This paper investigates the periodicity and synchronization of inertial memristive neural networks with supremums and time delays. The analysis in this paper employs the results from the theory of differential equations with discontinuous right-hand side as introduced by Filippov. Further, by employing the second order differential inclusion theory and then choosing suitable variable transformation, the original system can be transformed into first order differential equations. By using the matrix measure method and Halany inequality techniques, the sufficient condition that guarantee the global exponential synchronization of drive-response system of coupled inertial memristive neural networks via the state feedback controller is derived. Furthermore, the global exponential periodicity of the addressed neural networks is also discussed. Finally, numerical examples and simulations are given to demonstrate the validation of the proposed results.