Département de Physique Théorique, Université de Genève Genève, Switzerland.
Front Comput Neurosci. 2010 Sep 22;4. doi: 10.3389/fncom.2010.00132. eCollection 2010.
We present a theoretical framework using quorum percolation for describing the initiation of activity in a neural culture. The cultures are modeled as random graphs, whose nodes are excitatory neurons with k(in) inputs and k(out) outputs, and whose input degrees k(in) = k obey given distribution functions p(k). We examine the firing activity of the population of neurons according to their input degree (k) classes and calculate for each class its firing probability Φ(k)(t) as a function of t. The probability of a node to fire is found to be determined by its in-degree k, and the first-to-fire neurons are those that have a high k. A small minority of high-k-classes may be called "Leaders," as they form an interconnected sub-network that consistently fires much before the rest of the culture. Once initiated, the activity spreads from the Leaders to the less connected majority of the culture. We then use the distribution of in-degree of the Leaders to study the growth rate of the number of neurons active in a burst, which was experimentally measured to be initially exponential. We find that this kind of growth rate is best described by a population that has an in-degree distribution that is a Gaussian centered around k = 75 with width σ = 31 for the majority of the neurons, but also has a power law tail with exponent -2 for 10% of the population. Neurons in the tail may have as many as k = 4,700 inputs. We explore and discuss the correspondence between the degree distribution and a dynamic neuronal threshold, showing that from the functional point of view, structure and elementary dynamics are interchangeable. We discuss possible geometric origins of this distribution, and comment on the importance of size, or of having a large number of neurons, in the culture.
我们提出了一个使用群体感应渗滤理论框架来描述神经培养物中活性的起始。培养物被建模为随机图，其节点是兴奋性神经元，具有 k(in)个输入和 k(out)个输出，并且输入度 k(in)=k 服从给定的分布函数 p(k)。我们根据神经元的输入度 (k)类来检查神经元群体的发射活动，并计算每个类的发射概率 Φ(k)(t)作为 t 的函数。节点的发射概率被发现取决于其入度 k，而首先发射的神经元是那些具有高 k 的神经元。一小部分高 k 类神经元可以被称为“领导者”，因为它们形成了一个相互连接的子网络，在文化的其余部分之前很早就开始发射。一旦被启动，活动就会从领导者传播到文化中连接程度较低的大部分神经元。然后，我们使用领导者的入度分布来研究爆发中活跃神经元数量的增长率，该增长率在实验中被测量为最初呈指数增长。我们发现，这种增长率最好用具有以下分布的群体来描述：该群体的入度分布是中心位于 k=75 左右的高斯分布，宽度 σ=31，适用于大多数神经元，但也具有指数为-2的幂律尾部，适用于 10%的群体。尾部中的神经元的输入可能多达 k=4700。我们探讨并讨论了度分布和神经元动态阈值之间的对应关系，表明从功能的角度来看，结构和基本动力学是可以互换的。我们讨论了这种分布的可能几何起源，并评论了大小或文化中大量神经元的重要性。
