Synchronization of bursting neurons with delayed chemical synapses

The synchronization of bursting Hindmarsh-Rose neurons coupled by a time-delayed fast threshold modulation synapse was studied. It is shown that there is a domain of the coupling parameter and nonzero time-lag values such that the bursting neurons are exactly synchronized. Furthermore, and contrary to the case of electrical synapses, such synchronous bursting is stochastically stable.

Figure 1

Domains in $(g,\tau )$ parameter plane that imply stability of stationary state (light gray), instability of the stationary state with asynchronous bursting (gray), and instability of the stationary state with synchronous bursting (black).

Figure 2

Dynamics for $(g,\tau )$ parameters in Fig. 1, corresponding to stable stationary state (c), (e) and asynchronous bursting (a), (b), (d), (f). Values of $(g,\tau )$, and the plotted curves are (a) (1,0) $x_1$ (black), $x_2$ (light gray); (b) (1,0), $x_1-x_2$; (c) (1.45,0) $x_1$ (black), $x_2$ (light gray); (d) (1.45,30) $x_1$ (black), $x_2$ (light gray); (e) (1.7,35), $x_1$ (black), $x_2$ (light gray); (e) (1.7,60), $x_1$ (black), $x_2$ (light gray).

Figure 3

Synchronization by increase of the values of the time lag $\tau$. Values of $(g,\tau )$, and the plotted curves are (a) (2,65) $x_1$ (black), $x_2$ (light gray); (b) (2,85) $x_1$ (light gray), $x_1-x_2$ (black); (c) (2,95) $x_1$ (light gray), $x_1-x_2$ (black).

Figure 4

Effect of small noise on the time-delay-induced synchronization of bursting for chemical FTM coupling (a), (b) and for electrical [$g(x_1-x_2)$; see [10, 12]] coupling (c), (d). Presented are $x_1$ (black) and $x_1-x_2$ (light gray). Values of the parameters are (a) $D=0.001$, $g=2$, $\tau =95$; (b) $D=0.01$, $g=2$, $\tau =95$; (c) $D=0.001$, $g=0.1$, $\tau =8$; (d) $D=0.01$, $g=0.1$, $\tau =8$.