Master stability functions of networks of Izhikevich neurons

Synchronization has attracted interest in many areas where the systems under study can be described by complex networks. Among such areas is neuroscience, where it is hypothesized that synchronization plays a role in many functions and dysfunctions of the brain. We study the linear stability of synchronized states in networks of Izhikevich neurons using master stability functions (MSFs), and to accomplish that, we exploit the formalism of saltation matrices. Such a tool allows us to calculate the Lyapunov exponents of the MSF properly since the Izhikevich model displays a discontinuity within its spikes. We consider both electrical and chemical couplings as well as global and cluster synchronized states. The MSF calculations are compared with a measure of the synchronization error for simulated networks. We give special attention to the case of electric and chemical coupling, where a riddled basin of attraction makes the synchronized solution more sensitive to perturbations.

Figure 1

Maximum Lyapunov exponent transversal to the synchronization manifold for electrically coupled Izhikevich neurons as a function of $g_e\gamma$.

Figure 2

Undirected network with $N=4$ nodes, which is used to exemplify the stability of complete synchronization for identical Izhikevich neurons.

Figure 3

Synchronization error for a network of $N=4$ Izhikevich neurons with electrical coupling. It goes to zero at $g_e\approx 0.133$, in conformity with the MSF. The solid line represents the average of 100 simulations, the shaded area represents the first and third quartiles, and the gray dashes represent the maximum and minimum values.

Figure 4

Maximum Lyapunov exponent transversal to the synchronization manifold (${\gamma }_4=-2$) for $N=4$ for chemically coupled Izhikevich neurons.

Figure 5

Synchronization error for a network of $N=4$ Izhikevich neurons with chemical coupling. As expected from the MSF analysis, the system does not synchronize. The solid line represents the average of 100 simulations, the shaded area represents the first and third quartiles, and the gray dashes represent the maximum and minimum values.

Figure 6

Maximum Lyapunov exponent transversal to the synchronization manifold for $N=4$ Izhikevich neurons with electrical and chemical coupling.

Figure 7

Synchronization error for $N=4$ with electrical and chemical coupling. The solid line represents the average of 200 simulations, the shaded area represents the first and third quartiles, and the gray dashes represent the maximum and minimum values.

Figure 8

Riddled basin of attraction in $(x_1,x_2)$ plane for $g=0.155$, calculated with $10.24\times {10}^4$ initial conditions in (a) $x_1,x_2\in [1,-1]$ and (b) $x_1,x_2\in [0.01,-0.01]$, while $x_3=x_1$, $x_4=x_2$, and $y$ variables are set to $y_i=-101.5$. Solutions that reach a final state with a synchronization error Err $>0.05$ are marked as white dots; otherwise, they are marked as black.

Figure 9

The fraction of pairs of initial conditions that converge to different asymptotic solutions, $f$, as a function of the distance between initial condition $\varepsilon$ [Eq. (20)]. For $g=0.1555$, the slope of the line that best fits the points yields an uncertainty exponent $\alpha =0.005$.

Figure 10

Undirected network with N = 8 nodes, which is used to exemplify the stability of cluster synchronization, where the clusters are indicated with colors.

Figure 11

Maximum Lyapunov exponent transversal to the synchronization manifold for $N=8$ Izhikevich neurons with electrical coupling under cluster synchronization.

Figure 12

Synchronization error for Izhikevich model with electrical coupling and cluster synchronization. The gray lines represent maximum and minimum errors, the shaded regions represent the first and third quartiles, and the circles the average over 100 simulations.

Figure 13

Maximum Lyapunov exponent transversal to the synchronization manifold for $N=8$ Izhikevich neurons with chemical coupling and cluster synchronization.

Figure 14

Synchronization error for Izhikevich model with chemical coupling, andcluster synchronization. The gray lines represent maximum and minimum errors, the shaded regions represent the first and third quartiles, and the circles the average over 100 simulations.

Figure 15

Maximum Lyapunov exponent transversal to the synchronization manifold for $N=8$ Izhikevich neurons with chemical and electric coupling and cluster synchronization.

Figure 16

Synchronization error as a function of $g$ for Izhikevich model with electrical and chemical coupling. The gray lines represent maximum and minimum errors, the shaded regions represent the first and third quartiles, and the circles the average over 100 simulations.

Figure 17

Riddled basin of attraction in $(x_5,x_6)$ plane for $g=0.155$. Panel (a) is calculated with $10.24\times {10}^4$ initial conditions in $x_5,x_6\in [1,-1]$ and $y_{(5,6)}=-101.5$. The other nodes are initiated at ${\mathbf{x}}_{(1,2)}={[-59,-111]}^T$, ${\mathbf{x}}_3={[-60,-110]}^T$, ${\mathbf{x}}_4={[-58,-108]}^T$, ${\mathbf{x}}_{(7,8)}={[-58,-109]}^T$. (b) The same as (a) for $g=0.195$. We differentiate synchronized solutions from nonsynchronized solutions by setting a threshold at Err $=0.05$, which are represented by white and black dots, respectively.

Figure 18

Time series of the $x$ variable from nodes 5 (solid red) and 6 (dashed black) for different initial conditions, both with $g=0.155$. (a) Incomplete synchronization Err $\approx 1.91$, (b) $\mathrm{Err}\approx 0.50$.

Figure 19

The fraction of pairs of initial conditions that converge to different asymptotic solutions, $f$, as a function of the distance between initial condition $\varepsilon$ [Eq. (20)]. For $g=0.1555$, the slope of the line that best fits the points yields an uncertainty exponent $\alpha =0.007$.