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On the Hopfield Neural Networks and Mean Field Theory
Ken Ichi Funahashi1, Naoki Kurita
1The University of Aizu, Japan
Summary
This study mathematically links the mean field theory (MFT) network model and the continuous-time Hopfield neural network. We prove their equivalence regarding stable fixed points and equilibria using dynamical systems theory.
Area of Science:
- Computational Neuroscience
- Theoretical Neuroscience
- Dynamical Systems Theory
Background:
- The mean field theory (MFT) network model, derived from Boltzmann machines via mean field approximation, is a discrete-time recurrent neural network.
- Understanding the relationship between discrete and continuous neural network models is crucial for advancing computational neuroscience.
Purpose of the Study:
- To mathematically analyze the relationship between the mean field theory (MFT) network model and the continuous-time Hopfield neural network.
- To establish the equivalence of these models concerning their stable states.
Main Methods:
- Utilizing the theory of dynamical systems for mathematical analysis.
- Comparing the asymptotically stable fixed points of the asynchronous MFT model with the asymptotically stable equilibria of the continuous-time Hopfield neural network.
Main Results:
- Proved that the set of asymptotically stable fixed points of the asynchronous MFT model is identical to the set of asymptotically stable equilibria of the continuous-time Hopfield neural network.
- Demonstrated the equivalence between the asynchronous MFT model and the Hopfield neural network in terms of the nature of their fixed points or equilibria.
Conclusions:
- The asynchronous MFT model and the continuous-time Hopfield neural network share the same set of stable states.
- This equivalence provides a theoretical bridge between discrete and continuous dynamical models in neural networks.