Nonlinear dynamics can be used to identify relationships between different firing patterns, which play important roles in the information processing. The present study provides novel biological experimental findings regarding complex bifurcation scenarios from period-1 bursting to period-1 spiking with chaotic firing patterns. These bifurcations were found to be similar to those simulated using the Hindmarsh-Rose model across two separated chaotic regions. One chaotic region lay between period-1 and period-2 burstings. This region has not attracted much attention. The other region is a well-known comb-shaped chaotic region, and it appears after period-2 bursting. After period-2 bursting, the chaotic firings lay in a period-adding bifurcation scenario or in a period-doubling bifurcation cascade. The deterministic dynamics of the chaotic firing patterns were identified using a nonlinear prediction method. These results provided details regarding the processes and dynamics of bifurcation containing the chaotic bursting between period-1 and period-2 burstings and other chaotic firing patterns within the comb-shaped chaotic region. They also provided details regarding the relationships between different firing patterns in parameter space.

Nonlinear dynamics have been instrumental in the improvement of human understanding of the dynamics of neural firing patterns, which have been shown to play important roles in information processing [

Three-dimensional differential equations containing fast and slow variables have been used in investigations of bifurcations and chaos of neural firing patterns [

The largest Lyapunov exponent of firing patterns of the HR model in the

(Color online) the largest Lyapunov exponent in the

Theoretically, bursting and spiking patterns in the HR model can be distinguished using dissection of the fast and slow subsystems [

As shown in Figure

In the present study, the experimental neural pacemaker employed in previous studies served as the experimental model [

The rest of this paper is organized as follows. Simulation results of the HR model are reproduced in Section

As shown in Figure

The bifurcation scenario of example 1 lasts from period-1 bursting to chaotic bursting to period-2 bursting to chaotic bursting to period-3 bursting to chaotic bursting to period-4 bursting to chaotic bursting to chaotic spiking to period-2 spiking to period-1 spiking as

Bifurcation processes of the ISI series of firing patterns simulated using the HR model along lines shown in Figure

The bifurcation scenario shown in example 2 resembles the behavior along line L2 with the following equation:

The last example of bifurcation is similar to the behavior observed along line L1 with the following equation:

The spike trains of the chaotic bursting (

Chaotic firing pattern between period-1 bursting and period-2 bursting simulated in the HR model with

Bennet and Xie developed an animal model of chronic constriction injury (CCI) of the rat sciatic nerve [

An experimental neural pacemaker was formed at the site of injury of a rat sciatic nerve [

In the present study, a neural pacemaker capable of generating period-1 bursting under controlled conditions was selected. The solution was replaced with 0 mM

From a physiological perspective, multiple ionic currents including sodium, potassium, and calcium currents participate in the electrophysiology of nervous system. Potassium current that can induce the decrease or repolarization of the membrane potential participates in the generation of action potential through cooperation and competition with sodium current, which can induce the increase or depolarization of the membrane potential. Calcium current is a slow factor and can adjust the interval of continuous action potentials, that is, the ISIs, through the calcium-dependent potassium current which is related to the calcium concentration and/or the conductance of calcium-dependent potassium channel. The modulation in

To a certain extent,

In previous studies, most of experimental neural pacemakers exhibited bifurcation processes not invovling from period-1 bursting to period-1 spiking and only some pacemakers generated bifurcations from period-1 bursting to period-1 spiking [

The bifurcation process of example 1 was similar to that simulated in the HR model along L3. The details involve period-1 bursting to chaotic bursting to period-2 bursting to chaotic bursting to period-3 bursting to chaotic bursting to period-4 bursting to chaotic bursting to chaotic spiking to period-2 spiking to period-1 spiking, as shown in Figure

Example 1 of bifurcation process from period-1 bursting to period-1 spiking. (a) In detail, the process lasts from period-1 bursting to chaotic bursting to period-2 bursting to chaotic bursting to period-3 bursting to chaotic bursting to period-4 bursting to chaotic bursting to chaotic spiking to period-2 spiking to period-1 spiking. (b) Detail of Figure

The first return map of ISI series of chaotic firing within the bifurcation scenario shown in Figure

Example 2 of bifurcation scenario from period-1 bursting to period-1 spiking observed from a pacemaker with decreasing

Example 2 of bifurcation scenario as observed from a pacemaker with decreasing

The first return maps of ISI series of chaotic firings within the bifurcation scenario as shown in Figure

The bifurcation process shown in example 3 lasts from period-1 bursting to chaotic-bursting to chaotic spiking to period-2 spiking to period-1 spiking, as shown in Figure

Example 3 of bifurcation scenario and firing patterns observed from a pacemaker with decreasing

The deterministic property of chaotic firing patterns can be estimated by nonlinear time series analysis method [

With a reconstruction dimension

All experimental chaotic firings exhibited a short-term prediction when

NPE of the experimental chaotic firings. (a) The chaotic bursting between period-1 and period-2 burstings of example 1 (Figure

Bifurcation scenarios containing chaotic bursting between period-1 and period-2 burstings and chaotic firings appearing after period-2 bursting in period-adding sequences or period-doubling cascade were observed in biological experiments performed using different neural pacemakers. The deterministic dynamics within the chaotic bursting were identified. The experimental results manifested characteristics that were very similar to those of the HR model in parameter space. These characteristics included two chaotic regions; one is the well-known comb-shaped region appearing after period-2 bursting and the other is the chaotic region between period-1 and period-2 burstings. The results showed the chaotic firing pattern between period-1 and period-2 bursting and bifurcation scenarios to contain two chaotic regions, such as those observed in real nervous systems. With exception of the two separated chaotic regions, the process that transitioned from period-1 bursting to period-1 spiking through complex processes should be emphasized. Most bifurcation processes observed from the neural pacemakers terminated at a certain firing pattern before period-1 spiking [

As studied in a previous study [

Compared with many simulation results about the bifurcations and chaos of the HR model, there existed less theoretical investigations of the HR model. The generation of the chaotic bursting and the transition from bursting to spiking in excitable membrane models were analyzed in a theoretical model resembling HR model by Terman [

Neural firing patterns play important roles in neural information processing in different nervous systems [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China under Grant nos. 11372224 and 11072135 and the Fundamental Research Funds for Central Universities of Tongji University under Grant no. 1330219127.