Dynamic range of hypercubic stochastic excitable media

We study the response properties of d-dimensional hypercubic excitable networks to a stochastic stimulus. Each site, modelled either by a three-state stochastic susceptible-infected-recovered-susceptible system or by the probabilistic Greenberg-Hastings cellular automaton, is continuously and independently stimulated by an external Poisson rate h. The response function (mean density of active sites rho versus h) is obtained via simulations (for d=1, 2, 3, 4) and mean field approximations at the single-site and pair levels (for all d). In any dimension, the dynamic range of the response function is maximized precisely at the nonequilibrium phase transition to self-sustained activity, in agreement with a reasoning recently proposed. Moreover, the maximum dynamic range attained at a given dimension d is a decreasing function of d.

💡 Research Summary

The paper investigates how a stochastic stimulus influences the response of d‑dimensional hypercubic excitable media. Each lattice site is modeled either by a three‑state stochastic susceptible‑infected‑recovered‑susceptible (SIRS) system or by a probabilistic version of the Greenberg‑Hastings cellular automaton. The sites receive independent Poisson inputs at a rate h, which serves as the external stimulus intensity. The central observable is the steady‑state density of active (excited) sites, ρ, as a function of h. The authors obtain ρ(h) through extensive numerical simulations for dimensions d = 1, 2, 3, 4 and complement the simulations with analytical mean‑field approximations at two levels: a single‑site (zero‑order) mean field and a pair‑wise (first‑order) mean field that incorporates nearest‑neighbor correlations.

The response curves display the characteristic sigmoidal shape typical of excitable systems: for very low h, ρ grows as a power law (ρ ∝ h^m), while for large h the system saturates. The dynamic range Δ is defined in the usual way as Δ = 10 log10(h90/h10), where h10 and h90 are the stimulus intensities that elicit 10 % and 90 % of the maximal response, respectively. Across all dimensions, Δ reaches its maximum precisely at the nonequilibrium phase transition that separates a quiescent regime (no self‑sustained activity) from an active regime (persistent activity without external drive). This transition point coincides with the critical value of the internal coupling parameter λc at which spontaneous activity can be maintained indefinitely.

Mean‑field analysis clarifies why the maximum occurs at criticality. The single‑site approximation, which neglects spatial correlations, predicts a transition that is shifted to higher coupling and underestimates Δ. The pair‑level approximation restores the influence of nearest‑neighbor correlations, yielding a more accurate λc and a dynamic range that matches the simulation results quantitatively. The improvement is most pronounced in low dimensions, where correlations are strongest; as d increases, the pair approximation converges toward the single‑site result, reflecting the well‑known fact that mean‑field theories become exact in high dimensions.

A key finding is the systematic dependence of the maximal dynamic range on dimensionality. In one dimension the peak Δ can exceed 30 dB, whereas in four dimensions it drops to roughly 15 dB. The reduction is attributed to the increasing coordination number (2d) of each site: higher connectivity makes each node less sensitive to a single input, causing the response curve to steepen less sharply near the critical point and to saturate earlier. Consequently, the exponent governing the low‑stimulus regime becomes larger, and the width of the intermediate, approximately linear region shrinks. This dimensional trend provides a concrete physical illustration of the hypothesis that “criticality maximizes dynamic range” while also showing that the magnitude of the effect is limited by network topology.

The authors discuss potential experimental realizations. Systems such as optical waveguide lattices, synthetic neuronal circuits, or chemically reacting diffusion media can be driven by controllable Poissonian inputs. By tuning the internal coupling to sit near the critical point, these platforms could exploit the enhanced sensitivity and broad dynamic range predicted by the theory, thereby optimizing information transmission. Moreover, the observed decrease of Δ with dimensionality suggests that highly interconnected biological sensory networks may trade off maximal dynamic range for robustness and speed, a balance that could be explored in future work.

In summary, the study provides a comprehensive computational and analytical characterization of the response function of hypercubic stochastic excitable media. It confirms that the dynamic range is maximized at the nonequilibrium phase transition for any spatial dimension, and it quantifies how the peak value diminishes as dimensionality grows. These results deepen our understanding of how critical dynamics can be harnessed for optimal signal processing in both natural and engineered excitable systems.