When Weak Inhibition Synchronizes Strongly Desynchronizing Networks of Bursting Neurons

We show that weak common inhibition applied to a network of bursting neurons with strong desynchronizing connections can induce burst and complete synchronization. We demonstrate that the weak synchronizing inhibition from the same pacemaker neuron can win out over much stronger desynchronizing connections within the network, provided that the neuron’s duty cycle is sufficiently long. We also gain insight into how the changes in burst duty cycles can trigger unexpected clusters of synchrony in bursting networks.

Figure 1

Half-center oscillator composed of inhibitory neurons (1). The neurons are identical ($V_{1,2}^{\mathrm{shift}}=-0.02$); the inhibitory connections are strong ($g_{12}=g_{21}\equiv g_{\mathrm{S}}=2$). (Top) The uncoupled and inhibited nullclines are depicted by solid and dotted blue lines, respectively. Color-matching balls represent the instant phase points of the cells on the bursting orbit. The dark gray trajectory corresponds to the antiphase solution, while the light one is the reference trajectory of the uncoupled cell. As soon as the active (green) cell is above the threshold ${\Theta }_{\mathrm{syn}}$, the nulcline $M_{\mathrm{eq}}$ is shifted towards the slow nullcline $m^{\prime }=0$ to generate a stable equilibrium state near the lower knee through the saddle-node bifurcation. The inactive (blue) cell is trapped at it until the active cell falls down to $M_{\mathrm{eq}}$. (Bottom) Time series of the established antiphase dynamics. Note the delayed postinhibition firing of the inactive cell due to the slow passage throughout the vicinity of the disappeared saddle node.

Figure 2

Dynamics of the weak vs strong network (left inset) with weak $g_{31}=g_{32}\equiv g_w=0.02$ and strong coupling $g_{\mathrm{S}}=2$ for a long duty cycle. The control parameters are $V_{1,2}^{\mathrm{shift}}=-0.02$ and $V_3^{\mathrm{shift}}=-0.024$. Longer burst duration of the driving neuron allows the driven neurons 1 and 2 to get together at the lock-down state near the lower knee of the inhibited nullcline $M_{\mathrm{eq}}$. Neuron 1 (blue ball) is locked by the driving neuron 3, while the phase point of neuron 2 (green ball) moves along $M_{\mathrm{eq}}$ towards its lower knee and eventually catches up with the phase state of neuron 1. Having become inactive, the driving neuron 3 releases neurons 1 and 2 from inhibition simultaneously. Jumping up to the spiking manifold, they achieve complete synchronization, shown by the time-series of the established regimes. Note that, when synchronized, neurons 1 and 2 shorten their natural duty cycle.

Figure 3

Dependence of the threshold coupling strength $g_w$, inducing synchronization in the half-center network, on the duty cycle of the driving neuron 3. Other parameters are the same as in Fig. 2. Values of $V_3^{\mathrm{shift}}$ correspond to the indicated duty cycles.

Figure 4

Examples of the networks satisfying the assumed synchronization conditions. The neurons of same color form a cluster. The width of the links may be thought of as the coupling strength. Note strong uniform couplings within the cluster and weak connections from the driving neuron/neurons. (Left) Ring network driven by the central pacemaker. Reciprocal inhibitory connections, $g_r=1$, among neurons within the ring are depicted by links without arrows. Directional connections from the central pacemaker cell to the ring are uniform and weak, $g_d=0.02$. Two backward connections, $g_b=0.02$, from the ring are introduced to make the network asymmetric. (Right) Network with an irregular structure. The coupling strengths are set as follows: $g_{12}=g_{21}=2$, all other $g_{ij}=0.02$. Note that in contrast with the network of Fig. 2, the input to cells 1 and 2 comes from more than one driving cell. Moreover, the driving cells 3 and 4 are always desynchronized due to the network topology. For $V_i^{\mathrm{shift}}=-0.02$, $i=1,\dots ,7$, corresponding to the 50% duty cycle of all the neurons, no synchronization within the clusters of both networks is induced. The longer 80% duty cycle of the driving neuron/neurons arising at $V_{\mathrm{center}}^{\mathrm{shift}}=-0.024$ (left network) and $V_{3,4}^{\mathrm{shift}}=-0.024$ (right network) puts the neurons within the clusters in synchronization.