Theory of threshold-linear networks and combinatorial neural codes.

神经元之间的联系如何存储记忆并塑造大脑中神经活动的动态？神经元的放电模式如何代表我们的感觉体验？促进同时记录大量神经元的技术的出现为回答这些神经科学的经典问题提供了新的机会。有一些经常用于网络模拟和数据分析的数学模型可以使用，但对其数学属性仍知之甚少。为了指导这些努力，更好地理解循环网络和人口代码的理论模型是至关重要的。这项研究将集中在两个这样的例子：门限线性网络和组合神经码。目标是在这些模型的数学理论方面取得重大进展，着眼于神经科学的应用。部分研究将与实验者合作，分析大脑皮层和海马区的神经活动。尽管侧重于神经科学，但数学结果有可能足够普遍，从而在生物和社会科学的各种更广泛的背景下有用。门限线性网络是递归网络的常见发电率模型，具有门限非线性。这些网络通常表现出多个稳定的不动点，并且多稳定性使得它们作为记忆存储和检索的模型很有吸引力。初步结果表明，这些平衡点具有丰富的组合结构，可以用经典距离几何的思想进行分析。第一个项目将建立在这一理解的基础上，以便更全面地了解这些网络的固定点和高维吸引子的结构。组合神经编码是一组神经元的二进制模式的集合。第二个项目将使用最近开发的神经环框架来开发组合代码的代数分类。神经环以使诸如感受野组织等属性最透明的方式对关于神经编码的信息进行编码。由此产生的方法将通过对海马区定位细胞的电生理记录进行测试和改进。这项研究还将在神经科学与应用代数、组合学和几何学的交界处产生新的有趣的问题。

How do connections between neurons store memories and shape the dynamics of neural activity in the brain? How do firing patterns of neurons represent our sensory experiences? The advent of technologies that facilitate simultaneous recordings of large populations of neurons present new opportunities to answer these classical questions of neuroscience. There are mathematical models that are frequently used in network simulations and data analyses that can be employed, but whose mathematical properties are still poorly understood. To guide these efforts, a better understanding of theoretical models of recurrent networks and population codes is essential. This research will focus on two such examples: threshold-linear networks and combinatorial neural codes. The goal is to produce major advances in the mathematical theory of these models, with an eye towards neuroscience applications. Part of the research will involve the analyses of neural activity in the cortex and hippocampus, in collaboration with experimentalists. Despite the focus on neuroscience, the mathematical results have the potential to be sufficiently general so as to be useful in a variety of broader contexts in the biological and social sciences.A threshold-linear network is a common firing rate model for a recurrent network, with a threshold nonlinearity. These networks generically exhibit multiple stable fixed points, and multistability makes them attractive as models for memory storage and retrieval. Preliminary results have shown that the equilibria possess a rich combinatorial structure, and can be analyzed using ideas from classical distance geometry. The first project will build on this understanding in order to develop a more complete picture of the structure of fixed points and higher-dimensional attractors of these networks. A combinatorial neural code is a collection of binary patterns for a population of neurons. The second project will develop an algebraic classification of combinatorial codes, using the recently developed framework of the neural ring. The neural ring encodes information about a neural code in a manner that makes properties such as receptive field organization most transparent. The resulting methods will be tested and refined using electrophysiological recordings of place cells in the hippocampus. This research will also generate new and interesting problems at the interface of neuroscience with applied algebra, combinatorics, and geometry.