Networks of neurons in the cerebral cortex exhibit a balance between excitation (positive input current) and inhibition (negative input current). Balanced network theory provides a parsimonious mathematical model of this excitatory-inhibitory balance using randomly connected networks of model neurons in which balance is realized as a stable fixed point of network dynamics in the limit of large network size. Balanced network theory reproduces many salient features of cortical network dynamics such as asynchronous-irregular spiking activity. Early studies of balanced networks did not account for the spatial topology of cortical networks. Later works introduced spatial connectivity structure, but were restricted to networks with translationally invariant connectivity structure in which connection probability depends on distance alone and boundaries are assumed to be periodic. Spatial connectivity structure in cortical network does not always satisfy these assumptions. We use the mathematical theory of integral equations to extend the mean-field theory of balanced networks to account for more general dependence of connection probability on the spatial location of pre- and postsynaptic neurons. We compare our mathematical derivations to simulations of large networks of recurrently connected spiking neuron models.

中文翻译：

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空间扩展的平衡网络，没有平移不变的连接性。

大脑皮层中的神经元网络在激发（正输入电流）和抑制（负输入电流）之间表现出平衡。平衡网络理论使用模型神经元的随机连接网络提供了这种兴奋性-抑制性平衡的简化数学模型，其中平衡在大型网络规模的限制下实现为网络动力学的稳定固定点。平衡网络理论再现了皮质网络动力学的许多显着特征，例如异步-不规则尖峰活动。平衡网络的早期研究并未考虑皮质网络的空间拓扑。后来的作品介绍了空间连通性结构，但仅限于具有平移不变连接结构的网络，其中连接概率仅取决于距离，并且边界是周期性的。皮质网络中的空间连接结构并不总是满足这些假设。我们使用积分方程的数学理论来扩展平衡网络的均值理论，以说明连接概率对突触前和突触后神经元空间位置的更一般的依赖性。我们将我们的数学推导与循环连接的尖峰神经元模型的大型网络的仿真进行比较。我们使用积分方程的数学理论来扩展平衡网络的均值理论，以说明连接概率对突触前和突触后神经元空间位置的更一般的依赖性。我们将我们的数学推导与循环连接的尖峰神经元模型的大型网络的仿真进行比较。我们使用积分方程的数学理论来扩展平衡网络的均值理论，以说明连接概率对突触前和突触后神经元空间位置的更一般的依赖性。我们将我们的数学推导与循环连接的尖峰神经元模型的大型网络的仿真进行比较。