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A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems
Degree assortativity in networks of spiking neurons
School of Natural and Computational Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand |
Degree assortativity refers to the increased or decreased probability of connecting two neurons based on their in- or out-degrees, relative to what would be expected by chance. We investigate the effects of such assortativity in a network of theta neurons. The Ott/Antonsen ansatz is used to derive equations for the expected state of each neuron, and these equations are then coarse-grained in degree space. We generate families of effective connectivity matrices parametrised by assortativity coefficient and use SVD decompositions of these to efficiently perform numerical bifurcation analysis of the coarse-grained equations. We find that of the four possible types of degree assortativity, two have no effect on the networks' dynamics, while the other two can have a significant effect.
References:
[1] |
V. Avalos-Gaytán, J. A. Almendral, D. Papo, S. E. Schaeffer and S. Boccaletti,
Assortative and modular networks are shaped by adaptive synchronization processes, Phys. Rev. E, 86 (2012), 015101(R).
doi: 10.1103/PhysRevE.86.015101. |
[2] |
G. Barlev, T. M. Antonsen and E. Ott,
The dynamics of network coupled phase oscillators: An ensemble approach, Chaos, 21 (2011), 025103.
doi: 10.1063/1.3596711. |
[3] |
S. Chandra, D. Hathcock, K. Crain, T. M. Antonsen, M. Girvan and E. Ott,
Modeling the network dynamics of pulse-coupled neurons, Chaos, 27 (2017), 033102.
doi: 10.1063/1.4977514. |
[4] |
S.-N. Chow and X.-B. Lin,
Bifurcation of a homoclinic orbit with a saddle-node equilibrium, Differential Integral Equations, 3 (1990), 435-466.
|
[5] |
F. Chung and L. Lu,
Connected components in random graphs with given expected degree sequences, Ann. Comb., 6 (2002), 125-145.
doi: 10.1007/PL00012580. |
[6] |
B. C. Coutinho, A. V. Goltsev, S. N. Dorogovtsev and J. F. F. Mendes,
Kuramoto model with frequency-degree correlations on complex networks, Phys. Rev. E, 87 (2013), 032106.
doi: 10.1103/PhysRevE.87.032106. |
[7] |
S. De Franciscis, S. Johnson and J. J. Torres,
Enhancing neural-network performance via assortativity, Phys. Rev. E, 83 (2011), 036114.
doi: 10.1103/PhysRevE.83.036114. |
[8] |
D. de Santos-Sierra, I. Sendiña-Nadal, I. Leyva, J. A. Almendral and S. Anava, et al., Emergence of small-world anatomical networks in self-organizing clustered neuronal cultures, PLoS one, 9 (2014), e85828.
doi: 10.1371/journal.pone.0085828. |
[9] |
V. M. Eguíluz, D. R. Chialvo, G. A. Cecchi, M. Baliki and A. V. Apkarian,
Scale-free brain functional networks, Phys. Rev. Lett., 94 (2005), 018102.
doi: 10.1103/PhysRevLett.94.018102. |
[10] |
B. Ermentrout,
Type I membranes, phase resetting curves, and synchrony, Neural Comput., 8 (1996), 979-1001.
doi: 10.1162/neco.1996.8.5.979. |
[11] |
G. B. Ermentrout and N. Kopell,
Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math., 46 (1986), 233-253.
doi: 10.1137/0146017. |
[12] |
J. G. Foster, D. V. Foster, P. Grassberger and M. Paczuski,
Edge direction and the structure of networks, PNAS, 107 (2010), 10815-10820.
doi: 10.1073/pnas.0912671107. |
[13] |
W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[14] |
P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli and C. J. Honey, et al., Mapping the structural core of human cerebral cortex, PLoS Biology, 6 (2008), 1479-1493.
doi: 10.1371/journal.pbio.0060159. |
[15] |
T. Ichinomiya,
Frequency synchronization in a random oscillator network, Phys. Rev. E, 70 (2004), 026116.
doi: 10.1103/PhysRevE.70.026116. |
[16] |
M. Kähne, I. M. Sokolov and S. Rüdiger,
Population equations for degree-heterogenous neural networks, Phys. Rev. E, 96 (2017), 052306.
doi: 10.1103/PhysRevE.96.052306. |
[17] |
C. R. Laing,
Derivation of a neural field model from a network of theta neurons, Phys. Rev. E, 90 (2014), 010901(R).
doi: 10.1103/PhysRevE.90.010901. |
[18] |
C. R. Laing,
Numerical bifurcation theory for high-dimensional neural models, J. Math. Neurosci., 4 (2014), 13.
doi: 10.1186/2190-8567-4-13. |
[19] |
C. R. Laing,
Exact neural fields incorporating gap junctions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1899-1929.
doi: 10.1137/15M1011287. |
[20] |
M. D. LaMar and G. D. Smith,
Effect of node-degree correlation on synchronization of identical pulse-coupled oscillators, Phys. Rev. E, 81 (2010), 046206.
doi: 10.1103/PhysRevE.81.046206. |
[21] |
P. E. Latham, B. J. Richmond, P. G. Nelson and S. Nirenberg,
Intrinsic dynamics in neuronal networks. I. theory, J. Neurophysiology, 83 (2000), 808-827.
doi: 10.1152/jn.2000.83.2.808. |
[22] |
T. B. Luke, E. Barreto and P. So,
Complete classification of the macroscopic behavior of a heterogeneous network of theta neurons, Neural Comput., 25 (2013), 3207-3234.
doi: 10.1162/NECO_a_00525. |
[23] |
M. B. Martens, A. R. Houweling and P. H. E. Tiesinga,
Anti-correlations in the degree distribution increase stimulus detection performance in noisy spiking neural networks, J. Comput. Neuroscience, 42 (2017), 87-106.
doi: 10.1007/s10827-016-0629-1. |
[24] |
W. S. McCulloch and W. Pitts,
The statistical organization of nervous activity, Biometrics, 4 (1948), 91-99.
doi: 10.2307/3001453. |
[25] |
E. Montbrió, D. Pazó and A. Roxin,
Macroscopic description for networks of spiking neurons, Phys. Rev. X, 5 (2015), 021028.
doi: 10.1103/PhysRevX.5.021028. |
[26] |
M. E. J. Newman,
Assortative mixing in networks, Phys. Rev. Lett., 89 (2002), 208701.
doi: 10.1103/PhysRevLett.89.208701. |
[27] |
M. E. J. Newman,
The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[28] |
D. Q. Nykamp, D. Friedman, S. Shaker, M. Shinn and M. Vella, et al., Mean-field equations for neuronal networks with arbitrary degree distributions, Phys. Rev. E, 95 (2017), 042323.
doi: 10.1103/PhysRevE.95.042323. |
[29] |
E. Ott and T. M. Antonsen,
Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 037113.
doi: 10.1063/1.2930766. |
[30] |
J. G. Restrepo and E. Ott,
Mean-field theory of assortative networks of phase oscillators, Europhys. Lett., 107 (2014), 60006.
doi: 10.1209/0295-5075/107/60006. |
[31] |
C. Schmeltzer, A. H. Kihara, I. M. Sokolov and S. Rüdiger,
Degree correlations optimize neuronal network sensitivity to sub-threshold stimuli, PLoS One, 10 (2015), e0121794.
doi: 10.1371/journal.pone.0121794. |
[32] |
P. S. Skardal, J. G. Restrepo and E. Ott,
Frequency assortativity can induce chaos in oscillator networks, Phys. Rev. E, 91 (2015), 060902(R).
doi: 10.1103/PhysRevE.91.060902. |
[33] |
P. S. Skardal, J. Sun, D. Taylor and J. G. Restrepo,
Effects of degree-frequency correlations on network synchronization: Universality and full phase-locking, Europhys. Lett., 101 (2013), 20001.
doi: 10.1209/0295-5075/101/20001. |
[34] |
B. Sonnenschein, F. Sagués and L. Schimansky-Geier,
Networks of noisy oscillators with correlated degree and frequency dispersion, Eur. Phys. J. B, 86 (2013), 12.
doi: 10.1140/epjb/e2012-31026-x. |
[35] |
S. Teller, C. Granell, M. De Domenico, J. Soriano, S. Gómez and A. Arenas,
Emergence of assortative mixing between clusters of cultured neurons, PLoS Comput. Biology, 10 (2014), e1003796.
doi: 10.1371/journal.pcbi.1003796. |
[36] |
J. C. Vasquez, A. R. Houweling and P. Tiesinga,
Simultaneous stability and sensitivity in model cortical networks is achieved through anti-correlations between the in- and out-degree of connectivity, Frontiers Comput. Neuroscience, 7 (2013), 156.
doi: 10.3389/fncom.2013.00156. |
[37] |
M. Vegué and A. Roxin,
Firing rate distributions in spiking networks with heterogeneous connectivity, Phys. Rev. E, 100 (2019), 022208.
doi: 10.1103/PhysRevE.100.022208. |
show all references
References:
[1] |
V. Avalos-Gaytán, J. A. Almendral, D. Papo, S. E. Schaeffer and S. Boccaletti,
Assortative and modular networks are shaped by adaptive synchronization processes, Phys. Rev. E, 86 (2012), 015101(R).
doi: 10.1103/PhysRevE.86.015101. |
[2] |
G. Barlev, T. M. Antonsen and E. Ott,
The dynamics of network coupled phase oscillators: An ensemble approach, Chaos, 21 (2011), 025103.
doi: 10.1063/1.3596711. |
[3] |
S. Chandra, D. Hathcock, K. Crain, T. M. Antonsen, M. Girvan and E. Ott,
Modeling the network dynamics of pulse-coupled neurons, Chaos, 27 (2017), 033102.
doi: 10.1063/1.4977514. |
[4] |
S.-N. Chow and X.-B. Lin,
Bifurcation of a homoclinic orbit with a saddle-node equilibrium, Differential Integral Equations, 3 (1990), 435-466.
|
[5] |
F. Chung and L. Lu,
Connected components in random graphs with given expected degree sequences, Ann. Comb., 6 (2002), 125-145.
doi: 10.1007/PL00012580. |
[6] |
B. C. Coutinho, A. V. Goltsev, S. N. Dorogovtsev and J. F. F. Mendes,
Kuramoto model with frequency-degree correlations on complex networks, Phys. Rev. E, 87 (2013), 032106.
doi: 10.1103/PhysRevE.87.032106. |
[7] |
S. De Franciscis, S. Johnson and J. J. Torres,
Enhancing neural-network performance via assortativity, Phys. Rev. E, 83 (2011), 036114.
doi: 10.1103/PhysRevE.83.036114. |
[8] |
D. de Santos-Sierra, I. Sendiña-Nadal, I. Leyva, J. A. Almendral and S. Anava, et al., Emergence of small-world anatomical networks in self-organizing clustered neuronal cultures, PLoS one, 9 (2014), e85828.
doi: 10.1371/journal.pone.0085828. |
[9] |
V. M. Eguíluz, D. R. Chialvo, G. A. Cecchi, M. Baliki and A. V. Apkarian,
Scale-free brain functional networks, Phys. Rev. Lett., 94 (2005), 018102.
doi: 10.1103/PhysRevLett.94.018102. |
[10] |
B. Ermentrout,
Type I membranes, phase resetting curves, and synchrony, Neural Comput., 8 (1996), 979-1001.
doi: 10.1162/neco.1996.8.5.979. |
[11] |
G. B. Ermentrout and N. Kopell,
Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math., 46 (1986), 233-253.
doi: 10.1137/0146017. |
[12] |
J. G. Foster, D. V. Foster, P. Grassberger and M. Paczuski,
Edge direction and the structure of networks, PNAS, 107 (2010), 10815-10820.
doi: 10.1073/pnas.0912671107. |
[13] |
W. J. F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. |
[14] |
P. Hagmann, L. Cammoun, X. Gigandet, R. Meuli and C. J. Honey, et al., Mapping the structural core of human cerebral cortex, PLoS Biology, 6 (2008), 1479-1493.
doi: 10.1371/journal.pbio.0060159. |
[15] |
T. Ichinomiya,
Frequency synchronization in a random oscillator network, Phys. Rev. E, 70 (2004), 026116.
doi: 10.1103/PhysRevE.70.026116. |
[16] |
M. Kähne, I. M. Sokolov and S. Rüdiger,
Population equations for degree-heterogenous neural networks, Phys. Rev. E, 96 (2017), 052306.
doi: 10.1103/PhysRevE.96.052306. |
[17] |
C. R. Laing,
Derivation of a neural field model from a network of theta neurons, Phys. Rev. E, 90 (2014), 010901(R).
doi: 10.1103/PhysRevE.90.010901. |
[18] |
C. R. Laing,
Numerical bifurcation theory for high-dimensional neural models, J. Math. Neurosci., 4 (2014), 13.
doi: 10.1186/2190-8567-4-13. |
[19] |
C. R. Laing,
Exact neural fields incorporating gap junctions, SIAM J. Appl. Dyn. Syst., 14 (2015), 1899-1929.
doi: 10.1137/15M1011287. |
[20] |
M. D. LaMar and G. D. Smith,
Effect of node-degree correlation on synchronization of identical pulse-coupled oscillators, Phys. Rev. E, 81 (2010), 046206.
doi: 10.1103/PhysRevE.81.046206. |
[21] |
P. E. Latham, B. J. Richmond, P. G. Nelson and S. Nirenberg,
Intrinsic dynamics in neuronal networks. I. theory, J. Neurophysiology, 83 (2000), 808-827.
doi: 10.1152/jn.2000.83.2.808. |
[22] |
T. B. Luke, E. Barreto and P. So,
Complete classification of the macroscopic behavior of a heterogeneous network of theta neurons, Neural Comput., 25 (2013), 3207-3234.
doi: 10.1162/NECO_a_00525. |
[23] |
M. B. Martens, A. R. Houweling and P. H. E. Tiesinga,
Anti-correlations in the degree distribution increase stimulus detection performance in noisy spiking neural networks, J. Comput. Neuroscience, 42 (2017), 87-106.
doi: 10.1007/s10827-016-0629-1. |
[24] |
W. S. McCulloch and W. Pitts,
The statistical organization of nervous activity, Biometrics, 4 (1948), 91-99.
doi: 10.2307/3001453. |
[25] |
E. Montbrió, D. Pazó and A. Roxin,
Macroscopic description for networks of spiking neurons, Phys. Rev. X, 5 (2015), 021028.
doi: 10.1103/PhysRevX.5.021028. |
[26] |
M. E. J. Newman,
Assortative mixing in networks, Phys. Rev. Lett., 89 (2002), 208701.
doi: 10.1103/PhysRevLett.89.208701. |
[27] |
M. E. J. Newman,
The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256.
doi: 10.1137/S003614450342480. |
[28] |
D. Q. Nykamp, D. Friedman, S. Shaker, M. Shinn and M. Vella, et al., Mean-field equations for neuronal networks with arbitrary degree distributions, Phys. Rev. E, 95 (2017), 042323.
doi: 10.1103/PhysRevE.95.042323. |
[29] |
E. Ott and T. M. Antonsen,
Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18 (2008), 037113.
doi: 10.1063/1.2930766. |
[30] |
J. G. Restrepo and E. Ott,
Mean-field theory of assortative networks of phase oscillators, Europhys. Lett., 107 (2014), 60006.
doi: 10.1209/0295-5075/107/60006. |
[31] |
C. Schmeltzer, A. H. Kihara, I. M. Sokolov and S. Rüdiger,
Degree correlations optimize neuronal network sensitivity to sub-threshold stimuli, PLoS One, 10 (2015), e0121794.
doi: 10.1371/journal.pone.0121794. |
[32] |
P. S. Skardal, J. G. Restrepo and E. Ott,
Frequency assortativity can induce chaos in oscillator networks, Phys. Rev. E, 91 (2015), 060902(R).
doi: 10.1103/PhysRevE.91.060902. |
[33] |
P. S. Skardal, J. Sun, D. Taylor and J. G. Restrepo,
Effects of degree-frequency correlations on network synchronization: Universality and full phase-locking, Europhys. Lett., 101 (2013), 20001.
doi: 10.1209/0295-5075/101/20001. |
[34] |
B. Sonnenschein, F. Sagués and L. Schimansky-Geier,
Networks of noisy oscillators with correlated degree and frequency dispersion, Eur. Phys. J. B, 86 (2013), 12.
doi: 10.1140/epjb/e2012-31026-x. |
[35] |
S. Teller, C. Granell, M. De Domenico, J. Soriano, S. Gómez and A. Arenas,
Emergence of assortative mixing between clusters of cultured neurons, PLoS Comput. Biology, 10 (2014), e1003796.
doi: 10.1371/journal.pcbi.1003796. |
[36] |
J. C. Vasquez, A. R. Houweling and P. Tiesinga,
Simultaneous stability and sensitivity in model cortical networks is achieved through anti-correlations between the in- and out-degree of connectivity, Frontiers Comput. Neuroscience, 7 (2013), 156.
doi: 10.3389/fncom.2013.00156. |
[37] |
M. Vegué and A. Roxin,
Firing rate distributions in spiking networks with heterogeneous connectivity, Phys. Rev. E, 100 (2019), 022208.
doi: 10.1103/PhysRevE.100.022208. |









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