Cluster synchronization in networks of identical oscillators with $\alpha$-function pulse coupling

We study a network of $N$ identical leaky integrate-and-fire model neurons coupled by $\alpha$-function pulses, weighted by a coupling parameter $K$. Studies of the dynamics of this system have mostly focused on the stability of the fully synchronized and the fully asynchronous splay states, which naturally depends on the sign of $K$, i.e., excitation vs inhibition. We find that there is also a rich set of attractors consisting of clusters of fully synchronized oscillators, such as fixed $(N-1,1)$ states, which have synchronized clusters of sizes $N-1$ and 1, as well as splay states of clusters with equal sizes greater than 1. Additionally, we find limit cycles that clarify the stability of previously observed quasiperiodic behavior. Our framework exploits the neutrality of the dynamics for $K=0$ which allows us to implement a dimensional reduction strategy that simplifies the dynamics to a continuous flow on a codimension 3 subspace with the sign of $K$ determining the flow direction. This reduction framework naturally incorporates a hierarchy of partially synchronized subspaces in which the new attracting states lie. Using high-precision numerical simulations, we describe completely the sequence of bifurcations and the stability of all fixed points and limit cycles for $N=2\text{–}4$. The set of possible attracting states can be used to distinguish different classes of neuron models. For instance from our previous work [Chaos 24, 013114 (2014)] we know that of the types of partially synchronized states discussed here, only the $(N-1,1)$ states can be stable in systems of identical coupled sinusoidal (i.e., Kuramoto type) oscillators, such as $\theta$-neuron models. Upon introducing a small variation in individual neuron parameters, the attracting fixed points we discuss here generalize to equivalent fixed points in which neurons need not fire coincidently.

Figure 1

The case of $N=2$ with symmetric coupling $w_1=w_2$: PF for $K\to 0^{-}$; attractors in the $K\to 0$ limit for fixed $a$; pitchfork bifurcation for $K\to 0^{+}$. (The black circles are attractors, and the white circles are repellors.)

Figure 2

The case of $N=2$ with asymmetric coupling $w_1\ne w_2$: bifurcation diagram for $K\to 0^{-}$; SN for $K\to 0^{-}$; attractors in the $K\to 0$ limit for fixed $a$.

Figure 3

$N=3$ schematic Hopf bifurcation followed by double saddle-node (DSN) bifurcation; attractors in the $K\to 0$ limit for fixed $a$. (Crosses denote saddles; the darker shade indicates attracting direction.)

Figure 4

Schematic of a double saddle-node bifurcation for $K\to 0^{-}$.

Figure 5

The basins of attraction of the stable fixed points for large $\alpha$. For $K\to 0^{-}$ the attractors are sync, splay, and three equivalent (2,1) states; for $K\to 0^{+}$ the attractors are the other three equivalent (2,1) states.

Figure 6

$N=4$ flow on a fundamental tetrahedral subdomain for small $\alpha$ and $K\to 0^{-}$. Partially synchronized invariant subspaces include four (3,1) edges, two (2,2) edges, and four (2,1,1) triangular faces. The broken rings indicate face saddles (FSs) that repel in the face and attract transverse to it.

Figure 7

Bifurcation sequence for $N=4$ and $K\to 0^{-}$, starting upper left and moving clockwise.

Figure 8

Six face saddle bifurcation, viewed transverse to the (3,1) edge $0=x_4=x_3=x_2<x_1<1$; the center in the diagram is the edge fixed point, which is surrounded by six face saddles on the six faces meeting the edge.

Figure 9

The $N=4$ bifurcation sequence and resulting phase diagram, showing attractors for $K\to 0^{+}$ and $K\to 0^{-}$ for fixed $a$.

Figure 10

Orbits for $N=3,a=1.01,\left|K\right|=0.01$, and two values of $\alpha$ that are separated by a Hopf bifurcation and thus have different attractors; left panel: $\alpha =1.01<{\alpha }_H,K<0$ and right panel: ${\alpha }_H<\alpha =1.5<{\alpha }_{\mathrm{DSN}},K>0$.

Figure 11

Examples of orbits for $N=3,a=1.01,K=-0.01$, and ${\alpha }_{\mathrm{DSN}}<\alpha =3$. The main panel shows pairs of orbits originating from the three (2,1) edge sources that either diverge near the FSs or near the (2,1) ESs. The orbits near each saddle are magnified in the insets.

Figure 12

Velocity field in the top ($x_3=0$) and side ($x_1=x_2$) (2,1,1) faces of the tetrahedral subdomain for $\alpha =1.133,a=1.05$, and $K=-0.01$ just before the FEF-rPF bifurcation. The main panel: sync attractors at vertices plus three (2,2) edge fixed points. The inset: magnified (2,2) edge with a central splay (2,2) attractor, surrounded by two edge saddles labeled $A$ and $A^{\prime }$.

Figure 13

Velocity field in the top ($x_3=0$) 211 face relative to the (2,2) edge saddle labeled $A$ in the previous figure. At this magnification one can resolve the face saddle labeled $B$.

Figure 14

Quasiperiodic orbits approximating limit cycles for positive $K=0.01,a=1.05$ for four values of $\alpha$ with two slightly larger than ${\alpha }_H$ ($\alpha =1.38$ and 1.39) and two slightly smaller than ${\alpha }_{\mathrm{SN}}$ ($\alpha =2.1$ and 2.44). The orbits increase in size as $\alpha$ increases.

Figure 15

Velocity field in the $x_3=0$ face, relative to the (3,1) attractor for $K<0$ (labeled $C$) that lies on the $x_2=0$ edge. Here $K=-0.01,a=1.05$, and $\alpha =2.445$ which yields dynamics just after the 6xFS bifurcation that stabilizes the (3,1) attractor at the origin of the figure and in which the face saddle (labeled $D$) is born.