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

New Global Exponential Stability Criteria for Nonlinear Delay Differential Systems with Applications to BAM Neural Networks