Asymptotic Behavior of an Artificial Neural Network Defined on Multipartite Directed Graph

Problem statement: Artificial Neural Network (ANN) are simple models to mimic some essential features of the complex central nervous system. ANN models are realistic due to their inherent stochastic nature of neural computation and strong synchronicity. Different ANN models are associated with directed and signed graphs. The present study proceeded by relaxing certain simplifying assumptions in the ANN model. Approach: It was assumed that the connected graph associated with the ANN is a multipartite directed graph whose connection comprising of four blocks and two blocks are either both symmetric or both anti symmetric. The convergence of such network was studied in the present research with the help of Lyapunov functional. Results: Attractors (fixed points) of such ANN and also limit cycles of different orders are investigated. Bounds of transient length of the neural network were also calculated. Numerical simulation in support of the results was also depicted. Conclusion: It was shown that under synchronous updating rule such networks converge to a fixed point or to a limit cycle of period 2 or 4. The bound of transient length was discussed. Conclusions were drawn from the simulation studies carried out in support of the results.