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# Degree assortativity in networks of spiking neurons

• 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.

Mathematics Subject Classification: Primary: 92C20; Secondary: 92B25, 34B45, 34C15.

 Citation: • • 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

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

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: $N = 5000,\eta_0 = -2,\Delta = 0.1,K = 3,q = 2$

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: $\eta_0 = -2,\Delta = 0.1,K = 3$

Figure 5.  Six largest singular values of the SVD decomposition of $E$ as a function of assortativity coefficent, for 4 types of assortativity

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: $K = 3,\Delta = 0.1$

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

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

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

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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