Neural networks have been widely and deeply studied in the field of computational neurodynamics. However, coupled neural networks and their brain-like chaotic dynamics have not been noticed yet. In this article, we focus on the coupled neural network-based brain-like initial boosting coexisting hyperchaos and its application in biomedical image encryption. We first construct a memristive-coupled neural network (MCNN) model based on two subneural networks and one multistable memristor synapse. Then we investigate its coupling strength-related dynamical behaviors, initial states-related dynamical behaviors, and initial-boosted coexisting hyperchaos using bifurcation diagrams, phase portraits, Lyapunov exponents, and attraction basins. The numerical results demonstrate that the proposed MCNN not only can generate hyperchaotic attractors with high complexity but also can boost the attractor positions by switching their initial states. This makes the MCNN more suitable for many chaos-based engineering applications. Moreover, we design a biomedical image encryption scheme to explore the application of the MCNN. Performance evaluations show that the designed cryptosystem has several advantages in the keyspace, information entropy, and key sensitivity. Finally, we develop a field-programmable gate array test platform to verify the practicability of the presented MCNN and the designed medical image cryptosystem.