Memory encoding in spatially structured networks: dynamics, discrete geometry & topology

由于海马体在学习和记忆中的中心作用，以及相对较高水平的经常性连接和突触可塑性，海马体网络被认为是联想学习的主要中心。海马体缺乏拓扑结构，这使得它成为联想记忆模型的自然灵感来源，例如Hopfield模型，用于在非结构化递归网络中编码记忆。与此同时，对啮齿动物的研究揭示了海马体在空间导航以及最近的时间跟踪中的关键作用。与联想记忆编码不同，这些功能已经使用空间结构网络成功地建模。如何调和这些观点呢？这项研究的中心目标是发展空间结构网络中记忆编码的数学理论，并研究由这种网络产生的神经编码。具体地说，这项研究将发展数学理论来回答以下问题：(1)如何将重叠的记忆模式准确地编码为非结构化神经网络的吸引子，而不引入不想要的“虚假状态”？(2)如何在空间结构的网络(如凹凸吸引子网络)中对记忆进行编码，同时保持依赖于网络的空间组织的功能？(3)除了纠错之外，神经编码中的冗余还有什么优势，例如以严重重叠的感受野为特征的海马体位置场编码？最后一个问题也将通过对合作实验室提供的大脑皮层和海马体数据集的分析来探索。海马体通常被认为是大脑中的一把“瑞士刀”。几十年的实验工作揭示了它在学习和记忆以及空间导航中的重要作用。从理论上讲，令人费解的是，同一个神经网络怎么能实现如此不同的功能。特别是，记忆编码的数学模型与空间导航模型有很大的不同。这项工作将整合这两种主要类型的神经网络模型，目的是了解海马体如何支持多种重要功能。其核心是，这项研究将推动我们理解大脑中的网络级计算背后的数学理论。

Hippocampal networks are believed to be a major center of associative learning due to the central role of the hippocampus in learning and memory, as well as the relatively high levels of recurrent connectivity and synaptic plasticity. The lack of topographic structure in hippocampus has made it a natural inspiration for associative memory models, such as the Hopfield model, for encoding memories in unstructured recurrent networks. At the same time, studies in rodents have uncovered the critical role of hippocampus in spatial navigation and, more recently, time-tracking. In contrast to associative memory encoding, these functions have been successfully modeled using spatially structured networks. How can these viewpoints be reconciled? The central goal of this research is to develop a mathematical theory of memory encoding in spatially structured networks, and to study the neural codes that arise from such networks. Specifically, the research will develop mathematical theory to answer the following questions: (1) How can overlapping memory patterns be encoded precisely as attractors of an unstructured neural network, without introducing unwanted "spurious states"? (2) How can memories be encoded in a spatially structured network, such as a bump attractor network, while maintaining functions that depend on the network's spatial organization? (3) Aside from error correction, what are the advantages of redundancy in a neural code, such as the hippocampal place field code, that is characterized by heavily overlapping receptive fields? This last question will also be explored via the analysis of cortical and hippocampal data sets provided by collaborating labs.The hippocampus is often thought of as a "Swiss knife" in the brain. Decades of experimental work have uncovered its essential role in learning and memory, as well as in spatial navigation. From a theoretical standpoint, it is puzzling how the same neural network can achieve such disparate functions. In particular, mathematical models of memory encoding are fundamentally quite different from models of spatial navigation. This work will integrate these two major types of neural network models, with the goal of understanding how the hippocampus can support multiple important functions. At its core, the research will advance the mathematical theory behind our understanding of network-level computation in the brain.