We study pairwise Ising models for describing the statistics of multi-neuron
spike trains, using data from a simulated cortical network. We explore
efficient ways of finding the optimal couplings in these models and examine
their statistical properties. To do this, we extract the optimal couplings for
subsets of size up to 200 neurons, essentially exactly, using Boltzmann
learning. We then study the quality of several approximate methods for finding
the couplings by comparing their results with those found from Boltzmann
learning. Two of these methods- inversion of the TAP equations and an
approximation proposed by Sessak and Monasson- are remarkably accurate. Using
these approximations for larger subsets of neurons, we find that extracting
couplings using data from a subset smaller than the full network tends
systematically to overestimate their magnitude. This effect is described
qualitatively by infinite-range spin glass theory for the normal phase. We also
show that a globally-correlated input to the neurons in the network lead to a
small increase in the average coupling. However, the pair-to-pair variation of
the couplings is much larger than this and reflects intrinsic properties of the
network. Finally, we study the quality of these models by comparing their
entropies with that of the data. We find that they perform well for small
subsets of the neurons in the network, but the fit quality starts to
deteriorate as the subset size grows, signalling the need to include higher
order correlations to describe the statistics of large networks.

使用模拟皮层网络的数据，探究描述多神经元尖峰列统计学的成对 Ising 模型的最优耦合的优化方法和它们的统计特性，并通过比较不同的近似方法从一定程度上找到了这些最优耦合，但发现从小规模子集中提取耦合常常高估其大小，而且随着子集大小的增长，所用模型的拟合效果逐渐恶化，需要引入高阶相关性来描述大型网络的统计特性。