In this paper, we consider a 5-dimensional Hindmarsh–Rose neuron model. This improved version of the original model shows rich dynamical behaviors, including a chaotic super-bursting regime. This regime promises a greater information encoding capacity than the standard bursting activity. Based on the Krasovskii–Lyapunov stability theory, the sufficient conditions (on the synaptic strengths and magnetic gain parameters) for stable chaotic synchronization of the model are obtained. Based on Helmholtz’s theorem, the Hamilton function of the corresponding error dynamical system is also obtained. It is shown that the time variation of this Hamilton function along trajectories can play the role of the time variation of the Lyapunov function—in determining the stability of the synchronization manifold. Numerical computations indicate that as the synaptic strengths and the magnetic gain parameters change, the time variation of the Hamilton function is always nonzero (i.e., a relatively large positive or negative value) only when the time variation of the Lyapunov function is positive, and zero (or vanishingly small) only when the time variation of the Lyapunov function is also zero. This, therefore, paves an alternative way to determine the stability of synchronization manifolds and can be particularly useful for systems whose Lyapunov function is difficult to construct, but whose Hamilton function corresponding to the dynamic error system is easier to calculate.

在本文中，我们考虑了5维Hindmarsh-Rose神经元模型。原始模型的改进版本显示了丰富的动力学行为，包括混沌的超爆发状态。与标准的突发活动相比，这种机制保证了更大的信息编码能力。基于Krasovskii–Lyapunov稳定性理论，获得了用于模型稳定混沌同步的充分条件（关于突触强度和磁增益参数）。基于亥姆霍兹定理，还获得了相应误差动力学系统的汉密尔顿函数。结果表明，该汉密尔顿函数沿轨迹的时间变化可以在确定同步流形的稳定性方面发挥李雅普诺夫函数的时间变化的作用。数值计算表明，随着突触强度和磁增益参数的变化，汉密尔顿函数的时间变化始终为非零（即，相对较大的正值或负值），只有当李雅普诺夫函数的时间变化为正时为零。 （或消失得很小），只有当Lyapunov函数的时间变化也为零时。因此，这为确定同步流形的稳定性铺平了另一种途径，对于其Lyapunov函数难以构造但其动态误差系统所对应的Hamilton函数更易于计算的系统尤其有用。仅当Lyapunov函数的时间变化为正时才相对较大（正或负值），而只有当Lyapunov函数的时间变化也为零时才为零（或逐渐减小）。因此，这为确定同步流形的稳定性铺平了另一种途径，对于其Lyapunov函数难以构造但其动态误差系统所对应的Hamilton函数更易于计算的系统尤其有用。仅当Lyapunov函数的时间变化为正时才相对较大（正或负值），而只有当Lyapunov函数的时间变化也为零时才为零（或逐渐减小）。因此，这为确定同步流形的稳定性铺平了另一种途径，对于其Lyapunov函数难以构造但其动态误差系统所对应的Hamilton函数更易于计算的系统尤其有用。