S1 Text - Dendritic Pooling of Noisy Threshold Processes Can Explain Many Properties of a Collision-Sensitive Visual Neuron

Pooling of Noisy Threshold Units—Mathematical Considerations

In this section, a closed expression for Eq (14) is derived, which is used for fitting the n-ψ-model to neuronal recordings in Section D in S1 Text.

Noise in the Excitatory Pathway

In this section the impact of excitatory noise on the predictions of n-ψ is studied, where the same threshold smoothing mechanism is used for the excitatory and the inhibitory synaptic input. This is to say that threshold smoothing is applied simultaneously to the excitatory and inhibitory pathway. It turns out that the predictions of n-ψ are reasonably robust with this configuration.

Integration time constant dt and nrelax

In this section, the influence of the number of relaxation time steps nrelax and that of the integration time constant dt is studied. Specifically, corresponding values of nrelax and dt are determined such that n-ψ operates close to the equilibrium solution Eq (10). The exact values are important as they were shown to influence the location of the LGMD’s predicted response peak [19].

Fitting the n-ψ and the η-Function to Neuronal Recordings

The n-ψ-model is fit to several recording curves from different studies. The fits are juxtaposed with those of the η-function. Goodness of fit measures are provided as well, and some fitting results of the predecessor model Ψ are also shown. This section presents the results of previously published studies in the same fitting framework. A common fitting framework enables a meaningful comparison of the respective predictions of the η-function and the n-ψ-model.

List of Symbols

A list with mathematical symbols along with corresponding brief descriptions are provided in this section.

Pooling of Noisy Threshold Units—Mathematical Considerations
In this section, a closed expression for Eq (14) is derived, which is used for fitting the n-ψ-model to neuronal recordings in Section D in S1 Text.
Noise in the Excitatory Pathway
In this section the impact of excitatory noise on the predictions of n-ψ is studied, where the same threshold smoothing mechanism is used for the excitatory and the inhibitory synaptic input. This is to say that threshold smoothing is applied simultaneously to the excitatory and inhibitory pathway. It turns out that the predictions of n-ψ are reasonably robust with this configuration.
Integration time constant dt and nrelax
In this section, the influence of the number of relaxation time steps nrelax and that of the integration time constant dt is studied. Specifically, corresponding values of nrelax and dt are determined such that n-ψ operates close to the equilibrium solution Eq (10). The exact values are important as they were shown to influence the location of the LGMD’s predicted response peak [19].
Fitting the n-ψ and the η-Function to Neuronal Recordings
The n-ψ-model is fit to several recording curves from different studies. The fits are juxtaposed with those of the η-function. Goodness of fit measures are provided as well, and some fitting results of the predecessor model Ψ are also shown. This section presents the results of previously published studies in the same fitting framework. A common fitting framework enables a meaningful comparison of the respective predictions of the η-function and the n-ψ-model.
List of Symbols
A list with mathematical symbols along with corresponding brief descriptions are provided in this section.
(PDF)

带噪阈值单元的池化——数学考量
本节推导了式(14)的闭合表达式，该表达式将在补充材料S1文本的D节中用于将n-ψ模型拟合至神经元记录数据。

兴奋性通路中的噪声
本节研究兴奋性噪声对n-ψ模型预测结果的影响，其中兴奋性与抑制性突触输入采用相同的阈值平滑机制。换言之，阈值平滑同时应用于兴奋性与抑制性通路。研究发现，在此配置下n-ψ模型的预测结果具备较好的鲁棒性。

积分时间常数$dt$与$n_{relax}$
本节研究松弛时间步数量$n_{relax}$与积分时间常数$dt$对模型的影响。具体而言，我们将确定$n_{relax}$与$dt$的对应取值，使得n-ψ模型的运行状态接近式(10)的平衡解。精确取值至关重要，因为已有研究证实其会影响LGMD的预测响应峰值位置[19]。

将n-ψ模型与η函数拟合至神经元记录数据
本研究将n-ψ模型拟合至多组来自不同研究的记录曲线，并将拟合结果与η函数的拟合结果进行并列对比。同时提供了拟合优度指标，此外还展示了前代模型Ψ的部分拟合结果。本节将已发表的多项研究结果置于同一拟合框架下进行呈现，统一的拟合框架可实现η函数与n-ψ模型预测结果的有效对比。

符号列表
本节提供了数学符号及其对应简要说明的列表。
（PDF）