On Bootstrap Percolation in Living Neural Networks

TL;DR

This paper provides a rigorous mathematical proof for the behavior of neural activation in living neural networks modeled by bootstrap percolation on directed random graphs, explaining how initial stimuli lead to widespread activation.

Contribution

It introduces a theorem that determines the asymptotic final active neuron proportion in directed random networks, bridging experimental observations with mathematical rigor.

Findings

Mathematically characterizes neural activation thresholds

Proves the nonlinear relationship between initial stimuli and final activation

Validates experimental phenomena with rigorous proofs

Abstract

Recent experimental studies of living neural networks reveal that their global activation induced by electrical stimulation can be explained using the concept of bootstrap percolation on a directed random network. The experiment consists in activating externally an initial random fraction of the neurons and observe the process of firing until its equilibrium. The final portion of neurons that are active depends in a non linear way on the initial fraction. The main result of this paper is a theorem which enables us to find the asymptotic of final proportion of the fired neurons in the case of random directed graphs with given node degrees as the model for interacting network. This gives a rigorous mathematical proof of a phenomena observed by physicists in neural networks.