Understanding Activation Patterns in Artificial Neural Networks by
  Exploring Stochastic Processes
Stephan Johann Lehmler Muhammad Saif-ur-Rehman Tobias Glasmachers Ioannis Iossifidis
To gain a deeper understanding of the behavior and learning dynamics of
(deep) artificial neural networks, it is valuable to employ mathematical
abstractions and models. These tools provide a simplified perspective on
network performance and facilitate systematic investigations through
simulations. In this paper, we propose utilizing the framework of stochastic
processes, which has been underutilized thus far.
  Our approach models activation patterns of thresholded nodes in (deep)
artificial neural networks as stochastic processes. We focus solely on
activation frequency, leveraging neuroscience techniques used for real neuron
spike trains. During a classification task, we extract spiking activity and use
an arrival process following the Poisson distribution.
  We examine observed data from various artificial neural networks in image
recognition tasks, fitting the proposed model's assumptions. Through this, we
derive parameters describing activation patterns in each network. Our analysis
covers randomly initialized, generalizing, and memorizing networks, revealing
consistent differences across architectures and training sets.
  Calculating Mean Firing Rate, Mean Fano Factor, and Variances, we find stable
indicators of memorization during learning, providing valuable insights into
network behavior. The proposed model shows promise in describing activation
patterns and could serve as a general framework for future investigations. It
has potential applications in theoretical simulations, pruning, and transfer
learning.