Hostname: page-component-8448b6f56d-42gr6 Total loading time: 0 Render date: 2024-04-25T04:36:37.798Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for randomly connected neural networks: II. State-dependent interactions

Part of: Mathematical biology in general Limit theorems Time-dependent statistical mechanics (dynamic and nonequilibrium) Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  16 November 2018

Tanguy Cabana  and
Jonathan D. Touboul
Tanguy Cabana
Affiliation:
Collège de France
Jonathan D. Touboul*
Affiliation:
Collège de France and Brandeis University
*
* Postal address: Department of Mathematics and Volen National Center for Complex Systems, Brandeis University, 415 South Street, Waltham, MA 02453, USA. Email address: jtouboul@brandeis.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We continue the analysis of large deviations for randomly connected neural networks used as models of the brain. The originality of the model relies on the fact that the directed impact of one particle onto another depends on the state of both particles, and they have random Gaussian amplitude with mean and variance scaling as the inverse of the network size. Similarly to the spatially extended case (see Cabana and Touboul (2018)), we show that under sufficient regularity assumptions, the empirical measure satisfies a large deviations principle with a good rate function achieving its minimum at a unique probability measure, implying, in particular, its convergence in both averaged and quenched cases, as well as a propagation of a chaos property (in the averaged case only). The class of model we consider notably includes a stochastic version of the Kuramoto model with random connections.

Keywords

Large deviationconvergence of measureneural networkdisordered system

MSC classification

Primary: 60F10: Large deviations
Secondary: 60B10: Convergence of probability measures 92B20: Neural networks, artificial life and related topics 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 983 - 1004
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ben Arous, G. and Guionnet, A. (1995). Large deviations for Langevin spin glass dynamics. Prob. Theory Relat. Fields 102, 455–509.Google Scholar
[2]Ben Arous, G. and Guionnet, A. (1997). Symmetric Langevin spin glass dynamics. Ann. Prob. 25, 1367–1422.Google Scholar
[3]Ben Arous, G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Prob. Theory Relat. Fields 120, 1–67.Google Scholar
[4]Cabana, T. and Touboul, J. (2013). Large deviations, dynamics and phase transitions in large stochastic disordered neural networks. J. Statist. Phys. 153, 211–269.Google Scholar
[5]Cabana, T. and Touboul, J. (2018). Large deviations for randomly connected neural networks: I. Spatially extended systems. Adv. Appl. Prob. 50, 944–982.Google Scholar
[6]Carrillo, J. A., Fornasier, M., Toscani, G. and Vecil, F. (2010). Particle, kinetic, and hydrodynamic models of swarming. In Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Birkhäuser, Boston, MA, pp. 297–336.Google Scholar
[7]Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin.Google Scholar
[8]Dörfler, F. and Bullo, F. (2012). Synchronization and transient stability in power networks and nonuniform Kuramoto oscillators. SIAM J. Control Optimization 50, 1616–1642.Google Scholar
[9]Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Springer, New York.Google Scholar
[10]Gerstner, W. and Kistler, W. M. (2002). Mathematical formulations of Hebbian learning. Biol. Cybernet. 87, 404–415.Google Scholar
[11]Guionnet, A. (1997). Averaged and quenched propagation of chaos for spin glass dynamics. Prob. Theory Relat. Fields 109, 183–215.Google Scholar
[12]Kuramoto, Y. (1975). Self-entrainment of a population of coupled non-linear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics, Springer, Berlin, pp. 420–422.Google Scholar
[13]Luçon, E. (2012). Oscillateurs couplés, désordre et synchronisation. Doctoral thesis. Université Pierre et Marie Curie.Google Scholar
[14]Lynch, J. and Sethuraman, J. (1987). Large deviations for processes with independent increments. Ann. Prob. 15, 610–627.Google Scholar
[15]Moynot, O. and Samuelides, M. (2002). Large deviations and mean-field theory for asymmetric random recurrent neural networks. Prob. Theory Relat. Fields 123, 41–75.Google Scholar
[16]Neveu, J. (1968). Processus aléatoires gaussiens. Presses de l'Université de Montréal.Google Scholar
[17]Protter, P. E. (2005). Stochastic differential equations. In Stochastic Integration and Differential Equations, Springer, Berlin, pp. 249–361.Google Scholar
[18]Pukhalskii, A. A. (1991). On functional principle of large deviations. In New Trends in Probability and Statistics, VSP International Science Publishers, Moscow, pp. 198–218.Google Scholar
[19]Strogatz, S. (2003). Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life. Hyperion, New York.Google Scholar
[20]Strogatz, S. H. (2000). From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143, 1–20.Google Scholar
[21]Sznitman, A.-S. (1984). Équations de type de Boltzmann, spatialement homogènes. Z. Wahrscheinlichkeitsth. 66, 559–592.Google Scholar