The research addresses a long-standing blind spot in the modeling of criticality. Most spiking network models that exhibit self-organized critical behavior rely on random or all-to-all connectivity, an abstraction that ignores geometry entirely. Real neurons, however, live in physical space. Their axons and dendrites span measurable distances, connection probability falls off with separation, and signals take longer to travel between distant cells. Because both the wiring and the timing of inhibition are shaped by distance, the authors asked a deceptively simple question: if you constrain where inhibitory synapses can form, does the network still find its way to criticality, and if so, to which side of the critical point does it drift?

To find out, the team built two-dimensional spiking networks in which they systematically varied the reach and density of inhibitory connectivity. Excitatory and inhibitory neurons were distributed across a plane, and inhibitory connections were restricted to a maximum length scale, so that some networks contained only short-range inhibition while others allowed inhibitory signals to travel much farther. Crucially, the synaptic weights were not fixed by hand. Instead, they developed through activity-dependent plasticity rules for both excitatory and inhibitory synapses, driven by the network’s own firing and by a low level of stochastic intrinsic activity that acted as a background hum, nudging the system into motion even before strong collective dynamics emerged.

This developmental setup is what gives the study its power. Rather than imposing a critical state and observing it, the researchers let the networks grow toward whatever dynamical regime their structure and plasticity rules produced, then measured where they landed. Throughout development and after it, they tracked the relationships between inhibitory connectivity, the distribution of synaptic weights, and the resulting patterns of network activity. The comparison across spatial configurations allowed them to isolate the effect of inhibitory reach while holding the total number of inhibitory synapses roughly constant, a control that turns out to be essential for interpreting the results.

The headline finding is stark. Networks equipped with longer-range inhibitory synapses tended toward supercritical behavior, meaning their activity became more explosive and less self-limiting, compared with networks that had a similar number of shorter-range inhibitory connections. In the language of criticality, the long-range inhibition pushed the system past the tipping point rather than holding it at the edge. This is in many ways counterintuitive: inhibition is the brain’s braking system, so one might expect more far-reaching brakes to tame runaway activity. The simulations show the opposite, and the reason lies in what happens to the strength of those synapses during development.

The mechanism the authors uncovered centers on synaptic delays and the timing windows of inhibitory learning. Inhibitory plasticity rules typically strengthen a synapse when the pre- and postsynaptic spikes fall within a specific potentiation window in time. But signals traveling across longer distances arrive with longer conduction delays. Those delays shift most spike pairs outside the potentiation window of the learning rule, so the long-range inhibitory synapses never receive the same drive to strengthen that short-range synapses do. After development, the result is a population of long-range inhibitory connections that are systematically weaker than their short-range counterparts, even though they exist in comparable numbers.

Weak long-range inhibition has consequences that ripple through the entire network. Because distant inhibitory neurons cannot effectively rein in activity in far-flung parts of the network, the balance of excitation and inhibition, the key determinant of the critical point in spiking systems, tilts toward excitation at the global scale. The network therefore drifts into a supercritical regime, where activity cascades grow larger and more sustained than a critical system would allow. The study thus reframes the classical picture: the critical point is not set by connection counts alone but by the effective, developed weights of connections, and those weights are themselves shaped by the geometry of the tissue through the physics of signal travel time.

The implications reach well beyond simulation. The cortex is a layered, spatially extended sheet in which inhibitory interneurons form connections with characteristic spatial profiles, and axonal conduction delays of milliseconds to tens of milliseconds are ubiquitous. If the same timing-based weakening operates in biological tissue, then the spatial statistics of inhibitory wiring could be a fundamental ingredient in how real brains tune themselves to the vicinity of criticality. The findings suggest that models of self-organized criticality that ignore space may be systematically misestimating where a network sits relative to its critical point, and that realistic spatial constraints should be considered a feature of the theory rather than a complication to be abstracted away.

The work also speaks to a broader theme in neuroscience: structure and dynamics are two sides of the same coin. Activity-dependent plasticity is often studied as if it operated on abstract graphs, but the study demonstrates that physical distance enters the loop twice, first by determining which neurons can connect and second by determining when their signals arrive, which in turn decides whether plasticity strengthens or leaves a synapse alone. A learning rule, a set of conduction delays, and a spatial wiring profile together produce a dynamical fate. Change any one of them and the same network can settle into a critical, subcritical, or supercritical regime.

For the field of computational neuroscience, the study opens concrete avenues for follow-up. Natural next questions include how excitatory spatial constraints interact with inhibitory ones, how different inhibitory plasticity rules with wider or narrower timing windows would alter the distance dependence, and whether developing networks could compensate for weak long-range inhibition through homeostatic mechanisms not included in the current model. It also raises the possibility that biological brains exploit precisely this geometry-dependent weakening, using short-range inhibition to carve local structure while allowing excitation to bind distant regions, achieving a balance that no random network could replicate. What is clear from Kern, Date, and Chao’s results is that the road to criticality is paved not only with the number of synapses a network possesses but with the distances those synapses must cross, and with the milliseconds of delay that every long journey through neural tissue inevitably costs.

Subject of Research: Effects of spatially constrained inhibitory connectivity on the development of criticality in spiking neural networks

Article Title: Effects of spatial constraints of inhibitory connectivity on the dynamical development of criticality in spiking networks

Article References: Kern, F. B., Date, T., & Chao, Z. C. (2026). Effects of spatial constraints of inhibitory connectivity on the dynamical development of criticality in spiking networks. PLOS Computational Biology, 22(10), e1014798. https://doi.org/10.1371/journal.pcbi.1014798

Image Credits: AI Generated

DOI: 10.1371/journal.pcbi.1014798

Keywords: criticality, spiking networks, inhibitory connectivity, synaptic plasticity, synaptic delays, self-organization, neural dynamics, computational neuroscience, excitation-inhibition balance, spatial networks, network development, PLOS Computational Biology