Lagzi Fereshteh, Rotter Stefan
Bernstein Center Freiburg and Faculty of Biology, University of Freiburg Freiburg, Germany.
Front Comput Neurosci. 2014 Dec 3;8:142. doi: 10.3389/fncom.2014.00142. eCollection 2014.
The balanced state of recurrent networks of excitatory and inhibitory spiking neurons is characterized by fluctuations of population activity about an attractive fixed point. Numerical simulations show that these dynamics are essentially nonlinear, and the intrinsic noise (self-generated fluctuations) in networks of finite size is state-dependent. Therefore, stochastic differential equations with additive noise of fixed amplitude cannot provide an adequate description of the stochastic dynamics. The noise model should, rather, result from a self-consistent description of the network dynamics. Here, we consider a two-state Markovian neuron model, where spikes correspond to transitions from the active state to the refractory state. Excitatory and inhibitory input to this neuron affects the transition rates between the two states. The corresponding nonlinear dependencies can be identified directly from numerical simulations of networks of leaky integrate-and-fire neurons, discretized at a time resolution in the sub-millisecond range. Deterministic mean-field equations, and a noise component that depends on the dynamic state of the network, are obtained from this model. The resulting stochastic model reflects the behavior observed in numerical simulations quite well, irrespective of the size of the network. In particular, a strong temporal correlation between the two populations, a hallmark of the balanced state in random recurrent networks, are well represented by our model. Numerical simulations of such networks show that a log-normal distribution of short-term spike counts is a property of balanced random networks with fixed in-degree that has not been considered before, and our model shares this statistical property. Furthermore, the reconstruction of the flow from simulated time series suggests that the mean-field dynamics of finite-size networks are essentially of Wilson-Cowan type. We expect that this novel nonlinear stochastic model of the interaction between neuronal populations also opens new doors to analyze the joint dynamics of multiple interacting networks.
兴奋性和抑制性脉冲神经元的循环网络的平衡状态的特征是群体活动围绕一个吸引性固定点的波动。数值模拟表明，这些动力学本质上是非线性的，并且有限规模网络中的内在噪声（自生波动）是状态依赖的。因此，具有固定幅度加性噪声的随机微分方程不能充分描述随机动力学。相反，噪声模型应该来自对网络动力学的自洽描述。在这里，我们考虑一个两态马尔可夫神经元模型，其中脉冲对应于从活动状态到不应期状态的转变。该神经元的兴奋性和抑制性输入会影响这两个状态之间的转变速率。相应的非线性依赖关系可以直接从漏电积分发放神经元网络的数值模拟中识别出来，这些模拟在亚毫秒范围内的时间分辨率下进行离散化。从这个模型中可以得到确定性平均场方程以及一个依赖于网络动态状态的噪声分量。由此产生的随机模型很好地反映了在数值模拟中观察到的行为，而与网络的大小无关。特别是，两个群体之间强烈的时间相关性，这是随机循环网络平衡状态的一个标志，在我们的模型中得到了很好的体现。此类网络的数值模拟表明，短期脉冲计数的对数正态分布是具有固定入度的平衡随机网络的一个以前未被考虑过的特性，并且我们的模型具有这种统计特性。此外，从模拟时间序列重建流表明，有限规模网络的平均场动力学本质上是威尔逊 - 考恩类型的。我们期望这种神经元群体之间相互作用的新型非线性随机模型也为分析多个相互作用网络的联合动力学打开新的大门。
