Inferring effective neuronal circuits via network flux counting

Extracting circuit mechanisms from neuronal population activity is challenging due to the heterogeneous neuronal properties and diverse strengths in synaptic connections. Standard inference methods, such as Generalized Linear Models (GLMs), typically regress for parameters on all neuronal activity at once. Such a global fitting approach can face identifiability difficulties—for example, where the statistical estimation of strong, opposing weights becomes ill-conditioned in excitatory-inhibitory balanced networks. Here, we introduce FLux-based Effective Coupling (FLEC), a framework that maps spike trains directly to probability fluxes on network state space. Instead of enforcing a single global fit, FLEC infers connectivity and response heterogeneity by quantifying transition rates for each network configuration independently. We demonstrate that FLEC outperforms GLMs and Granger Causality in strongly coupled networks while matching GLM’s performance in standard regimes. Additionally, when combined with Maximum Caliber to construct a minimal dynamical model, the framework better captures temporal statistics—such as inter-spike intervals—than Maximum Entropy models. Robust to parameter variations and unobserved hidden units, and applied to multi-electrode recordings from the salamander retina, FLEC offers a systematic, counting-based tool for inference in nonlinear neuronal circuits.Author summary: The brain computes information through a vast network of interacting neurons. To understand these circuits, we need to determine how neurons connect in the network and measure their intrinsic activation properties. However, experiments often provide partial observations of the underlying process: time series of neuronal firing. When a neuron fires frequently, it is difficult to distinguish if it has high intrinsic excitability or if it is receiving strong input from the network. This problem worsens in highly active networks, where strong excitation and inhibition signals cancel each other out and cause standard mathematical methods of inference to fail. To address this, we developed a new statistical tool called FLux-based Effective Coupling (FLEC). Instead of solving complex equations for the entire network all at once, FLEC takes a direct and tractable counting-based approach. It measures how the network transitions between activity states over time and compares counts across different activity levels. This allows FLEC to separate a neuron’s intrinsic excitability from the influence of its neighbors. We demonstrate that FLEC succeeds in regimes where traditional methods break down, providing a more accurate way to infer brain wiring from experimental data.