Finite-size and correlation-induced effects in mean-field dynamics

Area of Science:

• Computational neuroscience
• Mathematical modeling of neural systems

Background:

• Brain activity involves numerous neurons affected by noise, necessitating mean-field approaches for large assemblies.
• Classical mean-field models often neglect stochasticity, focusing on deterministic average activity.

Purpose of the Study:

• To investigate the influence of stochastic effects, correlations, and higher-order moments on mean-field equations for neural networks.
• To introduce and analyze a new 'infinite model' for studying large, finite neural networks.

Main Methods:

• Building upon recent approaches that incorporate correlations and higher-order moments in mean-field equations.
• Developing and analyzing a rescaled 'infinite model' that is invertible for finite-size networks.
• Comparing solutions and stability properties between deterministic, stochastic, and finite-size network models.

Main Results:

• Stochastic effects and correlations modify the stability of limit cycles in neural network activity.
• New, non-trivial behaviors, including additional periodic orbits, emerge in stochastic models compared to deterministic ones.
• Finite-size networks exhibit distinct mesoscopic effects, differing from the singular limit of infinite-size systems.

Conclusions:

• Stochasticity and correlations are crucial for accurately modeling large-scale neural network dynamics.
• The proposed 'infinite model' offers a valuable tool for understanding qualitative behaviors in finite neural networks.
• Infinite-size systems serve as singular limits, but finite networks display unique emergent properties.