Gu Qing-Long L, Li Songting, Dai Wei P, Zhou Douglas, Cai David
School of Mathematical Sciences, MOE-LSC, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China.
Department of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China.
Front Comput Neurosci. 2019 Jan 29;12:109. doi: 10.3389/fncom.2018.00109. eCollection 2018.
It is hypothesized that cortical neuronal circuits operate in a global balanced state, i.e., the majority of neurons fire irregularly by receiving balanced inputs of excitation and inhibition. Meanwhile, it has been observed in experiments that sensory information is often sparsely encoded by only a small set of firing neurons, while neurons in the rest of the network are silent. The phenomenon of sparse coding challenges the hypothesis of a global balanced state in the brain. To reconcile this, here we address the issue of whether a balanced state can exist in a small number of firing neurons by taking account of the heterogeneity of network structure such as scale-free and small-world networks. We propose necessary conditions and show that, under these conditions, for sparsely but strongly connected heterogeneous networks with various types of single-neuron dynamics, despite the fact that the whole network receives external inputs, there is a small active subnetwork (active core) inherently embedded within it. The neurons in this active core have relatively high firing rates while the neurons in the rest of the network are quiescent. Surprisingly, although the whole network is heterogeneous and unbalanced, the active core possesses a balanced state and its connectivity structure is close to a homogeneous Erdös-Rényi network. The dynamics of the active core can be well-predicted using the Fokker-Planck equation. Our results suggest that the balanced state may be maintained by a small group of spiking neurons embedded in a large heterogeneous network in the brain. The existence of the small active core reconciles the balanced state and the sparse coding, and also provides a potential dynamical scenario underlying sparse coding in neuronal networks.
据推测，皮层神经元回路处于全局平衡状态，即大多数神经元通过接收平衡的兴奋和抑制输入而不规则地放电。同时，实验观察到感觉信息通常仅由一小部分放电神经元稀疏编码，而网络其余部分的神经元则处于沉默状态。稀疏编码现象挑战了大脑中全局平衡状态的假设。为了调和这一点，我们在此考虑无标度和小世界网络等网络结构的异质性，探讨少数放电神经元中是否能存在平衡状态的问题。我们提出了必要条件，并表明在这些条件下，对于具有各种类型单神经元动力学的稀疏但强连接的异质网络，尽管整个网络接收外部输入，但其中固有地嵌入着一个小的活跃子网（活跃核心）。这个活跃核心中的神经元具有相对较高的放电率，而网络其余部分的神经元则处于静止状态。令人惊讶的是，尽管整个网络是异质且不平衡的，但活跃核心具有平衡状态，其连接结构接近均匀的厄多斯 - 雷尼网络。活跃核心的动力学可以用福克 - 普朗克方程很好地预测。我们的结果表明，平衡状态可能由嵌入大脑中大型异质网络的一小群放电神经元维持。小活跃核心的存在调和了平衡状态和稀疏编码，也为神经网络中稀疏编码提供了潜在的动力学场景。
