Feng S, Holmes P
Department of Applied Mathematics and Sciences, Khalifa University of Science, Technology, and Research, Abu Dhabi, United Arab Emirates.
Program in Applied and Computational Mathematics, Department of Mechanical and Aerospace Engineering and Princeton Neuroscience Institute, Princeton University, NJ 08544.
IMA J Appl Math. 2016 Jun;81(3):432-456. doi: 10.1093/imamat/hxw026. Epub 2016 Jul 11.
New mathematics has often been inspired by new insights into the natural world. Here we describe some ongoing and possible future interactions among the massive data sets being collected in neuroscience, methods for their analysis and mathematical models of the underlying, still largely uncharted neural substrates that generate these data. We start by recalling events that occurred in turbulence modelling when substantial space-time velocity field measurements and numerical simulations allowed a new perspective on the governing equations of fluid mechanics. While no analogous global mathematical model of neural processes exists, we argue that big data may enable validation or at least rejection of models at cellular to brain area scales and may illuminate connections among models. We give examples of such models and survey some relatively new experimental technologies, including optogenetics and functional imaging, that can report neural activity in live animals performing complex tasks. The search for analytical techniques for these data is already yielding new mathematics, and we believe their multi-scale nature may help relate well-established models, such as the Hodgkin-Huxley equations for single neurons, to more abstract models of neural circuits, brain areas and larger networks within the brain. In brief, we envisage a closer liaison, if not a marriage, between neuroscience and mathematics.
新数学常常受到对自然世界新见解的启发。在此，我们描述神经科学中正在收集的海量数据集、其分析方法以及产生这些数据的潜在的、仍很大程度上未知的神经基质的数学模型之间一些正在进行的以及未来可能的相互作用。我们首先回顾湍流建模中发生的事件，当时大量时空速度场测量和数值模拟为流体力学的控制方程带来了新视角。虽然不存在类似的神经过程全局数学模型，但我们认为大数据可能在细胞到脑区尺度上验证或至少否定模型，并可能阐明模型之间的联系。我们给出此类模型的示例，并概述一些相对较新的实验技术，包括光遗传学和功能成像，这些技术可以报告执行复杂任务的活体动物的神经活动。对这些数据的分析技术的探索已经产生了新数学，并且我们相信它们的多尺度性质可能有助于将诸如单个神经元的霍奇金 - 赫胥黎方程等成熟模型与神经回路、脑区和大脑中更大网络的更抽象模型联系起来。简而言之，我们设想神经科学与数学之间建立更紧密的联系，即便不是联姻。
