The Annals of Applied Probability

Propagation of chaos in neural fields

Jonathan Touboul

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We consider the problem of the limit of bio-inspired spatially extended neuronal networks including an infinite number of neuronal types (space locations), with space-dependent propagation delays modeling neural fields. The propagation of chaos property is proved in this setting under mild assumptions on the neuronal dynamics, valid for most models used in neuroscience, in a mesoscopic limit, the neural-field limit, in which we can resolve the quite fine structure of the neuron’s activity in space and where averaging effects occur. The mean-field equations obtained are of a new type: they take the form of well-posed infinite-dimensional delayed integro-differential equations with a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We also show how these intricate equations can be used in practice to uncover mathematically the precise mesoscopic dynamics of the neural field in a particular model where the mean-field equations exactly reduce to deterministic nonlinear delayed integro-differential equations. These results have several theoretical implications in neuroscience we review in the discussion.

Article information

Ann. Appl. Probab., Volume 24, Number 3 (2014), 1298-1328.

First available in Project Euclid: 23 April 2014

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Zentralblatt MATH identifier

Primary: 60F99: None of the above, but in this section 60B10: Convergence of probability measures
Secondary: 34C15: Nonlinear oscillations, coupled oscillators

Mean-field limits propagation of chaos delayed stochastic differential equations infinite-dimensional stochastic processes neural fields


Touboul, Jonathan. Propagation of chaos in neural fields. Ann. Appl. Probab. 24 (2014), no. 3, 1298--1328. doi:10.1214/13-AAP950.

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  • [1] Amari, S. (1972). Characteristics of random nets of analog neuron-like elements. IEEE Trans. Syst. Man Cybern. 2 643–657.
  • [2] Bosking, W. H., Zhang, Y., Schofield, B. and Fitzpatrick, D. (1997). Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Comput. Neurosci. 17 2112–2127.
  • [3] Bressloff, P. C. (2012). Spatiotemporal dynamics of continuum neural fields. J. Phys. A 45 033001, 109.
  • [4] Buzsáki, G. (2006). Rhythms of the Brain. Oxford Univ. Press, Oxford.
  • [5] Coombes, S. and Laing, C. (2009). Delays in activity-based neural networks. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 367 1117–1129.
  • [6] Coombes, S. and Owen, M. R. (2005). Bumps, breathers, and waves in a neural network with spike frequency adaptation. Phys. Rev. Lett. 94 148102.
  • [7] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, Cambridge.
  • [8] Destexhe, A., Mainen, Z. F. and Sejnowski, T. J. (1994). An efficient method for computing synaptic conductances based on a kinetic model of receptor binding. Neural Comput. 6 14–18.
  • [9] Dobrushin, R. L. (1970). Prescribing a System of Random Variables by Conditional Distributions. Theory of Probability and Its Applications 15.
  • [10] Ecker, A. S., Berens, P., Keliris, G. A., Bethge, M., Logothetis, N. K. and Tolias, A. S. (2010). Decorrelated neuronal firing in cortical microcircuits. Science 327 584–587.
  • [11] Ermentrout, G. B. and Cowan, J. D. (1979). Temporal oscillations in neuronal nets. J. Math. Biol. 7 265–280.
  • [12] Ermentrout, G. B. and Cowan, J. D. (1980). Large scale spatially organized activity in neural nets. SIAM J. Appl. Math. 38 1–21.
  • [13] Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience. Springer, New York.
  • [14] FitzHugh, R. (1969). Mathematical Models of Excitation and Propagation in Nerve. McGraw-Hill, New York.
  • [15] Fregnac, Y., Blatow, M., Changeux, J. P., De Felipe, J., Lansner, A., Maass, W., Mc Cormick, D. A., Michel, C. M., Monyer, H., Szathmáry, E. and R., Yuste (2006). Ups and downs in cortical computation. In Microcircuits: The Interface Between Neurons and Global Brain Function 393–433. MIT Press, Cambridge, MA.
  • [16] Goldwyn, J. H., Imennov, N. S., Famulare, M. and Shea-Brown, E. (2011). Stochastic differential equation models for ion channel noise in Hodgkin–Huxley neurons. Phys. Rev. E (3) 83 041908.
  • [17] Hodgkin, A. L. and Huxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117 500–544.
  • [18] Hubel, D. H., Wiesel, T. N. and Stryker, M. P. (1978). Anatomical demonstration of orientation columns in macaque monkey. J. Comput. Neurosci. 177 361–380.
  • [19] Izhikevich, E. M. (2006). Polychronization: Computation with spikes. Neural Comput. 18 245–282.
  • [20] Kandel, E. R., Schwartz, J. H. and Jessel, T. M. (2000). Principles of Neural Science, 4th ed. McGraw-Hill, New York.
  • [21] Laing, C. R., Troy, W. C., Gutkin, B. and Ermentrout, G. B. (2002). Multiple bumps in a neuronal model of working memory. SIAM J. Appl. Math. 63 62–97 (electronic).
  • [22] Mao, X. (2002). A note on the LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 268 125–142.
  • [23] McKean, H. P. Jr. (1966). A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 1907–1911.
  • [24] Mountcastle, V. B. (1997). The columnar organization of the neocortex. Brain 120 701–722.
  • [25] Renart, A., De la Rocha, J., Bartho, P., Hollender, L., Parga, N., Reyes, A. and Harris, K. D. (2010). The asynchronous state in cortical circuits. Science 327 587–590.
  • [26] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin.
  • [27] Series, P., Georges, S., Lorenceau, J. and Frégnac, Y. (2002). Orientation dependent modulation of apparent speed: A model based on the dynamics of feed-forward and horizontal connectivity in v1 cortex. Vision Research 42 2781–2797.
  • [28] Sznitman, A.-S. (1984). Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56 311–336.
  • [29] Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX 165–251. Springer, Berlin.
  • [30] Tanaka, H. (1983). Some probabilistic problems in the spatially homogeneous Boltzmann equation. In Theory and Application of Random Fields (Bangalore, 1982) 258–267. Springer, Berlin.
  • [31] Thorpe, S., Delorme, A. and VanRullen, R. (2001). Spike based strategies for rapid processing. Neural Netw. 14 715–726.
  • [32] Touboul, J. (2012). Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions. Phys. D 241 1223–1244.
  • [33] Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics, Vol. I 71–305. North-Holland, Amsterdam.
  • [34] Wilson, H. R. and Cowan, J. D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12 1–24.
  • [35] Wilson, H. R. and Cowan, J. D. (1973). A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybernet. 13 55–80.