Martinos Center for Biomedical Imaging, Massachusetts General Hospital Charlestown, MA, USA ; Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology Cambridge, MA, USA.
Front Comput Neurosci. 2013 Jul 24;7:96. doi: 10.3389/fncom.2013.00096. eCollection 2013.
Ising models are routinely used to quantify the second order, functional structure of neural populations. With some recent exceptions, they generally do not include the influence of time varying stimulus drive. Yet if the dynamics of network function are to be understood, time varying stimuli must be taken into account. Inclusion of stimulus drive carries a heavy computational burden because the partition function becomes stimulus dependent and must be separately calculated for all unique stimuli observed. This potentially increases computation time by the length of the data set. Here we present an extremely fast, yet simply implemented, method for approximating the stimulus dependent partition function in minutes or seconds. Noting that the most probable spike patterns (which are few) occur in the training data, we sum partition function terms corresponding to those patterns explicitly. We then approximate the sum over the remaining patterns (which are improbable, but many) by casting it in terms of the stimulus modulated missing mass (total stimulus dependent probability of all patterns not observed in the training data). We use a product of conditioned logistic regression models to approximate the stimulus modulated missing mass. This method has complexity of roughly O(LNNpat) where is L the data length, N the number of neurons and N pat the number of unique patterns in the data, contrasting with the O(L2 (N) ) complexity of alternate methods. Using multiple unit recordings from rat hippocampus, macaque DLPFC and cat Area 18 we demonstrate our method requires orders of magnitude less computation time than Monte Carlo methods and can approximate the stimulus driven partition function more accurately than either Monte Carlo methods or deterministic approximations. This advance allows stimuli to be easily included in Ising models making them suitable for studying population based stimulus encoding.
伊辛模型常用于量化神经群体的二阶、功能结构。除了最近的一些例外，它们通常不包括随时间变化的刺激驱动的影响。然而，如果要理解网络功能的动态，就必须考虑随时间变化的刺激。包括刺激驱动会带来沉重的计算负担，因为配分函数变得依赖于刺激，并且必须为所有观察到的独特刺激分别计算。这可能会使计算时间增加到数据集的长度。在这里，我们提出了一种极其快速、简单实现的方法，可以在几分钟或几秒钟内近似刺激相关的配分函数。注意到最可能的尖峰模式（很少）出现在训练数据中，我们明确地对与这些模式相对应的配分函数项进行求和。然后，我们通过将剩余模式的和（不太可能但很多）表示为受刺激调制的缺失质量（所有未在训练数据中观察到的模式的总刺激相关概率）来近似该和。我们使用条件逻辑回归模型的乘积来近似受刺激调制的缺失质量。这种方法的复杂度大约为 O(LNNpat)，其中 L 是数据长度，N 是神经元数量，Npat 是数据中唯一模式的数量，与替代方法的 O(L2 (N) )复杂度形成对比。使用大鼠海马体、猕猴 DLPFC 和猫 Area 18 的多单位记录，我们证明我们的方法需要比蒙特卡罗方法少几个数量级的计算时间，并且可以比蒙特卡罗方法或确定性近似更准确地近似刺激驱动的配分函数。这一进展使得伊辛模型能够轻松地包含刺激，使其适合研究基于群体的刺激编码。
