Stochastic Systems with Variable Range Interactions: Modelling Temporal Evolution of Individually Recorded Neuronal Populations Using Generalised Integrate-and-Fire Models

The temporal evolution of neuronal populations represents one of the most complex dynamical systems in nature, characterised by intricate stochastic interactions that span multiple spatial and temporal scales. This study presents a comprehensive mathematical framework for modelling the dynamics of individually recorded neuronal populations through generalised integrate-and-fire models with variable range interactions. We develop a systematic approach that extends classical leaky integrate-and-fire formulations to incorporate stochastic elements, adaptive thresholds, and distance-dependent coupling mechanisms. Our methodology addresses the fundamental challenge of capturing both local and long-range interactions within neuronal networks whilst maintaining computational tractability. Through rigorous mathematical analysis and numerical simulations, we demonstrate that variable range interaction models can effectively reproduce the complex temporal patterns observed in experimental recordings of cortical neuronal populations. The proposed framework incorporates five distinct generalised leaky integrate-and-fire (GLIF) models of increasing complexity, each designed to capture specific aspects of neuronal dynamics including spike-triggered adaptation, voltage-dependent threshold modulation, and stochastic membrane fluctuations. Our results indicate that higherorder GLIF models achieve superior predictive accuracy for spike timing prediction, with improvements of approximately % over classical formulations when tested against in vivo-like stimuli. Furthermore, we establish that the model parameters serve as effective dimensionality reduction tools for neuronal classification, enabling the differentiation of distinct cell types based purely on electrophysiological responses. The implications of this work extend beyond theoretical neuroscience, providing practical tools for large-scale network simulations and advancing our understanding of how stochastic interactions at the cellular level give rise to emergent population dynamics in the mammalian cortex.