Artif Life Robotics (2009) 14:289–292 DOI 10.1007/s10015-009-0675-0

© ISAROB 2009

ORIGINAL ARTICLE

Yoshiyuki Sato · Kazuyuki Aihara

Integrative Bayesian model on two opposite types of sensory adaptation

Received and accepted: May 12, 2009

Abstract Adaptation is a fundamental property of human perception. Recently, it was found that there are two opposite types of adaptation to repetitive stimuli with a temporal difference. In this article, we construct an integrative model of adaptation. We model the perception as a Bayesian inference, and represent the two types of adaptation as changes in the likelihood function and the prior distribution in the Bayesian inference. We examine our model analytically and show how the types of adaptation depend on model parameters. Key words Bayesian inference · Lag adaptation · Bayesian calibration · Ventriloquism aftereffect

1 Introduction Our surrounding world is constantly changing. Our perception has to deal with these changes in our surroundings by adjusting their inner representations. In addition to those changes in the outer world, there are also changes in our bodies. For example, when we injure our eyes or ears, our perception would be impaired due to the change in the inner representation of the physically delivered stimuli in the brain. Such adaptation phenomena are important aspects of human perception, and they themselves are worth investigating. In addition, by investigating the properties of the adaptation of a particular type of perception or a motor system, its neural mechanism can often be deduced by psychophysical experiments and brain imaging experiments.1 Y. Sato (*) · K. Aihara Department of Complexity Science and Technology, Graduate School of Frontier Sciences, and Institute of Industrial Sciences, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan e-mail: [email protected] K. Aihara Aihara Complexity Modelling Project, ERATO, JST, Japan This work was presented in part at the 14th International Symposium on Artificial Life and Robotics, Oita, Japan, February 5–7, 2009

We showed in our previous work that the ventriloquism aftereffect, which is an adaptation phenomenon in audiovisual spatial perception, can be explained by updating the parameter that determines the mean value of the likelihood function that represents a noise distribution.2 By the ventriloquism aftereffect, the repeated stimuli are perceived to be presented at the same place. This type of adaptation is also observed in the adaptation to audiovisual temporal differences, i.e., subjects perceive the temporal differences in the adapting stimuli to be simultaneous.3 This type of adaptation is called the “lag adaptation.” Recently, however, an opposite type of adaptation has been found4 in tactile temporal adaptation; this adaptational effect was the opposite to lag adaptation, i.e., subjects were more unlikely to perceive simultaneity for the repeatedly presented stimuli. This result could be explained by assuming that subjects had learned the prior distribution of stimulus timing. This type of adaptation is called the “Bayesian calibration.” In our previous work,2 adaptation was modeled as the update of the mean values of likelihood functions. Therefore, lag adaptation can be considered to be changes in the likelihood functions. On the other hand, as Miyazaki et al.4 showed, Bayesian calibration can be considered to be changes in the prior distributions. In this work, we extend our model,2 investigate the interaction of these two types of adaptation, and show what parameters determine the types of adaptation.

2 Integrative Bayesian model of adaptation We consider an audiovisual localization task where sound and light with spatial disparity are presented, and the subjects determine which stimulus is presented on the right-hand side. If we plot the percentage of “sound right” response for various test stimulus disparities, we obtain a psychometric function. The center point of the psychometric function represents the disparity which subjects judged to be at the same location. During an adaptation period,

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stimuli with biased disparities are repeatedly presented according to a biased probability distribution. Then we measure the psychometric function again. The type of adaptation is represented as the difference in the center point of the psychometric function before and after the adaptation period. If the center point is shifted toward the adapting stimuli, it is the “lag adaptation” type, and if it is shifted in the opposite direction from the adapting stimuli, it is the “Bayesian calibration” type. We formalize the optimal observer that uses Bayesian inference to estimate the true positions of stimuli. We assume that the observer can only observe noisy positions of sound and light, denoted as yA and yV, respectively, that deviate from the true positions of the stimuli, denoted as xA and xV, respectively. The observer is supposed to determine estimators xˆA and xˆV from yA and yV by maximizing the posterior probability distribution P(xA, xV|yA, yV). We assume independence between the auditory and visual noises. Then, from Bayes’ theorem, it follows that P ( x A, xV yA, yV ) ∝ P ( yA x A ) P ( yV xV ) P ( x A, xV ) .

(1)

We model the adaptation by changing the mean values of the likelihood functions,2 P(yA|xA) and P(yV|xV), and the prior distribution P(xA, xV). We assume that the noises are Gaussian and that the prior probability of xA and xV depends only on their difference xA − xV. Thus, we assume that

relative adaptation effect at each step, and are assumed to satisfy 0 ≤ αi ≤ 1, where i ∈ {A, V, p}. We assume that the initial values of μˆ A and μˆ V are their true values, i.e., zero. We also assume that the initial value of μˆ p is zero.

3 Psychometric function Here, we derive the dependency of the center point of a psychometric function on model parameters. In our model, the observer’s task corresponds to judging the sign of xˆA − xˆV: if it is positive, the sound is on the right. Therefore, the probability that the observer’s response is “sound right” given a presented disparity Δx ≡ xA − xV is P(xˆA − xˆV > 0|xA − xV = Δx). As we will show later, this probability distribution does not depend on the absolute values of xA and xV but only on their difference Δx. Thus in our model, the psychometric function, denoted as Psycho(Δx), can be written as Psycho ( Δ x ) = P ( xˆ A − xˆ V > 0 x A − xV = Δ x ).

From previous experiments, it is known that the psychometric function can be approximated by a cumulative Gaussian distribution.4 Therefore, it can be written as Δx

Psycho ( Δ x ) = ∫ dΔ ′x N ( Δ ′x ; μ psycho, σ 2psycho ) −∞

P ( yV xV ) =

⎛ ( y − xV − μV ) ⎞ 1 exp ⎜ − V ⎟⎠ ⎝ 2σ V2 2 πσ V

(2)

P ( yA x A ) =

⎛ ( y − xA − μA ) ⎞ 1 exp ⎜ − A ⎟⎠ ⎝ 2σ 2A 2 πσ A

(3)

P ( x A, xV ) =

⎛ ( x A − xV − μ p ) ⎞ 1 exp ⎜ − ⎟ 2σ 2p 2 πσ p L ⎝ ⎠

(4)

2

2

2

where μA, μV, and μp represent the mean values of the distributions. We interpret the adaptational effect observed in psychophysical experiments as the false update of μA and μV due to the unnatural stimuli that subjects are exposed to, and the learning of μp of such unnatural stimuli. We assume that the real values of μA and μV are zero and are unchanged from their initial values, and that the observer knows the other parameters like σA, σV, and σp. Parameters σp and μp can be controlled by the experimenter. Each time the observer receives the adapting audiovisual stimulus, it estimates the corresponding parameters and updates its estimations of μV, μA, and μp based on observations and estimations. We denote these observer’s estimations of μV, μA, and μp as μˆ V, μˆ A, and μˆ p, respectively. The observer determines MAP estimators xˆV and xˆA from yV and yA, and updates μˆ V, μˆ A, and μˆ p as follows: (5)

μˆ V ( t + 1) = ( 1 − α V ) μˆ V ( t ) + α V ( yV − xˆ V )

(6)

μˆ p ( t + 1) = (1 − α p ) μˆ p ( t ) + α p ( xˆ A − xˆ V )

(7)

where μˆ A(t), μˆ V(t), and μˆ p(t) represent the observer’s estimations at time t. Parameters αA, αV, and αp determine the

(9)

where N(x; μ, σ2) represents a normal probability distribution of x with mean μ and variance σ2. Thus, by calculating P(xˆA − xˆV > 0|xA − xV = Δx) and comparing Eqs. 8 and 9, we can determine how the center point of the psychometric function, i.e., μpsycho, depends on the model parameters. By substituting Eqs. 2–4 into Eq. 1 and maximizing it, we obtain xˆ A =

1 σ 2 + σ 2p )( yA − μˆ A ) + σ 2A ( yV − μˆ V ) + σ 2Aμˆ p ) 2 (( V σ all

(10)

xˆ V =

1 σ 2 ( yA − μˆ A ) + (σ 2A + σ 2p )( yV − μˆ V ) − σ V2 μˆ p ) 2 ( V σ all

(11)

where σ2all ≡ σ2A + σ2V + σ2p. From Eqs. 10 and 11 we obtain σ 2p σ2 + σ2 Δˆ x = 2 Δ y − Δˆ μ + A 2 V μˆ p σ all σ all

(

)

(12)

where Δˆ x ≡ xˆA − xˆV, Δy ≡ yA − yV, and Δˆ μ ≡ μˆ A − μˆ V. Then we can calculate P(Δˆ x > 0|Δx) as follows: Δx ⎛ ⎞ σ2 + σ2 P Δˆ x > 0 Δ x = ∫ dΔ ′x N ⎜ Δ ′x ; Δˆ μ − A 2 V μˆ p, σ 2A + σ V2 ⎟ . −∞ σp ⎝ ⎠

(

)

(13) Thus, from Eqs. 9 and 13 we obtain σ +σ μ psycho = Δˆ μ − A 2 V μˆ p σp

(14)

σ 2psycho = σ 2A + σ V2 .

(15)

2

μˆ A ( t + 1) = (1 − α A ) μˆ A ( t ) + α A ( yA − xˆ A )

(8)

2

Now that we know how μpsycho depends on the model parameters and μˆ A, μˆ V, and μˆ p, we must next investigate the time course of μˆ s during the adaptation period and their

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converging values. Then we can show how the types of adaptation are determined.

4 Analysis of the behavior of the model It can be seen from Eqs. 5–7 that the update rules of μˆ A, μˆ V, and μˆ p are independent from each other given xˆA and xˆV. However, because xˆA and xˆV depend on μˆ A(t), μˆ V(t), and μˆ p(t), the values of μˆ s are not independently changed. By substituting Eqs. 10 and 11 into Eqs. 5–7, we obtain −a ⎞ ⎛ μˆ A ( t )⎞ ⎛ a ⎞ a ⎛ μˆ A ( t + 1)⎞ ⎛ 1 − a ⎜ μˆ V ( t + 1)⎟ = ⎜ v 1− v v ⎟ ⎜ μˆ V ( t )⎟ + ⎜ − v⎟ Δ y ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟⎜ p 1 − p⎠ ⎝ μˆ p ( t )⎠ ⎝ p ⎠ ⎝ μˆ p ( t + 1)⎠ ⎝ − p where a, v, and p are defined by a ≡ α A

(16)

σ 2A σ V2 v ≡ α , , and V 2 2 σ all σ all

σ 2p . 2 σ all Although, in reality, yA and yV are determined randomly from trial to trial, we can pursue the average behavior of the model by fixing each of yA and yV to its mean value during the adaptation period. We validate this assumption later by numerical simulations. From Eqs. 2–4 and our assumption that the true values of μA and μV are zero, the mean value of Δy ≡ yA − yV is Δadapt ≡ μp. We use the notation Δadapt to avoid confusion of μp with μˆ p. With this assumption, we can solve Eq. 16 explicitly with respect to t, which yields p ≡ αp

⎛ μˆ A ( t )⎞ ⎜ μˆ V ( t )⎟ ⎜ ⎟ ⎝ μˆ p ( t )⎠

⎛ a Δ − a Δ (1 − z)t ⎞ ⎜ z y z y ⎟ ⎜ ⎟ v v t = ⎜ − Δ y + Δ y ( 1 − z) ⎟ ⎜ z ⎟ z ⎜ p ⎟ p ⎜ Δ y − Δ y (1 − z)t ⎟ ⎝ z ⎠ z

5 Numerical simulations In deriving Eq. 17, we assumed Δy was constant with respect to t, and investigated the mean behavior of the model. Here, we validate this assumption using numerical simulations. The parameter values were set as follows: μp = 8°, σA = 8°, σV = 2.5°, σp = 1°, αA = 0.01, αV = 0.01, and αp = 0.05. At each time step, we sampled xA from a normal distribution with mean μp and variance σ2p, while xV was fixed to 0. We also sampled yA and yV according to the noise distributions in Eqs. 2 and 3. Then the model observer judged xˆA and xˆV based on Eqs. 10 and 11, and updated μˆ A, μˆ V, and μˆ A according to Eqs. 5–7. This procedure was repeated 1000 times. We also investigated the time course of μpsycho. At each time step, after updating all μˆ s, we measured the psychometric function using the updated μˆ s. We presented test stimuli with Δx from −30° to 30° in 1° steps, 1000 times each. Then we calculated μpsycho(t) by fitting the result to Eq. 9 by minimizing the mean squared error. Figure 1 shows an example of the simulation results for the time course of μˆ A, μˆ V, and μˆ p, together with the analytical results in Eq. 17. Figure 2 shows the simulation result for the time course of μpsycho, together with the analytical result

(17)

where z ≡ a + v + p. By definition, z satisfies 0 ≤ z ≤ 1, and we omit the case z = 0 because it means that all α s are zero, and the results are trivial. Then (1 − z)t converges to zero. From Eq. 17, we can also see that the converging speeds of all μˆ s are the same. Since αi represents the degree of adaptation at each step, it seems, at first sight, that the converging speeds are different if the αi’s are different. However, due to the interaction of all μˆ s, their converging speeds are the same. By substituting Eq. 17 into Eq. 14, we obtain μ psycho ( t ) = βΔ adapt − βΔ adapt (1 − z)

t

Fig. 1. Time course of μˆ A, μˆ V, and μˆ p. Lines without markers show numerical simulation results, and lines with markers show the corresponding analytical results

(18)

where μpsycho(t) is the center of the psychometric function measured with μˆ A(t), μˆ V(t), and μˆ p(t), and β is defined as follows: β≡

1 α σ 2 + α V σ V2 + α p (σ 2A + σ V2 )). 2 ( A A zσ all

(19)

Thus, the direction of the shift in the center point of the psychometric function relative to Δadapt is determined by the sign of β.

Fig. 2. Time course of μpsycho(t). The solid line shows a numerical simulation result, and the dashed line shows the corresponding analytical result

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in Eq. 18. These figures clearly show that the analytical results in Eqs. 17 and 18 correctly follow the average behavior of μˆ s or μpsycho.

Acknowledgments This work was partially supported by Grant-in-Aid for JSPS Fellows from JSPS (20·8988), and Grant-in-Aid for Scientific Research on Priority Areas – System study on higher-order brain functions – from MEXT, Japan (17022012).

6 Conclusion

References

In this article, we have constructed an integrative Bayesian model of adaptation, and investigated what factors determine the types of adaptation. We have shown that the types of adaptation are determined by the sign of β defined in Eq. 19. Parameters σA, σV, and σp can be measured or adjusted experimentally. Therefore, our model implies the possibility of experimentally controlling the types of adaptation by adjusting the parameters. However, it is not clear what determines the adaptation parameters αA, αV, and αp. The investigation of the meaning of these parameters is an important subject for future work.

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