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DYNAMICAL SYSTEMS AND BIOLOGICAL REGULATIONS

机译:动力系统和生物法规

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We consider in this tutorial particular features of many definitions of attractors given in the literature by looking at examples of attracting cycles. We prove by considering specific counterexamples, that there is in fact no order between definitions, only some of them being weaker than others. The chaotic case is also discussed. After which we study mathematical properties of the "natural measure" of a semi-orbit and show that this set function is not a measure. In spite of this fact, this definition may be used in order to define fractal dimensions and dimension functions of an attractor. We compare these dimensions, by studying for example the equality between the two dimension functions proposed by Grassberger, and by Hentschel and Procaccia. Then we study the distribution of the point-wise dimension in the special case of the Baker transformation. We propose one example of application in neuromodelling. We consider after the case of a universal minimal regulatory system (called a regulon), having a positive and a negative circuit in its interaction matrix, and we recall the main results related to the presence of such circuits. Finally, we give two examples of application of the interaction matrix: one concerns the dual problems of synchronization and de-synchronization of a neural model (susceptible to serve as a sketch for hippocampus memory evocation processes), and the second deals with the problem of occurrence of weak parts along the chromosomes, related to the ubiquitory genes expression.
机译:我们考虑在本教程中,通过查看吸引周期的示例,在文献中给出了许多吸引子定义的特征。我们证明了通过考虑特定的反例,实际上没有定义之间的命令,其中一些人只是比其他的一些弱点。还讨论了混沌案例。之后我们研究半轨的“自然度量”的数学特性,并表明该集合功能不是一种措施。尽管有了这一事实,可以使用该定义以限定吸引物的分形尺寸和尺寸函数。我们通过研究例如由草莓提出的两个维函数之间的平等来进行比较这些尺寸,以及HentsChel和Provaccia。然后我们研究了贝克变换的特殊情况下的点明智维度的分布。我们提出了一种在神经形状的应用中的一个例子。我们考虑在普遍最小的监管系统(称为稳态)的情况下,在其相互作用矩阵中具有正和负电路,我们回顾了与此类电路的存在相关的主要结果。最后,我们提供了两个应用程序的应用程序的示例:一个涉及神经模型的同步和解除同步的双重问题(易感用作海马存储器唤起过程的草图),第二个处理问题沿染色体的弱部件发生,与ubiquity基因表达有关。

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