We study macroscopic observables of large random matrices introduced by Pennington and Worah, defined by applying entry wise a non linear function on a product of matrices with independent entries. We allow the variance of the entries of the matrices to vary from entry to entry. We complement Péché perspective from [Electron. Commun. Probab. 24 (2019)] showing a decomposition of these matrices whose and traffic asymptotic traffic-equivalent for their ingredients, when the activation function belongs to the space of odd polynomials. This give a new interpretation of the linear plus chaos phenomenon observed for these matrices.