Angular Variables

For simplicity, we assume that a circular object with diameter 2l approaches an observer with a constant velocity v. If the object starts its approach at a distance x₀ then the time to collision (ttc) is at t_c = x₀/v (here we do not distinguish between time to contact and time to collision). At the time t < t_c, the object spans an angular size Θ(t) on the retina of the observer, (3) where x(t) = x₀ − vt. The angular velocity or rate of expansion is defined as the first temporal derivative, (4)

Maximum of the η-function

The time T_max = t_max + δ of the maximum of the η-function is obtained through . This is equivalent to and yields Eq (2). Thus, the relative time of the maximum T_rel ≡ t_c − T_max depends linearly on the halfsize-to-velocity ratio l/v with slope α and intercept δ. Together with Eq (3), Eq (2) also implies that the response maximum always occurs at a fixed angular size, (5) which only depends on α, but not on time.

Simulation details

The Matlab environment was used for all simulations. The default parameters of the n- ψ-model were as follows: β = 1, V_inh = −0.005, V_rest = 10⁻⁵, V_exc = 1, γ = 500, σ = 0.25, Δ₀ = 0.9, ζ₀ = ζ₁ = 0.95, N = 500 (nearly identical n- ψ-predictions are obtained already for N ⪆ 10). If different values from the latter were used, they would be indicated with each figure. Since V_inh ≈ V_rest, inhibitory input acted by increasing the effective leakage conductance β (silent or shunting inhibition). In other words, inhibitory input will not show unless excitatory input is present at the same time. The seed of the random number generator was chosen identical for each simulation run. This means that the same sequence of random numbers was generated. Eq (10) was integrated with the 4th-order Runge-Kutta (“RK”) method using the step size 500μs. The stimulation time scale was set to Δt_stim = 1ms. Or, expressed differently, angular size Θ Eq (3) and angular velocity Eq (4) were discretized with Δt_stim (cf. Eqs (11) and (12). As a consequence, the membrane potential V(t) is far from its steady state V_∞ Eq (10) at each time t. For this reason, and in order to maintain the consistency with its predecessor model (= ψ, see ref. [19]), n_relax = 250 relaxation time steps were intercalated at each t. A relaxation time step is just an RK step with frozen values of g_exc and g_inh. V(t) can be driven in this way closer to its equilibrium solution. The influence of n_relax on the predictions of n- ψ is broken down in Section C in S1 Text.

The default parameters for the object approach were diameter 2l = 0.12m, approach velocity v = 6m/s, initial distance x₀ = 0.3m, continuous stimulation (versus discrete stimulation, see [19]). Notice that Θ and were not computed from video sequences, but directly from Eqs (3) and (4), respectively.

Can a power law undo logarithmic encoding?

Logarithmic encoding of Eq (1) means [16, 23]. By doing so, the LGMD could basically multiply: log(x ⋅ y) = log(x) + log(y). The result of multiplication is obtained by applying exp(⋅). Accordingly, the LGMD’s membrane potential should be exponentially scaled, but figure 4d in reference [23] rather suggests a power law. At least mathematically it is clear that (logη)^p is a rather poor approximation of exp(logη). But how would the predictions from η look like when this approximation was used in the context of approaching objects? In order to answer that, we “undo” logarithmic encoding with a power law of order p (let g_p(t) = predicted LGMD response): (6) The extremum of this function is either a maximum (for power law exponents p = 1, 3, 5, …) or a minimum (p = 2, 4, 6, …) at , that is (7) Because of the factor the extrema of g_p will still occur at the same time points as those of the η-function, which is good news. The bad news, however, is that the extrema will be maxima only for odd values of p, where g_p(t) ≤ 0 for all t. Thus, if we wanted g_{1, 3, 5, …} to be positive, we would need to add an offset o to Eq (6) (say a such that g_p(t) + o ≥ 0). For even values of p, g_p has minima with g_p(t) ≥ 0 for all t. In the latter case, an operation such as o − g_p(t) (with o ≥ max_τ[g_p(τ)], τ ≤ t_c) would be needed to transform the minima into maxima.

If the exponent p is fixed, then the biophysical implementation of o + g_{1, 3, 5, …} and o − g_{2, 4, 6, …}, respectively, should be feasible. Nevertheless, a general drawback of “undoing” the logarithmic transformation with a power law would be that the maximum of the LGMD membrane potential is strongly flattened: The bigger p is, the more (Fig 1a). As a consequence, the reliability of detecting the time of peak firing drops already for small noise amplitudes (Fig 2b). The different phases of escape jumps seem to occur around the peak firing rate of the DCMD [25], although this does not automatically imply that the peak time is explicitly decoded by the motor system of the locust. Experimentally measured timings of DCMD peak firing have standard deviations of ≈ 50ms [20]. The comparatively high variability of the response maximum as predicted by g₃(t) (Fig 2b) stands in contrast to these observations, since standard deviations are higher than 100ms even for small noise levels. It seems therefore that the sole power law of Eq (6) cannot reproduce LGMD firing patterns. Is it possible to regain the undistorted, “LGMD-like” response curves from g_p(t)? Reference [17] holds a general mathematical solution in order to map the η-function to the firing rate of the LGMD. The mapping function is a (here simplified) sigmoid 𝒮(x) = [1 + exp(− ax)]⁻¹. The value of a determines whether the activation from zero to one proceeds in a more linear way (a ≪ 1) or according to step-function (a ≫ 1). With g_p = (logη)^p Eq (6) one obtains (8) and specifically 𝒮(g₁) = η^a/(1+η^a). With an appropriate choice of the parameters a and p, respectively, the function 𝒮(g_p) reduces the initial activity compared to the ordinary η-function. In this way the peak is more pronounced (see Fig 1c). However, with increasing p and a, respectively, the peak amplitudes of curves with higher l/v decrease relative to low halfsize-to-velocity ratios (see Fig 1b). The peak values for a set of l/v values could quickly span several orders of magnitude for already “moderate values” a > 1 (e.g. a = 5) and p > = 1.

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Fig 1. Logarithmic encoding of η(t).

(a)The logarithmically encoded η-function is “approximately exponentiated” by a power law with exponent p according to g_p(t) = [logη(t)]^p Eq (6). The corresponding curves of g_p (with p = 1,2,3,4,5 for l/v = 30ms) are denoted in the legend. For control, the sum (exponential series: dash-dotted curve), and the η-function without log-encoding (gray curve) are also shown. Note that the curves become flatter with increasing p (and also with increasing l/v—not shown), what could make the detection of the maxima more uncertain in the presence of noise (cf. Fig 2b). (b)Applying a sigmoidal function 𝒮(x) = [1 + exp(− ax)]⁻¹ (with a = 1) to the properly normalized curves g_p(t) (with p = 3) produces curves which resemble true LGMD responses Eq (8). The curves shown here correspond to different halfsize-to-velocity ratios l/v = 10,20,40,40,50ms (see legend). The peak amplitude of the curves decreases with increasing halfsize-to-velocity ratio, but also with increasing a and p, respectively (not shown). (c) Identical with figure panel b, but here all curves 𝒮(g₃) were re-scaled to the range from 0 to 1. For comparison, the corresponding η-functions are drawn alongside with dashed lines (α = 4.7, δ = 0, and t_c = 0.5s in all figure panels).

https://doi.org/10.1371/journal.pcbi.1004479.g001

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Fig 2. How noise interacts with maximum detection.

This figure shows how the numerical detection of the global maximum is affected by adding noise to the η-function, g₃(t), and 𝒮(g₃). To this end, the root mean square error (RMSE) between and the theoretical (i.e. noise-free) t_max is determined across 999 trials. The RMSE is computed as a function of noise amplitude σ (abscissae) and the halfsize-to-velocity ratio l/v (ordinates). Brighter (“hotter”) colors denote bigger RMSE values (see colorbar; units in milliseconds). Noise is added as follows. Let ξ be a normal-distributed random variable with mean zero and standard deviation one. Then, (a)η(t) + σξ is the “noisified” η-function (with δ = 0 and α = 4.7), (b)shows the RMSE for g₃(t) + σξ, and (c) 𝒮(g₃) + σξ (with a = 1 in Eq (8)). The curves of η(t), g_p(t) and 𝒮(g_p) become flatter for higher halfsize-to-velocity ratios (cf. Fig 1). In addition, the curves of g_p(t) become flatter for increasing p, and the curves of 𝒮(g_p) get flatter for decreasing values of a and p, respectively. Flatter curves are associated with an increased RMSE. Although g₃(t) + σξ is associated with higher RMSE values than η(t) + σξ, the completely “decoded” function 𝒮(g₃) has a better overall robustness against additive Gaussian noise than η(t). For each value of l/v, g₃(t) and 𝒮(g₃) were re-scaled before noise was added, in order to match the range of η(t), and thus normalize the RMSE. For all figure panels, t_c = 0.5s and l = 0.06m.

https://doi.org/10.1371/journal.pcbi.1004479.g002

What are the possible implications for biology of the mathematical exercise presented in this section? First, a direct biophysical implementation of η (with logarithmic decoding based on a power law) seems to be insufficient. At least mathematically, an additional (probably nonlinear) function is required (e.g. based on Eq (8)). In reference [17], it has been demonstrated that the linearity constraint Eq (2) implies a partial differential equation of the structure x(∂f/∂x) = −(∂f/∂y) with a general solution f = h(x exp(− y)). This means that η is defined up to a (monotonically increasing) function h that “characterizes the transformation between η and the firing rate of the LGMD” [17]. In [17] the response curves of the LGMD were fitted with a sigmoidal function whose parameters depended on the halfsize-to-velocity ratio and also on the rising phase (i.e. a sigmoid fitted to the curve until the response peak at t_max) and falling phase (a second sigmoid fitted to the curve after t_max). Nevertheless, to the best of my knowledge, this nonlinearity has not been identified biophysically so far. This prompted me to formulate an alternative model, which is presented below.

“n-ψ”—a model of the LGMD neuron

In this section, we introduce a new model (“n-ψ”) for the LGMD and other collision-sensitive neurons with η-like response. While n-ψ is biophysically more plausible than η, it is not a biophysically detailed model (compared with, e.g. [22]). The n-ψ-model is based on its predecessor model Ψ, which used an exact power law for re-scaling the inhibitory input [19]. An important component of n-ψ is noise: n-ψ generates an approximative power law by pooling of noisy and thresholded inhibitory inputs. The power law is only approximative because it turns to a linear relationship sufficiently far from threshold. The pooling process may be implemented in vivo by integrating synaptic inputs across the dendritic tree. n-ψ is based on a standard RC-circuit describing the membrane potential V of the LGMD neuron [26] (9) As we do not use a spiking mechanism, we assume that [V]⁺ ≡ max(V, 0) (the half-wave-rectified membrane potential) directly represents the LGMD’s mean firing rate. (For simplicity, we will omit physical units in what follows). The membrane capacitance C_m is set to unity in all simulations; β ≡ 1/R_m is the leakage conductance across the cell membrane and R_m is the membrane resistance; g_exc ≥ 0 and g_inh ≥ 0 are the excitatory and inhibitory synaptic inputs. The corresponding reversal potentials V_exc and V_inh, respectively, represent upper (V_exc > V_rest) or lower (V_inh < V_rest) limits to V if the neuron is driven by the associated input g_exc or g_inh. The reversal potentials are asymptotically approached for sufficiently high excitatory or inhibitory drive. Typically one sets V_inh ≤ V_rest ≤ V_exc, what implies V_inh ≤ V(t) ≤ V_exc. Shunting inhibition is defined by setting a reversal potential equal to the resting potential V_rest. Shunting inhibition essentially increases the leakage conductance, and thus becomes effective only when the neuron is driven away from V_rest. The time scale of Eq (9) is determined by the membrane time constant τ_m ≡ C_m/β. If the synaptic inputs vary on a slower time scale than τ_m, then the neuron will reach its equilibrium potential V_∞ at each instant t according to V(τ) = V_∞(1 − exp(− τ/τ_m)), where V_∞ is defined as: (10) Without synaptic input, V approaches V_rest with τ_m. Note that with C_m = 1, the time scale is set directly by β, where the synaptic inputs are subjected to a higher degree of lowpass filtering for lower values of β. Slowly varying inputs g_exc, g_inh > 0 modify the time scale approximately to τ_m/(1 + (g_exc + g_inh)/β). With highly dynamic inputs, τ_m also varies with time.

For the definition of synaptic input we first introduce the low-pass filtered angular size ϑ(t) and angular velocity , respectively (see section S8 in [21] for a brief introduction): (11) (12) Low-pass-filtering is supposed to model delays and filtering effects as they are introduced by the pre-synaptic layers (lamina and medulla). The stimulation time scale is fixed by Δt_stim, and 0 ≤ ζ_i < 1 (i = 1,2) determines the degree of low-pass filtering (no filtering would take place for ζ_i = 0). The lowpass filtered signals lag behind the original signals (the bigger ζ_i, the more). This behavior is important to explain experimental observations, where the LGMD’s activity continues to rise after ttc [14, 18, 19]. Lowpass filtering also reduces stimulus-dependent noise in the optical variables (Θ and , Eqs (3) and (4), repsectively), particularly when the angular size is small [21]. The n-ψ-model is defined by assigning the following synaptic inputs: (13) (14) Where [⋅]⁺ = max(⋅,0). Similar to η Eq (1), n-ψ uses lowpass-filtered angular size for excitation. Unlike η, however, n-ψ does not use an exponential function but instead sums threshold processes. The threshold is set by Δ₀. The ξ_i are random numbers drawn from the standard normal distribution (Fig 3b). Thus, σ⋅ξ_i adds Gaussian noise with standard deviation σ to ϑ. We furthermore require that corr(ξ_i, ξ_j) = δ_ij, where corr(⋅) denotes correlation, and δ_ij is the Kronecker delta. The number of threshold processes that are pooled is N, and γ is a synaptic weight.

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Fig 3. Threshold smoothing and power law (illustration of Eq (14)).

(a)The dependence of g_inh on 13 values of ϑ is shown at two different noise levels (σ = 1 and σ = 3, see legend) along with respective fit of a power law (exponents e = 2.69 and e = 1.73, see legend). For σ = 3, the theoretical prediction is shown as well (circles; cf. Equation A7 in S1 Text) (b)For ϑ₁₁ = 5 (eleventh data point), the distribution of ϑ₁₁ + σ ⋅ ξ_i − Δ₀ is shown from i = 1 to i = N. In order to compute g_inh, these values are first half-wave rectified (what zeros the values which correspond to the white bars in the histogram), and subsequently averaged (non-zero contribution from values represented by the colored bars). This eventually yields g_inh = 2.46 for ϑ₁₁. Thus, non-zero g_inh-responses for some ϑ < Δ₀ result from fluctuations with ξ_i > 0 such that ϑ + ξ_i > Δ₀. If ϑ is decreased, then whole distribution will shift to the left. (c) Power laws were fitted to the g_inh versus ϑ curves and exponents e (circles) along with R² (triangles) are plotted. These curves are well described by a power law only at intermediate noise levels, here 1.75 ≤ σ ≤ 5.25.

https://doi.org/10.1371/journal.pcbi.1004479.g003

Biophysical motivation for g_inh

The LGMD’s dendritic tree has one excitatory field and two smaller inhibitory ones. It receives about 15,000 retinotopically organized excitatory inputs from the medulla [22]. Each of the inhibitory fields is contacted by some 500 axons originating in the second optic chiasm [27]. Response dynamic and response characteristics are similar in both inhibitory pathways. They can be distinguished, however, by contrast polarity. The ON-pathway responds to increments in luminance, and the OFF-pathway responds to luminance decrements. Along the excitatory visual pathway (ommatidia, lamina, medulla and lobula), the signal-to-noise ratio decreases, being lowest in the LGMD [4]. In Eq (14), we assume that noise is present in the inhibitory pathway as well, and that it is uncorrelated for each of the N = 500 inhibitory inputs (“processes”) that are pooled by the LGMD. For the sake of clarity we did not incorporate noise in g_exc Eq (13), because the predictions made by n-ψ would not change much if we did so (Section B in S1 Text). Conversely, n-ψ would fail to predict the LGMD’s response characteristics with noise in the excitatory pathway but without noise in g_inh. Since the pooling of noisy threshold processes approximates a power law, this behaviour of n-ψ is consistent with the schematic proposed in [22], where both the excitatory and the inhibitory input are scaled according to a power law (with exponents 2–3 and 2, respectively).

The integrated (or pooled) inhibitory input to the LGMD is supposed to encode angular size [27, 28]. But Eq (14) rather suggests that ϑ is identical for each inhibitory process, and that only the noise varies from one input to the next. To see that this is nevertheless a reasonable simplification, assume that the (noise-free) ϑ(t) at each instant t is just the sum of many individual ϑ_i(t), that is ϑ(t) = ∑_i ϑ_i(t), with 1 ≤ i ≤ N. The index i could be retinotopically arranged, but this is not relevant. Then, Eq (14) would read [ϑ_i(t) + σ ⋅ ξ_i − Δ₀]⁺, what looks more decent. The problem is that we do not know the exact distribution of the ϑ_i(t). Therefore we suppose that it is uniform, meaning that all ϑ_i(t) contribute equally. But then, ϑ(t) ≡ Nϑ_i(t) if ϑ_i(t) = ϑ_j(t) for all i, j. Thus, with uniformly distributed ϑ_i(t), we can draw 1/N outside the rectification, rescale the noise level σ and the threshold Δ₀ by N, and readily arrive at Eq (14).

Threshold smoothing

The threshold operation [x]⁺ = max(x, 0) is zero for x ≤ 0 and x otherwise. Around x = 0, we have a discontinuity. If x is set to a suitable negative number, then by adding noise σ ⋅ ξ to x we may obtain non-zero responses. The probability of noise-elicited threshold crossing will increase as x approaches zero. By computing a mean value ∝ ∑_i[x + σ ⋅ ξ_i]⁺ across many channels at some instant in time, one may effectively smooth out the discontinuity at the threshold and create continuous responses (“threshold smoothing”: In ref. [10] averaging was carried out across successive trials). Fig 3a shows two representative curves g_inh vs. ϑ for the noise levels σ = 1 and σ = 3. The threshold is fixed at Δ₀ = 3. Bigger σ mean more noise, what in turn causes a higher degree of threshold smoothing, since non-zero g_inh-responses will be obtained already for ϑ < Δ₀. For ϑ > Δ₀, g_inh gradually approaches a linear response. Around Δ₀, a power law provides a good description of g_inh [11, 12]. The two curves in Fig 3a were fit by power laws with exponents e = 2.7 and e = 1.7, respectively. Fig 3c (circles) illustrates that higher noise levels result in smaller exponents. Nevertheless, a power law is only adequate within a certain range of σ: the red curve in Fig 3c (triangles) representatively shows the R² as a goodness of fit measure. The highest R² values are obtained around 1.75 ≤ σ ≤ 5.25. For smaller σ, the response curves show less smoothness, while for bigger σ, g_inh depends “more linearly” on ϑ. Mathematically, it can be shown that g_inh is composed of three additive terms (Equation A7 in S1 Text): A linear term plus an error function plus a Gaussian [11, 12]. As a function of ϑ, the error function and the Gaussian will only vary within a certain interval around Δ₀. Outside this interval, the Gaussian approaches zero and the error function approaches a constant value.

Predicting the response characteristics of the LGMD neuron

LGMD responses have two important properties: A response maximum before collision on the one hand [16], and linearity on the other [17]. The η-function and n-ψ’s predecessor model Ψ successfully predicts these properties (Methods Section). Below we study the corresponding predictions of n-ψ.

Fig 4a shows that n-ψ has an activity maximum. When the noise level σ is increased (while keeping the threshold Δ₀ constant), then the maximum shifts towards t_c, and the amplitude decreases. Fig 4b confirms the linear dependence of t_rel on l/v for the n-ψ-model Eq (2). The noise level σ is decisive for the slope α of the corresponding line fits. Interestingly, α adopts a maximum value at some noise level. Fig 5 shows the dependence of α on σ and Δ₀: The location of the maximum α as a function of σ depends also on Δ₀. Bigger Δ₀ mean that the maximum α is obtained at smaller σ. Furthermore, the maximum value of α will be higher if Δ₀ is bigger.

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Fig 4. Influence of the inhibitory noise level σ.

(a)The figure illustrates how an increase of the inhibitory noise level from σ = 0.25 to σ = 0.50 moves the maximum of n-ψ towards t_c = 500ms, thus t_rel (= the remaining time to collision after the peak) decreases from 133ms to 80ms. Furthermore, the amplitude of the n-ψ-response decreases. (b)The line fits represent the linear dependence of t_rel from l/v, where different symbols and line colors, respectively, correspond to different values of σ. The smallest line fit slopes α see Eq (2) were obtained for σ = 0 and σ = 0.75, that is in the absence of inhibitory noise (α = 1.92) and for the highest tested noise level (α = 1.13). For the intermediate noise levels σ = 0.25 and σ = 0.50, respectively, the biggest slopes were measured (cf. figure legend; “normal” means that the residuals were normal distributed according to a one sample Kolmogorov-Smirnov test).

https://doi.org/10.1371/journal.pcbi.1004479.g004

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Fig 5. Line fit slopes.

This figure shows how the line fit slopes α depend on σ and Δ₀. The proceeding is similar to Fig 4b (where Δ₀ was fixed): For each σ and Δ₀, t_rel was computed as a function of l/v. A line was then fit to these data (with a robust fit algorithm), and its slope α was recorded. If Δ₀ is held constant and we consider α as a function of σ only, then this function adopts a maximum slope. The maxima are situated along a crest, with the corresponding values being indicated in the figure. Hotter colors indicate higher slopes.

https://doi.org/10.1371/journal.pcbi.1004479.g005

One can ask how the predictions of n-ψ are influenced by excitatory noise. As it turns out, excitatory noise distorts the response curves of n-ψ somewhat, and shifts the response maxima slightly towards t_c (Fig. A1 in S1 Text). Nevertheless, excitatory noise has only a negligible effect on α, and thus does not alter the linear relationship between t_rel and l/v. (More details are provided in Section B in S1 Text).

How does n-ψ compare to the η-function when it comes to fitting response curves of the LGMD? We resampled 36 corresponding curves from the figures of several publications to this end [21], and fitted the two functions to them. Fig 6a illustrates that η and n-ψ describe the LGMD responses very well (all fits are shown in Section D.5 in S1 Text). Excellent predictions are also obtained for non-trial-averaged recording traces (Fig 6b; more examples are shown in Section D.6 in S1 Text). The 36 response curves were fit, on the average, with a median noise level σ = 0.4 ± 0.1 (Fig. D3a in S1 Text), and a median threshold Δ₀ = 0.9 ± 0.1 (Fig. D3b in S1 Text). These values should be compared with the median α = 3.1 ± 0.7 (D2a in S1 Text) of the η-function, and the exponent e = 2.7 ± 0.5 of the ψ-model (Fig. D2b in S1 Text).

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Fig 6. Prediction of LGMD responses by n-ψ.

(a)The figure shows the data from figure 3 of reference [17] (locust), where looming dark squares were used for stimulation (t_c = 0.5s: vertical red line). The η-function and n-ψ perfectly match these data. More examples are shown in Section D.5 in S1 Text. (b)DCMD spike trace in response to an approaching black square (t_c = 5s, l/∣v∣ = 30ms, ref. [29]) directed to the eye center of a gregarious locust (final visual angle 50^o). Data show the first stimulation so habituation is minimal. The spike trace (sampled at 10⁴Hz) was full wave rectified, lowpass filtered, and sub-sampled to 1ms resolution. Firing rate was estimated with Savitzky-Golay filtering (“sgolay”), and fit by the η-function and n-ψ, respectively. Both functions fit the firing rate estimates very nicely. More details and further examples are provided in Section D.6 in S1 Text.

https://doi.org/10.1371/journal.pcbi.1004479.g006

A “recession” is a stimulation protocol where the object moves away from the eye—just the opposite to an object approach. DCMD responses to receding objects are significantly smaller than those to approaching objects [14]. Once an object starts to recede, the DCMD response reaches an activity maximum, which is smaller than that of a corresponding approach. Subsequently, the activity decreases quickly. Thus, the directional sensitivity in DCMD responses is reflected in overall activity, but also in terms of curve shape. The η-function predicts symmetric responses (approach vs. recession) [16]. The n-ψ-model, on the other hand, predicts an asymmetry of 40ms in the timing of the response peaks for the configuration of Fig 4a, because of lowpass filtering. A perhaps more interesting kind of stimulation is shown in Fig 7, where . A constant angular velocity implies that angular size is linearly increasing with time. The experimentally obtained response curves of the LGMD to this “linear approach” are nevertheless decreasing [16]. In that case, n-ψ provides a much better description of the LGMD data than the η-function, both in terms of goodness-of-fit measures and subjective judgement (“Chi-by-eye”).

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Fig 7. LGMD stimulation with constant angular velocity.

The original data (legend label “HaGaLa95”) were resampled from ref. [16] and show DCMD responses to an object approach with , what is equivalent to that Θ increases linearly with time. The η-function (fitting function: Aη(t + δ) + o) and n-ψ Eq (10) were fitted to these data (see Section D in S1 Text for specifics). (a)(Figure 3 Di in [16]) Good fits for n-ψ are obtained with σ = 0.2 and Δ₀ = 0.7. n-ψ follows a sigmoid-like curve that (subjectively) appears to fit the original data better than η. The fitting parameters for η were {A, α, δ, o} (“TR:41”), and {σ, Δ₀} for n-ψ(“TR:444”). (b)(Figure 3 Dii in [16]) The fit of n-ψ agrees excellently with the data for σ = 0.15 and Δ₀ = 0.6. The fitting parameters for n-ψ were {β, σ, Δ₀} (“TR:44”). Figure D26 in S1 Text shows how the linear approach data could also be predicted by explicitly integrating n-ψ instead of fitting the equilibrium solution.

https://doi.org/10.1371/journal.pcbi.1004479.g007

Experimental evidence [23] and modelling [22] suggest a logarithmic scaling of angular velocity , which provides the excitatory input to the LGMD. The n-ψ-model also predicts such logarithmic encoding of mainly as a consequence of the reversal potential V_exc > V_rest in Eq (9). When the excitatory input g_exc increases, then the driving potential V_exc − V(t) decreases. Thus, V_exc represents an upper limit to V(t), which is approached asymptotically. The corresponding (saturating)“approach curve” V(g_exc) is approximately logarithmically. This mechanism is identical to the one proposed in [22].