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26 right-handed observers (female: 17, mean age: 22, range: 18–31) took part in this study, which was conducted with local ethics approval at the Department of Psychiatry and Psychotherapy, Charité Universitätsmedizin Berlin, Germany. All participants had normal or corrected-to-normal vision and provided informed written consent to participate in this study.
Due to technical problems, 7 out of 26 participants viewed the experiment at refresh rate of 75 Hz instead of 60 Hz, resulting in a slight overall increase in speed of presentation and consequently shorter block durations. For the sake of simplicity, we excluded these participants from analysis. Including these participants did not change the overall pattern of results. Given that one further participant had to be excluded for not following the experimental instructions correctly, the results presented here are based on 18 participants in total (female: 11, mean age: 22, range: 18–28).
Stimuli were generated using the Psychophysics Toolbox 3.0.9 [12] running under Matlab R2007b (Mathworks Inc., USA) and presented at 60 Hz for 19 participants, and at 75 Hz for the remaining 7 participants on a CRT monitor (SAMTRON 98 PDF, dimensions: 36.5 x 27.5cm, resolution: 1024 x 768 pixels) at a viewing distance of 60 cm. Participants observed moving Lissajous figures (Fig 1A), which were generated by the intersection of two sinusoids with perpendicular axes: x(t) = sin(at), y(t) = cos(bt+δ), with a = 3, b = 6, and δ increasing continuously from 0 to 2π.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0160772.g001 Lissajous figure and distribution of perceptual phase durations. (A) Lissajous are generated by the intersection of two sinusoids with perpendicular axes and increasing phase-shift whose frequency determines the speed of illusory 3D rotation. Following their introduction to experimental psychology, Lissajous figures were originally studied by means of twin-oscillators and analog cathode ray oscillographs in the 1940s and 1950s [6]. (B) Perceptual phase duration across all participants, runs and conditions. Distributions with a sharp rise and a long tail are typical of bistable perception.
The line width of the Lissajous figures was 0.10° visual angle. By presenting two identical figures separately to both eyes using a mirror stereoscope, we followed the experimental setup described by [9]. This enabled us to present a brief disambiguated training session at the beginning of the experiment, which we generated by introducing disparity cues (data not reported in results). We systematically varied shifting frequency and the size of the Lissajous figure in a 2 x 2 repeated measures design. The frequency at which the sinusoids were shifted against each other was set to 0.15 or 0.30 revolutions per second (rps). During one revolution, the sinusoids were shifted from 0 to 2π. Figure size was either 2.05° x 2.05° (250 x 250 pixels) or 4.10° x 4.10° (500 x 500 pixels). Our two experimental manipulations had distinct effects on five parameters (a-d) of the Lissajous figure (summarized in Table 1): A twofold increase in shifting frequency leads to a twofold increase in the number of self-occlusion events (a), the planar-speed of the sinusoids (d) and speed of illusory rotation of the Lissajous figure (e) as well as a twofold reduction of the duration of single self-occlusion events (b). Please note the total duration of self-occlusion events (c = a x b) remains unchanged by the manipulation in shifting frequency. Conversely, a twofold increase in stimulus size results in a twofold reduction in the duration of single self-occlusion events (b) and the total duration of self-occlusion events (c) as well as a twofold increase in planar speed of the sinusoids. The number of self-occlusion events and the speed of illusory rotation remain unchanged by the manipulation in stimulus size. In total, our two-factorial design yielded 2 x 2 = 4 conditions. During each run, which lasted 345 seconds, each condition was presented once in blocks of 80 seconds. Such blocks were separated by 5 seconds of fixation, and the order of conditions within each run was randomized. Participants completed 8–9 runs of the experiment, amounting to approximately 60 minutes of psychophysical testing per participant. Responses were recorded by a standard keyboard using the left arrow button for clock-wise (CW) and the right arrow button for counter-clock-wise (CCW) rotation of the Lissajous figure (viewed from top). The down arrow button was used to report unclear or mixed percepts. Participants were instructed to report their first perceived direction of rotation at stimulus onset with the first button press and indicate all further changes in perceived direction of rotation.
To analyse the effect of shifting frequency and stimulus size on perceptual transitions of the Lissajous figure, we first performed a conventional analysis. Next, we performed a novel Bayesian analysis in the aim to estimate the precision at which a current percept influences on perceptual decisions at the consecutive self-occlusion configuration of the Lissajous figure. In order to verify the common features of bistable Lissajous figures, we calculated the percentage of clear percepts (i.e., the number of CW and CCW responses divided by the total number of responses, including mixed percept responses), the distribution of perceptual phase durations across all conditions and participants and the distribution of button-presses relative to degrees of rotation of the Lissajous figure (ranging from 0° to 360°), i.e., to the phase shift of the two sinusoids (ranging from 0 to 2π). As our main dependent variable, we calculated mean transition probabilities, which reflect how fast percepts change [13]. Here, transition probabilities are defined by the number of perception transitions divided by the number of self-occluding events, thus denoting the probability that a self-occlusion is accompanied by a transition in perception. For every condition separately, these dependent variables were averaged first across runs and then across participants. Finally, the percentage of clear percepts and transition probabilities were submitted to repeated measures ANOVAs. We report partial eta squared (ηp2) as measure of effect size (SPSS 22.0 for Windows, IBM Corp. 2013).
For Bayesian analysis, we designed a generative model of bistable perception based on the prediction of perceptual outcomes on a trial-by-trial basis. (The term ‘trial’ refers to the interval between two consecutive self-occluding configurations of the Lissajous figure: Given that perceptual transitions of the Lissajous figure occur almost exclusively at self-occluding configurations, we defined this as the sampling rate in our model and down-sampled the participants’ responses accordingly. Thus, every trial t corresponds to the interval between the self-occluding configurations t and t + 1). This method frames perception as an inferential process in which perceptual decisions are based on posterior distributions. According to Bayes’ rule, such posterior distributions are derived from likelihood distributions representing the visual stimulation, and prior distributions reflecting a-priori knowledge about the visual world. Under Gaussian assumptions, the posterior distributions can be derived analytically [14]. Crucially, the impact of likelihood and prior on the posterior scales with their respective precision (i.e., the inverse variance). Bistable perception can be conceived to result from the sampling of a bimodal posterior distribution [15], which in turn originates from the combination of a bimodal likelihood with unimodal prior distributions. For the context of this experiment, we assume that the current percept constitutes a ‘perceptual stability’-prior for the visual system, the mean of which corresponds to the current percept, while its impact on visual perception is represented by its precision that we estimate for every condition separately. Importantly, we allow the prior precision to be affected by prediction error signals, which enables the modelling of a dynamic process eliciting transitions in perception. Furthermore, it allows for the estimation of the strength of the implicit prediction of perceptual stability (as represented by the precision of the prior distribution ‘perceptual stability’) and thus mirrors the idea of “representational momentum”. A schematic depiction of the modelling procedure—which we describe in detail below—can be found in Fig 2.
Figure data removed from full text. Figure identifier and caption: 10.1371/journal.pone.0160772.g002 Bayesian modeling procedure. (A) The prior distribution ‘perceptual stability’ is defined by its mean μstability (corresponding to the perceptual decision made at the last self-occluding configuration, see ‘Percept update’-arrow) and its precision πstability. If a new percept was reported at the preceding trial, πstability is set to πinit produce a weighted bimodal distribution P(θ), which is in a next step transformed by a unit-sigmoid function determined by parameter ζ and used to predict the perceptual outcome θ. The difference between P(θ = 1) and θ constitutes the prediction error (PE). (B) In this illustration with exemplary Gaussian probability distributions, the prior distribution ‘perceptual stability’ is defined by its mean μstability (blue line) and its precision πstability (the inverse of its variance depicted in green). This prior distribution is combined with a bimodal likelihood distribution. The weighted bimodal distribution is used to predict the percept indicated by the subject at that trial (defined by its mean θ depicted in cyan and the inverse of its variance shown in purple). The difference between the weighted bimodal distribution and the percept is highlighted in red and constitutes a prediction error signal, which is used to adjust the prior distribution ‘perceptual stability’.
In this experiment, we presented moving Lissajous figures which could be perceived as rotating either CW or CCW. In our modelling analysis, we represent this stimulus by a bimodal likelihood distribution, while the two alternative percepts are given by: θ=1:→(rotation)0:←(rotation)(1) The mean of the prior distribution ‘perceptual stability’ (μstability) at trial t was defined by the percept indicated by the participant at the preceding trial: μstability(t)=θ(t-1)(2) The precision of the prior distribution ‘perceptual stability’ (πstability) in trial t was defined as follows: If a new perceptual decision was made in trial t − 1 (i.e., t − 1 = t0), πstability was set to an intial precision πinit. In all other cases, πstability was calculated by subtracting a precision-weighted prediction error (PE) from the precision at the preceding trial: πstability(t=t0)=πinit(3) πstability(t≠t0)=πstability(t-1)*exp(-πsensoryπstability(t-1)|PE(t-1)|)(4) The precision-weight is given by the fraction between the precision of the sensory representation πsensory and the current πstability. We derived a posterior probability of CW rotation P(θ = 1) by weighting the bimodal likelihood distribution with the prior distribution πstability [15]: r=P(θ=0)P(θ=1)=exp((θ0-μstability)2-(θ1-μstability)2πstability-2)(5) P(θ=1)=1r+1(6) In order to predict participants’ responses θ(t), we applied a unit sigmoid function to P(θ = 1): θ(t)=P(θ=1)ζP(θ=1)ζ+(1-P(θ=1))ζ(7) The prediction error was calculated by subtracting P(θ = 1) from actual percept indicated by the participants’ θ: PE(t)=θ(t)-P(θ=1)(t)(8) We used Bayesian model inversion to estimate the precision of the initial precision of the prior distribution ‘perceptual stability’ (πinit) separately for all conditions. This method maximises the log-model evidence by minimising the surprise about the data of individual participants as approximated by negative free energy [16]. For model inversion, precisions were modelled as log-normal distributions. πinit was estimated as a free parameter for the four conditions separately, each of which had a prior mean of log(1) and a prior variance of 1. All other parameters were fixed (i.e., a prior variance of 0) and set to πsensory = 1 and ζ = 1 in both models. We compared this model (model A) to a control model (model B) in which πinit and πsensory were set fixed to zero and thus effectively removed from the model. Parameters were optimised using the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno minimisation algorithm as implemented in the HGF 4.0 toolbox (distributed as part of the TAPAS toolbox, http://www.translationalneuromodeling.org/tapas/). For model level inference, we calculated exceedance probabilities (i.e., the probability that model A is better at explaining the observed data than model B) using random effects Bayesian model selection [17] as implemented in SPM12 (http://www.fil.ion.ucl.ac.uk/spm/software/spm12/). From the winning model, we report posterior parameter estimates for πinit separately for each condition, averaged first across runs and then across participants.
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