SOC induced by diversity

The claim

A population of pulse-coupled integrate-and-fire oscillators does not show self-organized criticality when the units are identical — it tends instead to synchronize. Introducing diversity (a random spread of natural periods) is the mechanism that induces SOC in the long-time regime. As diversity grows the system sweeps through supercritical → SOC → subcritical, and — crucially — criticality holds over an extended range of the control parameter, not a single point (unlike percolation or equilibrium transitions).

The model

Each oscillator has a state $E_i$ with $dE_i/dt = S - \gamma E_i$ ; on reaching threshold it resets to zero and redistributes a fixed $\varepsilon$ to its neighbours, which can trigger a chain of relaxations — an avalanche of size $s$ . The reset-and-transfer rule makes it intrinsically non-conservative. On a 2D lattice with open boundaries (so edge units couple to fewer neighbours), it models spiking neurons (membrane as an RC circuit), cardiac pacemakers, and flashing fireflies; at $\gamma=0$ it is the Feder–Feder stick-slip earthquake model.

Diversity that touches only the slow scale

Diversity enters as a distribution of intrinsic periods $T = \gamma^{-1}\ln\!\big(S/(S-\gamma E_{\text{th}})\big)$ , implemented via a spread of input currents $S$ of width $\Delta$ . The key subtlety: this affects only the slow (driving) timescale, leaving the fast (avalanche) dynamics untouched — unlike a distribution of thresholds, which perturbs both scales and is known to destroy criticality in the dissipative case. The slow-vs-fast distinction is what makes diversity constructive here.

Results

Identical units ( $\Delta=0$ ): no SOC — avalanche sizes peak at multiples of $L$ (open-boundary effect) and at $L^2$ (synchronization), not a power law.
Increasing $\Delta$ : the peaks smooth out; a mild spread ( $\Delta\approx0.15$ ) is still supercritical (system-spanning events); around $\Delta\approx0.5$ the system reaches SOC — a scale-free power-law avalanche distribution ( $P(s)\sim s^{-\tau}$ , $\tau\approx1.6$ ); large $\Delta\gtrsim1.5$ turns subcritical — exponential cutoff, localized avalanches, finite correlation length.
Finite-size scaling ( $L^{\beta}P(s,L)$ vs. $s/L^{\nu}$ , with $\langle s\rangle\sim L^{2\nu-\beta}$ ) collapses across a band of $\Delta$ values — direct evidence of an extended critical region, not a critical point (e.g. $\nu\approx2.0,\beta\approx3.15$ at $\Delta=0.3$ ).
Robust to random (quenched or annealed) coupling strengths and to Olami–Feder–Christensen relaxation rules; the SOC-inducing regime sits at large $\varepsilon$ , small $\gamma$ .

Why it’s collected here

Three hooks for this lab. First, the substrate is exactly the node scale — integrate-and-fire units producing power-law avalanches — so its exponents and finite-size-scaling method are the same machinery used on reservoir spikes. Second, its headline is that heterogeneity is constructive: diversity among units creates criticality rather than washing it out, which bears directly on a reservoir of non-identical nodes or a swarm of non-identical agents. Third, the extended-critical-region finding lines up with the Griffiths-phase / generic-scale-invariance thread (Muñoz), the swarm extended-region model (González-Albaladejo), and the connectivity-band result (Bessone & Plantec) — and its insistence that diversity work only on the slow timescale is a concrete lesson for any homeostatic / rate-separated mechanism.

References

Corral, Pérez & Díaz-Guilera (1997), Phys. Rev. Lett. 78, 1492 · Bak, Tang & Wiesenfeld (1987), Phys. Rev. Lett. 59, 381 · Olami, Feder & Christensen (1992), Phys. Rev. Lett. 68, 1244 · Feder & Feder (1991), Phys. Rev. Lett. 66, 2669 · Kuramoto (1984), Chemical Oscillations, Waves and Turbulence · Mirollo & Strogatz (1990), SIAM J. Appl. Math. 50, 1645 · Corral et al. (1995), Phys. Rev. Lett. 74, 118 · Muñoz (2018), Rev. Mod. Phys. 90, 031001.