A theoretical study for the mechanism of information processing of chaos in central nervous systems

(1)“奇异非混沌”是一类新的混沌动力系统。在本研究中，我们成功地构建了它的数学模型，该模型可以作为分析的刻板印象，并阐明了奇异非混沌出现的机制。我们的模型是一个被一维混沌所迫的三维环面。它的构造使得系统是公理A。这个公理A系统在一般意义上具有不变流形不可微的性质。本研究再次强化了一类结构稳定动力系统中存在光滑不变流形和非光滑两种不同的动力系统，后者只有霍尔德连续性。同时阐明了这类吸引子的Hausdorff维数和拓扑维数相差不止一个的机理。(2)研究了混沌流动在神经网络中的可能作用。当网络产生混沌流动时，可以发展出更多的模式和额外的学习。(3)加州大学伯克利分校的沃尔特·弗里曼（Walter Freeman）在嗅球中发现了混沌和混沌流动。我们从半生物神经网络的角度研究了它们的产生机制。该网络由阻尼振荡器作为一个单元组成。只有在单元之间有兴奋性突触连接的情况下，神经网络才显示出混乱。当引入Hebbian学习时，观察到混沌的流动和行波，这两者在实验中都有观察到。此外，我们研究了一种新型的强迫振荡，它源于与呼吸周期的网络同步。强迫的函数形式有指数递增部分和指数递减部分，而在通常的强迫系统研究中，周期强迫是由正弦函数给出的。我们观察到典型的分岔，如周期加倍，间歇和危机，连续变化的功能形式的强迫项。(4)从动力系统的角度研究了大脑的宏观活动。我们从李雅普诺夫指数的估计和在耦合映射格中观察到的统计异常中得出结论，大小为10^<**>3至10^<**>10的神经网络必须被视为非线性系统，而不是线性系统，甚至是宏观系统。少

(1) It is known that 'strange nonchaos' is a new class of chaotic dynamical systems. In the present study, we succeeded in constructing its mathematical model that could be a stereotype for analyzes, and we clarified the mechanism for the appearance of strange nonchaos. Our model is a 3-d torus forced by 1-d chaos. It was constructed such that the system is Axim A.This Axiom A systems has the property in a generic sense that invariant manifold is nowhere-differentiable. This study strengthens again the statement that there coexist two distinct dynamical systems with smooth invariant manifold and without smoothness in a class of structural stable dynamical systems, the latter of which has only Hoelder continuity. Also the mechanism was clarified that the Hausdorff and the topological dimensions of this type of attractors differs by more than one.(2) We studied a possible role of chaotic itinerancy in neural networks. We concluded that the network abilities for nonlinear separability of … More patterns and additional learning can develop when the network creates chaotic itinerancy.(3) Walter Freeman in UC-Berkeley has found chaos and chaotic itinerancy in the olfactory bulb. We studied about the mechanism of their appearance in terms of semi-biological neural networks. The network is constituted of damped oscillators as a unit. The nets showed chaos only in the case with excitatory synaptic connections between units. When a Hebbian learning is introduced, chaotic itinerancy and traveling waves were observed, both of which have been observed in the experiment. Furthermore, we investigated a new type of forced oscillations which stems from a network synchronization with a period of respiration. The functional form of forcing has an exponential increasing and decreasing parts, while in usual forced systems' study a periodic forcing is given by a sinusoidal function. We observed typical bifurcations such as period-doublimg, intermittency, and crisis as continuously varing the functional form of forcing term.(4) We studied a macroscopic activities of brain from the dynamical systems viewpoint. We concluded from the estimation of the Lyapunov exponents and an observed statistical anormality in coupled map lattices that neural networks of the size of 10^<**>3 to 10^<**>10 must be viewed as a nonlinear systems, not linear ones as even a macroscopic system. Less