Data from: Bistability, non-ergodicity, and inhibition in pairwise maximum-entropy models

Pairwise maximum-entropy models have been used in neuroscience to predict the activity of neuronal populations, given only the time-averaged correlations of the neuron activities. This paper provides evidence that the pairwise model, applied to experimental recordings, would produce a bimodal distribution for the population-averaged activity, and for some population sizes the second mode would peak at high activities, that experimentally would be equivalent to 90% of the neuron population active within time-windows of few milliseconds. Several problems are connected with this bimodality: 1. The presence of the high-activity mode is unrealistic in view of observed neuronal activity and on neurobiological grounds. 2. Boltzmann learning becomes non-ergodic, hence the pairwise maximum-entropy distribution cannot be found: in fact, Boltzmann learning would produce an incorrect distribution; similarly, common variants of mean-field approximations also produce an incorrect distribution. 3. The Glauber dynamics associated with the model is unrealistically bistable and cannot be used to generate realistic surrogate data. This bimodality problem is first demonstrated for an experimental dataset from 159 neurons in the motor cortex of macaque monkey. Evidence is then provided that this problem affects typical neural recordings of population sizes of a couple of hundreds or more neurons. The cause of the bimodality problem is identified as the inability of standard maximum-entropy distributions with a uniform reference measure to model neuronal inhibition. To eliminate this problem a modified maximum-entropy model is presented, which reflects a basic effect of inhibition in the form of a simple but non-uniform reference measure. This model does not lead to unrealistic bimodalities, can be found with Boltzmann learning, and has an associated Glauber dynamics which incorporates a minimal asymmetric inhibition.

成对最大熵模型（Pairwise maximum-entropy models）已被应用于神经科学领域，用于仅基于神经元活动的时间平均相关性，预测神经元集群的活动。本论文提供证据表明，将成对模型应用于实验记录时，会得到集群平均活动的双峰分布；在部分集群规模下，第二峰会出现在高活动水平处，该水平在实验中对应于在数毫秒时间窗内，90%的神经元集群处于激活状态。该双峰分布存在若干问题：1. 从已观测到的神经元活动及神经生物学依据来看，高活动模式的存在并不符合实际。2. 玻尔兹曼学习（Boltzmann learning）会变为非遍历过程，因此无法得到成对最大熵分布；事实上，玻尔兹曼学习会生成错误的分布，同理，常见的平均场近似变体也会生成错误分布。3. 该模型对应的格拉乌贝尔动力学（Glauber dynamics）具有不符合实际的双稳态，无法用于生成真实的替代数据。该双峰性问题首先通过猕猴运动皮层中159个神经元的实验数据集得到验证，随后的证据表明，该问题会影响数百乃至更多神经元规模的典型神经记录。双峰性问题的根源被归结为：采用均匀参考测度的标准最大熵分布，无法对神经元抑制作用进行建模。为解决该问题，本文提出一种改进型最大熵模型，其通过简单但非均匀的参考测度，反映了抑制作用的基本效应。该模型不会产生不符合实际的双峰分布，可通过玻尔兹曼学习求解，且其对应的格拉乌贝尔动力学纳入了最小化的不对称抑制机制。