We analyze the model of nonlinear viscoelastic medium in the form of a non-linear viscoelastic element consisting of Maxwell elements, Jeffreys, the Voigt-Kelvin and consistently incorporated the non-linear element such as -non-time delay. This model is built integral-differential equation that describes the evolution of deformation from the moment of application to the equilibrium state (evolution of the attractor). Said integral-differential equation is constructed in the form of space-time transformation vectors viscosities in different elements of a viscoelastic medium (parameter vector), and describes the trajectory of changes in these parameters. These trajectories (portrait state dynamical system) in conjunction with the associative memory attractor define viscoelastic medium (unlike its hereditary memory). It constructed a block diagram which implements the equation obtained in the form of a dynamic-static neural network. Subject to certain restrictions on the synthesized model, using the Cohen-Grossberg theorem is given expression for the energy of the system (Lyapunov function) and prove the asymptotic stability of the global dynamics of the system. It is shown that by switching off of the synthesized model temporal filters with finite impulse response (FIR-filter), the proposed model goes into the Hopfield model. The FIR-filter is implemented hereditary viscoelastic memory elements. In the computer simulation of the Hopfield model in a two-dimensional parameter space viscosities constructed trajectory behavior of a viscoelastic material. It is shown that these trajectories converge to a fixed point (attractor), corresponding to the minimum energy environment. If the target vector contains a strain coordinate values different from the units, then the attractor does not coincide with the target vector, as the path passes by the target vector.