Large variability in dynamical transitions in biological systems with quenched disorder

Coherent oscillatory activity can arise spontaneously as a result of increased coupling in a system of excitable and passive cells, each being quiescent in isolation. This can potentially explain the appearance of spontaneous rhythmic contractions in the pregnant uterus close to term. We investigate the transition to periodic activity using a model system comprising a lattice of excitable cells, each being coupled to a variable number of passive cells whose distribution defines a quenched realization (replica) of spatial disorder. Close to the transition between quiescent state and sustained oscillations in the system we observe large fluctuations between different replicas induced by variations in the local density of passive cells around an excitable cell. We demonstrate that the disorder-induced fluctuations can be described in terms of a simple scaling relation which involves the strength of coupling between excitable cells, the mean passive cell density, as well as the logarithm of the system size. Our results can be interpreted as suggesting that larger organs will have greater variability in the onset of persistent activity.

💡 Research Summary

The paper investigates how spontaneous rhythmic activity can emerge in a heterogeneous tissue composed of excitable and passive cells, using a two‑dimensional lattice model. Each excitable cell follows a FitzHugh‑Nagumo type dynamics and is electrically coupled to its four nearest neighbours with strength G. In addition, every excitable cell is coupled to a variable number of passive cells; the number of passive cells attached to a given lattice site is drawn from a Poisson distribution with mean ρ. Once assigned, this distribution is fixed for the entire simulation, representing quenched spatial disorder (a “replica”).

The authors explore the system’s behavior as the coupling strength G is increased. For low G the network settles into a quiescent fixed point where all cells remain silent. Above a critical value Gc the network exhibits sustained, synchronized oscillations. Near the transition, however, different replicas display markedly different outcomes: some begin to oscillate immediately, others remain quiescent, and still others show mixed patterns with localized oscillatory islands. This replica‑to‑replica variability is traced to fluctuations in the local passive‑cell density ηi around each excitable cell. Sites with a higher-than‑average ηi experience stronger current sink effects, which lower the effective excitation threshold and promote early onset of oscillations. Conversely, regions with low ηi suppress activity, delaying the transition.

To quantify the effect of disorder, the authors measure the critical coupling Gc for many system sizes N = L² and for several values of ρ. They find that Gc obeys a simple scaling law:

 Gc(N, ρ) ≈ G0 + α · log(N) / ρ

where G0 and α are constants determined by the intrinsic cellular parameters (e.g., recovery time, membrane conductance) but are independent of N and ρ. The logarithmic dependence on N reflects the role of extreme‑value statistics: as the system grows, the probability of encountering a region with an unusually high passive‑cell density increases, and such rare “hot spots” dominate the onset of global oscillations. Consequently, larger tissues are far more sensitive to small variations in coupling strength, leading to amplified variability in the transition point.

The biological relevance of this finding is illustrated with the example of the pregnant uterus. Near term, hormonal changes increase gap‑junction coupling among myometrial cells, effectively raising G. The uterine wall contains a massive number of contractile smooth‑muscle cells interspersed with fibroblasts, immune cells, and other passive elements, creating a natural quenched disorder in passive‑cell density. According to the scaling law, a larger uterus (greater N) will exhibit a broader distribution of the gestational age at which sustained contractions begin, consistent with clinical observations that the timing of labor varies widely among individuals with otherwise similar physiological conditions.

Beyond obstetrics, the authors argue that the same principles apply to other excitable‑passive composites such as cardiac tissue (where fibroblasts act as passive loads), cortical networks with inhibitory interneurons, and engineered tissue constructs. By controlling the spatial arrangement or proportion of passive cells, one could deliberately shift Gc and modulate the robustness of synchronized activity, offering a potential design strategy for biomedical devices and synthetic organoids.

In summary, the study demonstrates that quenched spatial disorder introduces large replica‑dependent fluctuations in the dynamical transition from silence to rhythm. These fluctuations obey a universal scaling relation involving the coupling strength, mean passive‑cell density, and the logarithm of system size. The work provides a quantitative framework for understanding why larger biological organs display greater variability in the onset of persistent activity, and it suggests practical avenues for manipulating collective dynamics in heterogeneous excitable media.