A special chaotic firing pattern lying between period-1 and period-2 firing pattern simulated in theoretical neuronal firing model, Hindmarsh-Rose (HR) model, has not been adequately understood for a long time. The non-periodic neural firing patterns lying between period-1 and period-2 firing pattern discovered in the biological experiments on neural pacemakers of rats are identified to be chaotic bursting and divided into three styles in appearance, according to the deterministic structures of the first return map and the short-term predictability of nonlinear predication. One style of the experimental chaos exhibits characteristics similar to the numerical simulations of the theoretical model, verifying the theoretical participation of HR model, while other styles display different characteristics. The characteristics of the three styles and the relationship and distinction among 3 styles of the chaotic rhythms are identified, and compared with those lying between period-2 and period-3 firing pattern, and between period-3 and period-4 firing pattern. In addition, the distinction between the transition procedure from period-1 to chaos and then to period-2 and other bifurcation scenarios from period-1 to period-2 firing pattern is also identified. The results confirm the novel chaos lying between period-1 and period-2 and the corresponding novel bifurcation scenario, enriching the kinds of the chaotic rhythms and bifurcation scenarios of neural firing. Finally simulations of the diversity and non-smooth characteristics of the chaotic rhythms discovered in the experiment and identification of the routine to chaos are also discussed.