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192 nips-2006-Theory and Dynamics of Perceptual Bistability

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Author: Paul R. Schrater, Rashmi Sundareswara

Abstract: Perceptual Bistability refers to the phenomenon of spontaneously switching between two or more interpretations of an image under continuous viewing. Although switching behavior is increasingly well characterized, the origins remain elusive. We propose that perceptual switching naturally arises from the brain’s search for best interpretations while performing Bayesian inference. In particular, we propose that the brain explores a posterior distribution over image interpretations at a rapid time scale via a sampling-like process and updates its interpretation when a sampled interpretation is better than the discounted value of its current interpretation. We formalize the theory, explicitly derive switching rate distributions and discuss qualitative properties of the theory including the effect of changes in the posterior distribution on switching rates. Finally, predictions of the theory are shown to be consistent with measured changes in human switching dynamics to Necker cube stimuli induced by context.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Perceptual Bistability refers to the phenomenon of spontaneously switching between two or more interpretations of an image under continuous viewing. [sent-9, score-0.289]

2 Although switching behavior is increasingly well characterized, the origins remain elusive. [sent-10, score-0.222]

3 We propose that perceptual switching naturally arises from the brain’s search for best interpretations while performing Bayesian inference. [sent-11, score-0.517]

4 In particular, we propose that the brain explores a posterior distribution over image interpretations at a rapid time scale via a sampling-like process and updates its interpretation when a sampled interpretation is better than the discounted value of its current interpretation. [sent-12, score-0.611]

5 We formalize the theory, explicitly derive switching rate distributions and discuss qualitative properties of the theory including the effect of changes in the posterior distribution on switching rates. [sent-13, score-0.738]

6 Finally, predictions of the theory are shown to be consistent with measured changes in human switching dynamics to Necker cube stimuli induced by context. [sent-14, score-0.688]

7 1 Introduction Our visual system is remarkably good at producing consistent, crisp percepts of the world around us, in the process hiding interpretation uncertainty. [sent-15, score-0.331]

8 Perceptual bistability is one of the few circumstances where ambiguity in the visual processing is exposed to conscious awareness. [sent-16, score-0.409]

9 Spontaneous switching of perceptual states frequently occurs during continuously viewing an ambiguous image, and when a new interpretation of a previously stable stimuli is revealed (as in the sax/girl in ﬁgure ? [sent-17, score-0.722]

10 Moreover, although perceptual switching can be modulated by conscious effort[? [sent-21, score-0.481]

11 (b) can be interpreted as a cube viewed from two different viewpoints. [sent-25, score-0.217]

12 Stimuli that produce bistability are characterized by having several distinct interpretations that are in some sense equally plausible. [sent-26, score-0.44]

13 Given the successes of Bayesian inference as a model of perception ∗ http://www. [sent-27, score-0.179]

14 ]), these observations suggest that bistability is intimately connected with making perceptual decisions in the presence of a multi-modal posterior distribution, as previously noted by several authors[? [sent-32, score-0.673]

15 In fact, most explanations of bistability have been historically rooted in proposals about the nature of neural processing of visual stimuli, involving low-level visual processes like retinal adaptation and neural fatigue[? [sent-36, score-0.413]

16 However, the abundance of behavioral and brain imaging data that show high level inﬂuences on switching (like intentional control which can produce 3-fold changes in alternation rate[? [sent-40, score-0.441]

17 The goal of this paper is to provide a simple explanation for the origins of bistability based on general principles that can potentially handle both top-down and bottom-up effects. [sent-43, score-0.294]

18 2 Basic theory The basic ideas that constitute our theory are simple and partly form standard assumptions about perceptual processing. [sent-44, score-0.369]

19 Perception performs Bayesian inference by exploring and updating the posterior distribution across time by a kind of sampling process (e. [sent-46, score-0.337]

20 Conscious percepts result from a decision process that picks the interpretations by ﬁnding sample interpretations with the highest posterior probability (possibly weighted by the cost of making errors). [sent-51, score-0.635]

21 The results of these decisions and their associated posterior probabilities are stored in memory until a better interpretation is sampled. [sent-53, score-0.433]

22 The posterior probability associated with the interpretation in memory decays with time. [sent-55, score-0.473]

23 The intuition behind the model is that most percepts of objects in a scene are built up across a series of ﬁxations. [sent-56, score-0.106]

24 When an object previously ﬁxated is eccentrically viewed or occluded, the brain should store the previous interpretation in memory until better data comes along or the memory becomes too old to be trusted. [sent-57, score-0.524]

25 Finally, the interpretation space required for direct Bayesian inference is too large for even simple images, but sampling schemes may provide a simple way to perform approximate inference. [sent-58, score-0.211]

26 The theory provides a natural interface to interpret both high-level and low-level effects on bistability, because any event that has an impact on the relative heights or positions of the modes in the posterior can potentially inﬂuence durations. [sent-59, score-0.398]

27 For example, patterns of eye ﬁxations have long been known to inﬂuence the dominant percept[? [sent-60, score-0.094]

28 Because eye movement events create sudden changes in image information, it is natural that they should be associated with changes in the dominant mode. [sent-62, score-0.343]

29 Similarly, control of information via selective attention and changes in decision thresholds offer concrete loci for intentional effects on bistability. [sent-63, score-0.199]

30 3 Analysis To analyze the proposed theory, we need to develop temporal distributions for the maxima of a multimodal posterior based on a sampling process and describe circumstances under which a current sample will produce an interpretation better than the one in memory. [sent-64, score-0.655]

31 First we develop a general approximation to multi-modal posterior distributions that can vary over time, and analyze the probability that a sample from the posterior are close to maximal. [sent-66, score-0.386]

32 We then describe how the samples close to the max interact with a sample in memory with decay. [sent-67, score-0.178]

33 shape for Necker Cube) at time t, θt,i is the location of the maxima of the ith mode, Dt is the most recent data, D0:t−∆t is the data history, ∗ and Pi (θ|D0:t−∆t ; θt,i ) is the predictive distribution (prior) for the current data based on recent experience1. [sent-71, score-0.11]

34 Thus, samples from a posterior mode will be approximately χ2 distributed near the maxi−1 mum with effective degrees of freedom n given by the number of signiﬁcant eigenvalues of Ii . [sent-73, score-0.471]

35 2 Essentially n encodes the effective degrees of freedom in interpretation space. [sent-74, score-0.268]

36 Given these assumptions, we can approximate the probability distribution for update events. [sent-77, score-0.113]

37 Because variances add, the average effect on the distance ωt is a linear increase: ωt = ω0 + ρσt, where ρ is the sampling rate. [sent-85, score-0.146]

38 These disturbances could represent changes in the local of the maxima of the posterior due to the incorporation of new data, neural noise, or even discounting (note that linearly increasing ωt corresponds to exponential or multiplicative discounting in probability). [sent-86, score-0.322]

39 1 Because time is critical to our arguments, we assume that the posterior is updated across time (and hence new data) using a process that resembles Bayesian updating. [sent-87, score-0.23]

40 2 For the Necker cube, the interpretation space can be thought of as the depths of the vertices. [sent-88, score-0.104]

41 A strong prior assumption that world angles between vertices are close to 90deg produces two dominate modes in the posterior that correspond to the typical interpretations. [sent-89, score-0.203]

42 To understand the behavior of this memory process, notice that every mθ (0) must be within distance ξ of the maximum of the posterior for an update to occur. [sent-92, score-0.444]

43 Due to the properties of extrema of distributions of χ2 random variables, an mθ (0) will be (in expectation) a characteristic distance µm (ξ) below ξ and for t > 0 drifts with linear dynamics3. [sent-93, score-0.098]

44 This suggests the approximation, p(ωt < ξ) ≈ δ(µm + ρσt − ωt ), which can be formally justiﬁed because p(ωt < ξ) will be i highly peaked with respect to the distribution of the sampling process p(δτ ). [sent-94, score-0.276]

45 Finally assuming slow 4 drift means (1 − P (ωt < ξ)) ≈ 0 on the time scale that transitions occur . [sent-95, score-0.093]

46 reduces to: t P (Mti < min{ξ, ωt }) = 0 i p(δτ < ωt )δ(µm + ρσt − ωt )P (i|τ )dτ ≈ P (Mti < µm + ρσt)P (i) where P (i) is the average frequency of sampling from the i th (5) (6) mode. [sent-98, score-0.107]

47 Extrema of the posterior sampling process If the sampling process has no long-range temporal dependence, then under mild assumptions the distribution of extrema converge in distribution5 to one of three characteristic forms that depend only on the domain of the random variable[? [sent-99, score-0.672]

48 Set N = ρt and let ρ = 1 for 2 convenience, where ρ is the effective sampling rate, and equation ? [sent-102, score-0.142]

49 In particular, for n > 4 and µm (ξ) relatively small, the distribution has a gamma-like behavior, where new memory update transitions are suppressed near recent transitions. [sent-106, score-0.344]

50 This behavior shows the effect of the decision threshold, as without a decision threshold the asymptotic behavior of simple sampling schemes will generate approximately exponentially distributed update event times, as a consequence of extreme value theory. [sent-108, score-0.477]

51 Note that the time scale of events can be arbitrarily controlled by appropriately selecting ρ (controls the time scale of the sampling process) and σ (controls the time scale of the memory decay process). [sent-113, score-0.428]

52 Effects of posterior parameters on update events The memory update distributions are effected primarily by two factors, the log posterior heights and their difference ∆kij = ki − kj , and the effective number of degrees of freedom per mode n. [sent-114, score-1.256]

53 Effect of ki , ∆kij The variable ∆kij has possible effects both on the probability that a mode is sampled, and the temporal distributions. [sent-115, score-0.354]

54 When the modes are strongly peaked (and the sampling procedure is unbiased) log P (i) ≈ ∆kij . [sent-116, score-0.249]

55 Finally, if the posterior becomes more peaked while the threshold remains ﬁxed, the update rates should increase and the temporal distributions will move toward exponential. [sent-118, score-0.56]

56 If we assume increased viewing time makes the posterior more peaked, then our model predicts the common ﬁnding of increased transition rates with viewing duration. [sent-119, score-0.46]

57 3 In the simulations, µm is chosen as the expected value of the set of events below the threshold xi Conversely fast drift in the limit means P (ωt < ξ) ≈ 0, which results in transitions entirely determined by the minima of the sampling process and ξ. [sent-120, score-0.476]

58 Time to next update > t (P(T>t)) 1 Effective degrees of freedom n=8 n=4 n=2 0. [sent-122, score-0.202]

59 1 0 0 500 1000 1500 2000 2500 3000 3500 4000 Time(t) (in number of samples) Figure 2: Examples of cumulative distribution functions of memory update times. [sent-131, score-0.251]

60 Solid curves are generated by simulating the sampling with decision process described in the text. [sent-132, score-0.274]

61 Effect of n One of the surprising aspects of the theory above is the strong dependence on the effective number of degrees of freedom. [sent-136, score-0.147]

62 The theory makes a strong prediction that stimuli that have more interpretation degrees of freedom will have longer durations between transitions, which appears to be qualitatively true across both rivalry and bistability experiments[? [sent-137, score-0.786]

63 Relating theory to behavioral data via induced Semi-Markov Renewal Process Assuming that memory update events involving transitions to the same mode are not perceptually accessible, only update events that switch modes are potentially measurable. [sent-140, score-1.034]

64 However, the process described above ﬁts the description of a generator for a semi-Markov renewal process. [sent-141, score-0.283]

65 A semi-Markov renewal process involves Markov transitions between discrete states i and j determined by a matrix with entries Pij , coupled with random durations spent in that state sampled from time distributions Fij (t). [sent-142, score-0.58]

66 The product of these distributions Qij (t) = Pij Fij (t) is the generator of the process, that describes the conditional probability of ﬁrst transition between states i → j in time less than t, given ﬁrst entry into state i occurs at time t = 0. [sent-143, score-0.31]

67 In the theory above, Fij (t) = Fii (t) = P (Ti < t), while Pij = Pjj = P (j)6 The main reason for introducing the notion of a renewal process is that they can be used to express the relationship between the theoretical distributions and observable quantities. [sent-144, score-0.345]

68 The most commonly collected data are times between transitions and (possibly contingent) percept frequencies. [sent-145, score-0.213]

69 Let the state s(t) = i refer to when the memory process is in the support of mode i: mθ (t) ∈ Si at time t. [sent-148, score-0.455]

70 Let P (0), denote the probability of sampling from mode 0. [sent-151, score-0.303]

71 Moreover, 6 The independence relations are a consequence of an assumption of independence in the sampling procedure, and relaxing that assumption can produce state contingencies in Qij (t). [sent-153, score-0.235]

72 MCMC-like sampling with large steps) can create contingencies in the frequency of sampling from the ith mode that will produce a non-independent transition matrix Pij = P (θt ∈ Si |θt−ρ∆t ∈ Sj ). [sent-157, score-0.606]

73 for gamma-like distributions, the convolution integral tends to increase the shape parameter, which means that gamma parameter estimates produced by ﬁtting transition durations will overestimate the amount of ’memory’ in the process7 . [sent-158, score-0.369]

74 Finally note the limiting behavior as P (0) → 0, G01 (t) = P (T0 < t), so that direct measurement of the temporal distributions is possible but only for the (almost) supressed perceptual state. [sent-159, score-0.372]

75 To bias perception of a bistable stimuli, we had observers view a Necker cube ﬂanked with ’ﬁelds of cubes’ that are perceptually unambiguous and match one of the two percepts (see ﬁgure ? [sent-161, score-0.645]

76 Subjects are typically biased toward seeing the Necker cube in the “looking down” state (65-70% response rates), and the context stimuli shown in ﬁgure ? [sent-164, score-0.45]

77 We found that the looking up context, boosts “looking up” response rates from 30% to 55%. [sent-167, score-0.219]

78 1 Methods Subject’s perceptual state while viewing the stimuli in ﬁg. [sent-169, score-0.42]

79 Each observer viewed 100 randomly generated context stimuli and each stimulus was viewed long enough to acquire 10 responses (taking 10-12 sec on average). [sent-177, score-0.174]

80 (a) An instance of the “Looking down” con- (b) An instance of the “Looking up” context text with the Necker cube in the middle with the Necker cube in the middle Figure 3: The two ﬁgures are examples of the “Looking down” and “Looking up” context conditions. [sent-179, score-0.534]

81 2 Results We measured the effect of context on estimates of perceptual switching rates, Ri = P (s(t) = i), ﬁrst transition durations Gij , and survival probabilities Pii = P (s(t) = i|s(0) = i) by counting the number of events of each type. [sent-181, score-1.026]

82 Additionally, we ﬁt a semi-Markov renewal process Qij (t) = Pij Fij (t) to the data using a sampling based procedure. [sent-182, score-0.356]

83 For ease of sampling, Fij (t) were gamma with separate parameters for each of the four conditionals {00, 01, 10, 11}, resulting in 10 parameters 7 gamma shape parameters are frequently interpreted as the number of events in some abstract Poisson process that must occur before transition overall. [sent-184, score-0.541]

84 The process was ﬁt by iteratively choosing parameter values for Qij (t), simulating response data and measuring the mismatch between the simulated and human Gij and Pii distributions. [sent-185, score-0.17]

85 The effect of context on Gij and Pii is shown in Fig. [sent-186, score-0.089]

86 of ﬁrst transition and the survival probability of the “Looking down” percept. [sent-218, score-0.309]

87 of ﬁrst transition and conditional survival probability of the “Looking Up” percept. [sent-220, score-0.309]

88 A semi-Markov renewal process with transition paramters Pij , gamma means mij and gamma variances vij was ﬁt to all the data via max. [sent-221, score-0.533]

89 ] also presents a theory for average transitions times in bistability based on random processes and a multi-modal posterior distribution, their theory is fundamentally different as it derives switching events from tunneling probabilities that arise from input noise. [sent-248, score-1.0]

90 Moreover, their theory predicts increasing transition times as the posterior becomes increasingly peaked, exactly opposite our predictions. [sent-249, score-0.388]

91 In conclusion, we have presented a novel theory of perceptual bistability based on simple assumptions about how the brain makes perceptual decisions. [sent-250, score-0.903]

92 In addition, results from a simple experiment show that manipulations which change the dominance of a percept produce coupled changes in the probability of transition events as predicted by theory. [sent-251, score-0.543]

93 We believe the strength of the theory is that it can make a large set of qualitative predictions about the distribution of transition events by coupling transition times to simple properties of the posterior distribution. [sent-253, score-0.72]

94 Our theory suggests that the basic descriptive model sufﬁcient to capture perceptual bistability is a semi-Markov renewal process, which we showed could successfully simulate the temporal dynamics of human data for the Necker cube. [sent-254, score-0.898]

95 Distributions of alternation rates in various forms of bistable perception. [sent-272, score-0.155]

96 Are switches in perception of the Necker cube related to eye position? [sent-280, score-0.45]

97 T (1994) The generic viewpoint assumption in a framework for visual perception Nature vol. [sent-283, score-0.221]

98 (1992) Prime Time: Fatigue and set effects in the perception of reversible ﬁgures. [sent-313, score-0.256]

99 (2006) Noise characteristics and prior expectations in human visual speed perception Nature Neuroscience vol. [sent-333, score-0.265]

100 (2005) Dynamics of perceptual bi-stability for stereoscopic slant rivalry. [sent-365, score-0.277]

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