Biological experimental observations of an unnoticed chaos as simulated by the Hindmarsh-Rose model

Abstract

An unnoticed chaotic firing pattern, lying between period-1 and period-2 firing patterns, has received little attention over the past 20 years since it was first simulated in the Hindmarsh-Rose (HR) model. In the present study, the rat sciatic nerve model of chronic constriction injury (CCI) was used as an experimental neural pacemaker to investigate the transition regularities of spontaneous firing patterns. Chaotic firing lying between period-1 and period-2 firings was observed located in four bifurcation scenarios in different, isolated neural pacemakers. These bifurcation scenarios were induced by decreasing extracellular calcium concentrations. The behaviors after period-2 firing pattern in the four scenarios were period-doubling bifurcation not to chaos, period-doubling bifurcation to chaos, period-adding sequences with chaotic firings, and period-adding sequences with stochastic firings. The deterministic structure of the chaotic firing pattern was identified by the first return map of interspike intervals and a short-term prediction using nonlinear prediction. The experimental observations closely match those simulated in a two-dimensional parameter space using the HR model, providing strong evidences of the existence of chaotic firing lying between period-1 and period-2 firing patterns in the actual nervous system. The results also present relationships in the parameter space between this chaotic firing and other firing patterns, such as the chaotic firings that appear after period-2 firing pattern located within the well-known comb-shaped region, periodic firing patterns and stochastic firing patterns, as predicted by the HR model. We hope that this study can focus attention on and help to further the understanding of the unnoticed chaotic neural firing pattern.

Figure 1. The largest Lyapunov exponent.

The largest Lyapunov exponent in parameter space of the HR model (0.0010.035, 2.33.42). Colors shown in the right column are associated with the values of the largest Lyapunov exponent.

Figure 2. Bifurcation scenario in the HR model.

Bifurcation scenario of the ISI series dependently with decreasing and increasing . (A) ; (B) ; (C) ; (D) .

Figure 3. Chaotic firing in the HR model.

Chaotic firing lying between period-1 firing and period-2 firing with 2.53 and 0.0245. (A) Spike trains; (B) The first return map of ISI series

Figure 4. Bifurcation scenarios observed from different neural pacemakers with decreasing [Ca²⁺]_o.

The initial part was from period-1 firing, to chaotic firing, and to period-2 firing. Bifurcation scenario after period-2 firing was different. (A) From chaotic firing, to period-3 firing, to chaotic firing, to period-4 firing. The chaotic firing was divided into 3 parts; (B) A part of Fig. 4(a). Bifurcation scenario from period-1 firing to period-2 firing via chaotic firing; (C) From chaotic firing, to period-3 firing, to chaotic firing, to period-4 firing; (D) From period-4 firing, to chaotic firing, to period-3 firing; (E) From stochastic firing, to period-3 firing, to stochastic firing, to period-4 firing; (F) Period-1 firing.

Figure 5. Spike trains observed from different neural pacemakers.

Spike trains of the chaotic firing lying between period-1 and period-2 firings. (A) Part 1 of the first example; (B) Part 2 of the first example; (C) Part 3 of the first example; (D) The second example; (E) The third example; (F) The fourth example; (G) The fifth example.

Figure 6. The first return map of the chaotic firing lying between period-1 and period-2 firings.

The first return map of ISI series observed from different neural pacemakers. (A) Part 1 of the first example; (B) Part 2 of the first example; (C) Part 3 of the first example; (D) The second example; (E) The third example; (F) The fourth example; (G) The fifth example.

Figure 7. The .

The of chaotic firings lying between period-1 and period-2 firings observed from different neural pacemakers. Line with triangle, original data; line with circle, mean of 10 realizations of surrogate data. (A) Part 1 of the first example; (B) Part 2 of the first example; (C) Part 3 of the first example; (D) The second example; (E) The third example; (F) The fourth example; (G) The fifth example.

Figure 8. Bifurcation scenarios observed from different neural pacemakers.

(A) Inverse bifurcation scenario corresponding to Fig. 4(D) with increasing [Ca²⁺]_o from 0 mM to 1.2 mM, whose process was from period-3 firing, to chaotic firing, to period-4 firing, to period-2 firing, to chaotic firing, and to period-1 firing; (B) Inverse bifurcation scenario corresponding to Fig. 4(E) with increasing [Ca²⁺]_o from 0 mM to 1.2 mM, whose process was from period-4 firing, to stochastic firing, to period-3 firing, to stochastic firing, to period-2 firing, to chaotic firing, and to period-1 firing; (C) Bifurcation scenario observed in a neural pacemaker with decreasing [Ca²⁺]_o from 1.2 mM to 0 mM, whose process was from period-1 firing to period-2 firing via chaotic firing, to chaotic firing, to period-3 firing, and to chaotic firing; (D) Inverse bifurcation scenario corresponding to Fig. 8(C) with increasing [Ca²⁺]_o from 0 mM to 1.2 mM, whose process was from chaotic firing, to period-3 firing, to chaotic firing, to period-2 firing, to chaotic firing, and to period-1 firing.