We analyze the energy aspects of single and coupled Hindmarsh–Rose (HR) neuron models with a quadratic flux controlled memristor. The energy function for HR neuron with memristor has been derived and the dynamics have been analyzed in the presence of various exteal stimuli. We found that the bursting mode of the system changes with exteal forcing. The negative feedback in Hamilton energy function effectively stabilizes the chaotic trajectories and controls the phase space. The Lyapunov exponents have been plotted to verify the stabilization of trajectories. The energy aspects during the synchronous dynamics of electrically coupled neurons have been analyzed. As the coupling strength increases, the average energy fluctuates and stabilizes at the point of synchronization. When the neurons are coupled via chemical synapse, the average energy variations show three important regimes: a fluctuating regime corresponding to the desynchronized, a stable region indicating synchronized and a linearly increasing regime corresponding to the amplitude death states have been observed. The synchronization transitions are verified by plotting the transverse Lyapunov exponents. The proposed method has a large number of applications in controlling coupled chaotic systems and in analyzing the energy change during various metabolic processes.