Aburn Matthew J, Holmes C A, Roberts James A, Boonstra Tjeerd W, Breakspear Michael
School of Mathematics and Physics, The University of Queensland Brisbane, QLD, Australia.
Front Physiol. 2012 Aug 20;3:331. doi: 10.3389/fphys.2012.00331. eCollection 2012.
Computational studies often proceed from the premise that cortical dynamics operate in a linearly stable domain, where fluctuations dissipate quickly and show only short memory. Studies of human electroencephalography (EEG), however, have shown significant autocorrelation at time lags on the scale of minutes, indicating the need to consider regimes where non-linearities influence the dynamics. Statistical properties such as increased autocorrelation length, increased variance, power law scaling, and bistable switching have been suggested as generic indicators of the approach to bifurcation in non-linear dynamical systems. We study temporal fluctuations in a widely-employed computational model (the Jansen-Rit model) of cortical activity, examining the statistical signatures that accompany bifurcations. Approaching supercritical Hopf bifurcations through tuning of the background excitatory input, we find a dramatic increase in the autocorrelation length that depends sensitively on the direction in phase space of the input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. Similar dependence on the input direction is found in the distribution of fluctuation size and duration, which show power law scaling that extends over four orders of magnitude at the Hopf bifurcation. We conjecture that the alignment in phase space between the input noise vector and the center manifold of the Hopf bifurcation is directly linked to these changes. These results are consistent with the possibility of statistical indicators of linear instability being detectable in real EEG time series. However, even in a simple cortical model, we find that these indicators may not necessarily be visible even when bifurcations are present because their expression can depend sensitively on the neuronal pathway of incoming fluctuations.
计算研究通常基于这样一个前提，即皮层动力学在一个线性稳定的范围内运行，在这个范围内波动会迅速消散，并且只表现出短期记忆。然而，对人类脑电图（EEG）的研究表明，在数分钟尺度的时间滞后上存在显著的自相关性，这表明需要考虑非线性影响动力学的情况。诸如自相关长度增加、方差增加、幂律缩放和双稳态切换等统计特性已被认为是非线性动力系统中接近分岔的通用指标。我们研究了一个广泛使用的皮层活动计算模型（Jansen-Rit模型）中的时间波动，检查了伴随分岔的统计特征。通过调整背景兴奋性输入来接近超临界霍普夫分岔，我们发现自相关长度急剧增加，这敏感地取决于输入波动在相空间中的方向，因此取决于哪个神经元亚群受到随机扰动。在波动大小和持续时间的分布中也发现了对输入方向的类似依赖，在霍普夫分岔处，它们呈现出跨越四个数量级的幂律缩放。我们推测，输入噪声向量与霍普夫分岔的中心流形在相空间中的对齐与这些变化直接相关。这些结果与在实际EEG时间序列中可检测到线性不稳定性的统计指标的可能性是一致的。然而，即使在一个简单的皮层模型中，我们发现即使存在分岔，这些指标也不一定可见，因为它们的表现可能敏感地取决于传入波动的神经元通路。
