Nontrivial amplification below the threshold for excitable cell signaling

In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.

Figure 1

Interaction patterns typically found in excitable systems. Each node represents a dynamical variable, while directed links represent activatory and inhibitory interactions between these variables (magenta arrows and cyan barred lines, respectively). Pattern (a) is asymptotically stable and contains only negative feedback loops. Patterns (b) and (c) are the only two-node patterns that simultaneously allow asymptotic stability and self-activation, a strong form of positive feedback, and are typically found near a bifurcation from excitable to oscillatory dynamics. As discussed in the main text, systems with these features tend to amplify noise.

Figure 2

Noise amplification in the FitzHugh-Nagumo model of neural excitation. The amplification ratio $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0$ (black contour lines), which compares the total fluctuation variance $\langle {\eta }^2\rangle$ to the variance ${\langle {\eta }^2\rangle }_0$ predicted by linear stability theory, increases by orders of magnitude as $\epsilon$ decreases and the system approaches a Hopf bifurcation (magenta curve at upper boundary of shaded region). Below the cyan curve (lower boundary of shaded region), the system is nonreactive: the distance $\eta$ to the steady state decays monotonically along all nearby phase trajectories (Appendix pp1).

Figure 3

Variation of the angle $\theta$ between the FitzHugh-Nagumo model's eigenvectors (black contour lines) establishes that this system is never close to normal and, moreover, is far from normal precisely where Fig. 2 shows dramatic amplification. The shaded region shown here matches that of Fig. 2, bounded above by a Hopf bifurcation (magenta curve) and bounded below by a transition to nonreactive dynamics (cyan curve, discussed in Appendix pp1).

Figure 4

Schematic representation of physiological processes found in an influential quantitative model of calcium signaling [17]. These processes together create an excitable system in which relatively small signals can trigger explosive bursts in cytosolic calcium levels.

Figure 5

Noise amplification in a quantitative model of calcium signaling. (a–c) Three different interaction patterns arise in the model at different values of the control parameter $\beta$. Panels (d) and (e) show how the amplification ratio $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0$ and angle $\theta$ between eigenvectors, respectively, vary with the control parameter $\beta$ for an ensemble of quantitatively similar copies of the system (see main text). Each color is associated with one of the three interaction patterns found in this system. Note that the amplification is greatest in regions 2 and 3, as the system approaches one of its Hopf bifurcation.

Figure 6

Cumulative distributions of the amplification scale $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ for excitable calcium signaling systems found in regions 2 and 3 (orange and yellow curves, respectively), showing a statistically significant shift relative to random systems with the same interaction patterns (purple curves, discussed in Sec. 4).

Figure 7

The total fluctuation variance $\langle {\eta }^2\rangle$ and the variance ${\langle {\eta }^2\rangle }_0$ predicted by linear stability theory for excitable calcium signaling systems near a Hopf bifurcation (orange and blue dots, respectively). For normal systems with complex eigenvalues, the fluctuation-dissipation theorem predicts ${\langle {\eta }^2\rangle }_0=-2/\tau$ (black line). The gap between $\langle {\eta }^2\rangle$ and ${\langle {\eta }^2\rangle }_0$ matches the amplification scale visible in Fig. 5.

Figure 8

Schematic representation of cellular processes found in an influential quantitative model of cAMP signaling dynamics used in Dictyostelium colonies [25, 26]. These processes together create an excitable system in which the detection of extracellular cAMP signals can trigger explosive bursts in intracellular cAMP synthesis.

Figure 9

Noise amplification in a quantitative model of excitable cAMP signaling in Dictyostelium colonies. (a) Three-variable interaction pattern found near the resting state. (b) Cumulative distributions of $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ for ensembles of quantitatively similar copies of this three-variable system (interaction pattern shown above). The blue, orange, and yellow curves are associated with parameter sets B, D, and E, respectively (see main text). All three parameter sets show a statistically significant shift relative to comparable random systems (purple curve, discussed in Sec. 4).

Figure 10

Noise amplification in a reduced model of excitable cAMP signaling in Dictyostelium colonies. The blue, orange, and yellow curves are cumulative distributions of $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ for parameter sets B, D, and E, respectively (see main text). In both panels, all three parameter sets show a statistically significant shift relative to comparable random systems (purple curves, discussed in Sec. 4).

Figure 11

The total fluctuation variance $\langle {\eta }^2\rangle$ and the variance ${\langle {\eta }^2\rangle }_0$ predicted by linear stability theory (orange and blue points, respectively) for excitable cAMP signaling systems near a Hopf bifurcation, drawn from ensemble D. For normal systems with complex eigenvalues, the fluctuation-dissipation theorem predicts ${\langle {\eta }^2\rangle }_0=-2/\tau$ (black line). Ensemble B produces statistically indistinguishable results, while E produces similar results.

Figure 12

Noise amplification in systems with random interaction strengths and complex eigenvalues. Cumulative distributions of $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ for systems with and without self-activation loops (orange and blue curves, respectively) reveal that, even when interactions strengths are assigned randomly, self-activation typically leads to greater amplification. Random systems with real eigenvalues generate very similar distributions.

Figure 13

Positive feedback in excitable cAMP signaling drives stronger amplification of noise, with the amplification scale $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ increasing roughly exponentially as the feedback strength increases. The trend for ensemble B is statistically indistinguishable from that for ensemble D, shown here. Ensemble E shows a similar trend.

Figure 14

Positive feedback in calcium signaling drives stronger amplification of noise, with the amplification scale $\langle {\eta }^2\rangle /{\langle {\eta }^2\rangle }_0-1$ increasing roughly exponentially as the feedback strength increases. The three regions identified here refer to specific interaction patterns identified in Fig. 5.

Figure 15

Reactive dynamics in the FitzHugh-Nagumo model of neural excitation. For any fixed value of $\epsilon$, the peak value ${\rho }_{\max }$ of the amplification envelope (black contour lines) increases with increasing $I$, i.e., as the system moves toward a Hopf bifurcation (magenta curve at upper boundary of shaded region). Note that ${\rho }_{\max }$ exceeds 1 throughout the shaded region and, for sufficiently small $\epsilon$, can be orders of magnitude greater than 1. Below the cyan curve (lower boundary of shaded region), the system is nonreactive.