Boundary conditions and phase transitions in neural networks. Theoretical results

Authors:
Jacques Demongeot;Christelle Jézéquel;Sylvain Sené
Affiliations:
TIMC-IMAG, Faculté de Médecine, UJF Grenoble, 38706 La Tronche, France;TIMC-IMAG, Faculté de Médecine, UJF Grenoble, 38706 La Tronche, France;TIMC-IMAG, Faculté de Médecine, UJF Grenoble, 38706 La Tronche, France and Institut rhône-alpin des systèmes complexes (IXXI), 5 rue du Vercors, 69007 Lyon, France
Venue:
Neural Networks
Year:
2008

Citing 9
Cited 3

Phase transitions in artificial intelligence systems

Artificial Intelligence
Constructing associative memories using neural networks

Neural Networks
Phase transitions in connectionist models having rapidly varying connection strengths

Neural Computation
Original Contribution: Convergence of a self-organizing stochastic neural network

Neural Networks
Original Contribution: Local conditions for phase transitions in neural networks with variable connection strengths

Neural Networks
Fault Tolerant Hopfield Associative Memory on Torus

DFT '03 Proceedings of the 18th IEEE International Symposium on Defect and Fault Tolerance in VLSI Systems
Boundary conditions and phase transitions in neural networks. Simulation results

Neural Networks
Stability of fully asynchronous discrete-time discrete-state dynamic networks

IEEE Transactions on Neural Networks
Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority

MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science

Boundary conditions and phase transitions in neural networks. Simulation results

Neural Networks
The singular power of the environment on stochastic nonlinear threshold Boolean automata networks

Proceedings of the 9th International Conference on Computational Methods in Systems Biology
Combinatorics of Boolean automata circuits dynamics

Discrete Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

Abstract

The purpose of this paper is to present some relevant theoretical results on the asymptotic behaviour of finite neural networks (on lattices) when they are subjected to fixed boundary conditions. This work focuses on two different topics of interest from the biological point of view. First, it exhibits a link between the possible updating iteration modes in these networks, whatever the number of dimensions is. It proves that the effects of boundary conditions on neural networks do not depend on the updating iteration mode under the hypothesis of synaptic weight symmetry. Thus, if the asymptotic behaviour admits phase transitions, these phase transitions are observable for many updating iteration modes (from synchrony to asynchrony). Then, it shows that boundaries have no significant impact on one-dimensional neural networks. In order to prove this property, we present a new general mathematical approach based on the use of a projectivity matrix in order to simplify the problem. This approach allows the theoretical study of the asymptotic dynamics and of the boundary influence in neural networks. We will also introduce the numerical tools generalising the method in order to study phase transitions in more complex cases.