Mapping spiking network simulations to mathematically tractable rate models

Abstract

Biological neural systems give rise to complex cognitive functions by processing information through interactions between neurons in the form of discrete spikes. While neuroscientists have developed a solid understanding of neuronal structure, network wiring, and information transmission in the brain, the mechanisms underlying cognitive functions remain largely elusive. Algorithmic characterizations of neural circuits have only been achieved for the simplest systems. Due to the brain’s complexity, two main approaches are used to study neural activity: detailed spiking simulations and rate models. Spiking simulations provide biologically realistic insights but are computationally demanding and difficult to generalize, while rate models offer simplified, interpretable mathematical frameworks that often impose restrictive assumptions on network behavior. <br /> In this work, I aim to bridge these two approaches by focusing on simulations of Excitatory-Inhibitory (E-I) networks of Leaky Integrate-and-Fire (LIF) neurons. These spiking models are widely used to simulate neural systems, but their mathematical analysis often requires complex, intractable equations. I demonstrate that the LIF neuron’s activation function can be approximated by a power law within the range of biologically relevant firing rates. This approximation enables the LIF spiking network to be described using a simplified Stabilized Supralinear Network (SSN) rate model, with parameters derived from fitting the neuronal transfer function. The activity regimes of the LIF simulations align closely with predictions of the SSN model, both qualitatively and quantitatively. The mathematical simplicity of the SSN model allows for deep analytical exploration of nonlinear dynamics in local neural circuits and shows how these operational regimes can support functionally relevant information processing. <br /> Overall, the SSN rate model provides a powerful, flexible, and mathematically tractable tool for studying neural networks. It reduces the complexity of their analysis by shifting the focus to a higher level, emphasizing transformations in local networks rather than the behavior of individual neurons.