Ebsch Christopher, Rosenbaum Robert
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, USA.
Interdisciplinary Center for Network Science and Applications, University of Notre Dame, Notre Dame, USA.
J Math Neurosci. 2020 May 13;10(1):8. doi: 10.1186/s13408-020-00085-w.
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.
大脑皮层中的神经元网络在兴奋（正输入电流）和抑制（负输入电流）之间呈现出一种平衡。平衡网络理论使用模型神经元的随机连接网络，为这种兴奋性 - 抑制性平衡提供了一个简洁的数学模型，其中平衡在大网络规模的极限情况下被实现为网络动力学的一个稳定不动点。平衡网络理论再现了皮层网络动力学的许多显著特征，如异步不规则放电活动。早期对平衡网络的研究没有考虑皮层网络的空间拓扑结构。后来的工作引入了空间连接结构，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，并且假设边界是周期性的。皮层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将平衡网络的平均场理论进行扩展，以考虑连接概率对突触前和突触后神经元空间位置的更一般依赖性。我们将我们的数学推导与循环连接的脉冲神经元模型的大型网络模拟进行比较。
