Novel Kuramoto model with inhibition dynamics modeling scale-free avalanches and synchronization in neuronal cultures

Neuronal cultures exhibit a complex activity, bursts, or avalanches, characterized by the coexistence of scale invariance and synchronization, quite stable with the percentage of inhibitory neurons. While this bistable behavior has been already observed in the past, the characterization of the statistical properties of avalanche activity and their temporal organization is still lacking, as well as a model able to reproduce these dynamics. Here, we analyze experimental data of human neuronal cultures with controlled percentage of inhibitory neurons and characterize their statistical properties and dynamical organization. In order to model the experimental data, we propose a novel version of the Kuramoto model for two populations of oscillators, excitatory and inhibitory, implementing successfully the inhibition dynamics. The model can fully reproduce the experimental results, confirming the existence of correlations in the temporal organization of avalanche activity and the presence of an amplification - attenuation regime, as found in the human brain.

💡 Research Summary

The paper investigates the coexistence of scale‑free neuronal avalanches and synchronized bursting in human induced pluripotent stem cell (hiPSC)‑derived cortical cultures, focusing on how the proportion of inhibitory neurons shapes these dynamics. Two experimental conditions were examined: a physiological excitatory‑to‑inhibitory ratio of 75 % E : 25 % I (75E:25I) and a fully excitatory network (100E). Cultures were grown for 70 days in vitro and recorded with 60‑channel micro‑electrode arrays (MEAs) at 10 kHz for 15 minutes. Spike detection employed a threshold of eight times the standard deviation of background noise, and inter‑spike intervals (ISIs) were used to define a time bin (t_b) (the average ISI) that sets the temporal resolution for avalanche detection. An avalanche is defined as a sequence of consecutive active bins (bins containing at least one spike) bounded by at least three silent bins on each side. Avalanche size (S) is the total number of active electrodes during the event, and duration (T) is the number of active bins multiplied by (t_b).

Statistical analysis of the experimental data shows that both size and duration distributions follow power‑law forms (P(S)\propto S^{-\tau}) and (P(T)\propto T^{-\alpha}) with exponents (\tau\approx1.5) and (\alpha\approx2.0). Moreover, the conditional average size given a duration obeys (\langle S|T\rangle\propto T^{\gamma}) with (\gamma) satisfying Sethna’s scaling relation (\gamma=(\alpha-1)/(\tau-1)), indicating that the system operates near a critical point. Inter‑avalanche intervals (\Delta t) display a heavy‑tailed distribution (P(\Delta t)\propto \Delta t^{-\mu}) with (\mu\approx1.3)–(1.5), deviating from a Poisson process and revealing temporal correlations. Conditional probability analysis (P(s_0,t_0)=P(\Delta S\le s_0|\Delta t\le t_0)) compared with surrogate sequences (size order shuffled) uncovers a systematic amplification‑attenuation pattern: short (\Delta t) tends to be followed by either larger or smaller subsequent avalanches, a signature of a feedback mechanism that alternately amplifies or damps activity.

To reproduce these findings theoretically, the authors extend the active Kuramoto model, which traditionally describes each neuron as a phase oscillator (\theta_j) driven by its natural frequency (\omega_j), Gaussian noise (\xi_j), a conservative sinusoidal term (a\sin\theta_j), and a mean‑field coupling (-K R\sin(\theta_j-\Psi)). In the classic formulation, inhibition is modeled simply by flipping the sign of the coupling constant, an approach that can paradoxically increase overall firing when inhibitory neurons fire. To overcome this, the authors introduce a second population of inhibitory oscillators and let the amplitude of the conservative term depend on the collective state of the inhibitory group. The resulting equations are:

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