Synergistic effect of repulsive inhibition in synchronization of excitatory networks

We show that the addition of pairwise repulsive inhibition to excitatory networks of bursting neurons induces synchrony, in contrast to one's expectations. Through stability analysis, we reveal the mechanism underlying this purely synergistic phenomenon and demonstrate that it originates from the transition between different types of bursting, caused by excitatory-inhibitory synaptic coupling. This effect is generic and observed in different models of bursting neurons and fast synaptic interactions. We also find a universal scaling law for the synchronization stability condition for large networks in terms of the number of excitatory and inhibitory inputs each neuron receives, regardless of the network size and topology. This general law is in sharp contrast with linearly coupled networks with positive (attractive) and negative (repulsive) coupling where the placement and structure of negative connections heavily affect synchronization.

Figure 1

(Left) Possible interactions between two excitatory neurons 1 and 2 with direct excitatory and tertiary synapses. The tertiary synapses mediate inhibition by exciting the presynaptic terminals of inhibitory interneurons at their somas. This network can be viewed as a pair of neurons effectively coupled through both excitatory and inhibitory connections (right). Excitatory (inhibitory) connections are depicted by arrows (circles). The dynamics of the two-cell network is studied in Fig. 2.

Figure 2

Synchronization in the two-cell network (1) as a function of excitation ($g_{\mathrm{exc}}$) and inhibiton ($g_{\mathrm{inh}}$). Top panel: The color bar indicates the voltage difference $|x_1-x_2|,$ averaged over the last three bursting periods. The blue (dark) zone (c) corresponds to the zero voltage difference (complete synchronization), appearing from random initial conditions. Observe the effect when a small increase of inhibition from 0 dramatically lowers the synchronization threshold from $1.28$ to $0.11.$ Note that the inhibition desynchronizes the cells in the absence of excitation ($g_{\mathrm{exc}}=0$), independently from the coupling strength and initial conditions. Bifurcation curve HB (white dotted line) corresponds to the transition to synchronized plateau bursting. Bottom panel: Burst synchronization. The color bar indicates the phase difference between the bursts, $\mathrm{\Delta }\varphi ={\varphi }_1-{\varphi }_2,$ averaged over the last three bursting periods. The normalized phase $0\le {\varphi }_i\le 1$ of the $i\mathrm{th}$ bursting cell ($i=1,2$) is initiated and reset every cycle at the beginning of the burst. The normalized phase difference $\mathrm{\Delta }\varphi$ ranges from 0 [burst synchrony, blue (dark gray) color] to $0.5$ [antiphase bursting, red (light gray) color]. Notice a similar effect of burst synchronization, induced by repulsive inhibition.

Figure 3

Transition from square-wave to plateau bursting in the self-coupled system (2), controlling the type of synchronous bursting. Top: Square-wave burster in the uncoupled network (1). The right branch of the fast nullcline $z=h\left(x\right)$ contains two points $AH1$ and $AH2$ corresponding to supercritical Andronov-Hopf bifurcations. A limit cycle of the fast system ($\mu =0$) is born from the Andronov-Hopf bifurcation $AH2$ and grows in size as $z$ increases. This family of limit cycles constitutes the spiking manifold which terminates at the homoclininc bifurcation HB of the saddle point of the fast system, located on the middle branch of $z=h\left(x\right).$ The red (dotted) curve schematically indicates the route for bursting trajectories. The plane $x={\mathrm{\Theta }}_s$ displays the synaptic threshold. Bottom: Plateau bursting induced by the combination of excitatory and inhibitory coupling ($g_{\mathrm{exc}}=0.6$ and $g_{\mathrm{inh}}=0.25$), corresponding to point (c) in Fig. 2. The added inhibition leads to the disappearance of the homoclinic bifurcation such that the spiking manifold extends further up and disappears as the limit cycle shrinks to zero amplitude and disappears via the reverse Andronov-Hopf bifurcation $AH1.$

Figure 4

Stability function $\mathrm{\Omega }\left(x\right)$ for synchronized bursting. Panels (a), (b), (c), and (d) correspond to the points (a), (b), (c), and (d) in Fig. 2. (a) $g_{\mathrm{exc}}=0.6,$ $g_{\mathrm{inh}}=0$: Unstable square-wave synchronous bursting [brown (gray)] and the fast nullcline $h\left(x\right)$ of the self-coupled system, together with $\mathrm{\Omega }\left(x\right)$ superimposed on its own scale. The impact of $\mathrm{\Omega }\left(x\right)$ subthreshold where the coupling is insignificant (to the left from the threshold ${\mathrm{\Theta }}_s$). (b) $g_{\mathrm{exc}}=1.28,$ $g_{\mathrm{inh}}=0$: The increased excitation makes the impact of $\mathrm{\Omega }\left(x\right)$ stronger; more importantly, it changes the type of synchronous bursting. Notice that the spikes have shifted to the right and moved to the region where the strong coupling is present. (c) $g_{\mathrm{exc}}=0.6,$ $g_{\mathrm{inh}}=0.25$: The red (upper) curve represents the contribution of the excitatory coupling ${\mathrm{\Omega }}_{\mathrm{exc}}=g_{\mathrm{exc}}\mathrm{\Gamma }\left(x\right)+g_{\mathrm{exc}}(V_{\mathrm{exc}}-x){\mathrm{\Gamma }}_x^{\prime }\left(x\right),$ the green (light gray) curve corresponds to that of the inhibitory coupling ${\mathrm{\Omega }}_{\mathrm{inh}}=g_{\mathrm{inh}}\mathrm{\Gamma }\left(x\right)+g_{\mathrm{inh}}(V_{\mathrm{inh}}-x){\mathrm{\Gamma }}_x^{\prime }\left(x\right),$ and the thick black line indicates the combined curve $\mathrm{\Omega }\left(x\right)={\mathrm{\Omega }}_{\mathrm{exc}}+{\mathrm{\Omega }}_{\mathrm{inh}}.$ Adding the inhibition decreases the impact of $\mathrm{\Omega }\left(x\right)$ [cf. with (a) where $\mathrm{\Omega }\left(x\right)$ equals ${\mathrm{\Omega }}_{\mathrm{exc}}$ in (c)]. At the same time, it induces plateau bursting, with the spikes in the region above the threshold, where the coupling is sufficiently strong to synchronize them. (d) $g_{\mathrm{exc}}=0.6$, $g_{\mathrm{inh}}=0.9$: Strong inhibition destabilizes synchronous plateau bursting. $\mathrm{\Omega }\left(x\right)$ has a drop in the region, covering the upper knee of the nullcline. As a result, the cells diverge when slowly crawling up this part of the nullcline. Note that synchronous plateau bursting of the self-coupled system is unstable and does not represent the dynamics observed in the network; the cells become locked into antiphase square-wave bursting [cf. Fig. 2].

Figure 5

Stability diagrams for network synchronization, similarly to that of Fig. 2. The color bar indicates the mean voltage difference ${\sum }_{i=1}^{n-1}{\sum }_{j>i}^n\frac{2}{n(n-1)}(x_i-x_j),$ calculated and averaged over the last three bursting periods. Notice the nearly identical diagrams for pairs of 10-cell irregular and 5-cell regular networks with $k_{\mathrm{exc}}=4$ and $k_{\mathrm{inh}}=4$ (left pair) and $k_{\mathrm{exc}}=2$ and $k_{\mathrm{inh}}=4$ (right pair). Excitatory (inhibitory) connections are depicted by arrows (circles). Excitatory (inhibitory) neurons in the 10-cell irregular networks [with only outgoing excitatory (inhibitory) connections] are denoted by light (dark) circles. The height and width of the left instability zone, adjacent to the $g_{\mathrm{exc}}$ axis and corresponding to desynchronized square-wave bursting, are inversely proportional to $k_{\mathrm{exc}}$ and $k_{\mathrm{inh}},$ respectively (also compare with Fig. 2).

Figure 6

Ratio of the synchronization threshold in an excitatory network without inhibition and the minimum synchronization threshold achieved by adding inhibition, as a function of the network size $n,$ for different values of $k_{\mathrm{exc}}$ and $k_{\mathrm{inh}}$. The ratio of the synchronization threshold reduction, induced by added inhibition is as large as 12 for the two-cell network (compare with Fig. 2). The four curves represent four types of network topology: rings of cells with local excitatory and inhibitory connections ($k_{\mathrm{exc}}=2$ and $k_{\mathrm{inh}}=2$); all-to-all networks with both global excitatory and inhibitory connections ($k_{\mathrm{exc}}=n-1$ and $k_{\mathrm{inh}}=n-1$); networks with global excitatory and local inhibitory connections ($k_{\mathrm{exc}}=n-1$ and $k_{\mathrm{inh}}=2$); and rings of cells with local excitatory and all-to-all inhibitory connections ($k_{\mathrm{exc}}=2$ and $k_{\mathrm{inh}}=n-1$). Notice that the addition of global inhibition to a locally coupled excitatory network (local excitation-global inhibition) yields the smallest reduction in the synchronization threshold for $n>3$ (lowest line) and therefore has the worst synchronization properties. At the same time, the addition of local inhibition to the same locally coupled excitatory network yields the highest reduction ratio for $n>3$ (top line) and indicates a nontrivial synergistic effect of the combined inhibitory and excitatory topologies. Also observe that global inhibition promotes synchronization more significantly than local inhibition when added to a globally coupled excitatory network, as the global excitation-global inhibition configuration has a higher synchronization threshold reduction ratio (second line from the top) compared to that of the global excitation-local inhibition configuration (third line from the top).

Figure 7

Top: Induced synchronization in a 100-cell randomly generated network with uniform $k_{\mathrm{exc}}=4$ and $k_{\mathrm{inh}}=4.$ Bottom: The network has 80 excitatory [red (light gray)] and 20 inhibitory [blue (dark gray)] cells. The excitatory connections are marked by red (light gray) arrowed lines; the inhibitory coupling is indicated by blue (dark gray) arrows. Both excitatory and inhibitory coupling strengths are heterogeneous, with randomly distributed mismatch up to 10%. The color bar indicates the mean voltage difference as in Fig. 5. The stability diagram is similar to those of the two left diagrams in Fig. 5, corresponding to the 5- and 10-cell networks with $k_{\mathrm{exc}}=4$ and $k_{\mathrm{inh}}=4$. Complete spike synchronization is impossible in this mismatched network; however, an approximate synchronization with small voltage differences (offsets between the spikes) is robustly present. Various shades of blue (black) and the nonhomogeneous structure of the synchronization stability zone correspond to slight voltage offsets due to the parameter mismatch.

Figure 8

Role of $E_{\mathrm{syn}}$ in synchronization of the two-cell network (A1). The stability diagram and color coding are similar to those of Fig. 2. Decreasing the reversal potential $E_{\mathrm{syn}}$ from 2 first dramatically lowers the synchronization threshold. Dropping $E_{\mathrm{syn}}$ below $-0.25$ makes the connection essentially inhibitory such that synchronization cannot be achieved for any value of $g_{\mathrm{syn}}$: Note the vertically rising stability boundary around $E_{\mathrm{syn}}=-0.25.$