随机与混沌神经放电节律的时间序列分析
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摘要
神经系统可以通过丰富的神经放电节律接收、传递和加工信息,因此,神经放电节律的识别是正确理解神经系统动力学行为的关键,随机、混沌、周期神经放电节律都是其基础形式之一。非线性科学尤其是混沌理论的发展为神经放电节律的识别提供了丰富的理论知识和分析方法,但过度依赖混沌时间序列分析方法容易对非混沌放电节律,尤其是随机节律,造成误判。同时,从非线性特征之外的角度考察混沌的细致特征,对于神经放电节律的鉴别也会起到积极作用。
本文针对当前神经放电节律研究中存在的一些实际问题,采用生物学实验、数学模型数值仿真以及时间序列分析结合的研究方法,对实验中以及数学模型数值仿真中产生的多种随机神经放电节律进行了节律特征、产生机制、随机性强弱的分析：同时对一类阵发混沌从光滑性角度进行了分析,并对其阵发类型进行了鉴别。该研究对于随机和混沌神经放电节律的识别和理解,都有重要的借鉴意义,并提供了一定理论价值和实用方法。具体研究内容如下：
1、利用同一数学模型在不同参数配置下,成功仿真了位于静息态与周期1放电态之间的随机整数倍节律和随机on-off节律,通过对峰峰间期(interspike interval, ISI)的多指标综合分析,展示了由这些结果所体现的两种节律的随机性：在将两种节律考虑为典型的Markov过程的基础上,将节律的ISI序列转换为01序列进行概率统计分析,进一步分层次细致说明了两种节律的随机性,展示了on-off节律连续放电或静息之间的概率依赖性；经分析功率谱与信噪比随不同噪声强度的变化,证实了两种节律均为噪声在平衡点与极限环之间的Hopf分岔点附近通过随机自共振机制诱发的随机节律。对应数学模型的数值仿真,利用相同时间序列方法分析实验性神经起步点中获得的与上述仿真结果类似的两种节律,验证了仿真节律分析结果。
2、利用随机Chay模型仿真了出现于加周期分岔点附近的两类随机节律,两种节律对应着系统在分岔点两侧极限环之间的跃迁行为,一种呈短串簇交替出现,另一种呈长串簇交替出现。综合多项时间序列指标对两种节律进行分析显示,二者ISI序列表观上呈明显的确定性。通过对事件间期(inter-event interval, IEI)的分析发现,“短串簇交替节律”具有内在整数倍节律特征,“长串簇交替节律”具有内在on-off节律特征。通过分析不同噪声强度对交替节律功率谱、信噪比等各指标的影响,证实二者均为噪声在加周期分岔点附近通过随机自共振机制诱发的随机节律,确定性结构来自簇内放电的ISI顺序。对于实验中产生的与上述仿真结果相似的两种节律,同样进行了相关指标的综合分析,结果与仿真节律一致,证实了在实验中出现的两种节律其发生机制,亦均为噪声在分岔点附近诱发的随机节律。其中呈现内在on-off节律特征的长串簇交替节律,是新的实验发现。
3、利用本实验室的数据,研究了实验性神经起步点放电实验中产生的阵发混沌节律,以周期3簇放电经混沌放电到周期2簇放电过程中靠近周期3簇的阵发混沌为例,通过时间序列分析展示了该阵发混沌以周期3簇节律为主要组成的内部特征,并结合定性分析和定量计算展示了其ISI回归映射的非光滑性。利用确定性Chay模型仿真了上述混沌节律并分析,结果与实验一致。通过计算不同参数下平均层流相长度随参数变化的规律,发现该类阵发混沌节律的标度率介于I型阵发和V型阵发之间,且随着Chay模型中表征慢变量时间尺度作用的参数λn的增大逐渐偏离I型阵发,而偏向V型阵发,为一类新型阵发。我们认为这种特殊标度率是由于Chay模型具有多个时间尺度造成,而标度率随λn的变化呈现的变化趋势,应当归因于慢变量作用强度的不同。
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.
本文针对当前神经放电节律研究中存在的一些实际问题,采用生物学实验、数学模型数值仿真以及时间序列分析结合的研究方法,对实验中以及数学模型数值仿真中产生的多种随机神经放电节律进行了节律特征、产生机制、随机性强弱的分析：同时对一类阵发混沌从光滑性角度进行了分析,并对其阵发类型进行了鉴别。该研究对于随机和混沌神经放电节律的识别和理解,都有重要的借鉴意义,并提供了一定理论价值和实用方法。具体研究内容如下：
1、利用同一数学模型在不同参数配置下,成功仿真了位于静息态与周期1放电态之间的随机整数倍节律和随机on-off节律,通过对峰峰间期(interspike interval, ISI)的多指标综合分析,展示了由这些结果所体现的两种节律的随机性：在将两种节律考虑为典型的Markov过程的基础上,将节律的ISI序列转换为01序列进行概率统计分析,进一步分层次细致说明了两种节律的随机性,展示了on-off节律连续放电或静息之间的概率依赖性；经分析功率谱与信噪比随不同噪声强度的变化,证实了两种节律均为噪声在平衡点与极限环之间的Hopf分岔点附近通过随机自共振机制诱发的随机节律。对应数学模型的数值仿真,利用相同时间序列方法分析实验性神经起步点中获得的与上述仿真结果类似的两种节律,验证了仿真节律分析结果。
2、利用随机Chay模型仿真了出现于加周期分岔点附近的两类随机节律,两种节律对应着系统在分岔点两侧极限环之间的跃迁行为,一种呈短串簇交替出现,另一种呈长串簇交替出现。综合多项时间序列指标对两种节律进行分析显示,二者ISI序列表观上呈明显的确定性。通过对事件间期(inter-event interval, IEI)的分析发现,“短串簇交替节律”具有内在整数倍节律特征,“长串簇交替节律”具有内在on-off节律特征。通过分析不同噪声强度对交替节律功率谱、信噪比等各指标的影响,证实二者均为噪声在加周期分岔点附近通过随机自共振机制诱发的随机节律,确定性结构来自簇内放电的ISI顺序。对于实验中产生的与上述仿真结果相似的两种节律,同样进行了相关指标的综合分析,结果与仿真节律一致,证实了在实验中出现的两种节律其发生机制,亦均为噪声在分岔点附近诱发的随机节律。其中呈现内在on-off节律特征的长串簇交替节律,是新的实验发现。
3、利用本实验室的数据,研究了实验性神经起步点放电实验中产生的阵发混沌节律,以周期3簇放电经混沌放电到周期2簇放电过程中靠近周期3簇的阵发混沌为例,通过时间序列分析展示了该阵发混沌以周期3簇节律为主要组成的内部特征,并结合定性分析和定量计算展示了其ISI回归映射的非光滑性。利用确定性Chay模型仿真了上述混沌节律并分析,结果与实验一致。通过计算不同参数下平均层流相长度随参数变化的规律,发现该类阵发混沌节律的标度率介于I型阵发和V型阵发之间,且随着Chay模型中表征慢变量时间尺度作用的参数λn的增大逐渐偏离I型阵发,而偏向V型阵发,为一类新型阵发。我们认为这种特殊标度率是由于Chay模型具有多个时间尺度造成,而标度率随λn的变化呈现的变化趋势,应当归因于慢变量作用强度的不同。
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.
引文
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