Computation of synchronized periodic solution in a BAM network with two delays

Area of Science:

• Dynamical Systems
• Computational Neuroscience
• Nonlinear Dynamics

Background:

• Bidirectional associative memory (BAM) neural networks are crucial for associative learning.
• Analyzing periodic solutions in delayed dynamical systems, particularly neural networks, is complex.
• Hopf bifurcation is a key phenomenon leading to oscillations in such systems.

Purpose of the Study:

• To introduce and validate the perturbation-incremental scheme (PIS) for analyzing periodic solutions in a 4-dimensional BAM neural network with discrete delays.
• To quantitatively assess the accuracy of PIS compared to the center manifold reduction (CMR) with normal form.
• To derive analytical conditions and expressions for synchronized periodic solutions arising from Hopf bifurcation.

Main Methods:

• Application of the perturbation-incremental scheme (PIS) to a 4-neuron BAM neural network model with two discrete delays.
• Derivation of analytical expressions for periodic solutions using PIS.
• Comparison of PIS accuracy with the center manifold reduction (CMR) method.
• Obtaining necessary and sufficient conditions for synchronized periodic solutions.

Main Results:

• The PIS provides accurate analytical expressions for periodic solutions derived from Hopf bifurcation.
• PIS exhibits higher accuracy than CMR for time delays near the Hopf bifurcation point.
• Necessary and sufficient conditions for synchronized periodic solutions were obtained analytically.
• Theoretical findings were validated through numerical simulations, showing good agreement.

Conclusions:

• The perturbation-incremental scheme (PIS) is a valid and accurate method for studying periodic solutions in delayed dynamical systems like BAM neural networks.
• This study is the first to quantitatively apply PIS to analyze Hopf bifurcation-induced periodic solutions in a 4-D delayed system.
• The derived analytical results offer valuable insights into the synchronization dynamics of neural networks.