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 non-linear neuronal circuits.