  We consider a nonlinear non-autonomous system with time-varying delays $$
\dot{x_i}(t)=-a_i(t)x_{i}(h_i(t))+\sum_{j=1}^mF_{ij}(t,x_j(g_{ij}(t))) $$ which
has a large number of applications in the theory of artificial neural networks.
Via the M-matrix method, easily verifiable sufficient stability conditions for
the nonlinear system and its linear version are obtained. Application of the
main theorem requires just to check whether a matrix, which is explicitly
constructed by the system's parameters, is an $M$-matrix. Comparison with the
tests obtained by K. Gopalsamy (2007) and B. Liu (2013) for BAM neural networks
illustrates novelty of the stability theorems. Some open problems conclude the
paper.
