Signal integration enhances the dynamic range in neuronal systems

The dynamic range measures the capacity of a system to discriminate the intensity of an external stimulus. Such an ability is fundamental for living beings to survive: to leverage resources and to avoid danger. Consequently, the larger is the dynamic range, the greater is the probability of survival. We investigate how the integration of different input signals affects the dynamic range, and in general the collective behavior of a network of excitable units. By means of numerical simulations and a mean-field approach, we explore the nonequilibrium phase transition in the presence of integration. We show that the firing rate in random and scale-free networks undergoes a discontinuous phase transition depending on both the integration time and the density of integrator units. Moreover, in the presence of external stimuli, we find that a system of excitable integrator units operating in a bistable regime largely enhances its dynamic range.

💡 Research Summary

In this work the authors investigate how the integration of multiple input signals influences the dynamic range and collective behavior of networks of excitable units, a problem of central relevance for sensory processing in the brain. Starting from the well‑known Kinouchi‑Copelli model of excitable media, they extend the dynamics by allowing each node to accumulate incoming active‑neighbor spikes within a time window τ and to fire only when the accumulated count Λi reaches a threshold θ. The model thus captures two limiting integration regimes: (i) coincidence detection (τ = 1 ms) where only simultaneous inputs matter, and (ii) infinite integration (τ → ∞) where the entire recent history is summed. Nodes can also be externally driven by a Poisson process of rate h, and they evolve through three states (quiescent, active, refractory) with a refractory recovery probability pγ = ½.

Simulations are performed on sparse random (Erdős‑Rényi) and scale‑free (Barabási‑Albert) networks of size N = 5 000 with average degree K = 50. When θ = 1 (no integration) the system exhibits the classic continuous (second‑order) phase transition: the average firing rate F grows smoothly as the coupling strength pλ exceeds the critical value pcλ ≈ K − 1. In contrast, for any θ > 1 the transition becomes discontinuous (first‑order). The firing rate jumps abruptly at a lower pcλ, and a hysteresis loop appears, indicating the coexistence of a quiescent and a self‑sustained active state – a bistable regime. The authors map out the dependence of the transition order on the density d of integrator nodes and on the integration time τ, showing that longer τ and higher d favor the discontinuous transition. Scale‑free networks require a smaller d to display bistability because hub nodes amplify the effect of integrators.

The dynamic range Δ is defined in the usual way as the logarithmic span of external stimulus intensities h that drive the response from 10 % to 90 % of the maximal firing rate. In the bistable region the response curves become history‑dependent: starting from low activity the system follows a low‑firing branch that rises gradually with h, while starting from high activity it follows a high‑firing branch that saturates quickly. The low‑activity branch yields a much larger Δ because it can discriminate a broader range of input intensities before saturation. Importantly, the maximal dynamic range Δmax grows dramatically with τ and with the proportion of integrators. For τ = ∞ and d ≈ 0.7–0.9 the authors report Δmax values exceeding 30 dB, i.e. more than a four‑fold increase compared with the non‑integrating case. This enhancement is observed both in random and scale‑free topologies, as summarized in Table I.

A mean‑field analysis reproduces the main findings. The authors derive a discrete‑time map for the average firing rate:

δt F_{t+1}=Q_t p_h+Q_t(1−p_h)