Limits and Dynamics of Randomly Connected Neuronal Networks

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• 作者：Cristobal Qui帽inao (1) (3)
  Jonathan Touboul (1) (2)
  
  1. Mathematical Neuroscience Team ; CIRB-Coll猫ge de France ; 11 ; place Marcelin Berthelot ; 75005 ; Paris ; France
  3. Laboratoire Jacques-Louis Lions ; Sorbonne Universit茅s ; UPMC Univ Paris 06 ; CNRS UMR 7598 ; 75005 ; Paris ; France
  2. INRIA Mycenae Team ; Domaine de Voluceau ; B.P. 105 ; Rocquencourt ; France
  
• 关键词：Heterogeneous neuronal networks ; Mean ; field limits ; Delay differential equations ; Bifurcations ; 82C22 ; 82C44 ; 37N25
• 刊名：Acta Applicandae Mathematicae
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：136
• 期：1
• 页码：167-192
• 全文大小：7,489 KB
• 参考文献：1. Aradi, I., Soltesz, I. (2002) Modulation of network behaviour by changes in variance in interneuronal properties. J. Physiol. 538: pp. 227 CrossRef
  2. Babb, T., Pretorius, J., Kupfer, W., Crandall, P. (1989) Glutamate decarboxylase-immunoreactive neurons are preserved in human epileptic hippocampus. J. Neurosci. 9: pp. 2562-2574
  3. Bassett, D.S., Bullmore, E. (2006) Small-world brain networks. Neuroscientist 12: pp. 512-523 CrossRef
  4. Bettus, G., Wendling, F., Guye, M., Valton, L., R茅gis, J., Chauvel, P., Bartolomei, F. (2008) Enhanced eeg functional connectivity in mesial temporal lobe epilepsy. Epilepsy Res. 81: pp. 58-68 CrossRef
  5. Bosking, W., Zhang, Y., Schofield, B., Fitzpatrick, D. (1997) Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17: pp. 2112-2127
  6. Bressloff, P.C. (2012) Spatiotemporal dynamics of continuum neural fields. J. Phys. A, Math. Theor. 45: CrossRef
  7. Bullmore, E., Sporns, O. (2009) Complex brain networks: graph theoretical analysis of structural and functional systems. Nat. Rev. Neurosci. 10: pp. 186-198 CrossRef
  8. Buzsaki, G. (2004) Rhythms of the Brain. Oxford University Press, Oxford
  9. Prato, G., Zabczyk, J. (1992) Stochastic Equations in Infinite Dimensions. Cambridge Univ Press, Cambridge CrossRef
  10. Dobrushin, R. (1970) Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15: pp. 458-486 CrossRef
  11. Ermentrout, G., Cowan, J. (1979) Temporal oscillations in neuronal nets. J. Math. Biol. 7: pp. 265-280 CrossRef
  12. Ermentrout, G., Terman, D. (2010) Mathematical Foundations of Neuroscience. CrossRef
  13. Ermentrout, G.B., Terman, D. (2010) Foundations of Mathematical Neuroscience. Springer, Berlin CrossRef
  14. FitzHugh, R. (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biol. 17: pp. 257-278
  15. Gray, C.M., K枚nig, P., Engel, A.K., Singer, W. (1989) Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties. Nature 338: pp. 334-337 CrossRef
  16. Hodgkin, A., Huxley, A. (1939) Action potentials recorded from inside a nerve fibre. Nature 144: pp. 710-711 CrossRef
  17. Hodgkin, A., Huxley, A. (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: pp. 500-544 CrossRef
  18. Jansen, B.H., Rit, V.G. (1995) Electroencephalogram and visual evoked potential generation in a mathematical model of coupled cortical columns. Biol. Cybern. 73: pp. 357-366 CrossRef
  19. Mao, X. (2008) Stochastic Differential Equations and Applications. Horwood, Chichester CrossRef
  20. Munoz, A., Mendez, P., DeFelipe, J., Alvarez-Leefmans, F. (2007) Cation-chloride cotransporters and gaba-ergic innervation in the human epileptic hippocampus. Epilepsia 48: pp. 663-673 CrossRef
  21. Noebels, J. (1996) Targeting epilepsy genes minireview. Neuron 16: pp. 241-244 CrossRef
  22. Ohki, K., Chung, S., Ch鈥檔g, Y., Kara, P., Reid, R. (2005) Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature 433: pp. 597-603 CrossRef
  23. Schnitzler, A., Gross, J. (2005) Normal and pathological oscillatory communication in the brain. Nat. Rev. Neurosci. 6: pp. 285-296 CrossRef
  24. Shpiro, A., Curtu, R., Rinzel, J., Rubin, N. (2007) Dynamical characteristics common to neuronal competition models. J. Neurophysiol. 97: pp. 462-473 CrossRef
  25. Sznitman, A. (1984) Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56: pp. 311-336 CrossRef
  26. Sznitman, A.: Topics in propagation of chaos. Ecole d鈥橢t茅 de Probabilit茅s de Saint-Flour XIX, pp. 165鈥?51 (1989)
  27. Tanaka, H. (1978) Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Probab. Theory Relat. Fields 46: pp. 67-105
  28. Touboul, J. (2012) Limits and dynamics of stochastic neuronal networks with random delays. J. Stat. Phys. 149: pp. 569-597 CrossRef
  29. Touboul, J. (2014) The propagation of chaos in neural fields. Ann. Appl. Probab. 24: pp. 1298-1327 CrossRef
  30. Touboul, J.: On the dynamics of mean-field equations for stochastic neural fields with delays
  31. Touboul, J., Hermann, G., Faugeras, O. (2011) Noise-induced behaviors in neural mean field dynamics. SIAM J. Appl. Dyn. Syst. 11: pp. 49-81 CrossRef
  32. Wilson, H., Cowan, J. (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys. J. 12: pp. 1-24 CrossRef
  33. Wilson, H., Cowan, J. (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern. 13: pp. 55-80
  
• 刊物主题：Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics;
• 出版者：Springer Netherlands
• ISSN：1572-9036

文摘

Networks of the brain are composed of a very large number of neurons connected through a random graph and interacting after random delays that both depend on the anatomical distance between cells. In order to comprehend the role of these random architectures on the dynamics of such networks, we analyze the mesoscopic and macroscopic limits of networks with random correlated connectivity weights and delays. We address both averaged and quenched limits, and show propagation of chaos and convergence to a complex integral McKean-Vlasov equations with distributed delays. We then instantiate a completely solvable model illustrating the role of such random architectures in the emerging macroscopic activity. We particularly focus on the role of connectivity levels in the emergence of periodic solutions.