The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition

Summary

The paper studies the hydrodynamic limit of a balanced neural network with excitatory and inhibitory neurons. It proves that the system's activity concentrates on a Gaussian distribution in the large-size limit, as long as the initial conditions lie on a stable "balanced manifold". The balanced manifold is defined as the set of states where the net excitatory and inhibitory inputs to each neuron approximately cancel out.

Highlights

• The paper studies the hydrodynamic limit of a balanced neural network with excitatory and inhibitory neurons.
• The system's activity concentrates on a Gaussian distribution in the large-size limit.
• The initial conditions must lie on a stable "balanced manifold".
• The balanced manifold is defined as the set of states where the net excitatory and inhibitory inputs to each neuron approximately cancel out.
• The paper proves the existence and uniqueness of the hydrodynamic limit.
• The system's activity is described by a set of ordinary differential equations (ODEs) in the large-size limit.
• The ODEs are derived using a mean-field approximation.

Key Insights

• The paper provides a rigorous mathematical framework for understanding the behavior of balanced neural networks in the large-size limit.
• The balanced manifold plays a crucial role in determining the stability of the system's activity.
• The system's activity is characterized by a Gaussian distribution, indicating that the fluctuations around the mean activity are random and uncorrelated.
• The mean-field approximation used in the paper is a common technique for studying the behavior of large systems of interacting particles.
• The ODEs derived in the paper provide a simplified description of the system's activity, which can be used to study the behavior of the system in different scenarios.
• The paper's results have implications for understanding the behavior of neural networks in the brain, where balanced neural networks are thought to play a crucial role in information processing.
• The paper's mathematical framework can be used to study other types of neural networks and their behavior in the large-size limit.

Mindmap

Citation

MacLaurin, J., & Vilanova, P. (2024). The Hydrodynamic Limit of Neural Networks with Balanced Excitation and Inhibition (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.17273