Non-normal amplification of stochastic quasicycles

Stochastic quasicycles for a two-species model of the excitatory-inhibitory type, arranged on a triangular loop, are studied. By increasing the strength of the internode coupling, one moves the system towards the Hopf bifurcation, and the amplitude of the stochastic oscillations is consequently magnified. When the system is instead constrained to evolve on specific manifolds, selected so as to return a constant rate of deterministic damping for the perturbations, the observed amplification correlates with the degree of non-normal reactivity, here quantified by the numerical abscissa. The thermodynamics of the reactive loop is also investigated and the degree of inherent reactivity shown to facilitate the out-of-equilibrium exploration of the available phase space.

Figure 1

The scheme of the model is illustrated. Two populations, labeled, respectively, $X$ and $Y$, are distributed on three distinct nodes of a triangular loop and therein interact via an activator-inhibitor cycle. The nodes of the collection are coupled together, through a nonlinear sigmoidal function. $D$ controls the strength of the internode interaction, while $\epsilon \in [1/2,1]$ sets the degree of the coupling asymmetry.

Figure 2

The domain of stability of the homogeneous equilibrium is depicted in the reference plane ($\epsilon , D$) and for $r=50$. The upper dashed curve stands for the Hopf bifurcation. Above the lower dashed curve the non-normal Jacobian matrix $\mathcal{J}$ is found to be reactive. Iso-$\alpha$ and iso-$\omega \left(\mathcal{J}\right)$ curves are traced and colored with relevant codes which reflect their associated level, as specified in the bars on either side.

Figure 3

(a) the asymptotic norm of the fluctuations ${\langle \left|\right|\mathbf{\zeta }\left|\right|}^2\rangle /3={\sum }_{i=1}^3({\xi }_i^2+{\eta }_i^2)/3$, as displayed on each individual node, is plotted versus $\omega \left(\mathcal{J}\right)-\left|\alpha \right|$, moving along isolines $\overline{\alpha }$. Different curves refer to different choices of $\overline{\alpha }$ ($=-0.8,-0.6,-0.4$, from bottom to top). Solid lines stand for the analytical solution after Eqs. (53). Symbols are obtained from direct simulations of the nonlinear Langevin equations (14), averaging over $M=300$ independent realizations. Here $V_1=V_2=V_3={10}^6$ and $r=50$: (b) stochastic trajectories are displayed, relative to the inhibitors on the first node, i.e., species $y_1$, for different values of the numerical abscissa. The red (small-amplitude) trajectory is obtained for $\omega \left(\mathcal{J}\right)=-0.24$ ($\epsilon =1$), while the blue (large-amplitude) trajectory refers to $\omega \left(\mathcal{J}\right)=5.18$ ($\epsilon =0.506$). Here $r=50$ and $V_1=V_2=V_3={10}^6$.

Figure 4

The theoretical power spectrum of the inhibitory species, ${\mathcal{P}}_{22}$, is plotted with a solid line against $\nu$. Different curves refer to different choices of ($\epsilon ,D$) constrained to move across the iso-$\alpha$ line $\overline{\alpha }=-0.4$. The degree of reactivity, as quantified by the numerical abscissa $\omega \left(\mathcal{J}\right)$, increases from right to left $\left[\omega \right(\mathcal{J})=0.1,4.6,5.2]$: the peak of the power spectrum gains correspondingly in power. Symbols refer to direct numerical simulations, based on Eq. (14), averaging over $M=200$ independent realizations. Here $V_1=V_2=V_3={10}^6$ and $r=50$. Notice that ${\mathcal{P}}_{22}={\mathcal{P}}_{44}={\mathcal{P}}_{66}$, due to translational invariance across the loop.

Figure 5

The theoretical power spectrum of the inhibitory species, ${\mathcal{P}}_{22}$, is plotted with a solid line against $\nu$, for different choices of $D$ (from right to left $D=0,6,8$). Symbols refer to direct numerical simulations, based on Eq. (14), averaging over $M=200$ independent realizations. Here $V_1=V_2=V_3={10}^6$; ${\mathcal{P}}_{22}={\mathcal{P}}_{44}={\mathcal{P}}_{66}$, due to translational invariance across the loop and $r=50$.

Figure 6

The stationary entropy ${\mathcal{S}}_{\infty }$ is plotted against $\omega \left(\mathcal{J}\right)-\left|\alpha \right|$ moving on iso-$\alpha$ lines. Here $\overline{\alpha }$ ($=-0.8,-0.6,-0.4$, from bottom to top). Solid lines stand for the analytical solutions. Symbols follow from direct simulations of the Langevin equations (14), on averaging over $M=300$ independent realizations. Here $V_1=V_2=V_3={10}^6$ and $r=50$.

Figure 7

Main panel: ${\Pi }_S-{\mathrm{\Phi }}_S$ is plotted in lin-log scale against $\tau$, for two choices of the parameters $D,\epsilon$ constrained to return constant $\alpha =\overline{\alpha }=-0.6$. The solid (red) line refers to $\omega \left(\mathcal{J}\right)=5.25$, while the dashed (blue) line stands for $\omega \left(\mathcal{J}\right)=-0.24$. In the inset ${\Pi }_S$ and ${\mathrm{\Phi }}_S$ are reported for the choices of parameters, as given in the main panel. Here $r=50$.