The human brain is a really complex organization of connectivity whose principal elements are neurons, synapses and brain regions. Up to now this connectivity is not fully understood, and recent impulse in investigating its structure has been given by Graph Theory. However, some points remain unclear, mainly due to possible mismatching between the Mathematical and the Neuroscientific approach. It is known that neural connectivity is classified into three categories: structural (or anatomical) connectivity, functional connectivity and effective connectivity. The point is that these categories demand different kinds of graphs, except in the case of the resting state, and sometimes topological and metrical parameters are involved simultaneously, without a specific distinction of their roles. In this paper we propose a mathematical model for treating the functional connectivity, based on directed graphs with weighted edges. The function W(i,j,t), representing the weight of the edge connecting nodes i,j at time t, is obtained by splitting the model in two parts, where different parameters have been introduced step by step and rigorously motivated. In particular, there is a double role played by the notion of distance, which, according to the different parts of the model, assumes a discrete or an Euclidean meaning. Analogously, the time t appears both from a local and from a global perspective. The local aspect relates to a specific task submitted to an health volunteer (in view of possible future applications also to subjects affected by neurological diseases), while the global one concerns the different periods in the human life that characterize the main changes in the neural brain network. In the particular case of the resting state, we have shown that W reduces to the usually employed probabilistic growth laws for the edge formation. We tested the correctness of our model by means of synthetic data, where the selection of all involved parameters has been motivated according to what is known from the available literature. It turns out that simulated outputs fit well with the expected results, which encourages further analysis on real data, and possible future applications to neurological pathologies.

中文翻译：

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用于评估健康人的功能连接强度的数学模型。

人脑是一个非常复杂的连通性组织，其主要元素是神经元，突触和大脑区域。到现在为止，这种连通性还没有被完全理解，并且图论已经给出了研究其结构的最新动力。但是，有些要点仍然不清楚，主要是由于数学方法和神经科学方法之间可能不匹配。众所周知，神经连接分为三类：结构（或解剖）连接，功能连接和有效连接。关键是这些类别需要不同类型的图，除了处于静止状态时，有时还同时涉及拓扑和度量参数，而没有明确区分它们的作用。在本文中，我们基于具有加权边的有向图，提出了一种用于处理功能连通性的数学模型。通过将模型分为两部分获得函数W（i，j，t），该函数代表时间t的边缘连接节点i，j的权重，其中逐步引入了不同的参数并严格激励了这些参数。特别地，距离的概念起着双重作用，根据模型的不同部分，距离的概念具有离散或欧几里得的含义。类似地，从局部和全局角度来看，时间t都出现了。当地方面涉及提交给卫生志愿者的特定任务（鉴于将来可能还会应用于受神经系统疾病影响的受试者），而全球范围关注的是人类生命中不同时期，这些时期表征了神经脑网络的主要变化。在静止状态的特定情况下，我们已经表明W减小到边缘形成通常采用的概率增长定律。我们通过综合数据测试了模型的正确性，其中根据现有文献中已知的知识来激励所有相关参数的选择。事实证明，模拟输出与预期结果非常吻合，这鼓励了对实际数据的进一步分析，并鼓励将来在神经病理学中的应用。我们通过综合数据测试了模型的正确性，其中根据现有文献中已知的知识来激励所有相关参数的选择。事实证明，模拟输出与预期结果非常吻合，这鼓励了对实际数据的进一步分析，并鼓励将来在神经病理学中的应用。我们通过综合数据测试了模型的正确性，其中根据现有文献中已知的知识来激励所有相关参数的选择。事实证明，模拟输出与预期结果非常吻合，这鼓励了对实际数据的进一步分析，并鼓励将来在神经病理学中的应用。