Rafati Ali H, Joca Sâmia, Vontell Regina T, Mallard Carina, Wegener Gregers, Ardalan Maryam
Translational Neuropsychiatry Unit, Department of Clinical Medicine, Aarhus University, 8200 Aarhus, Denmark.
Center of Functionally Integrative Neuroscience-SKS, Department of Clinical Medicine, Aarhus University, 8200 Aarhus, Denmark.
Methods Protoc. 2025 Jul 7;8(4):76. doi: 10.3390/mps8040076.
The study of neuronal electrical activity and spatial organization is essential for uncovering the mechanisms that regulate neuronal electrophysiology and function. Mathematical models have been utilized to analyze the structural properties of neuronal networks, predict connectivity patterns, and examine how morphological changes impact neural network function. In this study, we aimed to explore the role of the actin cytoskeleton in neuronal signaling via primary cilia and to elucidate the role of the actin network in conjunction with neuronal electrical activity in shaping spatial neuronal formation and organization, as demonstrated by relevant mathematical models. Our proposed model is based on the polygamma function, a mathematical application of ramification, and a geometrical definition of the actin cytoskeleton via complex numbers, ring polynomials, homogeneous polynomials, characteristic polynomials, gradients, the Dirac delta function, the vector Laplacian, the Goldman equation, and the Lie bracket of vector fields. We were able to reflect the effects of neuronal electrical activity, as modeled by the Van der Pol equation in combination with the actin cytoskeleton, on neuronal morphology in a 2D model. In the next step, we converted the 2D model into a 3D model of neuronal electrical activity, known as a core-shell model, in which our generated membrane potential is compatible with the neuronal membrane potential (in millivolts, mV). The generated neurons can grow and develop like an organoid brain based on the developed mathematical equations. Furthermore, we mathematically introduced the signal transduction of primary cilia in neurons. Additionally, we proposed a geometrical model of the neuronal branching pattern, which we described as ramification, that could serve as an alternative mathematical explanation for the branching pattern emanating from the neuronal soma. In conclusion, we highlighted the relationship between the actin cytoskeleton and the signaling processes of primary cilia. We also developed a 3D model that integrates the geometric organization unique to neurons, which contains soma and branches, such that the mathematical model represents the interaction between the actin cytoskeleton and neuronal electrical activity in generating action potentials. Next, we could generalize the model into a cluster of neurons, similar to an organoid brain model. This mathematical framework offers promising applications in artificial intelligence and advancements in neural networks.
对神经元电活动和空间组织的研究对于揭示调节神经元电生理学和功能的机制至关重要。数学模型已被用于分析神经元网络的结构特性、预测连接模式，并研究形态变化如何影响神经网络功能。在本研究中，我们旨在通过初级纤毛探索肌动蛋白细胞骨架在神经元信号传导中的作用，并阐明肌动蛋白网络与神经元电活动在塑造空间神经元形成和组织中的协同作用，相关数学模型已对此进行了证明。我们提出的模型基于多伽马函数、一种分支的数学应用，以及通过复数、环多项式、齐次多项式、特征多项式、梯度、狄拉克δ函数、向量拉普拉斯算子、戈德曼方程和向量场的李括号对肌动蛋白细胞骨架进行的几何定义。在二维模型中，我们能够反映出由范德波尔方程与肌动蛋白细胞骨架相结合所模拟的神经元电活动对神经元形态的影响。下一步，我们将二维模型转换为神经元电活动的三维模型，即核壳模型，其中我们生成的膜电位与神经元膜电位（以毫伏，mV为单位）兼容。基于所开发的数学方程，生成的神经元可以像类器官大脑一样生长和发育。此外，我们在数学上引入了神经元中初级纤毛的信号转导。此外，我们提出了一种神经元分支模式的几何模型，我们将其描述为分支，它可以作为对源自神经元胞体的分支模式的另一种数学解释。总之，我们强调了肌动蛋白细胞骨架与初级纤毛信号传导过程之间的关系。我们还开发了一个三维模型，该模型整合了神经元特有的几何组织，其中包含胞体和分支，使得数学模型能够代表肌动蛋白细胞骨架与神经元电活动在产生动作电位时的相互作用。接下来，可以将该模型推广到一群神经元，类似于类器官大脑模型。这个数学框架在人工智能和神经网络发展方面具有广阔的应用前景。
