Stability and bifurcation in a simplified four-neuron bam neural network with multiple delays (Q2491476)

In the first part, the paper examines in great detail the roots of a class of fourth-order exponential polynomials of the form \(\lambda^4+a_3\lambda^3+a_2\lambda^2+a_1\lambda+a_0+\exp(-\lambda\tau)(b_2\lambda^2+b_1\lambda+b_0)\). This is motivated by the study of a network of four neurons of so-called bidirectional associative memory (BAM) type with coupling delay. The linearization has a characteristic equation involving the above type of exponential polynomials. The authors derive criteria on the coefficients \(a_j\) and \(b_j\), classifying when the neural system is absolutely stable (that is, independent of the delay \(\tau\)), conditionally stable or unstable. In the second part, a normal form analysis of the Hopf bifurcation leading to the linear instability is performed giving a long sequence of expressions leading to the criticality coefficient that determines the stability of the periodic solution in the center direction.