The observation of apparent power laws in neuronal systems has led to the suggestion that the brain is at, or close to, a critical state and may be a self-organised critical system. Within the framework of self-organised criticality a separation of timescales is thought to be crucial for the observation of power-law dynamics and computational models are often constructed with this property. However, this is not necessarily a characteristic of physiological neural networks-external input does not only occur when the network is at rest/a steady state. In this paper we study a simple neuronal network model driven by a continuous external input (i.e. the model does not have an explicit separation of timescales from seeding the system only when in the quiescent state) and analytically tuned to operate in the region of a critical state (it reaches the critical regime exactly in the absence of input-the case studied in the companion paper to this article). The system displays avalanche dynamics in the form of cascades of neuronal firing separated by periods of silence. We observe partial scale-free behaviour in the distribution of avalanche size for low levels of external input. We analytically derive the distributions of waiting times and investigate their temporal behaviour in relation to different levels of external input, showing that the system's dynamics can exhibit partial long-range temporal correlations. We further show that as the system approaches the critical state by two alternative 'routes', different markers of criticality (partial scale-free behaviour and long-range temporal correlations) are displayed. This suggests that signatures of criticality exhibited by a particular system in close proximity to a critical state are dependent on the region in parameter space at which the system (currently) resides.

中文翻译：

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确定神经元雪崩的严重性：II。驱动案例的理论和实证研究。

对神经元系统中表观幂律的观察表明，大脑处于或接近临界状态，并且可能是一个自组织的临界系统。在自组织临界的框架内，时间尺度的分离被认为对于观察幂律动力学是至关重要的，并且计算模型通常是用这个属性构建的。然而，这不一定是生理神经网络的特征——外部输入不仅仅发生在网络处于静止/稳定状态时。在本文中，我们研究了一个由连续外部输入驱动的简单神经网络模型（即 该模型没有明确地将时间尺度与仅在静止状态下播种系统分开）并且经过分析调整以在临界状态区域中运行（它在没有输入的情况下准确地达到临界状态 - 研究的案例在本文的配套文件）。该系统以由静默期分隔的神经元放电级联形式显示雪崩动力学。对于低水平的外部输入，我们观察到雪崩大小分布的部分无标度行为。我们分析推导了等待时间的分布，并研究了它们与不同外部输入水平相关的时间行为，表明系统的动力学可以表现出部分的长期时间相关性。我们进一步表明，当系统通过两条替代“路线”接近临界状态时，会显示不同的临界状态标记（部分无标度行为和长期时间相关性）。这表明紧邻临界状态的特定系统表现出的临界特征取决于系统（当前）所在的参数空间区域。