Attractor and saddle node dynamics in heterogeneous neural fields

详细信息    查看全文

• 作者：Peter beim Graben (7) (8)
  Axel Hutt (9)
  
  7. Dept. of German Studies and Linguistics ; Humboldt-Universit盲t zu Berlin ; Unter den Linden 6 ; 10099 ; Berlin ; Germany
  8. Team Neurosys ; INRIA Grand Est - Nancy ; 615 rue du Jardin Botanique ; 54602 ; Villers-les-Nancy ; France
  9. Bernstein Center for Computational Neuroscience Berlin ; Berlin ; Germany
  
• 关键词：Chaotic itinerancy ; Linear stability ; Heteroclinic orbits ; Lotka ; Volterra model
• 刊名：EPJ Nonlinear Biomedical Physics
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：2
• 期：1
• 全文大小：568 KB
• 参考文献：1. Wilson, H, Cowan, J (1973) A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Kybernetik 13: pp. 55-80 CrossRef
  2. Amari, SI (1977) Dynamics of pattern formation in lateral-inhibition type neural fields. Biol Cybern 27: pp. 77-87 CrossRef
  3. Jancke, D, Erlhagen, W, Dinse, HR, Akhavan, AC, Giese, M, Steinhage, A, Sch枚ner, G (1999) Parametric population representation of retinal location: neuronal interaction dynamics in cat primary visual cortex. J Neurosci 19: pp. 9016-9028
  4. Bressloff, PC (2012) Spatiotemporal dynamics of continuum neural fields. J Phys A 45: pp. 033001 CrossRef
  5. Coombes, S (2010) Large-scale neural dynamics: simple and complex. NeuroImage 52: pp. 731-739 CrossRef
  6. Coombes, S, Venkov, NA, Shiau, L, Bojak, I, Liley, DTJ, Laing, CR (2007) Modeling electrocortical activity through improved local approximations of integral neural field equations. Phys Rev E 76: pp. 051901 CrossRef
  7. Coombes, S, Lord, G, Owen, M (2003) Waves and bumps in neuronal networks with axo-dendritic synaptic interactions. Physica D 178: pp. 219-241 CrossRef
  8. Folias, S, Bressloff, P (2005) Stimulus-locked waves and breathers in an excitatory neural network. SIAM J Appl Math 65: pp. 2067-2092 615171" target="_blank" title="It opens in new window">CrossRef
  9. Hutt, A, Rougier, N (2010) Activity spread and breathers induced by finite transmission speeds in two-dimensional neural fields. Phys Rev E 82: pp. R055701 CrossRef
  10. Coombes, S, Owen, M (2005) Bumps, breathers, and waves in a neural network with spike frequency adaptation. Phys Rev Lett 94: pp. 148102 CrossRef
  11. Ermentrout, GB, McLeod, JB (1993) Existence and uniqueness of travelling waves for a neural network. Proc R Soc E 123A: pp. 461-478 CrossRef
  12. Jirsa, VK, Haken, H (1996) Field theory of electromagnetic brain activity. Phys Rev Lett 77: pp. 960-963 CrossRef
  13. Hutt, A (2007) Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds. Phys Rev E 75: pp. 026214 CrossRef
  14. Bressloff, PC (2001) Traveling fronts and wave propagation failure in an inhomogeneous neural network. Physica D 155: pp. 83-100 CrossRef
  15. Jirsa, VK, Kelso, JAS (2000) Spatiotemporal pattern formation in neural systems with heterogeneous connection toplogies. Phys Rev E 62: pp. 8462-8465 CrossRef
  16. Kilpatrick, ZP, Folias, SE, Bressloff, PC (2008) Traveling pulses and wave propagation failure in inhomogeneous neural media. SIAM J Appl Dynanmical Syst 7: pp. 161-185 CrossRef
  17. Schmidt, H, Hutt, A, Schimansky-Geier, L (2009) Wave fronts in inhomogeneous neural field models. Physica D 238: pp. 1101-1112 CrossRef
  18. Potthast, R, beim Graben, P (2009) Inverse problems in neural field theory. SIAM J Appl Dynamical Syst 8: pp. 1405-1433 CrossRef
  19. Potthast, R, beim Graben, P (2010) Existence and properties of solutions for neural field equations. Math Methods Appl Sci 33: pp. 935-949
  20. Coombes, S, Laing, C, Schmidt, H, Svanstedt, N, Wyller, J (2012) Waves in random neural media. Discrete Contin Dyn Syst A 32: pp. 2951-2970 CrossRef
  21. Coombes, S, Laing, C (2011) Pulsating fronts in periodically modulated neural field models. Phys Rev E 83: pp. 011912 CrossRef
  22. Brackley, C, Turner, M (2009) Persistent fluctuations of activity in undriven continuum neural field models with power-law connections. Phys Rev E 79: pp. 011918 CrossRef
  23. beim Graben, P, Potthast, R (2009) Inverse problems in dynamic cognitive modeling. Chaos 19: pp. 015103 CrossRef
  24. Hutt, A, Riedel, H (2003) Analysis and modeling of quasi-stationary multivariate time series and their application to middle latency auditory evoked potentials. Physica D 177: pp. 203-232 CrossRef
  25. Yildiz, I, Kiebel, SJ (2011) A hierarchical neuronal model for generation and online recognition of birdsongs. PloS Comput Biol 7: pp. e1002303 CrossRef
  26. Veltz, R, Faugeras, O (2010) Local/global analysis of the stationary solutions of some neural field equations. SIAM J Appl Dynamical Syst 9: pp. 954-998 CrossRef
  27. Afraimovich, VS, Zhigulin, VP, Rabinovich, MI (2004) On the origin of reproducible sequential activity in neural circuits. Chaos 14: pp. 1123-1129 CrossRef
  28. Rabinovich, MI, Huerta, R, Varona, P, Afraimovichs, VS (2008) Transient cognitive dynamics, metastability, and decision making. PLoS Comput Biolog 4: pp. e1000072 CrossRef
  29. Hammerstein, A (1930) Nichtlineare Integralgleichungen nebst Anwendungen. Acta Math 54: pp. 117-176 CrossRef
  30. Kosko, B (1988) Bidirectional associated memories. IEEE Trans Syst Man Cybernet 18: pp. 49-60 CrossRef
  31. Hellwig, B (2000) A quantitative analysis of the local connectivity between pyramidal neurons in layers 2/3 of the rat visual cortex. Biol Cybernet 82: pp. 11-121 CrossRef
  32. Mazor, O, Laurent, G (2005) Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons. Neuron 48: pp. 661-673 CrossRef
  33. Rabinovich, MI, Huerta, R, Laurent, G (2008) Transient dynamics for neural processing. Science 321: pp. 48-50 CrossRef
  34. Kiebel, SJ, von Kriegstein, K, Daunizeau, J, Friston, KJ (2009) Recognizing sequences of sequences. Plos Comp Biol 5: pp. e1000464 CrossRef
  35. Desroches, M, Guckenheimer, J, Krauskopf, B, Kuehn, C, Osinga, H, Wechselberger, M (2012) Mixed-mode oscillations with multiple time scales. SIAM Rev 54: pp. 211-288 CrossRef
  36. Tsuda, I (2001) Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systems. Behav Brain Sci 24: pp. 793-847 CrossRef
  37. Freeman, W (2003) Evidence from human scalp EEG of global chaotic itinerancy. Chaos 13: pp. 1069 CrossRef
  38. Appell, J, Chen, CJ (2006) How to solve Hammerstein equations. J Integr Equat Appl 18: pp. 287-296 CrossRef
  39. Banas, J (1989) Integrable solutions of Hammerstein and Urysohn integral equations. J Austral Math Soc (Series A) 46: pp. 61-68 CrossRef
  40. Lakestani, M, Razzaghi, M, Dehghan, M (2005) Solution of nonlinear Fredholm-Hammerstein integral equations by using semiorthogonal spline wavelets. Math Problems Eng. pp. 113-121
  41. Djitteab, N, Senea, M (2013) An iterative algorithm for approximating solutions of Hammerstein integral equations. Numerical Funct Anal Optimization 34: pp. 1299-1316 CrossRef
  42. Hutt, A, Longtin, A (2010) Effects of the anesthetic agent propofol on neural populations. Cogn Neurodyn 4: pp. 37-59 CrossRef
  43. Bressloff, PC, Coombes, S (1997) Physics of the extended neuron. Int J Mod Phys B 11: pp. 2343-2392 CrossRef
  44. beim Graben, P, Potthast, R A dynamic field account to language-related brain potentials. In: Rabinovich, M, Friston, K, Varona, P eds. (2012) Principles of Brain Dynamics: Global State Interactions. MIT Press, Cambridge (MA), pp. 93-112
  45. Haken, H (1983) Synergetics. An Introduction Volume 1 of Springer Series in Synergetics. Springer, Berlin
  46. Fukai, T, Tanaka, S (1997) A simple neural network exhibiting selective activation of neuronal ensembles: from winner-take-all to winners-share-all. Neural Comp 9: pp. 77-97 CrossRef
  47. Wilson, H, Cowan, J (1972) Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12: pp. 1-24 CrossRef
  
• 刊物主题：Biological Networks, Systems Biology; Systems Biology; Statistical Physics, Dynamical Systems and Complexity;
• 出版者：Springer Berlin Heidelberg
• ISSN：2195-0008

文摘

Background We present analytical and numerical studies on the linear stability of spatially non-constant stationary states in heterogeneous neural fields for specific synaptic interaction kernels. Methods The work shows the linear stabiliy analysis of stationary states and the implementation of a nonlinear heteroclinic orbit. Results We find that the stationary state obeys the Hammerstein equation and that the neural field dynamics may obey a saddle-node bifurcation. Moreover our work takes up this finding and shows how to construct heteroclinic orbits built on a sequence of saddle nodes on multiple hierarchical levels on the basis of a Lotka-Volterra population dynamics. Conclusions The work represents the basis for future implementation of meta-stable attractor dynamics observed experimentally in neural population activity, such as Local Field Potentials and EEG.