Intertangled stochastic motifs in networks of excitatory-inhibitory units

A stochastic model of excitatory and inhibitory interactions which bears universality traits is introduced and studied. The endogenous component of noise, stemming from finite size corrections, drives robust internode correlations that persist at large distances. Antiphase synchrony at small frequencies is resolved on adjacent nodes and found to promote the spontaneous generation of long-ranged stochastic patterns that invade the network as a whole. These patterns are lacking under the idealized deterministic scenario, and could provide hints on how living systems implement and handle a large gallery of delicate computational tasks.

Figure 1

The theoretical power spectrum $P_{11}\left(\omega \right)=P_{22}\left(\omega \right)$ (3) is depicted with a solid line. Symbols refer to the power spectra computed from averaging independent realizations of the Gillespie dynamics. The squares refer to excitators ($X$), while the circles stand for inhibitors ($Y$). The vertical dotted line is traced at $\omega ={\lambda }_{\mathrm{Im}}=r/4$. The theoretical dashed line represents $|C_{12}|=|C_{21}|$. A phase lag equal to $\pi /2$ is predicted. Here $r=50$ so as to allow for the isolated peak in the power spectra to be distinctively revealed.

Figure 2

(a) The power spectrum $P_{11}$ is plotted as a function of $\omega$, for different choices of $D$. Lines refer to the theoretical predictions. Symbols are obtained by averaging over many realizations of the stochastic simulations. (b) $|C_{13}|$ is plotted in the plane $(\omega ,D)$. Two regions can be identified where the synchronization takes place. These are separated by the dashed (white) line, obtained by setting $|C_{13}|=0$. The synchronization at small frequencies occurs in antiphase ($\varphi =\pi$), while at high frequencies the theory predicts $\varphi =0$. The stochastic trajectories ($n_{X_1}$ and $n_{X_3}$ vs time $t$) confirm the adequacy of the LNA. In phase and antiphase regimes of synchronization are highlighted in the boxes. Here, $r=50$ and $V=20000$.

Figure 3

Snapshot of the stochastic dynamics for two network topologies: The stochastic density of the excitators is plotted. (a) A chain of four nodes is considered. In each node the density of excitators (upper circles) and inhibitors (lower circles) is depicted. (b) A snapshot of the stochastic pattern of activation of the excitators is shown.

Figure 4

Stochastic trajectories ($n_X$ and $n_Y$ versus time $t$). The fixed point of the underlying deterministic model is depicted with a dashed black line.

Figure 5

Bifurcation diagram for the two-node system, in its deterministic version. Above the bifurcation point ($D>D_c$), the solid black lines refer to the excitators $x_{1,2}$, while the dashed-dotted lines stand for the inhibitors $y_{1,2}$.

Figure 6

(a) Map describing the magnitude of $C_{12}$ in the reference plane $(D,\omega )$. This parameter allows one to resolve the degree of synchronization between excitators and inhibitors in the same node. (b) $|C_{14}|$ is plotted in the plane $(D,\omega )$.

Figure 7

(a) $|C_{1,2i+1}|$ vs $\omega$ (vertical axis) is plotted, for $i=1,2,3$ (horizontal axis). The system is made up of four nodes organized in a closed linear ring. The synchronization occurs for roughly two values of $\omega$, the modulus of the complex coherence function being more significant at low frequencies. The phase lag predicted by the theory is also displayed in the figure. The values reported apply inside the regions delimited by the (white) dashed lines. (b) Same for $|C_{1,2i}|$.