<p>Studies reveal that a locally active memristive Hopfield neural network can produce complex, chaotic dynamic behaviours, including hyperchaotic, multi-scroll, multi-stability, and hidden dynamical behaviours. There is no neural network-based hyperchaotic system with multi-double-scroll and double-multi-scroll attractors. This study aims to address the issue. In this paper, we build three memristive Hopfield neural networks with local activation, which is affected by external current. For the first time, double-multi-scroll and multi-double-scroll hyperchaotic attractors are found in the memristive Hopfield neural network. Analysis of equilibrium points demonstrates that the attractors are of self-excited nature. Further analysis shows that the memristive state equation controls the number of scrolls, and the attractor structure is also affected by external current. Simultaneously, by altering the system’s initial value, the memristive Hopfield neural network exhibits a controllable number of coexisting attractors, including a coexisting double-scroll and multi-double-scroll attractor that exhibits extreme multi-stability while remaining hyperchaotic under the right parameter conditions. Finally, an encryption method built on the suggested memristive Hopfield neural network is created to show the attractors’ potential for use.</p>