Mathematical Neuroscience Lab, CIRB – Collège de France
and Institute Jacques Monod, Université Paris Diderot
15, rue Hélène Brion
75013 Paris, FRANCE
office: 385 B
phone: +33 (0)157278054
email: garciadelmolino (at) ijm.univ-paris-diderot (dot) fr
Keywords : Complex Systems, Neuroscience, Random Matrices, Network Theory, Stochastic Processes, Coulomb gases
Random matrices and log-gases
Random matrices appear naturally in the modelling of systems where (i) a large number of interacting agents are involved and (ii) a detailed knowledge of the connections’ properties is lacking. In such cases a reasonable first order approximation is to assume that the connections’ matrix is a random matrix given by a suitable law.
The eigenvalues of certain random matrices have been shown to be distributed as particles interacting with logarithmic repulsion under an external confining potential. We use this analogy in order to better understand the properties of random matrices and their spectra.
Characterizing the influence of network properties on the global emerging behavior of interacting elements constitutes a central question in many areas, from physical to social sciences.
I focus on the study of the dynamics of agents interacting on weighted disordered networks with strong correlations in the connectivity. In particular I consider the case where the connectivity matrix is the sum of a deterministic (structure) part plus a random (disorder) part. In order to understand such systems I principally use results from random matrix theory and dynamical systems together with intensive numerical simulations.
Publications and preprints:
- L.C. García del Molino, K. Pakdaman, J. Touboul, G. Wainrib. The real Ginibre ensemble with k=O(n) real eigenvalues. [arXiv preprint]
- L.C. García del Molino, K. Pakdaman, J. Touboul. The heterogeneous gas with singular interaction: Generalized circular law and heterogeneous renormalized energy, J. Phys. A: Math. Theor. 48 045208 (2015) [arXiv preprint]
- L.C. García del Molino, K. Pakdaman, J. Touboul, G. Wainrib. Synchronization in random balanced networks, Phys. Rev. E 88, 042824 (2013). [arXiv preprint]
- G. Wainrib, L.C. García del Molino. Optimal system size for complex dynamics in random neural networks near criticality, Chaos 23, 043134 (2013). [arXiv preprint]
- L.C. García del Molino, P. Chleboun, S. Grossinsky. Condensation in disordered zero-range processes, J. Phys. A: Math. Theor. 45 205001 (2012). [arXiv preprint]
Talks and posters:
- The heterogeneous gas with singular interaction (Poster), Journées de l’Ecole Doctoral Pierre Luis Lions, Saint-Malo, October 2014.
- Finite size effects and synchronization in random networks, Symposium on open issues in experimental and theoretical neuroscience, Paris V, March 2014.
- Random matrices in models of neural networks, Institute Jacques Monod, January 2014.
- Dynamics of randomly connected systems, Santa Fe Institute Complex Systems Summer School, Zapallar (Chile), November 2013.
- Synchronization in random balanced networks (Poster), Journées de l’Ecole Doctorale Pierre Luis Lions, Saint-Malo, October 2013. Price best poster.
- Synchronization in random balanced networks, Collège de France, June 2013.