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December  2020, 7(2): 401-423. doi: 10.3934/jcd.2020016

Degree assortativity in networks of spiking neurons

Christian Bläsche , Shawn Means and  Carlo R. Laing ,

School of Natural and Computational Sciences, Massey University, Private Bag 102-904 NSMC, Auckland, New Zealand

* Corresponding author: c.r.laing@massey.ac.nz

Received  October 2019 Published  July 2020

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.

Keywords: Neuronal networks, degree assortativity, theta neurons, Ott/Antonsen, bifurcation.
Mathematics Subject Classification: Primary: 92C20; Secondary: 92B25, 34B45, 34C15.
Citation: Christian Bläsche, Shawn Means, Carlo R. Laing. Degree assortativity in networks of spiking neurons. Journal of Computational Dynamics, 2020, 7 (2) : 401-423. doi: 10.3934/jcd.2020016
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.

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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.

Figure 1.  Assortativity in undirected and directed networks. An undirected network (left column) is assortative if high degree nodes are more likely to be connected to high degree nodes, and low to low, than by chance (top left). Such a network is disassortative if the opposite occurs (bottom left). In directed networks (right column) there are four possible kinds of assortativity. The probability of a connection (red) is thus influenced by the number of red shaded links of the sending (left) and receiving (right) node
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Figure 2.  Orange circles: steady state of (19)-(20) for 20 different default networks. Blue circles: results from 50 different realisations of the $ \eta_i $ for (1)-(2), for each network. Parameters: $ \eta_0 = -2,\Delta = 0.1,K = 3 $. The orange line marks the ensemble mean value
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Figure 3.  Comparison of steady-state values of the order parameter (top: magnitude of $ R $; bottom: $ \mbox{Re}(R) $) over a suite of adjacency matrices with varied densities of multi-edge connections ranging from none to 97%. Higher densities of multi-edges were obtained, but assortativities exceeded the target neutral values of $ \pm 0.005 $. Values shown are from simulations of (1)-(2) after initial transients decay (i.e., time t $ \geq $ 40). Each of these 10 curves correspond to a unique realisation of default $ \eta $s from the distribution $ g(\eta) $. Parameters: $ N = 5000,\eta_0 = -2,\Delta = 0.1,K = 3,q = 2 $
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Figure 4.  Real part of $ z $ at steady state for 20 different default adjacency matrices (indicated by different colors), as the number of clusters in degree space is varied. Parameters: $ \eta_0 = -2,\Delta = 0.1,K = 3 $
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Figure 5.  Six largest singular values of the SVD decomposition of $ E $ as a function of assortativity coefficent, for 4 types of assortativity
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Figure 6.  Average firing rate at fixed points of (30)-(31) as a function of $ \eta_0 $, for the 4 types of assortativity. For each type of assortativity curves are plotted for $ r = 0 $ (black), $ r = -0.2 $ (blue) and $ r = 0.2 $ (green). Solid lines indicate stable and dashed lines unstable fixed points. Parameters: $ K = 3,\Delta = 0.1 $
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Figure 7.  Continuation of the saddle-node bifurcations seen in the upper two panels of Fig. 6 as $ r $ is varied. Curves in Figure 6 correspond to vertical slices at $ r = 0,\pm 0.2 $. The network is bistable in region $ B $ and has a single stable fixed point in regions A and C
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Figure 8.  Average firing rate at fixed points of (30)-(31) as a function of $ \eta_0 $, for the 4 types of assortativity. For each type of assortativity curves are plotted for $ r = 0 $ (black), $ r = -0.2 $ (blue) and $ r = 0.2 $ (green). In addition there are oscillations in certain regions and dash-dotted lines outline the minimal and maximal firing rate over one period of oscillation. The (in, in)-plot in the top left corner contains a zoom of rest of the panel, and the (out, in)-plot contains a subplot with the oscillation's period for $ r = 0 $ and which is aligned with the outer $ \eta_0 $ axis
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Figure 9.  Continuation of bifurcations seen in upper panels of Fig. 8. Solid black lines indicate saddle-node bifurcations, dashed blue is a Hopf bifurcation and dashed red a homoclinic bifurcation. Curves in Figure 8 can be understood as vertical slices through the respective plot at $ r = 0,\pm 0.2 $. See text for explanation of labels
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Figure 10.  Permutation method initial matrices illustration. Top: $ A^{(0)} $ showing arrangement of edge entries (all solo connections: $ \rho_{m}^{+} = 0 $) for each row aligned left where row sums add up to $ k_{in} $. If multi-edges are desired, we simply distribute them in the rows of $ A^{(0)} $ satisfying the proportion, $ \rho_{m}^{+} = 0 $ and the row sum. Bottom: permutation of rows in $ A^{(0)} $ into this example $ A^{(1)} $. Note row sums still add up to $ k_{in} $, with column sums adding to a current $ k_{out}^{(1)} $ — likely violating the designated $ k_{out} $
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