Whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. This seemingly worthless sentence is an informal description of the hairy ball theorem, an invaluable mathematical weapon that has been proven useful to describe a variety of physical/biological processes/phenomena in terms of topology, rather than classical cause/effect relationships. In this paper we will focus on the electrical brain field—electroencephalogram (EEG). As a starting point we consider the recently-raised observation that, when electromagnetic oscillations propagate with a spherical wave front, there must be at least one point of the tangential components of the vector fields where the electromagnetic field vanishes. We show how this description holds also for the electric waves produced by the brain and detectable by EEG. Once located these zero-points in EEG traces, we confirm that they are able to modify the electric wave fronts detectable in the brain. This sheds new light on the functional features of a nonlinear, metastable nervous system at the edge of chaos, based on the neuroscientific model of Operational Architectonics of brain-mind functioning. As an example of practical application of this theorem, we provide testable previsions, suggesting the proper location of transcranial magnetic stimulation’s coils to improve the clinical outcomes of drug-resistant epilepsy.

每当有人试图将一个毛球梳平时，毛球上的某一点上总会有至少一簇头发。这个看似毫无价值的句子是对毛球定理的非正式描述，这是一种无价的数学武器，已被证明可用于描述各种物理/生物过程/现象的拓扑结构，而不是经典的因果关系。在本文中，我们将重点关注脑电场——脑电图 (EEG)。作为起点，我们考虑最近提出的观察，即当电磁振荡以球面波前传播时，必须至少有一个点的矢量场的切向分量在该点处电磁场消失。我们展示了这种描述如何也适用于由大脑产生并可由 EEG 检测到的电波。一旦在 EEG 轨迹中找到这些零点，我们确认它们能够修改大脑中可检测到的电波前。基于脑-心功能操作架构的神经科学模型，这为处于混沌边缘的非线性、亚稳态神经系统的功能特征提供了新的思路。作为该定理实际应用的一个例子，我们提供了可测试的预测，建议正确放置经颅磁刺激线圈以改善耐药性癫痫的临床结果。基于脑-心功能操作架构的神经科学模型，这为处于混沌边缘的非线性、亚稳态神经系统的功能特征提供了新的思路。作为该定理实际应用的一个例子，我们提供了可测试的预测，建议正确放置经颅磁刺激线圈以改善耐药性癫痫的临床结果。基于脑-心功能操作架构的神经科学模型，这为处于混沌边缘的非线性、亚稳态神经系统的功能特征提供了新的思路。作为该定理实际应用的一个例子，我们提供了可测试的预测，建议正确放置经颅磁刺激线圈以改善耐药性癫痫的临床结果。