Time Series Analysis On Stochastic And Chaotic Neural Firing Rhythms

As the characteristic of nervous system that can receive, transmit and process the information by abundant neural firing rhythms, identification of various neural rhythm is essential for correctly understand of dynamic behaviors of the nervous system. Random, chaotic, periodic neural discharge rhythms are typical forms of neuronal firing rhythms. The development of nonlinear science, especially chaos theory, provides a rich theoretical knowledge and analytical methods for the identification of neural firing rhythms, but over-reliance on the chaotic time series analysis method may easily lead to misapprehend in non-chaotic discharge rhythm, particularly in random rhythms. Besides, it will be helpful for identification of chaotic neural firing rhythms when deeper study of them are performed from perspectives other than those of non-linear characteristics.For the current practical problems in the study of rhythm of neural firing, this paper, using the combinative methods of biological experiments, mathematical modelling and time series analysis, analyses rhythmic characteristics, mechanism and randomnrss on a variety of random neural firing rhythm produced from experiments and theoretical models. Non-smooth characteristics of some intermittenly chaotic firing rhythms are also studied. The results of the present study are helpful for recognition and understanding of the random and chaotic neural firing rhythm, and also provide some theoretical and practical methods. Specific contents are as follows:1. The integer multiple bursting and on-off firing lying between period 1 butsting and rest condition are numerically simulated with the same theoretical model. These two firing patterns exhibit stochastic characters by multi-method comprehensive analysis on interspike interval (ISI). Considering as Markov process, ISI series of the two patterns are transferred into 0.1 series. Then, the stochastic characters can exhibit significantly on two levels. The two firing pattern are suggested to be stochastic firing pattern generated near super-critical and sub-critical Hopf bifurcation, respectively. The experimental observation holds the same characters with simulated results.2. Two special stochastic neural firing patterns, generating near the bifurcation point in a period adding bifurcation scenario, are simulated in stochastic Chay model. The behavior of these two firing patterns is is transition between period n burst and period n+1 burst (n=1,2,3). On one hand, the firing patterns are found to show deterministic characteristics. On the other hand, when a period n burst (or period n+1 burst) is defined as an event, stochastic components can then be identified in the inter-event interval (IEI) series, one of which is integer multiple like characteristics, the other is on-off like characters, and the latter is a new discovery in experiment. The results of numerically simulation suggest the two patterns are all the stochastic behavior induced by stochastic noise near the bifurcation points in the period adding bifurcation scenario.3. A chaotic firing pattern, which was near the period 3 and then to period 2 bursting, was observed in experiments on a neural firing pacemaker. By the time series analysis, we found that most part of the firing train is composed of period 3 burst. And non-smooth like characteristics of the first return map of ISI was shown by the qualitative analysis and quantitative calculation. The experimentally observed intermittent chaos can be reproduced with the Chay model. It was shown that the intermittency is similar to both type I and type V intermittency, by computing the average length of laminar phase under different parameters. At the same time, with the increases of slow variable time-scale, the intermittency deviated from the type I intermittency to type V gradually. We suggest that it is due to the neuronal system with multiple time scale, including slow variable time-scale, which plays its important roles via parameterλ.