# Apparent Motion Perception in the Praying Mantis: Psychophysics and Modelling

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Results

#### 2.1. Experimental Findings

#### 2.2. Modeling and Simulation

#### 2.2.1. Model 1: Fourier Energy

#### 2.2.2. Model 2: Reichardt Detector

#### 2.2.3. Model Comparison

## 3. Discussion

## 4. Methods

#### 4.1. Insects

#### 4.2. Experimental Setup

#### 4.3. Experimental Procedure

#### 4.4. Visual Stimulus

#### 4.5. Calculating Dmax

#### 4.6. Simulation and Model Fitting

#### 4.6.1. Model 1

`fft`and

`trapz`in Matlab (Mathworks Inc, Natick, MA, USA). The threshold level (T) was fitted in Matlab using the function

`fminsearch`.

#### 4.6.2. Model 2

`fminsearch`in Matlab (Mathworks, Inc., Natick, MA, USA). Repeated fitting attempts from randomly selecting initial solutions resulted in the same final solution (within numerical error) in the overwhelming majority of cases, confirming that the algorithm had located the global optimum rather than being stuck at a local minimum.

## 5. Data Availability

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Examples of the random chequerboard pattern stimuli used in the experiment. Each column of panels shows still frames of a random chequerboard pattern that moves across a screen. The pattern is displaced by a fixed step ($\Delta x$) on $\Delta t$ intervals where the speed $v=\Delta x/\Delta t$ is constant (the red arrow at the top of each panel points to a reference point for easier visualization). The perception of motion in these patterns is strongly dependent on the displacement step ($\Delta x$) relative to the pattern’s chequer/element size. When $\Delta x$ is comparable to the element size, humans perceive the pattern as moving smoothly and can identify its direction reliably (

**column 1**). Increasing $\Delta x$ causes the pattern motion to appear more “jerky” and makes its direction harder to identify (

**column 2**). Because the perception of motion is dependent on the ratio between the element and step size, increasing the element size to match the step size makes the patterns appear to be moving smoothly again (

**column 3**).

**Figure 2.**Spacetime plots and Fourier spectra for stimuli with different step sizes (10 deg elements). The top panels show space–time plots and the bottom panels show corresponding spatiotemporal Fourier spectra for a number of step sizes (the element size is 10 deg in all plots). When the step size is much less than the element size (left column), the stimulus is perceived by humans as moving smoothly and its motion energy is situated in quadrants 1 and 3 in the Fourier spectra (i.e., motion in the rightwards direction). In the middle column (step size equals element size), humans see the stimulus as moving less smoothly and part of its motion energy is now distributed in quadrants 2 and 4 in the Fourier domain (leftward motion). Finally, when the step size is much larger than the element size (right column), motion energy is more evenly distributed across the four Fourier quadrants, and humans find it significantly more difficult to identify the motion direction.

**Figure 3.**Optomotor response rates against step size for various element sizes. Error bars are 95% confidence intervals calculated using simple binomial statistics. Each panel shows the pooled responses of 13 mantids to a moving chequerboard stimulus of a given element size. The data and corresponding psychometric fits show that motion detection becomes increasingly more difficult (i.e., the response rate decreases) as the step size increases and the step size corresponding to a 50% response rate ($\mathrm{Dmax}$) increases with element size.

**Figure 4.**Relationship between Dmax and element size. Calculated $\mathrm{Dmax}$ values and power law fits for individual mantids (

**left**) and pooled data (

**right**). The points appear to lie on a straight line on this log-log plot indicating that $\mathrm{Dmax}$ can be described accurately as a power law function of element size x, determined via fitting as $\mathrm{Dmax}\left(x\right)=5.36\times {x}^{0.462}$ (standard error of the regression $S=0.318$${}^{\circ}$) for pooled data. Individual power law fits (

**left**) had a mean S of 1.12${}^{\circ}$ with $\sigma =$ 0.714${}^{\circ}$.

**Figure 5.**Effect of aliasing on the distribution of motion energy in the Fourier domain. A pattern moving coherently at a constant speed (v) has spatial and temporal frequency components along the line ${f}_{t}/{f}_{s}=v$, and its motion energy content is in quadrants 1 and 3 that signify positive motion (

**left**panel). When the pattern is moved in steps of $\Delta x$ at the same speed, components with temporal frequencies higher than $v/\Delta x$ get aliased and cast towards lower frequencies (

**right**panel). Aliased components are distributed in the four quadrants, and their net motion energy is therefore close to zero, leaving the energy of unaliased components as a close approximation of what remains in quadrants 1 and 3.

**Figure 6.**Apparent motion detection model based on Fourier Energy (Model 1). In this model, the spatiotemporal Fourier transform of the stimulus is filtered by a $\mathrm{Dmax}$ aliasing window (whose extent is $\Delta f=1/\Delta x$ cpd, where $\Delta x$ is presentation step size) and multiplied with the spatiotemporal sensitivity of an (ideal) motion detector. The net energy is then summed and passed through a ternary threshold to obtain a decision on the motion direction. The red and blue rectangles in filter blocks represent the spatiotemporal regions with positive and negative sensitivities, respectively.

**Figure 7.**Simple motion energy-based model of $\mathrm{Dmax}$ in the praying mantis (Model 1). (

**Left**) the total motion energy of unaliased pattern components contained within a frequency window ($[-1/\Delta x,1/\Delta x]$) decreases as the step size ($\Delta x$) is increased. Patterns of larger elements have a larger portion of their motion energy within lower frequencies and therefore contain more energy at large step sizes. According to this model, $\mathrm{Dmax}$ is the step size corresponding to a given motion energy threshold (chosen here to be ∼${10}^{5}$ units); (

**Right**) This simple model is in good agreement with our experimental results for the mantis (standard error of the regression, $S=0.613$${}^{\circ}$).

**Figure 8.**The Reichardt Detector. The spatial input from two identical Gaussian filters (standard deviation $\sigma $) separated by $\Delta x$ is passed through high and low pass temporal filters (HP and LP, respectively). The LP output in each subunit is cross-correlated with the HP output from the other subunit using a multiplication stage (M), and the two products are then subtracted to produce a direction-sensitive measure of motion. In the RD-based Model 2 (Figure 9), the spatial filter parameters are ${\sigma}_{1}$ = 1.07${}^{\circ}$, $\Delta {x}_{1}$ = 1.0${}^{\circ}$ for detector class 1, and ${\sigma}_{2}$ = 3.42${}^{\circ}$, $\Delta {x}_{2}$ = 2.5${}^{\circ}$ for detector class 2, while the low and high pass filter time constants are ${\tau}_{L}$ = 13 ms, ${\tau}_{H}$ = 40 ms, respectively.

**Figure 9.**Optomotor response model based on the Reichardt Detector (Model 2). The model has two arrays of detectors that are positioned in random locations across a simulated 1D retina. Detectors within each array share the same subunit separation and spatial filter extent (and are thus all tuned to the same spatial frequency). Having two detector classes enables the model to respond to stimuli across a range of spatial frequencies broader than that of a single detector. The output of each detector is temporally integrated and then converted to a “vote” for leftward, rightward, or no motion by a hard threshold. The votes are subsequently combined, thresholded, and then passed through another temporal integrator and hard threshold blocks that model the decision process of the human observer in the experiment.

**Figure 10.**Predictions of Model 2 against experimental results. The mean (solid line) and interquartile range of Model 2 (dashed lines, based on 50 simulated trials per data point), showed good agreement with our experimental observations (symbols, with error-bars showing 95% confidence intervals).

**Figure 11.**Comparison between Models 1 and 2. Plots of $\mathrm{Dmax}$ vs. element size for Models 1 and 2 show that both are in good agreement with experimental results (Model 1 has a standard error of the regression of $S=0.613$${}^{\circ}$ and Model 2 has $S=0.967$${}^{\circ}$).

**Figure 12.**Comparison between $\mathrm{Dmax}$ in humans and mantids. Mantis $\mathrm{Dmax}$ increases with the element size, similar to humans, but does not appear to have a lower limit below which it remains constant (in humans, this limit is about 15 arcmin = 0.25 deg). The relationship between mantis $\mathrm{Dmax}$ and element size is well accounted for by a power law function with an exponent of 0.462 (blue line). Human data from [3] is plotted for comparison. Morgan interpreted the $\mathrm{Dmax}$ values he observed as scaling proportionately for element sizes larger than 0.25 deg (i.e., assumes a power law exponent of 1, green line) although his data appears to be accounted for equally well by a power law with an exponent of 0.462 (red line), similar to mantids. It is, therefore, unclear as to whether a true difference in power law exponents exists between humans and mantids.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Tarawneh, G.; Jones, L.; Nityananda, V.; Rosner, R.; Rind, C.; Read, J.C.A.
Apparent Motion Perception in the Praying Mantis: Psychophysics and Modelling. *Vision* **2018**, *2*, 32.
https://doi.org/10.3390/vision2030032

**AMA Style**

Tarawneh G, Jones L, Nityananda V, Rosner R, Rind C, Read JCA.
Apparent Motion Perception in the Praying Mantis: Psychophysics and Modelling. *Vision*. 2018; 2(3):32.
https://doi.org/10.3390/vision2030032

**Chicago/Turabian Style**

Tarawneh, Ghaith, Lisa Jones, Vivek Nityananda, Ronny Rosner, Claire Rind, and Jenny C. A. Read.
2018. "Apparent Motion Perception in the Praying Mantis: Psychophysics and Modelling" *Vision* 2, no. 3: 32.
https://doi.org/10.3390/vision2030032