The brain displays both the anatomical features of a vast amount of interconnected topological mappings as well as the functional features of a nonlinear, metastable system at the edge of chaos, equipped with a phase space where mental random walks tend towards lower energetic basins. Nevertheless, with the exception of some advanced neuro-anatomic descriptions and present-day connectomic research, very few studies have been addressing the topological path of a brain embedded or embodied in its external and internal environment. Herein, by using new formal tools derived from algebraic topology, we provide an account of the metastable brain, based on the neuro-scientific model of Operational Architectonics of brain–mind functioning. We introduce a “topodynamic” description that shows how the relationships among the countless intertwined spatio-temporal levels of brain functioning can be assessed in terms of projections and mappings that take place on abstract structures, equipped with different dimensions, curvatures and energetic constraints. Such a topodynamical approach, apart from providing a biologically plausible model of brain function that can be operationalized, is also able to tackle the issue of a long-standing dichotomy: it throws indeed a bridge between the subjective, immediate datum of the naïve complex of sensations and mentations and the objective, quantitative, data extracted from experimental neuro-scientific procedures. Importantly, it opens the door to a series of new predictions and future directions of advancement for neuroscientific research.

大脑既显示大量相互关联的拓扑图的解剖特征，又显示混沌边缘处的非线性，亚稳态系统的功能特征，并配备了一个相空间，在该相空间中，精神随机行走趋向于能量较低的盆地。但是，除了一些高级的神经解剖学描述和当今的连接体研究以外，很少有研究涉及嵌入或体现在其外部和内部环境中的大脑的拓扑路径。在此，通过使用从代数拓扑派生的新形式化工具，我们基于大脑思维功能的操作构造学的神经科学模型，对亚稳大脑进行了描述。我们引入了“拓扑动力学”描述，该描述显示了如何根据在具有不同尺寸，曲率和能量约束的抽象结构上进行的投影和映射来评估无数相互交织的时空大脑功能之间的关系。这种拓扑动力学方法，除了提供可以操作的生物学上合理的脑功能模型外，还能够解决长期存在的二分法问题：的确，它在天真复杂的主观立即数据之间架起了一座桥梁。感觉和思想，以及从实验性神经科学程序中提取的客观，定量的数据。重要的是，它为神经科学研究的一系列新的预测和未来的发展方向打开了大门。