Citation: Keil MS, López-Moliner J (2012) Unifying Time to Contact Estimation and Collision Avoidance across Species. PLoS Comput Biol 8(8):
e1002625.
https://doi.org/10.1371/journal.pcbi.1002625
Editor: Lyle J. Graham,
Université Paris Descartes, Centre National de la Recherche Scientifique, France
Received: September 21, 2011; Accepted: June 8, 2012; Published: August 16, 2012
Copyright: © Keil, López-Moliner. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: MSK acknowledges support from a Ramon & Cajal grant from the Ministry of Science and Innovation of the Spanish government, and from the national grant DPI2010-21513. JLM acknowledges support by grant PSI2010-15867 from the Ministry of Science and Innovation of the Spanish government and an ICREA Academia Distinguished Professorship award. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Monocular presentation of a looming object elicits escape or avoidance reactions in many species, including humans [1]–[4]. When a planar object travels perpendicular to a surface toward an observer (i.e. the object approaches the observer on a direct collision course), it projects a symmetrically expanding image on the retina. Notice that in the present paper we only focus on a subset of approaches where the approaching object eventually collides with the observer. We assume that collision happens at time 
(time to contact, “ttc”). At time 
before 
, the image subtends an angle 
, and its outer contours expand with angular velocity 
. Both angular variables grow nearly exponentially with decreasing distance 
between object and eye (assuming a constant velocity 
). With knowledge of a predator's or object's typical size [5], it is therefore possible to trigger a behavioral response as soon as 
or 
, respectively, crosses a threshold [1], [6], [7].
The visual systems of various species are also known to “compute” functions of 
and 
(see e.g. [8] for a recent review). The Tau-function (“
”) is defined by 
. Under the assumption that the object is a rigid sphere that approaches with 
, 
has several interesting properties [9], [10]: First, 
provides a running estimation of ttc during the approach. Second, the ttc estimation is largely independent of physical object size, provided that 
and 
are noise-free. Third, 
decreases approximately linearly with time with a constant slope of 
, but eventually linearity is compromised, as 
has a minimum shortly before ttc. It therefore would be comparatively easy to track the remaining time 
until impact, and to precisely time avoidance reactions, for example as soon as 
is below a certain threshold value.
These three properties, however, are valid only for “sufficiently small” angular sizes 
. Any quantitative criterion for “sufficiently small” implicates an error threshold for the deviation of 
from linearity, that is 
. For example, according to Text S6 a corresponding threshold for the visual angle can be defined as 
with some constant 
. Notice that the 
-criterion is independent from stimulus parameters such as object diameter or approach velocity.
Because 
is well suited for the estimation of 
, it could in principal serve as a universal mechanism for guiding motor actions during object approaches or during self-motion towards static objects. Indeed, several studies related 
to behavioral responses in this context, thus asserting that many organisms, including humans, rely on 
for their timing of motor actions (e.g. [10]–[12]). But a critical re-evaluation of the 
-hypothesis arrived at the conclusion that 
does not necessarily play a unique role for ttc estimation [13], [14]. For example, humans also rely on the rate of change of relative disparity, particularly in the late phase of an approach, for small object sizes [15]–[18], for low speeds [19], [20], or if knowledge of object size is available [7]. In addition, the task at hand (e.g. catching a ball or eluding a meteorite) seems to dictate the information that will eventually be used for action timing [14], [18], [21]. Further inconsistencies with respect to 
were reported with psychophysical results, where 
tended to be underestimated [16]. In addition, ttc -estimation reveals a certain dependence on object size [22], which is also not predicted by 
at “sufficiently small” angular sizes.
The Tau-function is often studied in the context of ttc -estimation. It appears, however, that in order to describe the responses of collision-sensitive neurons in certain species 
is inadequate. For example, the Lobula Giant Movement Detector (LGMD) neuron in locusts responds with increasing activity to a stimulation with a symmetrically expanding image, if the expansion pattern is consistent with an approaching object [23], [24]. The response curve of the LGMD neuron gradually increases to a maximum and then abruptly ceases (often to a nonzero baseline response). Because 
does not have a maximum, a different function has been proposed for modeling LGMD responses: The Eta-function (“
”). It is defined as 
, with a constant 
[25]. Theoretically, the time when the activity peak occurs depends linearly on the ratio 
of object half-size 
to object velocity 
. The peak will shift closer to 
for smaller or faster objects, and always occurs at angular size 
, independent of 
[26]. The LGMD activity peak could in principle signal a critical angular size for escaping. Indeed, a recent study with freely behaving locust suggests that the time of peak firing rate of the Descending Contralateral Movement Detector (DCMD) predicts that of jump [27] (each LGMD spike triggers a spike in the postsynaptic DCMD as well, because the LGMD is strongly coupled to the DCMD by a combined electrical and chemical synapse [28], [29]).
It has nevertheless been argued that – in some ecologically meaningful situations (small 
) – there is no guarantee for the peak to occur before 
[2], [5]. This statement may be true to the extent that in freely behaving locusts, a reliable escape jump is triggered before collision only in the range of 
to 
[30]. For 
, the jump would occur after projected collision, and this value thus may reflect the typical sizes and speeds of predators.
Apart from the locust, other species have collision-sensitive neurons with 
-like properties, for instance fruitflies [31] and bullfrogs [32]. In pigeons, the response properties of one of three classes of neurons in the dorsal posterior zone of the nucleus rotundus also seems to be compatible with the 
-function [1]. (The two remaining classes seem to compute 
and 
, respectively). In the goldfish, responses of the M-cell to looming stimuli also appear to follow a version of the 
-function, in which 
replaces 
, such that the new function does only depend on 
[33].
The Tau-function and the Eta-function are the two prevailing models for studying ttc -perception and (interceptive) action timing on the one hand, and escape behavior and collision avoidance on the other. In other words, we have two different models for two seemingly separated contexts. Each model brings about some hitherto unresolved issues, which are subsequently described.
From a computational point of view, 
is numerically unstable: In the presence of noise, we have to reckon with the fact that 
can get very small – or even reach zero – at certain instants during the initial phase of the approach (cf. [17]). As a consequence, fluctuations of 
with large amplitudes may occur. If, however, noise levels are constant in time, and noise is not multiplicative, the signal to noise ratio continuously improves as 
is approached. It is furthermore not entirely clear how 
could be biophysically implemented in a neuron.
As for the 
-function, the LGMD neuron seems to bypass a direct multiplication or division by computing 
with subsequent exponentiation of the result [34]. From a mathematical viewpoint, however, taking the logarithm introduces an instability for 
, although neuronal circuits with divisive inhibition can be adjusted such that no stability problems occur [35]. Moreover, Gabbiani et al. [34] found that a third-order power law fitted the mean instantaneous firing rate of the LGMD better than an exponential or a linear function (see also reference [36]).
Our original motivation was to improve the stability of 
with a simple modification. This modification led us to the modified Tau function 
. Similar to the 
-function, the 
-function also reveals a maximum before ttc. We were able to fit the response curves of 
-type neurons with 
(Text S4). Our 
-function represents the equilibrium solution of an equation for describing neuronal firing rate. Because of this, 
is based on a biophysically plausible mechanism.
But 
comes with a disadvantage: Unlike 
, it no longer provides a running value of ttc. In order to recover the ttc prediction, we needed to add a correction term to 
. This so-defined corrected modified Tau function (
) recovers the ttc prediction of the original 
-function, but suppresses noise better than 
. Most importantly, the corrected m-Tau function predicts the results of a psychophysical experiment, requiring subjects to estimate ttc.
Theoretically, we therefore can explain 
-type and 
-type responses within the 
framework, which contains 
(but also 
!) as a special case. Until now, 
and 
did not have any obvious relationship with each other (although we show in Text S6 how 
could formally be related to 
). The 
-function could thus serve to explain why 
-type and 
-type neurons could be found alongside each other in the pigeon brain [1].
Results
The corrected modified Tau function “
” (equation 5) contains the modified Tau function “
” (equation 1) as a special case. We nevertheless first introduce the 
model, as this makes its relation to the original 
-function much easier understood.
The modified 
model (“
”)
Behavioral and neural responses to optical variables (e.g., 
, 
, 
, 
) in the initial part of a trajectory are very noisy signals. Signal fluctuations may occur as a consequence of the discrete structure of the retinal photoreceptor array and its limited spatial resolution. The signal-to-noise ratio continuously improves as ttc is approached (Text S3).
Our first step adds computational stability to the 
model. Let 
be a constant (in units of 
). The modified Tau model is defined as:
(1)Biophysically, 
can be interpreted as leakage conductance (equation S2 in Text S1). According to equation (1), 
can formally be expressed in terms of 
multiplied with a gain control factor 
, which depends only on angular velocity. Notice, however, that the multiplicative version “
” would again compromise stability, because 
appears as one of the factors in the product. Figure 1 a juxtaposes 
and the factors 
and 
, respectively.
Let the initial distance between the eye and a circular object (diameter 
) be denoted by 
. Then, choosing 
will create a maximum of 
at time 
(i.e., a maximum before 
):
(2)(the previous equation is derived in the Methods Section). The time 
when 
assumes its maximum can thus be controlled by specifying 
, where bigger values will place the maximum closer to 
. The maximum depends as follows on approach velocity and object diameter, respectively.
Assume fixed values for 
and 
. Then, 
will have an activity maximum at 
(default case). Now increase approach velocity and initial distance, such that 
remains constant. As a consequence, the peak will shift closer to 
with respect to the default case (triangle symbols in Figure 1a; further figures in Text S2). This is the velocity effect.
Now increase the object diameter. The maximum of 
will then occur earlier compared to the default case (circle symbols in Figure 1). This is the size effect.
Assuming that the peak signals an imminent collision, this shifting behavior is consistent with larger objects being perceived to have an earlier ttc than smaller ones [22]. Note that the original 
-function (i.e. 
and noise-free angular variables) does not show a strong dependence on object size where 
holds (but see Text S6).
The 
-function is the prevailing model for describing responses from collision sensitive neurons to object approaches with constant velocity. Its characteristic feature is its maximum. Because 
also has a maximum, we fit 
previously published neuronal response curves with the 
-function and 
(Text S4). Figure 2 summarizes these fits by comparing the response maxima of the experimental curves (“
”) with the maxima predicted by the best fits achieved with the two functions (“
”). Predictions of 
are slightly better with 
-fits, both in terms of mean and median of absolute differences (
). With respect to goodness of fit measures (root-mean-square-errors, 
, F-statistics), both functions perform again on par with each other. Therefore, both 
and the 
-function describe neuronal responses of object approaches with constant velocity.
The experimental maxima at time 
depend linearly on 
[26]. The 
-function predicts this linear relationship (equation S5 in Text S2), where slope is identified by 
, and intercept by a temporal delay 
of corresponding line fits (Figure 3a).
The maximum of the 
-function 
, however, depends in a nonlinear way on 
(equation 2 & equation S6 in Text S2; illustration: Figure 4). (Nonlinearity means that the slope depends on 
, and linearity means that it does not). Linearity is approached with increasing values of 
, eventually reaching a slope of one for 
(equation S9 in Text S2). This is nevertheless inconsistent with experimental evidence, as the experimental values for 
are underestimated (typically 
).
We thus explored a different possibility: Can the nonlinear function 
be hidden by noise? Figure 3b suggests that it nearly can, as seen when fitting a line to a version of 
with additive Gaussian noise. Noise levels were set as reported in [26]. This hide-and-seek works quite well, and the fitting statistics (
, KS-test on residuals, F-statistics) are consistent with linearity in many random trials (detailed analysis: Text S2).
Figure 4 suggests a correlation between intercept and slope of line fits for different values of 
. We thus fit lines to the noisified version of 
for various values of 
. As before, noise levels were set as reported, and we again identified intercept and slope of the line fits to 
with 
and 
, respectively. The result of this procedure is shown in Figure 5, and agrees well with Figure 4 in [26]. Thus, 
consistently predicts a good correlation between intercepts and slopes both in the presence and in the absence of noise.
The corrected modified 
model (“
”)
Maximum detection of 
in the initial phase of an object approach (i.e., for small values of 
) is problematic, due to the signal's poor signal-to-noise ratio and the rather “shallow” curvature around the maximum. The situation gets progressively better if we place the maximum closer to 
, that is for bigger values of 
: The signal-to-noise ratio is better, and curvature is higher. With 
, however, we fell short of explaining the results of our psychophysical experiment (which is below described further). This led us to modify 
as follows.
Observe that 
for all 
, and thus
(3)is a positive correction factor to 
, such that 
. As with 
, the correction factor 
per se is again susceptible to fluctuations in the angular variable 
, and we would have gained no improvement by simply adding it to 
.
Now, the crucial idea is to render 
insensitive to such fluctuations. This is achieved with a first order low-pass filter (a short introduction is given in Text S8). Low-pass filtering of 
and 
transforms 
into a slowly varying signal, which is eventually added to 
:
(4)
and 
are low-pass filtered visual angle and angular velocity, respectively, and 
is the system's integration time constant. In order to avoid initial filter transients, the filter variables were initialized with 
and 
, respectively. The 
are filter memory coefficients with 
for 
. No filtering would take place for 
(no memory), and the filters would never change their initial state for 
(infinite memory).
The corrected, modified 
model (“corrected m-Tau”) is then defined as:
(5)where 
is a small constant, such that possible division-by-zero errors are avoided in the simulation. Nevertheless,in the presence of noise, division-by-zero errors do not typically represent a problem during an approach with 
, because 
if the following two conditions hold: (i) appropriate initialization of 
, and (ii) “sufficiently strong” lowpass filtering. The offset 
is included for the sake of completeness. It was only considered for simulating our psychophysical experiment (described below), where it turned out to be negligibly small. In general, therefore, it is safe to assume 
.
Similar to 
, the new corrected m-Tau-model also computes an estimation of ttc for “sufficiently small” angular sizes. But the principal advantage of 
over 
is that it is less sensitive to noise. The noise suppression of the corrected m-Tau-model is constrained by the noise suppression performance of two “limit functions”, which are approached dependent on the values of 
, 
, and 
(Figure 6). For the derivation of these limit functions, assume (to simplify matters) that in equation (5) 
with 
(and 
). Then, as we will show subsequently, the constraining functions are the ordinary 
function for 
, on the one hand (equation 6), and for 
a version of 
with lowpass-filtered angular variables, on the other (equation 8). Thus, 
, where 
, provided that we exclude the case 
, 
, which would imply that 
is unbounded.
Case I: 
.
For very small 
(more precisely 
), the first term of the equation (5) is approximately
(6)which is just the ordinary 
function. For the second term 
, which implies that it can be neglected because its denominator is approximately equal to 
. Furthermore, during an object approach with constant velocity, angular size 
and angular velocity 
are increasing, and 
, as 
is monotonically decreasing (except at times very close to ttc, see Text S6). The last arguments hold also for 
and 
, respectively, which are the lowpass-filtered optical variables, where 
. We eventually arrive at the approximation
(7)Summarizing the above, if 
, then the noise suppression performance of the corrected m-Tau -model is comparable with that of ordinary 
(Figure 6a).
Predicting psychophysical performance
Details on our psychophysical experiment are spelled out in the Methods Section. In a nutshell, subjects viewed approaching balls on a monitor. The balls had two different sizes (big & small, corresponding to object diameters 
& 
, respectively), and disappeared after 
(presentation time) until 
. A beep sounded always at the same time, 
, in order to indicate a reference time to the subjects. Approaches with different values of 
were presented, where 
could occur before or after 
. Subjects were asked to judge whether they were hit by the ball before or after 
. Responses were pooled, and the “proportion of later responses” for each presentation time (corresponding to “ball hit me after 
”) was computed as a function of ttc. Figure 7a shows the corresponding data points for 
, along with the best matching Gaussian cumulative density function (“GCDF”-fit) for each object diameter. The GCDF-fits represent an estimate of the underlying psychometric curves or psychometric functions, respectively. Figure 7b suggests that subjects did not respond to the average of the stimulus set, because the mean of the distribution (point of subjective simultaneity) shifted with presentation time. In addition, the variance of the distribution decreased with increasing presentation time. The small object diameter is furthermore associated with a higher variance than the big one.
The full set of data points is shown in Figure 8, where each figure panel corresponds to a different presentation time (small object size: circles; big: triangles). The curves shown in Figure 8 do not represent GCDF-fits (as in Figure 7a), but rather display simulation results from the 
-model. For short presentation times, subjects show near-random performance across ttc (Figure 8a, b), thereby revealing a bias towards later responses (i.e. “ball hit me after 
”). The GCDF-fits reveal a higher bias for the small object diameter (Figure 7b). The corresponding psychometric functions (not shown) and 
-predictions for the shortest presentation time (
; Figure 8a) are thus rather flat and noisy. This bias is progressively reduced with increasing 
, indicating improving performance: For 
, the point of subjective simultaneity approaches 
for both object diameters, and psychometric functions get closer to a step-wise increase at 
(Figure 7a).
We already mentioned that we simulated the psychometric functions with the corrected m-Tau -model (equation 5), at which we added noise to angular size and angular velocity (equation 9). By assuming a constant approach velocity, one could compute an estimation of ttc with equation (12). Note that this estimation should be constant throughout the approach in a noise-free situation and for “sufficiently small” angular sizes. As a consequence of having noise, however, the ttc estimation fluctuates. We therefore computed an average estimation with equation (14), by taking the mean value across a time interval (the interval contained the last 
estimates). The average ttc estimation was evaluated at presentation time 
, and compared with the reference 
. With a total number 
of such trials, we then counted 
occurrences where the average estimate occurred after 
. The simulated proportion of later responses is then obtained by dividing 
by 
(equation 13).
In order to find the appropriate 
-parameters for predicting psychophysical performance, the error between 
-predictions and psychophysical data points was minimized. We refer to this procedure as optimization. Optimization was carried out separately for object diameters big and small. The first step of the optimization procedure consisted in parsing the parameter space, and recording the error associated with each set of 
-parameters. The error was determined with two measures (“score measures”): The root mean square error (
), and an outlier-insensitive robust error (
). In the second step, the parameter sets were sorted in ascending order with respect to their associated score measure. Sorting took place separately for 
and 
, leading to corresponding tables where the best set of parameters was assigned rank one (1st table row), the second best rank two (2nd table row), and so on (Tables S1 & S2 in Text S5).
A third table of 
-parameters was then computed which was optimal for both object diameters simultaneously (combined; Table S3 in Text S5). This could be done in a straightforward way, simply by averaging the score measures of big and small of corresponding parameter sets, and subsequently sorting the averaged errors (more details on finding parameters are given in Text S5).
For the computation of 
and 
, all psychophysical data points that represent the proportion of later responses entered equivalently, in the sense that no weighting coefficients were used to bias the optimization process toward longer presentation times (as GCDF-fits at longer presentation times have a smaller variance, see Figure 7b). Notice that parameter optimization for the combined diameter naturally implicates a trade off – the errors with respect to big and small will be bigger compared to individual parameter optimization.
Figure 8 shows that the corrected m-Tau -model adjusts fairly well to the psychophysical data of both object diameters. Nevertheless, the 
-predictions for 
are somewhat worse with the combined parameter optimization (Figure 8e) when compared to a separate optimization for big and small (corresponding figures in Text S7). The most likely explanation for this discrepancy (individual versus combined parametrizations) is that each object size is associated with a different noise level (noise levels are represented by the 
-parameters 
with 
; see equation 9). We investigated this hypothesis by comparing the corresponding values of 
for big and small, as a function of their rank. Figure 9 shows that the 
for small are consistently higher than for big. Therefore, the corrected m-Tau -model generally supports the notion that smaller object diameters imply higher noise levels in angular size and angular velocity, respectively.
We also studied two models with less degrees of freedom than corrected m-Tau : The first was 
, and the second was 
with 
for 
(
). The best (i.e. smallest) score measures achieved with these reduced models were consistently higher than the best values achieved by the corrected m-Tau -model (Text S5), and their best-ranked parameter sets resulted in psychometric curve predictions that were also inferior by visual inspection (not shown).
Discussion
With the corrected m-Tau -model equation (5), we proposed a general framework that comprises the 
-function and several properties of the 
-function. By means of adjusting only a single parameter (
), the corrected m-Tau -model can approximate 
and 
, respectively. Moreover, the 
-approximation is less sensitive to noise than the original 
-function, and accounts well for the performance of the psychophysical experiment that we carried out.
In the experiment, subjects had to decide whether a (displayed) ball reached them before or after a reference signal at time 
. However, balls were only presented until 
, and disappeared afterwards. In other words, subjects had to estimate 
(
could occur before or after 
). With respect to our experiment, the corrected m-Tau -model suggests the following conclusions:
- Subjects relied on a
-based mechanism for judging ttc (
). We use the term “
-based” as a synonym for any timing-based mechanism. The full corrected m-Tau -model better predicted our psychophysical results than any of the two alternative models that we considered (
and 
).
- The decision about whether perceived
occurs before or after 
is based on information at (or immediately around) 
, as the only information used was from 
until 
for predicting ttc (see equation 14).
- Subjects' performance improves with increasing
, indicating that the signal-to-noise ratio (SNR) that is associated with the computation of (perceived) 
improves during an object approach. Such an improvement can be brought about by two mechanisms. First, the noise level is signal-independent and thus stays the same during an object approach. As angular size 
and angular velocity 
increase monotonically with time, the SNR would improve accordingly. Second, noise may increase with the signal [37], but is concurrently suppressed by low-pass filtering. Low-pass filtering may be adaptive, such that it adjusts to signal variability in each moment. We are not aware of any such signal-dependent noise suppression, and we therefore deem the first mechanism to be the more likely. Accordingly, we propose that approaching objects with smaller size lead to decreased SNRs in the signals that represent 
and 
, respectively.
- The perception of ttc in humans reveals a certain dependence on object size [22]. Thus, one might argue that
-based mechanisms are not an adequate model for ttc perception, because they are largely object-size-independent in the early phase of an object approach when 
is still “sufficiently small”. However, this argument ignores noise. As long as the noise-induced fluctuations in 
and 
do not cancel (“correlated noise”), the SNR of 
will depend on object size (Figure 6). Therefore, any decision based on computing 
with a 
-based mechanisms will be limited by the SNR at time 
(ttc can be computed by adding 
to 
, because in the early phase of an approach 
decreases linearly with time for 
, see equation 12). The SNR improves with increasing object size and with decreasing (initial) distance between object and observer. Thus, bigger objects will imply better accuracy in estimating 
. Similarly, smaller distances will imply better estimation accuracy. Both effects are observed in our psychophysical experiment, where a better “estimation accuracy” translates into psychometric curves that adjust better to a step-wise increase from zero to one at 
(because of 
; Figure 7a). Without noise, however, 
-based mechanisms cannot predict such dependence on object size for small angular sizes.
The modified 
-model (“
”) constitutes a special case of 
. It is obtained from equation (5) for 
(
by default). Its distinguishing feature is a maximum before 
, which can be shifted via 
(equation 2). The 
-maximum decreases as it is positioned closer to 
, because this implies bigger values of 
. The time 
of the 
-maximum depends furthermore on size and velocity (Figure 1). The curve shape of 
is reminiscent of the 
-function, since both functions have a maximum. We thus decided to fit previously published response curves from collision sensitive neurons to both functions, and observed that both functions fit the neural curves well in terms of goodness-of-fit criteria (Text S4). We must not forget, however, two important differences between 
and 
.
First, since 
reveals a minimum shortly before 
(Text S6) and 
derives from 
, the 
-response is more precisely biphasic. The biphasic structure gets pronounced in some of the curve fits, especially when 
is close to 
(see corresponding figures in Text S4). Then, the amplitude of the 
-maximum is small, and consequently the fitting algorithm has to scale it to the maximum of the neuronal recording data. In this way, the minimum is also scaled.
Second, 
depends in a nonlinear way on the size-to-velocity ratio 
(see Figure 4 for an illustration). This is contradictory to several studies that found a linear dependence. A linear dependence is also predicted by the 
-function (equation S5 in Text S2). The contradiction can be alleviated by adding noise to relative time of the 
-maximum (
; equation S10 in Text S2), with noise amplitudes as reported in [26]. As a consequence of noise, the nonlinear relationship can be literally hidden (Figure 3), such that statistical tests would affirm an underlying linear process (Text S2). Masking by noise is more effective for bigger values of 
, because the noise level is proportional to 
.
The 
-function in its original form cannot explain the neuronal response curves for an approach with 
(“linear approach”) [25]: Rather than predicting a decreasing response with time, the 
-function would linearly increase. In contrast, the 
-function makes correct predictions. Correct predictions with 
can nevertheless be made by including an additional inhibitory process in the firing rate equation of 
(equation S3 in Text S1, where a full proof of concept is described). Important, this extension of 
(i) is based on a power function with an exponent between 
and 
, but not on an exponential function as with 
, and in this regard it may hence be considered as being biophysically more plausible than 
(see also reference [36]); (ii) does not interfere with the “normal” 
behavior (i.e. normal object approaches are not affected); and (iii) tolerates high noise levels (i.e., the mechanism is robust).
What about alternative models which also have a response peak? In Text S6 we studied two such functions, namely “inverse 
” (
), and angular acceleration (
). Both of them reveal a linear dependence of 
on 
(equations S24 & S26, respectively, in Text S6). The maximum of 
always precedes that of 
. However, 
does not make correct predictions for the “linear approach”, as we would obtain 
ab initio for 
(although a dynamical version may predict the decreasing LGMD-activity on the basis of temporal filtering effects).
In contrast, 
would make consistent predictions in that case. Without further modifications, though, neither 
nor 
seems to be adequate for fitting the response curves of collision sensitive neurons, because there is no free model parameter to shift their respective maximum. Although the occurrence of their maxima could principally be controlled by a global shift of the time scale 
, the corresponding values (obtained by fitting the neuronal response curves) would overestimate experimental values (Text S6). Similarly, when “fitting” the 
-function to 
and 
the so obtained values of 
would underestimate experimental values: The 
-maximum would coincide with the maximum of 
for 
, and with the maximum of 
for 
.
In conclusion, 
is no replacement for the 
-function, at least for describing neuronal responses of collision sensitive neurons in insects. However, in the nucleus rotundus of pigeons three classes of neurons were reported [1], [38]. They conform to 
-like, 
-like, and 
-like responses. The fact that 
is just a special case of 
could possibly explain why neurons with 
-like and 
-like properties can be found in a single brain. Within the 
-framework, the 
function corresponds to 
, and 
is obtained for choosing 
. Thus, the adjustment of only a single weight (
) is necessary to go from one function to the other. The corrected m-Tau -framework could thus offer a parsimonious yet full-fledged explanation of the implementation of 
-like and 
-like neurons at the circuit level.
Methods
Psychophysical experiment
Subjects.
Four subjects that were members of the Basic Psychology Department of the University of Barcelona participated in the experiment. All had normal or corrected-to-normal vision and were naive with respect to the aims of the experiment. Two of the subjects were well-trained psychophysical subjects in similar tasks. None of the subjects was stereo blind (StereoFly test, Stereo Optical Co.). They all signed an informed consent. The psychophysical experiment was approved by the Ethics Committee of the Faculty for Psychology of the University of Barcelona, in agreement with the ethical guidelines of the Declaration of Helsinki in 1954.
Simulation of our psychophysical experiment
We simulated our psychophysical experiment with the corrected m-Tau -model (equation 3), where we plugged in noisified versions of the optical variables (i.e. 
),
(9)with noise probabilities 
(
), and with the dot denoting the time derivative. The 
are random variables, which at each instant 
return a value that is drawn from a centered normal distribution. In the last equations, we used the explicit expression for angular size,
(10)and angular velocity (
rate of expansion)
(11)with 
and 
. The values of 
and 
are the psychophysical stimulus parameters. Simulations were carried out with a temporal resolution of 
.
The corrected m-Tau -model is constrained by two limit functions: Ordinary 
on the one hand (equation 6), and 
on the other (equation 8). Both limit functions decrease approximately as 
(illustration: Figure 6). Thus, a ttc estimation at time 
can be computed as
(12)(Nomenclature: 
is the model prediction for ttc at time 
, and 
is the experimentally set parameter). In the psychophysical study, subjects were asked to estimate whether they were hit by the approaching object before or after 
. We accordingly define their proportion of later responses 
as the number of trials 
(where subjects responded with being struck after 
) divided by the total number of trials 
:
(13)
is represented by circle and triangle symbols in Figure 7 and 8. The corresponding predictions from the model are denoted by 
. Specifically, 
with 
and 
, and analogous for 
. Computation of 
is required for 
, which we did with equation (12) as per
(14)Notice that, due to noise (equation 9), 
will be subjected to random jitter with each trial 
. Therefore, in order to obtain a more robust estimate of ttc , we do not use only 
: The integral in the last equation computes – in the discrete case – the mean value across the last 
time steps until 
(typically 
, what amounts to a time interval for averaging of 
, cf. first figure in Text S7). In order to illustrate the noise level at each 
, we also computed the standard deviation 
of the 
last values of 
. The shaded areas in the figures which visualize 
& 
correspond to 
. Predictions of the corrected m-Tau -model are shown as curves in Figure 8, as well as in Text S7.
Parameters of the corrected m-Tau model
The corrected m-Tau -model has eight free parameters: 
, 
, 
, 
, 
, 
, 
, 
. The parameter space was parsed with constant step widths. For each set of parameter values 
, 
-predictions for the proportion-of-later-response curves were computed according to the procedure described in the previous section. The corresponding “goodness of prediction” (or “prediction performance”) was evaluated with the root mean square error (rmse, 
), and the outlier insensitive, robust error (robe, 
), see equation S18 in Text S5. The “goodness of prediction” measures are referred to as score-measures (rmse-scores & robe-scores, respectively). Parameter values were sorted according to their scores. In this way we ended up with several score tables, which list the best set of parameters, according to object size: Table S1 in Text S5 for small object diameter (
), Table S2 in Text S5 for big object diameter (
), & Table S3 in Text S5 for combined object diameter. The scores for the combined size were computed by averaging the scores of big & small for corresponding parameter values, and then sorting the averaged scores in ascending order. More details on parameter finding and analysis are given in Text S5.
Supporting Information
Properties and extension of modified Tau. Text S1 presents additional mathematical details of the 
-function. Specifically, it is shown how the 
-function could be extended to a model which predicts the so-called “linear approach” data. Corresponding simulation results from this model are also shown.
https://doi.org/10.1371/journal.pcbi.1002625.s001
(PDF)
Nonlinearity of the m-Tau function. Text S2 is dedicated to the nonlinear character of 
and how it could be successfully hidden behind noise. The section presents additional figures with random trials (analogous to Figure 3b), and corresponding scatter plots with goodness-of-fit measures as a function of 
.
https://doi.org/10.1371/journal.pcbi.1002625.s002
(PDF)
Noise suppression. Text S3 considers the numerical robustness of 
, 
, 
and 
, by adding correlated and uncorrelated noise to the angular variables. Similar to Figure 6, it is shown how noise affects these functions (e.g. bigger object diameters are associated with correspondingly less fluctuations), and thus the results presented in this section help to understand the simulation and the interpretation of our psychophysical experiment.
https://doi.org/10.1371/journal.pcbi.1002625.s003
(PDF)
Fitting m-Tau and 
-function to neuronal recordings. Text S4 juxtaposes the individual fitting results of 
and 
to a variety of previously published neural recording data, which served to compile Figure 2. Further summary results are presented along with fitting results of individual recording traces.
https://doi.org/10.1371/journal.pcbi.1002625.s004
(PDF)
Finding parameter values for corrected m-Tau. Text S5 describes the optimization procedure for the corrected m-Tau -model “
”, with which we obtained the parameter values for the simulation of our psychophysical experiment (e.g. Figure 8). The 
-parameters were optimized in three different ways: For achieving a good prediction performance of the psychophysical data corresponding to (i) the small object diameter (Table S1 in Text S5), (ii) the big object diameter (Table S2 in Text S5), and (iii) both diameters at the same time (combined; Table S3 in Text S5). The best ten values are listed in their respective tables according to their psychophysical prediction performance (as quantified by score measures 
and 
, respectively): The best parameter set (smallest score measure) was assigned rank one, the second best rank two, etc. Several figures were compiled that show an additional analysis of the parameter ranking.
https://doi.org/10.1371/journal.pcbi.1002625.s005
(PDF)
Predictions of corrected m-Tau for the psychophysical experiment. Text S7 shows the full set of figures with simulation results of our psychophysical experiment. Whereas Figure 8 shows 
-predictions that were obtained with the parameter set optimized for the “combined” object diameter according to 
-score, Text S7 shows analogous figures for the remaining parameter optimizations (big and small object diameter, and 
and 
-scores, respectively).
https://doi.org/10.1371/journal.pcbi.1002625.s007
(PDF)
Acknowledgments
The authors like to thank Eric Johnson for his help in proofreading our manuscript.
Author Contributions
Conceived and designed the experiments: MSK JLM. Performed the experiments: MSK JLM. Analyzed the data: MSK. Contributed reagents/materials/analysis tools: MSK. Wrote the paper: MSK. Designed and performed the psychophysical experiment: JLM. Developed the model and performed mathematical analysis: MSK.
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