Finitesize and correlationinduced effects in meanfield dynamics
 Jonathan D. Touboul,
 G. Bard Ermentrout
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The brain’s activity is characterized by the interaction of a very large number of neurons that are strongly affected by noise. However, signals often arise at macroscopic scales integrating the effect of many neurons into a reliable pattern of activity. In order to study such large neuronal assemblies, one is often led to derive meanfield limits summarizing the effect of the interaction of a large number of neurons into an effective signal. Classical meanfield approaches consider the evolution of a deterministic variable, the mean activity, thus neglecting the stochastic nature of neural behavior. In this article, we build upon two recent approaches that include correlations and higher order moments in meanfield equations, and study how these stochastic effects influence the solutions of the meanfield equations, both in the limit of an infinite number of neurons and for large yet finite networks. We introduce a new model, the infinite model, which arises from both equations by a rescaling of the variables and, which is invertible for finitesize networks, and hence, provides equivalent equations to those previously derived models. The study of this model allows us to understand qualitative behavior of such largescale networks. We show that, though the solutions of the deterministic meanfield equation constitute uncorrelated solutions of the new meanfield equations, the stability properties of limit cycles are modified by the presence of correlations, and additional nontrivial behaviors including periodic orbits appear when there were none in the mean field. The origin of all these behaviors is then explored in finitesize networks where interesting mesoscopic scale effects appear. This study leads us to show that the infinitesize system appears as a singular limit of the network equations, and for any finite network, the system will differ from the infinite system.
 Abbott, LF, Vreeswijk, CA (1993) Asynchronous states in networks of pulsecoupled neuron. Physical Review 48: pp. 14831490 CrossRef
 Amari, S. (1972). Characteristics of random nets of analog neuronlike elements. Syst. Man Cybernet., SMC2.
 Amari, SI (1977) Dynamics of pattern formation in lateralinhibition type neural fields. Biological Cybernetics 27: pp. 7787 CrossRef
 Amit, DJ, Brunel, N (1997) Model of global spontaneous activity and local structured delay activity during delay periods in the cerebral cortex. Cerebral Cortex 7: pp. 237252 CrossRef
 Arnold, V.I. (1981). Ordinary differential equations (Chap. 5). MIT Press.
 Arnold, L. (1995). Random dynamical systems. Springer.
 Beggs, JM, Plenz, D (2004) Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. Journal of Neuroscience 24: pp. 52165229 CrossRef
 Benayoun, M, Cowan, JD, Drongelen, W, Wallace, E (2010) Avalanches in a stochastic model of spiking neurons. PLoS Computational Biology 6: pp. e1000846 CrossRef
 Boland, RP, Galla, T, McKane, AJ (2008) How limit cycles and quasicycles are related in systems with intrinsic noise. Journal of Statistical Mechanics: Theory and Experiment 9: pp. P09001 CrossRef
 Bressfloff, P (2010) Stochastic neural field theory and the systemsize expansion. SIAM Journal on Applied Mathematics 70: pp. 14881521 CrossRef
 Bressfloff, P (2010) Metastable states and quasicycles in a stochastic Wilson–Cowan model. Physical Review E 82: pp. 051903 CrossRef
 Brewer, J. (1978). Kronecker products and matrix calculus in system theory. IEEE Transactions on Circuits and Systems, CAS25.
 Brunel, N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons. Journal of Computational Neuroscience 8: pp. 183208 CrossRef
 Brunel, N, Hakim, V (1999) Fast global oscillations in networks of integrateandfire neurons with low firing rates. Neural Computation 11: pp. 16211671 CrossRef
 Brunel, N, Latham, P (2003) Firing rate of noisy quadratic integrateandfire neurons. Neural Computation 15: pp. 22812306 CrossRef
 Buice, MA, Cowan, JD (2007) Field theoretic approach to fluctuation effects for neural networks. Physical Review E 75: pp. 051919 CrossRef
 Buice, MA, Cowan, JD, Chow, CC (2010) Systematic fluctuation expansion for neural network activity equations. Neural Computation 22: pp. 377426 CrossRef
 Cai, D, Tao, L, Shelley, M, McLaughlin, DW (2004) An effective kinetic representation of fluctuationdriven neuronal networks with application to simple and complex cells in visual cortex. Proceedings of the National Academy of Sciences 101: pp. 77577762 CrossRef
 Coombes, S, Owen, MR (2005) Bumps, breathers, and waves in a neural network with spike frequency adaptation. Physical Review Letters 94: pp. 14810211481024 CrossRef
 Doob, JL (1945) Markoff chains–denumerable case. Transactions of the American Mathematical Society 58: pp. 455473
 Dykman, MI, Mori, E, Ross, J, Hunt, PM (1994) Large fluctuations and optimal paths in chemical kinetics. Journal of Chemical Physics 100: pp. 57355750 CrossRef
 Boustani, S, Destexhe, A (2009) A master equation formalism for macroscopic modeling of asynchronous irregular activity states. Neural Computation 21: pp. 46100 CrossRef
 Boustani, S, Destexhe, A (2010) Brain dynamics at multiple scales: can one reconcile the apparent lowdimensional chaos of macroscopic variables with the seemingly stochastic behavior of single neurons?. International Journal of Bifurcation and Chaos 20: pp. 116 CrossRef
 Ermentrout, B (1998) Neural networks as spatiotemporal patternforming systems. Reports on Progress in Physics 61: pp. 353430 CrossRef
 Ermentrout, B. (2002). Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students. Society for Industrial Mathematics.
 Ermentrout, GB, Cowan, JD (1979) A mathematical theory of visual hallucination patterns. Biological Cybernetics 34: pp. 137150 CrossRef
 Faugeras, O, Touboul, J, Cessac, B (2009) A constructive meanfield analysis of multipopulation neural networks with random synaptic weights and stochastic inputs. Frontiers in Neuroscience 3: pp. 1
 Freidlin, M. I., & Wentzell, A. D. (1998). Random perturbations of dynamical systems. Springer Verlag.
 Gaspard, P (2002) Correlation time of mesoscopic chemical clocks. Journal of Chemical Physics 117: pp. 89058916 CrossRef
 Gillespie, DT (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics 22: pp. 403434 CrossRef
 Gillespie, DT (1977) Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81: pp. 23402361 CrossRef
 Guckenheimer, J., & Holmes, P. J. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Applied mathematical sciences (Vol. 42). Springer.
 Horsthemke, W., & Lefever, R. (2006). Noiseinduced transitions. Springer.
 Kurtz, TG (1976) Limit theorems and diffusion approximations for density dependent Markov chains. Mathematical Programming Studies 5: pp. 67
 Kuznetsov, Y. A. (1998). Elements of applied bifurcation theory. Applied Mathematical Sciences (2nd ed.). Springer.
 Laing, CL, Troy, WC, Gutkin, B, Ermentrout, GB (2002) Multiple bumps in a neuronal model of working memory. SIAM Journal on Applied Mathematics 63: pp. 6297 CrossRef
 Levina, A, Herrmann, JM, Geisel, T (2009) Phase transitions towards criticality in a neural system with adaptive interactions. Physical Review Letters 102: pp. 11811011181104 CrossRef
 Ly, C, Tranchina, D (2007) Critical analysis of dimension reduction by a moment closure method in a population density approach to neural network modeling. Neural Computation 19: pp. 20322092 CrossRef
 McKane, AJ, Nagy, JD, Newman, TJ, Stefanini, MO (2007) Amplified biochemical oscillations in cellular systems. Journal of Statistical Physics 71: pp. 165 CrossRef
 Mattia, M, Giudice, P (2002) Population dynamics of interacting spiking neurons. Physical Review E 66: pp. 51917 CrossRef
 Neudecker, H (1969) Some theorems on matrix differentiation with special reference to kronecker matrix products. Journal of the American Statistical Association 64: pp. 953963 CrossRef
 Ohira, T, Cowan, JD (1993) Masterequation approach to stochastic neurodynamics. Physical Review E 48: pp. 22592266 CrossRef
 Plesser, H. E. (1999). Aspects of signal processing in noisy neurons. PhD thesis, GeorgAugustUniversität.
 Rodriguez, R, Tuckwell, HC (1996) A dynamical system for the approximate moments of nonlinear stochastic models of spiking neurons and networks. Mathematical and Computer Modeling 31: pp. 175180 CrossRef
 Rodriguez, R, Tuckwell, HC (1998) Noisy spiking neurons and networks: Useful approximations for firing probabilities and global behavior. Biosystems 48: pp. 187194 CrossRef
 Rolls, E. T., & Deco, G. (2010). The noisy brain: Stochastic dynamics as a principle of brain function. Oxford University Press.
 Softky, WR, Koch, C (1993) The highly irregular firing of cortical cells is inconsistent with temporal integration of random epsps. Journal of Neuroscience 13: pp. 334350
 Teramae, J, Nakao, H, Ermentrout, GB (2009) Stochastic phase reduction for a general class of noisy limit cycle oscillators. Physical Review Letters 102: pp. 194102 CrossRef
 Touboul, J, Destexhe, A (2010) Can powerlaw scaling and neuronal avalanches arise from stochastic dynamics?. PLoS ONE 5: pp. 8982 CrossRef
 Touboul, J, Faugeras, O (2007) The spikes trains probability distributions: a stochastic calculus approach. Journal of Physiology 101: pp. 7898
 Touboul, J, Faugeras, O (2008) First hitting time of double integral processes to curved boundaries. Advances in Applied Probability 40: pp. 501528 CrossRef
 Wainrib, G. (2010). Randomness in neurons: A multiscale probabilistic analysis. PhD thesis, Ecole Polytechnique.
 Wilson, HR, Cowan, JD (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal 12: pp. 124 CrossRef
 Wilson, HR, Cowan, JD (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biological Cybernetics 13: pp. 5580
 Title
 Finitesize and correlationinduced effects in meanfield dynamics
 Journal

Journal of Computational Neuroscience
Volume 31, Issue 3 , pp 453484
 Cover Date
 20111101
 DOI
 10.1007/s1082701103205
 Print ISSN
 09295313
 Online ISSN
 15736873
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Neural mass equations
 Dynamical systems
 Markov process
 Master equation
 Moment equations
 Bifurcations
 Wilson and Cowan system
 Industry Sectors
 Authors

 Jonathan D. Touboul ^{(1)} ^{(2)}
 G. Bard Ermentrout ^{(2)}
 Author Affiliations

 1. NeuroMathComp Laboratory, INRIA/ENS Paris, 23 Avenue d’Italie, 75013, Paris, France
 2. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA