Department of Biomedical Engineering, University of Florida, Gainesville, FL 32611, U.S.A.
Neural Comput. 2012 Aug;24(8):2223-50. doi: 10.1162/NECO_a_00309. Epub 2012 Apr 17.
Exploratory tools that are sensitive to arbitrary statistical variations in spike train observations open up the possibility of novel neuroscientific discoveries. Developing such tools, however, is difficult due to the lack of Euclidean structure of the spike train space, and an experimenter usually prefers simpler tools that capture only limited statistical features of the spike train, such as mean spike count or mean firing rate. We explore strictly positive-definite kernels on the space of spike trains to offer both a structural representation of this space and a platform for developing statistical measures that explore features beyond count or rate. We apply these kernels to construct measures of divergence between two point processes and use them for hypothesis testing, that is, to observe if two sets of spike trains originate from the same underlying probability law. Although there exist positive-definite spike train kernels in the literature, we establish that these kernels are not strictly definite and thus do not induce measures of divergence. We discuss the properties of both of these existing nonstrict kernels and the novel strict kernels in terms of their computational complexity, choice of free parameters, and performance on both synthetic and real data through kernel principal component analysis and hypothesis testing.
探索性工具对尖峰火车观测中的任意统计变化敏感，为新的神经科学发现开辟了可能性。然而，由于缺乏尖峰火车空间的欧几里得结构，开发这样的工具是困难的，实验者通常更喜欢只捕捉尖峰火车有限的统计特征的简单工具，如平均尖峰计数或平均发放率。我们探索了尖峰火车空间上的严格正定核，以提供该空间的结构表示和开发探索计数或速率之外的特征的统计度量的平台。我们应用这些核来构建两个点过程之间的散度度量，并将其用于假设检验，即观察两组尖峰火车是否来自相同的潜在概率定律。尽管文献中存在正定尖峰火车核，但我们确定这些核不是严格正定的，因此不会诱导散度度量。我们讨论了这两种现有的非严格核和新的严格核的性质，包括它们的计算复杂性、自由参数的选择，以及通过核主成分分析和假设检验在合成数据和真实数据上的性能。
