## 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.*

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

*Illustration 3 : An epileptic seizure in Jansen and Rit model, with simulation of the injection of convulsant drug.*

## Selected Publications

- 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.

*26, 43–81.*

**J. Nonlinear Sci.**- Ribot, J.*, Romagnoni, A.*, Milleret, C., Bennequin, D. & Touboul, J. (2015), Pinwheel-dipole configuration in cat early visual cortex.

*128, 63–73 (*co-senior authors).*

**Neuroimage**- 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.

*48, 045208.*

**J. Phys. A-Math. Theor.**- 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.

- 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*].

- Godinho D. & Quiñinao C. (2014), Propagation of chaos for a subcritical Keller-Segel Model,

**Annales de l'Institut Henri Poincaré***(in press).*

- Touboul J. (2014), Spatially extended networks with singular multi-scale connectivity patterns,

*(online first, may 2014)*

**Journal of Statistical Physics***156*, 546–573.

- Touboul J. (2014), The propagation of chaos in neural fields,

*, 24 (3), 1298-1328.*

**The Annals of Applied Probability**- Wainrib G. & Touboul J. (2013), Topological and Dynamical Complexity of Random Neural Networks.

**110, 118101.**

*Physical Review Letters*- 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,

**109 (1), 018702.**

*Physical Review Letters*- Touboul J., Hermann G. & Faugeras O. (2012), Noise-induced behaviors in neural mean field dynamics.

**, vol. 11, number 1, pp. 49-81.**

*SIAM Journal on Applied Dynamical Systems*- Touboul J., Wendling F., Chauvel P. & Faugeras O. (2011), Neural Mass Activity, Bifurcations, and Epilepsy.

**, vol.23, number 12, pp. 3232-3286.**

*Neural Computation*- Touboul J. & Ermentrout B. (2011), Finite-size and correlation-induced effects in Mean-field

**, Mar 8.**

*Dynamics Journal of Computational Neuroscience*- Touboul J. & Brette R. (2009), Spiking dynamics of bidimensional integrate-and-fire neurons

**, vol. 8, pages 1462-1506.**

*SIAM Journal on Applied Dynamical Systems*## People

** Group leader: **Touboul Jonathan, CR INRIA

**Associate Members**

Wainrib Gilles, Univ. Paris 13

Krupa Maciej, INRIA Rocquencourt

Sarti Alessandro, DR CNRS, EHESS

**Ribot Jérôme**

**Postdoctoral fellows &****PhD****Students:**Cabana Tanguy, PhD student

Cui Yi, PhD student

Bailleul Richard, PhD student

**Master Students (M1-M2):**

Vignoud Gaëtan, M2, ENS

Piette Charlotte, M2, ENS

Aubin Benjamin, M2, ENS

**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