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264 nips-2013-Reciprocally Coupled Local Estimators Implement Bayesian Information Integration Distributively

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Author: Wenhao Zhang, Si Wu

Abstract: Psychophysical experiments have demonstrated that the brain integrates information from multiple sensory cues in a near Bayesian optimal manner. The present study proposes a novel mechanism to achieve this. We consider two reciprocally connected networks, mimicking the integration of heading direction information between the dorsal medial superior temporal (MSTd) and the ventral intraparietal (VIP) areas. Each network serves as a local estimator and receives an independent cue, either the visual or the vestibular, as direct input for the external stimulus. We ﬁnd that positive reciprocal interactions can improve the decoding accuracy of each individual network as if it implements Bayesian inference from two cues. Our model successfully explains the experimental ﬁnding that both MSTd and VIP achieve Bayesian multisensory integration, though each of them only receives a single cue as direct external input. Our result suggests that the brain may implement optimal information integration distributively at each local estimator through the reciprocal connections between cortical regions. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 cn 1 Abstract Psychophysical experiments have demonstrated that the brain integrates information from multiple sensory cues in a near Bayesian optimal manner. [sent-8, score-0.35]

2 We consider two reciprocally connected networks, mimicking the integration of heading direction information between the dorsal medial superior temporal (MSTd) and the ventral intraparietal (VIP) areas. [sent-10, score-0.563]

3 Each network serves as a local estimator and receives an independent cue, either the visual or the vestibular, as direct input for the external stimulus. [sent-11, score-0.432]

4 We ﬁnd that positive reciprocal interactions can improve the decoding accuracy of each individual network as if it implements Bayesian inference from two cues. [sent-12, score-0.607]

5 Our model successfully explains the experimental ﬁnding that both MSTd and VIP achieve Bayesian multisensory integration, though each of them only receives a single cue as direct external input. [sent-13, score-0.409]

6 Our result suggests that the brain may implement optimal information integration distributively at each local estimator through the reciprocal connections between cortical regions. [sent-14, score-0.631]

7 For instance, while walking, we perceive heading direction through either the visual cue (optic ﬂow), or the vestibular cue generated by body movement, or both of them [1, 2]. [sent-16, score-1.092]

8 In order to achieve an accurate or improved representation of the input information, it is critical for the brain to integrate information from multiple sensory modalities. [sent-18, score-0.211]

9 Consider the task of inferring heading direction θ based on the visual and vestibular cues. [sent-20, score-0.604]

10 Suppose that with a single cue cl (l = vi, ve correspond to the visual and the vestibular cues, respectively), the estimation of heading direction satisﬁes the 2 Gaussian distribution p(cl |θ), which has the mean µl and the variance σl . [sent-21, score-0.949]

11 Under the condition that noises from different cues are independent to each other, the Bayes’ theorem states that p(θ|cvi , cve ) ∝ p(cvi |θ)p(cve |θ)p(θ), (1) where p(θ|cvi , cve ) is the posterior distribution of the stimulus when two cues are presented, and p(θ) the prior distribution. [sent-22, score-0.961]

12 , p(θ) is uniform, p(θ|cvi , cve ) also 1 satisﬁes the Gaussian distribution with the mean and variance given by µb = 1 2 σb = 2 σ2 σve µvi + 2 vi 2 µve , 2 + σve σvi + σve 1 1 2 + σ2 . [sent-25, score-0.227]

13 In particular, in their model, the improved decoding accuracy after combining input cues (i. [sent-33, score-0.438]

14 Moreover, it is not clear where this centralized network responsible for information integration locates in the cortex. [sent-38, score-0.294]

15 In this work, we propose a novel mechanism to implement Bayesian information integration, which relies on the excitatory reciprocal interactions between local estimators, with each local estimator receiving an independent cue as external input. [sent-39, score-0.804]

16 Although our idea may be applicable to general cases, the present study focuses on two reciprocally connected networks, mimicking the integration of heading direction information between the dorsal medial superior temporal (MSTd) area and ventral intraparietal (VIP) area. [sent-40, score-0.563]

17 It is known that MSTd and VIP receive the visual and the vestibular cues as external input, respectively. [sent-41, score-0.756]

18 We model each network as a continuous attractor neural network (CANN), reﬂecting the property that neurons in MSTd and VTP are widely tuned by heading direction [10, 11]. [sent-42, score-0.638]

19 Interestingly, we ﬁnd that with positive reciprocal interactions, both networks read out heading direction optimally in Bayesian sense, despite the fact that each network only receives a single cue as directly external input. [sent-43, score-1.197]

20 This agrees well with the experimental ﬁnding that both MSTd and VIP integrate the visual and vestibular cues optimally [6, 7]. [sent-44, score-0.712]

21 Our result suggests that the brain may implement Bayesian information integration distributively at each local area through reciprocal connections between cortical regions. [sent-45, score-0.631]

22 2 The Model We consider two reciprocally connected networks, each of which receives the stimulus information from an independent sensory cue (see Fig. [sent-46, score-0.507]

23 Anatomical and fMRI data have revealed that there exist abundant reciprocal interactions between MSTd and VIP [12–14]. [sent-49, score-0.344]

24 Neurons in MSTd and VIP are tuned to heading direction, relying on the visual and the vestibular cues [10, 15]. [sent-50, score-0.808]

25 the heading direction) encoded by both networks, and the neuronal preferred stimuli are in the range of −π < θ ≤ π with periodic boundary condition. [sent-55, score-0.21]

26 Denote Ul (θ, t), for l = 1, 2, the synaptic input at time t to the neurons having the preferred stimulus θ in the l-th network. [sent-56, score-0.275]

27 The dynamics of Ul (θ, t) is determined by the recurrent inputs from other neurons in the same network, the reciprocal inputs from neurons in the other network, the external input Ilext (θ, t), and its own relaxation. [sent-57, score-0.823]

28 It is written as [ ] ][ ] [ ext ] [ ] ∫ [ ∂ U1 (θ, t) W11 W12 r1 (θ′ , t) I1 (θ, t) U1 (θ, t) ′ τ =ρ dθ + − , (4) ext W21 W22 r2 (θ′ , t) U2 (θ, t) I2 (θ, t) ∂t U2 (θ, t) where τ is the time constant for synaptic current, which is typically in the order of 2-5ms. [sent-58, score-0.203]

29 rl (θ, t) is the ﬁring rate of neurons, which increases with the synaptic input but saturates when the synaptic input is sufﬁciently large. [sent-60, score-0.2]

30 The two networks are reciprocally connected and each of them forms a CANN. [sent-63, score-0.21]

31 Each disk represents an excitatory neuron with its preferred heading direction indicated by the arrow inside. [sent-64, score-0.312]

32 The gray disk in the middle of the network represents the inhibitory neuron pool which sums the total activities of excitatory neurons and generates divisive normalization (Eq. [sent-65, score-0.351]

33 Wlm (θ, θ′ ) denotes the connection from the neurons θ′ in the network m to the neurons θ in the network l. [sent-79, score-0.486]

34 W11 (θ, θ′ ) and W22 (θ, θ′ ) are the recurrent connections within the same network, and W12 (θ, θ′ ) and W21 (θ, θ′ ) the reciprocal connections between the networks. [sent-80, score-0.489]

35 We choose Jlm > 0, for l, m = 1, 2, implying excitatory recurrent and reciprocal neuronal interactions. [sent-85, score-0.392]

36 The external inputs to two networks are given by [ Ilext (θ, t) = 2 (θ − µl ) αl exp − 4(all )2 ] + ηl ξl (θ, t), (7) where µl denotes the stimulus value conveyed to the network l by the corresponding sensory cue. [sent-87, score-0.519]

37 This can be understood as Ilext drives the network l to be stable at µl when no reciprocal interaction and noise exist. [sent-88, score-0.457]

38 The noise term causes the uncertainty of the input information, which induces ﬂuctuations of the network state. [sent-90, score-0.208]

39 1 The dynamics of uncoupled networks It is instructive to ﬁrst review the dynamics of two networks without reciprocal connection (by setting Wlm = 0 for l ̸= m in Eq. [sent-93, score-0.749]

40 In this case, the dynamics of each network is independent 3 from the other. [sent-95, score-0.234]

41 Because of the translation-invariance of the recurrent connections Wll (θ, θ′ ), each network can support a continuous family of active stationary states even when the external input is removed [19]. [sent-96, score-0.45]

42 These attractor states are of Gaussian-shape, called bumps, which are given by, ] [ 2 ˜l (x) = Ul0 exp − (θ − zl ) , U (8) 4(all )2 where zl is a free parameter, representing the peak position of the bump, and Ul0 = [1 + (1 − √ √ √ Jc /Jll )1/2 ]Jll /(4all k π). [sent-97, score-0.36]

43 In response to external inputs, the bump position zl is interpreted as the population decoding result of the network. [sent-99, score-0.664]

44 It has been proven that for a strong transient or a weak constant input, the network bump will move toward and be stable at a position having the maximum overlap with the noisy input, realizing the so called template-matching operation [17, 18]. [sent-100, score-0.535]

45 For temporally ﬂuctuating inputs, the bump position also ﬂuctuates in time, and the variance of bump position measures the network decoding uncertainty. [sent-101, score-0.977]

46 In a CANN, its stationary states form a continuous manifold in which the network is neutrally stable, i. [sent-102, score-0.181]

47 , the network state can translate smoothly when the external input changes continuously [18, 20]. [sent-104, score-0.347]

48 Due to the special structure of a CANN, it has been proved that the dynamics of a CANN is dominated by a few motion modes, corresponding to distortions in the height, position and other higher order features of the Gaussian bump [19]. [sent-106, score-0.483]

49 In the weak input limit, it is enough to project the network dynamics onto the ﬁrst few dominating motion modes and neglect the higher order ones then simplify the network dynamics signiﬁcantly. [sent-107, score-0.696]

50 The ﬁrst two dominating motion modes we are going to use are, [ ] (θ − z)2 height : ϕ0 (θ|z) = exp − , (9) 4a2 ( ) [ ] θ−z (θ − z)2 position : ϕ1 (θ|z) = (10) exp − , a 4a2 where a is the width of the basis function, whose value is determined by the bump width the network holds. [sent-108, score-0.673]

51 θ When reciprocal connections are included, the dynamics of the two networks interact with each other. [sent-110, score-0.57]

52 The bump position of each network is no longer solely determined by its own input, but is also affected by the input to the other network, enabling both networks to integrate two sensory cues via reciprocal connections. [sent-111, score-1.319]

53 We consider the reciprocal connections, Wlm (θ, θ′ ), for l ̸= m, also translation-invariant (Eq. [sent-112, score-0.312]

54 In the text below, we will consider the weak input limit and use a projection method to simplify the network dynamics. [sent-117, score-0.283]

55 The simpliﬁed model allows us to solve the network decoding performances analytically and gives us insight into the understanding of how reciprocal connections help both networks to integrate information optimally from independent cues. [sent-118, score-0.829]

56 , J11 = J22 ≡ Jrc , J12 = J21 ≡ Jrp , and alm = a; and they receive the same mean input value and input strength, i. [sent-123, score-0.201]

57 , ⟨ξ1 ξ2 ⟩ = 0, implying that two cues are independent to each other given the stimulus. [sent-128, score-0.257]

58 Two networks receive the same external input, whose value jumps from −1 to 1 abruptly. [sent-133, score-0.225]

59 The network states move smoothly from the initial to the target position, and their main changes are the height and the position of the Gaussian bumps. [sent-134, score-0.372]

60 (C) The simpliﬁed network dynamics after projecting onto the two dominating motion modes. [sent-136, score-0.321]

61 (13)), we can get the necessary condition for the networks holding self-sustained bump states, which is (see Supplemental information 2) √ √ Jrc + Jrp ≥ 2 2(2π)1/4 ka/ρ. [sent-149, score-0.352]

62 (16) It indicates that positive reciprocal interactions Jrp help the networks to retain attractor states. [sent-150, score-0.475]

63 To get clear understanding of the effect of reciprocal connections, we decouple the dynamics of z1 and z2 by studying the dynamics of their their difference, zd = z1 − z2 , and their summation, zs = z1 + z2 . [sent-151, score-0.677]

64 (14) and (15), we obtain √ √ ˜ α + 2Jrp B 2 2η a dzd = − zd + τ ϵd (t), (17) dt A (2π)1/4 A √ √ dzs α 2α 2 2η a τ ϵs (t), (18) = − zs + µ+ dt A A (2π)1/4 A 5 where ϵd (t) and ϵs (t) are independent Gaussian white noises re-organized from ξ1 (t) and ξ2 (t) √ ( 2ϵ = ξ1 ± ξ2 ). [sent-153, score-0.297]

65 (20) indicates that the positive re˜ ciprocal connections Jrp tend to decrease the variance of zd , i. [sent-156, score-0.205]

66 The decoding error of each network, measured by the variance of zl , is calculated to be (two networks have the same result due to the symmetry), ⟨zl ⟩ Var(zl ) = µ, for l = 1, 2, = [Var(zd ) + Var(zs )] /4, ( ) η2 a 1 1 = √ + . [sent-158, score-0.343]

67 ˜ 2πτ A α α + 2Jrp B (22) (23) We see that the network decoding is unbiased and their errors tend to decrease with the reciprocal ˜ connection strength Jrp (see the second term in the right-hand of Eq. [sent-159, score-0.651]

68 It is easy to check that in the extreme cases and assuming the bump shape is unchanged (which is not true but is still a good ˜ indication), the network decoding variance with vanishing reciprocal interaction (Jrp = 0) is two˜ fold of that with inﬁnitely strong reciprocal interactions (Jrp = ∞). [sent-161, score-1.229]

69 Thus, reciprocal connections between networks do provide an effective way to integrate information from independent input cues. [sent-162, score-0.599]

70 To further display the advantage of reciprocal connection, we also calculate the situation when a single network receives both input cues. [sent-163, score-0.554]

71 This equals to setting the external input to a single CANN √ 2 2 to be I ext (x, t) = 2αe−(x−µ) /4a + 2ηξ(x, t) (see Eq. [sent-164, score-0.248]

72 In the weak input limit, the decoding errors in general situations when two networks are not symmetric can also be calculated, (see Supplemental information 3) 2 2 ˜ ˜ ˜ ˜ 2a [(J12 B2 α2 + J21 B1 α1 + α1 α2 )A2 /A1 + (J21 B1 + α2 )2 ]η1 + (J12 B2 )2 η2 Var(z1 ) = √ . [sent-169, score-0.324]

73 Mimicking the experimental setting for exploring the integration of visual and vestibular cues in the inference of heading direction [6, 7], we apply three input conditions to two networks (see Fig. [sent-172, score-1.139]

74 3A), which are: • Only visual cue: • Only vestibular cue: • Combined cues: α1 = α, α1 = 0, α1 = α, α2 = 0. [sent-173, score-0.37]

75 In three conditions, the noise amplitude is unchanged and the reciprocal connections are intact. [sent-176, score-0.411]

76 6 A Input condition Both cues Vestibular cue Visual cue I Net 2 (VIP) Iext 2 ext 2 Net 1 (MSTd) Iext 1 B ext (z1|cvi) (z1|c ve) (z1|c b ) I1 z1 0. [sent-177, score-0.911]

77 The bump position of the network 1 ﬂuctuates around the true stimulus value 0. [sent-189, score-0.555]

78 The right panel displays the bump position distributions in three input conditions, from which we estimate the mean and variance of the decoding results. [sent-190, score-0.552]

79 Compared the network decoding results with two cues with the predictions of Bayesian inference. [sent-192, score-0.52]

80 Different combinations of the input strengths αl and the reciprocal connection strengths Jrp are chosen. [sent-194, score-0.487]

81 Considering the symmetric structures of two networks and ignoring the mild changes in the bump shape in the weak input limit, we can obtain from Eq. [sent-203, score-0.462]

82 We run the network dynamics under three input conditions for many trials, and calculate the means and variances of the bump positions in each condition. [sent-210, score-0.553]

83 3B shows that the bump position ﬂuctuations become 7 ( µ vi ) 0. [sent-212, score-0.393]

84 5 Var(z1|c vi )/Var(z1|c ve) 2 Figure 4: The decoding mean of the network 1 shifts toward the more reliable cue. [sent-219, score-0.355]

85 The color encodes the ratio of the input strengths to two networks α1 /α2 , which generates varied reliability for two cues. [sent-220, score-0.234]

86 , the vestibular cue becomes more reliable than the visual one, the network estimation shifts to the stimulus value µ2 conveyed by the vestibular cue. [sent-223, score-1.177]

87 narrower in the combined cue input condition, indicating greater accuracy in the decoding. [sent-229, score-0.307]

88 We compare the result when both cues are presented with the prediction of the Bayesian inference, obtained by using Eqs. [sent-230, score-0.257]

89 3C and D show that two networks indeed achieve near Bayesian optimal inference for a wide range of input amplitudes and reciprocal connection strengths. [sent-233, score-0.509]

90 The reliability of cues is quantiﬁed by their variance ratio, e. [sent-235, score-0.323]

91 , (σvi )2 < (σve )2 means that visual cue is more reliable than vestibular cue. [sent-237, score-0.656]

92 This property has been used as a criterion in experiment to check the implementation of Bayesian inference, called “reliability based cue weighting” [23]. [sent-240, score-0.244]

93 To achieve different reliability of the cues, we adjust the input strength α1 , and keep the other input parameters unchanged, mimicking the experimental ﬁnding that the ﬁring rate of MT neuron, the earlier stage before MSTd, increases with the input coherence for its preferred stimuli [24]. [sent-242, score-0.357]

94 With varying input strengths α1 , and hence varied ratios Var(z1 |cvi )/Var(z1 |cve ), we calculate the mean of the network decoding. [sent-243, score-0.245]

95 4 shows that the decoded mean in the combined cues condition indeed shifts towards to the more reliable cue, agreeing with the experimental ﬁnding and the property of Bayesian inference. [sent-245, score-0.299]

96 We consider two networks which are reciprocally connected, and each of them is modeled as a CANN receiving the stimulus information from an independent cue. [sent-247, score-0.251]

97 Our network model may be regarded as mimicking the information integration on heading direction between the neural circuits in MSTd and VIP. [sent-248, score-0.561]

98 Experimental data has revealed that the two areas are densely connected in reciprocity and that neurons in both areas are widely tuned by heading direction, favoring our model assumptions. [sent-249, score-0.286]

99 We use a projection method to solve the network dynamics in the weak input limit analytically and get insights into how positive reciprocal connections enable one network to effectively integrate information from the other. [sent-250, score-0.957]

100 It suggests that the brain can implement efﬁcient information integration in a distributive manner through reciprocal connections between cortical regions. [sent-254, score-0.614]

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