Anomalous random neural network's guide to Hopfield neural networks

Area of Science:

• Computational neuroscience
• Artificial neural networks
• Complex systems

Background:

• Neural networks exhibit heterogeneity, impacting their function.
• Existing random neural network models often assume exponential waiting times.
• Understanding signal flow and neuron dynamics is crucial for neural computation.

Purpose of the Study:

• To propose a generalized neural network model capturing heterogeneity.
• To investigate signal propagation and neuron state evolution in the proposed model.
• To establish connections between the generalized model and existing neural network architectures.

Main Methods:

• Development of the anomalous random neural network (ARNN) model.
• Application of renewal processes to analyze signal flow.
• Derivation of generalized master and rate equations for neuron state evolution.
• Analysis of waiting time distributions (exponential and power-law).

Main Results:

• Generalized master and rate equations for ARNNs were derived.
• Exponential waiting times in ARNNs reduce to Hopfield networks.
• Power-law waiting times in ARNNs lead to fractional-order Hopfield networks.
• A derived power-law firing rate for single neurons matches experimental data.

Conclusions:

• ARNNs provide a flexible framework for modeling neural heterogeneity.
• The model bridges standard and fractional-order neural network concepts.
• ARNNs offer a potential explanation for experimentally observed power-law firing rates.