3 Reduced Phase Models of Oscillatory Neural Networks
53
36. A. Ghosh, Y. Rho, A.R. McIntosh, R. Kötter, V.K. Jirsa, Noise during rest enables the exploration of the brain’s dynamic repertoire. PLoS Comput. Biol. 4(10), e1000196 (2008)
37. C.M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions. J. Comput. Neurosci. 1(1–2), 11 (1994)
38. S. Grossberg, Nonlinear neural networks: principles, mechanisms, and architectures. Neural
Netw. 1(1), 17 (1988)
39. P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli, C.J. Honey, V.J. Wedeen, O. Sporns, Mapping the structural core of human cerebral cortex. PLoS Biol. 6(7), e159 (2008)
40. H. Haken, Advanced Synergetics: Instability Hierarchies of Self-organizing Systems and
Devices. Springer Series in Synergetics (Springer, Berlin, 1983)
41. H. Haken, Synergetics: Introduction and Advanced Topics (Springer Science & Business Media,
2013)
42. H. Haken, Synergetics: Introduction and Advanced Topics (Springer, Berlin, 2004)
43. J. Hlinka, S. Coombes, Using computational models to relate structural and functional brain
connectivity. Eur. J. Neurosci. 36(2), 2137 (2012)
44. C.J. Honey, R. Kötter, M. Breakspear, O. Sporns, Network structure of cerebral cortex shapes
functional connectivity on multiple time scales. Proc. Natl. Acad. Sci. 104(24), 10240 (2007)
45. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks (Springer, New York,
1997)
46. E.M. Izhikevich, Dynamical Systems in Neuroscience (MIT press, 2007)
47. V.K. Jirsa, Neural field dynamics with local and global connectivity and time delay. Philos.
Trans. R. Soc. Lond. A: Math., Phys. Eng. Sci. 367(1891), 1131 (2009)
48. A.N. Khambhati, A.E. Sizemore, R.F. Betzel, D.S. Bassett, Modeling and interpreting
mesoscale network dynamics. NeuroImage 180, 337 (2018)
49. H. Kori, Y. Kuramoto, S. Jain, I.Z. Kiss, J.L. Hudson, Clustering in globally coupled oscillators
near a Hopf bifurcation: theory and experiments. Phys. Rev. E 89(6), 062906 (2014)
50. A. Koseska, E. Volkov, J. Kurths, Oscillation quenching mechanisms: amplitude versus oscillation death. Phys. Rep. 531(4), 173 (2013)
51. Y. Kuramoto, Chemical Oscillations, Turbulence and Waves (Springer, Berlin, 1984)
52. W. Kurebayashi, S. Shirasaka, H. Nakao, Phase reduction method for strongly perturbed limit
cycle oscillators. Phys. Rev. Lett. 111(21), 214101 (2013)
53. Y.A. Kuznetsov, Elements of applied bifurcation theory (Springer, New York, 1998)
54. C.W. Lynn, D.S. Bassett, The physics of brain network structure, function and control. Nat.
Rev. Phys. 1(5), 318 (2019)
55. W.A. MacKay, Synchronized neuronal oscillations and their role in motor processes. Trends
Cogn. Sci. 1(5), 176 (1997)
56. I. Malkin, Methods of Poincaré and Liapunov in Theory of Non-Linear Oscillations (Gostexizdat, Moscow, 1949)
57. B. Monga, D. Wilson, T. Matchen, J. Moehlis, Phase reduction and phase-based optimal control
for biological systems: a tutorial. Biol. Cybern. (2018)
58. J.Y. Moon, J. Kim, T.W. Ko, M. Kim, Y. Iturria-Medina, J.H. Choi, J. Lee, G.A. Mashour,
U. Lee, Structure shapes dynamics and directionality in diverse brain networks: mathematical
principles and empirical confirmation in three species. Sci. Rep. 7, 46606 EP (2017)
59. J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems (Springer Science &
Business Media, 2006)
60. H. Nakao, Phase reduction approach to synchronisation of nonlinear oscillators. Contemp.
Phys. 57(2), 188 (2016)
61. M. Newman, Networks (Oxford University Press, Oxford, 2018)
62. M. Newman, A.L. Barabási, D.J. Watts, The Structure and Dynamics of Networks, vol. 19
(Princeton University Press, Princeton, 2011)
63. V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, V. Latora, Remote synchronization
reveals network symmetries and functional modules. Phys. Rev. Lett. 110(17), 174102 (2013)
64. J.M. Palva, S. Palva, K. Kaila, Phase synchrony among neuronal oscillations in the human
cortex. J. Neurosci. 25(15), 3962 (2005)
53
36. A. Ghosh, Y. Rho, A.R. McIntosh, R. Kötter, V.K. Jirsa, Noise during rest enables the exploration of the brain’s dynamic repertoire. PLoS Comput. Biol. 4(10), e1000196 (2008)
37. C.M. Gray, Synchronous oscillations in neuronal systems: mechanisms and functions. J. Comput. Neurosci. 1(1–2), 11 (1994)
38. S. Grossberg, Nonlinear neural networks: principles, mechanisms, and architectures. Neural
Netw. 1(1), 17 (1988)
39. P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli, C.J. Honey, V.J. Wedeen, O. Sporns, Mapping the structural core of human cerebral cortex. PLoS Biol. 6(7), e159 (2008)
40. H. Haken, Advanced Synergetics: Instability Hierarchies of Self-organizing Systems and
Devices. Springer Series in Synergetics (Springer, Berlin, 1983)
41. H. Haken, Synergetics: Introduction and Advanced Topics (Springer Science & Business Media,
2013)
42. H. Haken, Synergetics: Introduction and Advanced Topics (Springer, Berlin, 2004)
43. J. Hlinka, S. Coombes, Using computational models to relate structural and functional brain
connectivity. Eur. J. Neurosci. 36(2), 2137 (2012)
44. C.J. Honey, R. Kötter, M. Breakspear, O. Sporns, Network structure of cerebral cortex shapes
functional connectivity on multiple time scales. Proc. Natl. Acad. Sci. 104(24), 10240 (2007)
45. F.C. Hoppensteadt, E.M. Izhikevich, Weakly Connected Neural Networks (Springer, New York,
1997)
46. E.M. Izhikevich, Dynamical Systems in Neuroscience (MIT press, 2007)
47. V.K. Jirsa, Neural field dynamics with local and global connectivity and time delay. Philos.
Trans. R. Soc. Lond. A: Math., Phys. Eng. Sci. 367(1891), 1131 (2009)
48. A.N. Khambhati, A.E. Sizemore, R.F. Betzel, D.S. Bassett, Modeling and interpreting
mesoscale network dynamics. NeuroImage 180, 337 (2018)
49. H. Kori, Y. Kuramoto, S. Jain, I.Z. Kiss, J.L. Hudson, Clustering in globally coupled oscillators
near a Hopf bifurcation: theory and experiments. Phys. Rev. E 89(6), 062906 (2014)
50. A. Koseska, E. Volkov, J. Kurths, Oscillation quenching mechanisms: amplitude versus oscillation death. Phys. Rep. 531(4), 173 (2013)
51. Y. Kuramoto, Chemical Oscillations, Turbulence and Waves (Springer, Berlin, 1984)
52. W. Kurebayashi, S. Shirasaka, H. Nakao, Phase reduction method for strongly perturbed limit
cycle oscillators. Phys. Rev. Lett. 111(21), 214101 (2013)
53. Y.A. Kuznetsov, Elements of applied bifurcation theory (Springer, New York, 1998)
54. C.W. Lynn, D.S. Bassett, The physics of brain network structure, function and control. Nat.
Rev. Phys. 1(5), 318 (2019)
55. W.A. MacKay, Synchronized neuronal oscillations and their role in motor processes. Trends
Cogn. Sci. 1(5), 176 (1997)
56. I. Malkin, Methods of Poincaré and Liapunov in Theory of Non-Linear Oscillations (Gostexizdat, Moscow, 1949)
57. B. Monga, D. Wilson, T. Matchen, J. Moehlis, Phase reduction and phase-based optimal control
for biological systems: a tutorial. Biol. Cybern. (2018)
58. J.Y. Moon, J. Kim, T.W. Ko, M. Kim, Y. Iturria-Medina, J.H. Choi, J. Lee, G.A. Mashour,
U. Lee, Structure shapes dynamics and directionality in diverse brain networks: mathematical
principles and empirical confirmation in three species. Sci. Rep. 7, 46606 EP (2017)
59. J. Murdock, Normal Forms and Unfoldings for Local Dynamical Systems (Springer Science &
Business Media, 2006)
60. H. Nakao, Phase reduction approach to synchronisation of nonlinear oscillators. Contemp.
Phys. 57(2), 188 (2016)
61. M. Newman, Networks (Oxford University Press, Oxford, 2018)
62. M. Newman, A.L. Barabási, D.J. Watts, The Structure and Dynamics of Networks, vol. 19
(Princeton University Press, Princeton, 2011)
63. V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, V. Latora, Remote synchronization
reveals network symmetries and functional modules. Phys. Rev. Lett. 110(17), 174102 (2013)
64. J.M. Palva, S. Palva, K. Kaila, Phase synchrony among neuronal oscillations in the human
cortex. J. Neurosci. 25(15), 3962 (2005)