The mutual information between a stimulus signal and the spike count of a stochastic neuron is in many cases difficult to determine. Therefore, it is often approximated by a lower bound formula that involves linear correlations between input and output only. Here, we improve the linear lower bound for the mutual information by incorporating nonlinear correlations. For the special case of a Gaussian output variable with nonlinear signal dependencies of mean and variance we also derive an exact integral formula for the full mutual information. In our numerical analysis, we first compare the linear and nonlinear lower bounds and the exact integral formula for two different Gaussian models and show under which conditions the nonlinear lower bound provides a significant improvement to the linear approximation. We then inspect two neuron models, the leaky integrate-and-fire model with white Gaussian noise and the Na-K model with channel noise. We show that for certain firing regimes and for intermediate signal strengths the nonlinear lower bound can provide a substantial improvement compared to the linear lower bound. Our results demonstrate the importance of nonlinear input-output correlations for neural information transmission and provide a simple nonlinear approximation for the mutual information that can be applied to more complicated neuron models as well as to experimental data.

中文翻译：

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改进了信号和神经尖峰计数之间的互信息下限。

在许多情况下，很难确定刺激信号和随机神经元的尖峰计数之间的相互信息。因此，通常可以通过一个下限公式来近似该公式，该公式仅涉及输入和输出之间的线性关系。在这里，我们通过合并非线性相关性来改善互信息的线性下界。对于具有均值和方差非线性信号相关性的高斯输出变量的特殊情况，我们还导出了完整互信息的精确积分公式。在我们的数值分析中，我们首先比较了两个不同的高斯模型的线性和非线性下界以及精确的积分公式，并显示了在哪些条件下非线性下界对线性逼近提供了显着改进。然后，我们检查两个神经元模型，具有高斯白噪声的泄漏集成发射模型和具有通道噪声的Na-K模型。我们表明，对于某些发射方式和中间信号强度，与线性下限相比，非线性下限可以提供实质性的改进。我们的结果证明了非线性输入输出相关性对于神经信息传输的重要性，并为相互信息提供了简单的非线性近似，可以将其应用于更复杂的神经元模型以及实验数据。