本文为美国科罗拉多大学（作者：Y. S. Virkar）的博士论文，共115页。

可激励单位的网络存在于社会科学、神经科学、遗传学、流行病学等多个学科中。先前的研究表明，当网络在“临界状态”运行时，网络功能的某些方面可以得到优化，即，在有序与无序的边界上，节点激发的统计信息对应于一个经典的分支过程。在这篇论文中，我们介绍并研究一个神经网路的数学模型，目的是理解长期存在的问题，即决定一个神经网路调节其活动的机制，以便在临界状态下运作。特别地，我们研究了由一个可激发的节点网络和一个提供节点触发所需资源的互补网络组成的两层网络模型的动力学。更具体地说，我们研究了由神经元（节点）通过突触（边缘）连接而成的可兴奋神经网络的动力学。突触强度由互补性胶质细胞网络提供的资源介导。血液中的资源以一定的速率供给神经胶质网络，资源在胶质细胞网络内扩散传输，最终到达突触，神经元每一次激发出的所有突触资源都以一定的速率被消耗掉。我们证明了这种自然和非常令人信服的反馈控制机制可以稳定临界状态。另外，神经网络可以学习、记忆和恢复学习后的临界状态。临界状态的特征是幂律分布的雪崩大小，对供给、消耗和扩散速率的变化具有鲁棒性。最后，我们证明我们的发现对于模型参数或网络结构的异质性是相当稳健的。

Networks of excitable units are found in varied disciplines such as social science, neuroscience, genetics, epidemiology, etc. Previous studies have shown that some aspects of network function can be optimized when the network operates in the ‘critical regime’, i.e., at the boundary between order and disorder where the statistics of node excitations correspond to those of a classical branching process. In this thesis, we introduce and study a mathematical model of a neural network with the goal of understanding the long-standing problem of determining the mechanisms by which a neural network regulates its activity so as to operate in the critical regime. In particular, we study the dynamics of a two-layered network model consisting of an excitable node network and a complementary network that supplies resources required for node firing. More specifically, we study the dynamics of an excitable neural network consisting of neurons (nodes) connected via synapses (edges). Synaptic strengths are mediated by resources supplied by the complementary glial cell network. Resources from the bloodstream are supplied to the glial network at some fixed rate, resources transport diffusively within the glial cell network and ultimately to the synapses, and each time a presynaptic neuron fires the resources for all outgoing synapses get consumed at some fixed rate. We show that this natural and very compelling mechanism for feedback control can stabilize the critical state. Additionally, the neural network can learn, remember and recover the critical state after learning. The critical state is characterized by power-law distributed avalanche sizes that are robust to changes in the supply, consumption and diffusion rates. Finally, we show that our findings are fairly robust to heterogeneity in model parameters or network structure.

1.  引言

2.  文献回顾

3.  两层网络模型

4.  数值实验

5.  结论与展望

更多精彩文章请关注公众号：