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

Power laws describe brain functions at many levels (from biophysics to psychophysics). It is therefore possible that they are generated by similar underlying mechanisms. Previously, the response properties of a collision-sensitive neuron were reproduced by a model which used a power law for scaling its inhibitory input. A common characteristic of such neurons is that they integrate information across a large part of the visual field. Here we present a biophysically plausible model of collision-sensitive neurons with η-like response properties, in which we assume that each information channel is noisy and has a response threshold. Then, an approximative power law is obtained as a result of pooling these channels. We show that with this mechanism one can successfully predict many response characteristics of the Lobula Giant Movement Detector Neuron (LGMD). Moreover, the results depend critically on noise in the inhibitory pathway, but they are fairly robust against noise in the excitatory pathway.


Integral Expression
We pool N channels (or processes) with independent Gaussian noise in each channel, where [·] + ≡ max (·, 0) denotes half-wave rectification, and ∆ 0 is a threshold. The ξ i are random numbers drawn from the standard normal distribution. It is scaled by σ and shifted by voltage V 0 , such that the resulting distribution has an effective mean at x ≡ V 0 − ∆ 0 with standard deviation σ. The result for V from pooling can be calculated by Because of exp(·) ≥ 0 we can drop half-wave rectification, where we substituted the integration variable z ≡ ξ − x and furthermore defined a := 1/2σ 2 . We split this integral in two parts, Now, continuing with solving I(x), where erf(x) = 2 √ π x 0 exp(−t 2 )dt is the error function, and where the power series definition of the exponential function was used. For numerical computation, it is sufficient to sum the first terms of the series. Finally, the complete result: Where the last equation is from references [2,3], and can be obtained via a software for symbolic mathematics (e.g. Mathematica). The equality of the last two equations was verified by means of computer simulations. Excitatory noise distorts the n-ψ response curves and shifts their maxima slightly towards t c (by 2, 3 and 14ms, respectively). With twice the excitatory noise σ e = 0.50, the maxima would shift towards t c by 0, 7 and 19ms, respectively, relative to having no excitatory noise, but only inhibitory noise. Corresponding fits of the η-function are shown as well, with α, the root mean squared error (rmse), and the degree-offreedom adjusted coefficient of determination (R 2 ) being indicated in the figure legends.

B Noise in the Excitatory Pathway
In the locust visual system, noise increases along the excitatory pathway from the photoreceptors to the LGMD [4]. But to the best of our knowledge there are no data available about the noise levels in the inhibitory pathway. The results of the simulations shown in this study, however, depend critically on noise in the inhibitory pathway. This leads to the question about the impact of excitatory noise on our results. We verified that excitatory noise will not interfere strongly with the predictions of n-ψ. In order to illustrate, we replace Equation (10) by where the subscript "e" denotes the corresponding noise parameters for the excitatory pathway. Supplementary Figure A1 shows that n-ψ peaks slightly later in the presence of excitatory and inhibitory noise, in contrast to having only inhibitory noise. In addition, the n-ψ response gets slightly distorted when having noise in the excitatory pathway. One important signature of the LGMD is the linear dependence of t rel ≡ t c −t max on l/v [5] (Eq. 2). Supplementary Figure A2a shows that the linear relationship remains intact in the presence of excitatory noise. In addition, the line fit slopes α vary only weakly as a function of the excitatory noise level σ e , compared with similar variations of the inhibitory noise level (Fig. 4b). Supplementary Figure  Setting a different value for the threshold ∆ e = 0.9 does not compromise the linearity either, but causes only a decrease of the overall slopes α relative to those with ∆ e = 0.25.
threshold ∆ e . This causes an overall decrease of the α, but does not affect linearity.
If the equilibrium solution Equation (7) is used instead of explicitly integrating the differential Equation (6), then the linear dependence of t rel on l/v starts to deteriorate for either ∆ e > 0.6 (with fixed σ e = 0.5) or σ e > 1.6 (with fixed ∆ e = 0.25).
Supplementary Figure B1

C Integration time constant dt and n relax
After each integration step of the membrane potential (=n-ψ -response; Eq. 6) with integration step size dt, a number of relaxation time steps n relax is intercalated. During this "relaxation phase", the excitatory and inhibitory synaptic input (Eqs. 10 and 11, respectively) are held constant. If a suitably small inte- Supplementary Figure B3 | Influence of relaxation steps n relax on the linear dependence of t rel from l/v. Different symbols correspond to different values of n relax as indicated in the figure legends. The continuous lines represent corresponding linear regressions to the data points. The resulting line slopes α are indicated in the legends as well ("normal" means that the residuals are normal-distributed). (a) With an integration step size of dt = 100µs, the number of relaxation time steps n relax have a notable influence on the line slopes, because the n-ψ model is far from its equilibrium solution. Taking more relaxation time steps will move the response maxima of n-ψ away from t c (i.e., to earlier times), thus t rel increases. (b) With the default integration step size dt = 500µs, n-ψ is already close to its steady-state, and therefore the slope values α depend only weakly on n relax . gration time constant dt is chosen, it is therefore possible to study the responses of n-ψ close or far from the equilibrium state. In reference [1], stimulation far away from the equilibrium state has been put forward as an explanation of why the response peak of the LGMD neuron could occur after collision time t c .
Here we do not repeat those numerical experiments, but are rather interested to select dt (and n relax ) such that we are reasonably close to the equilibrium. In Supplementary Figure B1 the n-ψ-responses (n relax = 250) with dt = 100µs (i.e., somewhat away from equilibrium) are compared with dt = 500µs (close to equilibrium). Supplementary Figure B2 compares the n-ψ-responses (default: dt = 500µs and n relax = 250) with the steady-state solution (V ∞ ), and confirms that the default values are close to the equilibrium state. Supplementary Figure B3 illustrates the influence of n relax on the maxima of n-ψ -responses for dt = 100µs and dt = 500µs, respectively, and confirms once more that the latter (default) value is close to the equilibrium state.
Supplementary Figure  D Fitting the n-ψ and the η-Function to Neuronal Recordings Collision-sensitive neurons with similar response properties are found in the brains of many mammals and insects [13]. A particular class of these neurons reveal a response maximum when an object approaches the eye with constant velocity. The η-function was the first model for describing the response properties of these neurons [11]. It is commonly used for fitting corresponding neuronal recording curves (of "η-type" neurons). The purpose of this section is to juxtapose fitting results of the η-function with those of n-ψ. To this end, 36 data sets from eight publications were scanned and fitted ( Supplementary Fig. C1a), analogous to Supplementary Section S4 in reference [13]. Thus, many previously s8 label(s) fitted function fitting parameters Table D1 | Labels and fitting parameters The table relates the set of fitted parameters to the corresponding labels as they are indicated in the legends. TR means that the Trust region algorithm was used for fitting [14,15]. LM means that the Levenberg-Marquardt algorithm was used [16]) published data sets are united with a common fitting framework, what allows for an unequivocal and unbiased evaluation of the fitted functions. The data sets covered halfsize-to-velocity ratios from l/v = 5ms to 50ms.

D.1 Details for Fitting the η-Function
The fitting model for the η-function was (cf. Eq. 1) with A = amplitude, δ = temporal delay, o = offset, = Θ angular size,Θ = angular velocity, and α is a weight. For each experimentally obtained response curve from the LGMD, two parameter sets combined with two fitting algorithms were tested. The best result in terms of goodness of fit measures was then selected from the four combinations. For Supplementary Equation (D1), the parameter set to be determined was either {A, δ, o, α} or {A, o, α} (with δ = 0). The fitting method was either Trust-Region (legend label "TR", [14,15]), or Levenberg-Marquardt ("LM ", [16]). The respective fitting procedures are indicated in the figure legends by corresponding labels which are listed in Table D1.

D.2 Details for Fitting the ψ-Model
For the purpose of comparison, fitting results for the ψ-model (see reference [1]) are shown in Supplementary Figure D2. The fitting model for ψ is based on Equation (7): with A = amplitude and o = offset. The ψ-model is defined by assigning excitatory and inhibitory input, respectively, in Equation (7) as follows: where γ = synaptic weight and e is an exponent (notice that ψ was fit without temporal delay). Both the Trust-Region algorithm [14,15] and the Levenberg-Marquardt algorithm were used [16].

D.3 Details for Fitting the n-ψ-Model
For fitting n-ψ, only the Trust-Region algorithm was used [14,15]. The fitting model for n-ψ is based on Equation (7): with A = amplitude and o = offset. No lowpass filtering was employed for fitting, thus ζ 0 = ζ 1 = 0 in Equations (8) and (9), respectively. The n-ψ-model is defined by assigning excitatory and inhibitory input, respectively, according to Equations (10) and (11). However, Equation (11) cannot be used readily with a conventional fitting algorithm, because of thresholding. For this reason, we used the explicit solution instead, which is smooth (Supplementary Eq. A7: V ≡ g inh and x ≡ ϑ). The legend labels in the fitting results relate to the fitted parameters as shown in Table D1. For example, "TR50 " means that parameters {A, β, γ, σ, ∆ 0 , o} were determined by the Trust-Region algorithm, and "TR40 " refers to the same set of parameters but excluding ∆ 0 . In the latter case, ∆ 0 was held constant at 0.9. Why fitting the noise level σ and not keeping it constant instead? The underlying idea is to use the n-ψ-model to measure the noise level directly from the different data sets. The noise level ideally should settle around some value then. However, different factors such as body temperature or stimulation conditions (light level, stimulus velocity) could influence the effective noise level across different studies [4,13], so some spread could be expected around the average value of σ.

D.4 Summary Results
We fitted the η-function, the ψ-model, and the n-ψ-model to the data sets referenced by Supplementary Figure C1. We then plotted the 'important" fitting parameters of these models versus l/v ("important" are those model parameters which determine the location of the activity maximum). Supplementary Figure  D2a shows the dependence of α of the η-function from l/v. The median value ±1σ rob is 3.1 ± 0.7, where the α were more scattered for smaller l/v. Supplementary Figure D2b shows the exponent e of the ψ-model versus l/v, where e = 2.7 ± 0.5 (median ±1σ rob ). Although most of the values fall approximately between 2 and 3.5, occasionally much bigger values were obtained across the whole l/v domain. For the n-ψ -model, we get the median noise level σ = 0.4 ± 0.1 ( Supplementary  Fig. D3a), and the median threshold ∆ 0 = 0.9 ± 0.1 (Supplementary Fig. D3b). Note, however, that ∆ 0 was held constant 0.9 while fitting as far as possible. If ∆ 0 was to be determined by the fitting algorithm, then these values are often different from 0.9. This variation in ∆ 0 is reflected also in the σ: The highest deviations of σ from its median value are observed when ∆ 0 was fitted at the same time. We asked whether some parameters of the different models correlated across the fitted data. Correlations between two parameters could be suggestive of that they play similar roles in the considered models. Supplementary Figure D4 visualizes the correlations between most variables. In order to achieve a better visualization, positive correlations are displayed above the diagonal (=self-correlations), and negative values below (cf. figure The broken lines denote the median value if more than one value of α (or e) was available at some l/v. Green symbols mark values which were smaller than two times the robust estimation of spread ("σ rob ") via the median absolute deviation ("MAD"), that is σ rob ≡ MAD/0.6745 (e.g. [17]). For both function, the green symbols appear to be reasonably independent from l/v. Since the α and e, respectively, determine the location of the LGMD's response maximum, their approximate independence from l/v may indicate a specific tuning of the LGMD to certain ecological conditions. In agreement with the latter interpretation, the bullfrog-data (six-pointed stars; [6]) seem to suggest much bigger values -but this has to be confirmed with a bigger sample size.
legend for more details). No correlation value was sufficiently high in order to suggest a possible link between any parameter pair. The highest correlation was −0.51 between e and σ, and the correlation between σ and ∆ 0 was 0.77. The relative high correlation between σ and ∆ 0 can be explained by the fitting procedure, where strong co-variations can be observed between both values in Supplementary Figure D3 if ∆ 0 and σ were fitted together (compared to holding ∆ 0 constant at 0.9). Furthermore, it is likely that the low correlation values are a consequence of our sample size being too small (36 data sets), because statistical variations in the fitting results did not average out. The noise level σ and the threshold ∆ 0 . σ was fitted rather than being set to some fixed value in order to "measure" noise levels from the the neuronal data sets (identifiable by their symbols, see Fig. C1). ∆ 0 was only fitted if goodness of fit measures with a fixed ∆ 0 were poor. (a) Noise level σ versus l/v (n-ψ-function fit, Eq. 11). The median noise level (here dimensionless) is σ = 0.4 ± 0.1 (mean ± 1 SD), what should be compared with the experimentally found values from ref. [4] (p. 1071: mean spontaneous noise levels relative to rest in photoreceptors = 0.19mV, LMCs = 0.4mV, and LGMD=1.05mV). (b) Threshold ∆ 0 versus l/v (n-ψ-function fit, Eq. 11). The broken lines denote the median value if more than one value of σ (or ∆ 0 ) was available at some l/v. Green symbols mark values which were smaller than two times the robust estimation of spread ("σ rob ") via the median absolute deviation ("MAD"), that is σ rob ≡ MAD/0.6745 (e.g. [17]). Labels: LV = l/v, diameter dia = 2l; β, A and γ refer to the ψ-model, while n-β, n-A and n-γ refer to n-ψ. The highest correlation (0.9) occurs between β and A of the ψ model. For n-ψ, the correlation between n-β and n-A is 0.7, the correlation between σ and ∆ 0 is 0.77, and between n-A and LV it is 0.72. Furthermore, across the "important" model parameters, we found −0.51 between e and σ, −0.42 between α and σ, and finally 0.13 between α and e.

D.5 Detailed Results
(This section is analogous to Supplementary Text S4 in reference [13]). The Supplementary Figures D5 to D23 juxtapose the detailed fitting results of the η-function with those of n-ψ . Each figure legend has a label that identifies the data set via Figure Supplementary C1. Goodness of fit measures and model parameters are indicated as well.
The target data for fitting are always plotted with a thick and dark gray line.
Occasionally the noisy neuronal data were smoothed. In that case, the original neuronal data are drawn with a thin and light gray line, and the smoothing method (along with its corresponding parameter value) is indicated as follows: "robLR1st" robust local regression with weighted linear least squares and a 1st degree polynomial model. The parameter specifies the span (= number of data points for computing a smoothed value) in terms of percentage of total number of data points.
"robLR2nd " robust local regression with weighted linear least squares and a 2nd degree polynomial model. The parameter specifies the span in terms of percentage of total number of data points.
"sgolay" Savitzky-Golay method with polynomial degree 2. Parameter value p means that the span in terms of number of data points is 2p + 1.
If no clear response peak could be detected in the neuronal data, or if successive data points were too separated in time, we proceeded with selecting a smoothing algorithm. In that case, the tuning criteria were (i) to leave the original data as less distorted as possible, and (ii) to assure, by visual inspection, a sound location of the response peak in the smoothed data. This is to say that t max of the smooth data should coincide with where an experienced observer would place it. The label "fit 2 smooth" means that η or n-ψ were fit to the smoothed data, and "fit 2 raw " means that the corresponding models was fitted to the nonsmoothed (= raw) data.
Supplementary Figure D5 | NakHon10 Bullfrog rana catesbeiana, l/v = 10ms, figure 8 and 9, respectively, from reference [6]. The stimulus was a 35mm × 35mm black square (l ≈ 2cm), t max at around 2600ms (400ms before t c ). (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.  Table D1 relates the legend labels to the set of fitted parameters.  Table D1 relates the legend labels to the set of fitted parameters.  Table D1 relates the legend labels to the set of fitted parameters.
Supplementary Figure D10 | GaMoLa01 II Locust. Figure 1c from reference [10], looming squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.  Table D1 relates the legend labels to the set of fitted parameters.
Supplementary Figure D12 | GaKrLa99 I Locust. Figure 3 from reference [5], dark squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.
Supplementary Figure D13 | GaKrLa99 II Locust. Figure 3 from reference [5], dark squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.
Supplementary Figure D14 | GaKrLa99 III Locust. Figure 3 from reference [5], dark squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.  Table D1 relates the legend labels to the set of fitted parameters. Figure 2, (b) Figure 3, both from reference [5], dark squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.
Supplementary Figure D17 | GaKrLa99 VI Locust, l/v = 50ms. Figure  3 from reference [5], dark squares. (Abscissa in units of seconds). Table D1 relates the legend labels to the set of fitted parameters.

D.6 Fit to Recording Traces
Figures 6, D24 and D25 show recording traces from the Descending Contralateral Movement Detector neuron (DCMD). All five recording traces were kindly provided by Steve Rogers [18]. The DCMD replicates the firing patterns of the LGMD (cf. [19,20,21]). The spike traces (sampled at 10 4 Hz) were full wave rectified, lowpass filtered, and sub-sampled to 1ms resolution. Firing rates were estimated with Savitzky-Golay filtering ("sgolay"). The η-function and n-ψ were fitted to these firing rate estimates with the Trust-Region algorithm [14,15]. The respective parameter sets were {A, α, o} for the η-function ("TR:3 ") and {A, β, γ, σ, o} for n-ψ ("TR:32 "). For all fits, the temporal delay was set to zero (δ = 0). The results can be summarized in just a few words: Both functions describe the data excellently, and η fits them marginally better than n-ψ.  Figure 7 in the main text. The difference is that instead of fitting n-ψ ∞ , here the dynamic version of n-ψ (equation 6) was used. This of course implies transient effects, which are reflected at the beginning by an increase in activity to a maximum. For this reason, the (dynamic) n-ψ-predictions were manually adjusted (trial and error) in order achieve a good match with the smoothed version of the recording curve was achieved (the smoothed curves are not shown, though). The optimization criterion was the maximization of the coefficient of determination (R 2 ). Even in this way n-ψ matches the original (i.e. unsmoothed) data better than the η-function. All goodness of fit measures in this figure refer to the unsmoothed data.