. Rev. E 95, 012413.

- Touboul, J., Romagnoni, A. & Schwartz, R. (2017), On the Dynamical Interplay of Positive and Negative Affects. Neural Comput 29, 897–936.

- Mischler, S., Quininao, C. & Touboul, J. (2016), On a Kinetic Fitzhugh-Nagumo Model of Neuronal Network. Commun. Math. Phys. 342, 1001–1042. 

- Krupa, M. & Touboul, J.D. (2016), Complex Oscillations in the Delayed FitzHugh-Nagumo Equation. J. Nonlinear Sci. 26, 43–81.

- Ribot, J.*, Romagnoni, A.*, Milleret, C., Bennequin, D. & Touboul, J. (2015), Pinwheel-dipole configuration in cat early visual cortex. Neuroimage 128, 63–73 (*co-senior authors).

- Romagnoni, A., Ribot, J., Bennequin, D. & Touboul, J. (2015), Parsimony, Exhaustivity and Balanced Detection in Neocortex. PLoS Comput. Biol. 11, e1004623.

- Hsu, L.C.-L., Nam, S., Cui, Y., Chang, C.-P., Wang, C.-F., Kuo, H.-C., Touboul, J.D.& Chou, S.-J. (2015), Lhx2 regulates the timing of beta-catenin-dependent cortical neurogenesis. Proc. Natl. Acad. Sci. U. S. A. 112, 12199–12204.

- Perthame B., Quininao C. & Touboul J. (2015), Competition and boundary formation in heterogeneous media: Application to neuronal differentiation. Math. Models Meth. Appl. Sci. 25, 2477–2502. 

- Del Molino L.C.G., Pakdaman K. & Touboul J. (2015), The heterogeneous gas with singular interaction: generalized circular law and heterogeneous renormalized energy. J. Phys. A-Math. Theor. 48, 045208.

- Quininao C. & Touboul J., (2015), Limits and Dynamics of Randomly Connected Neuronal Networks. Acta Appl. Math. 136, 167–192.

- Quiñinao C., Prochiantz A. & Touboul J. (2015), Local homeoprotein diffusion can stabilize boundaries generated by graded positional cues. Development 142, 1860–1868. 

- Godinho D. & Quiñinao C. (2015), Propagation of chaos for a subcritical Keller-Segel Model, Ann. Inst. H. Poincaré Probab. Statist. 51, 965–992. 

- Allez, R., Touboul, J., & Wainrib, G. (2014), Index Distribution of the Ginibre Ensemble. Journal of Physics A: Mathematical and Theoretical 47, 042001.

- Faye, G., & Touboul, J. (2014), Pulsatile localized dynamics in delayed neural-field equations in arbitrary dimension. [onlin, Q-Bio].

- Touboul J. (2014), Spatially extended networks with singular multi-scale connectivity patterns, Journal of Statistical Physics (online first, may 2014) 156, 546–573.

- Touboul J. (2014), The propagation of chaos in neural fields, The Annals of Applied Probability, 24 (3), 1298-1328.

- Wainrib G. & Touboul J. (2013), Topological and Dynamical Complexity of Random Neural Networks. Physical Review Letters 110, 118101.

- Touboul J. (2012b), Limits and dynamics of stochastic neuronal networks with random delays. Journal of Statistical Physics 149, 569-597.

- Hermann G. & Touboul J. (2012), Heterogeneous connections induce oscillations in large scale networks, Physical Review Letters 109 (1), 018702.

- Touboul J., Hermann G. & Faugeras O. (2012), Noise-induced behaviors in neural mean field dynamics. SIAM Journal on Applied Dynamical Systems, vol. 11, number 1, pp. 49-81.

- Touboul J., Wendling F., Chauvel P. & Faugeras O. (2011), Neural Mass Activity, Bifurcations, and Epilepsy. Neural Computation, vol.23, number 12, pp. 3232-3286.

- Touboul J. & Ermentrout B. (2011), Finite-size and correlation-induced effects in Mean-field Dynamics Journal of Computational Neuroscience, Mar 8.

- Touboul J. & Brette R. (2009), Spiking dynamics of bidimensional integrate-and-fire neurons SIAM Journal on Applied Dynamical Systems, vol. 8, pages 1462-1506.


Group leader:
Touboul Jonathan, CR INRIA

Associate Members
Wainrib Gilles, Univ. Paris 13
Krupa Maciej, INRIA Rocquencourt
Sarti Alessandro, DR CNRS, EHESS

Postdoctoral fellows & PhD Students:
Ribot Jérôme
Cui Yi, PhD student
Bailleul Richard, PhD student

Master Students (M1-M2):
Vignoud Gaëtan, M2, ENS
Piette Charlotte, M2, ENS
Aubin Benjamin, M2, ENS
Habis Julien, M2

External Collaborators
Ermentrout Bard, Univ. Pittsburgh (USA)
Taillefumier Thibaud, Univ. Princeton (USA)
Faye Gregory, Univ. Minnesota (USA)
Prochiantz Alain (Collège de France)
Robert Philippe, INRIA Rocquencourt (Paris)
Perthame Benoît, UPMC (Paris)
Krikorian Raphaël, UPMC (Paris)
Mischler Stéphane, Univ. Paris-Dauphine (Paris)
Pakdaman Khashayar, Institut Jacques Monod (Paris)

Former Members
Hermann Geoffroy (PhD obtained in 2012) 
Scher Justine (aggrégation of mathematics obtained in 2012)
Quan Shi, Ecole Polytechnique
De Souza Flavio, Ecole Polytechnique 
Moreau Nathan, Ecole Polytechnique
Bailleul Richard, ENS Cachan