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156 nips-2008-Nonparametric sparse hierarchical models describe V1 fMRI responses to natural images

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Author: Vincent Q. Vu, Bin Yu, Thomas Naselaris, Kendrick Kay, Jack Gallant, Pradeep K. Ravikumar

Abstract: We propose a novel hierarchical, nonlinear model that predicts brain activity in area V1 evoked by natural images. In the study reported here brain activity was measured by means of functional magnetic resonance imaging (fMRI), a noninvasive technique that provides an indirect measure of neural activity pooled over a small volume (≈ 2mm cube) of brain tissue. Our model, which we call the V-SPAM model, is based on the reasonable assumption that fMRI measurements reﬂect the (possibly nonlinearly) pooled, rectiﬁed output of a large population of simple and complex cells in V1. It has a hierarchical ﬁltering stage that consists of three layers: model simple cells, model complex cells, and a third layer in which the complex cells are linearly pooled (called “pooled-complex” cells). The pooling stage then obtains the measured fMRI signals as a sparse additive model (SpAM) in which a sparse nonparametric (nonlinear) combination of model complex cell and model pooled-complex cell outputs are summed. Our results show that the V-SPAM model predicts fMRI responses evoked by natural images better than a benchmark model that only provides linear pooling of model complex cells. Furthermore, the spatial receptive ﬁelds, frequency tuning and orientation tuning curves of the V-SPAM model estimated for each voxel appears to be consistent with the known properties of V1, and with previous analyses of this data set. A visualization procedure applied to the V-SPAM model shows that most of the nonlinear pooling consists of simple compressive or saturating nonlinearities. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Nonparametric sparse hierarchical models describe V1 fMRI responses to natural images Pradeep Ravikumar, Vincent Q. [sent-1, score-0.181]

2 Gallant Department of Psychology University of California, Berkeley Berkeley, CA Abstract We propose a novel hierarchical, nonlinear model that predicts brain activity in area V1 evoked by natural images. [sent-4, score-0.292]

3 In the study reported here brain activity was measured by means of functional magnetic resonance imaging (fMRI), a noninvasive technique that provides an indirect measure of neural activity pooled over a small volume (≈ 2mm cube) of brain tissue. [sent-5, score-0.227]

4 Our model, which we call the V-SPAM model, is based on the reasonable assumption that fMRI measurements reﬂect the (possibly nonlinearly) pooled, rectiﬁed output of a large population of simple and complex cells in V1. [sent-6, score-0.24]

5 It has a hierarchical ﬁltering stage that consists of three layers: model simple cells, model complex cells, and a third layer in which the complex cells are linearly pooled (called “pooled-complex” cells). [sent-7, score-0.521]

6 The pooling stage then obtains the measured fMRI signals as a sparse additive model (SpAM) in which a sparse nonparametric (nonlinear) combination of model complex cell and model pooled-complex cell outputs are summed. [sent-8, score-1.048]

7 Our results show that the V-SPAM model predicts fMRI responses evoked by natural images better than a benchmark model that only provides linear pooling of model complex cells. [sent-9, score-0.499]

8 Furthermore, the spatial receptive ﬁelds, frequency tuning and orientation tuning curves of the V-SPAM model estimated for each voxel appears to be consistent with the known properties of V1, and with previous analyses of this data set. [sent-10, score-0.613]

9 A visualization procedure applied to the V-SPAM model shows that most of the nonlinear pooling consists of simple compressive or saturating nonlinearities. [sent-11, score-0.206]

10 1 Introduction An important step toward understanding the neural basis of vision is to develop computational models that describe how complex visual stimuli are mapping onto evoked neuronal responses. [sent-12, score-0.219]

11 Although the relationship between measured fMRI activity and the spiking activity of neurons is rather complex, as a ﬁrst-order approximation the fMRI signal can be considered to be monotonically related to the pooled activity of the underlying neural population. [sent-16, score-0.198]

12 1 In this paper we consider the task of predicting fMRI brain activity evoked by a series of grayscale natural images. [sent-17, score-0.209]

13 Natural images are a useful stimulus set for efﬁciently probing the visual system, because they are likely to evoke response from both early visual areas and from more central, highly nonlinear visual areas. [sent-18, score-0.24]

14 The fMRI scanner provides a three-dimensional image of the brain with a spatial resolution of a few cubic millimeters and fairly low temporal resolution (about 0. [sent-19, score-0.162]

15 Here we restrict our analysis to voxels sampled from visual area V1, the primary visual area in humans. [sent-22, score-0.229]

16 There are two problems that make predicting evoked responses of fMRI voxels difﬁcult. [sent-23, score-0.257]

17 Second, each voxel reﬂects the combined inﬂuence of hundreds of thousands of neurons [4]. [sent-25, score-0.226]

18 fMRI scans of a single voxel in human V1 likely reﬂect the nonlinearly-pooled, rectiﬁed outputs of two functionally distinct classes of neurons: simple cells that are sensitive to spatial phase, and phase-invariant complex cells [2]. [sent-26, score-0.695]

19 In order for a predictive model to advance our understanding of the brain, the function of any predictive model must be conceptually interpretable. [sent-31, score-0.13]

20 Our V-SPAM model is a hierarchical and sparse nonparametric additive model. [sent-33, score-0.203]

21 It combines a biologically-inspired hierarchical ﬁltering scheme with a nonlinear (nonparametric) pooling of the outputs from various levels of the hierarchical ﬁltering stage. [sent-34, score-0.312]

22 The model is estimated separately for each recorded fMRI voxel using a ﬁt data set, and then its predictions are evaluated against an entirely separate data set reserved for this purpose. [sent-35, score-0.253]

23 The ﬁltering component of the model consists of three distinct layers: simple cells, complex cells, and linear combinations of the complex cells (here called pooled-complex cells). [sent-36, score-0.329]

24 The fMRI response is then modeled as a sparse additive combination of nonlinear (nonparametric) functions of the complex and pooled-complex cell model outputs. [sent-37, score-0.533]

25 This last step automatically learns the optimal combinatorial output nonlinearity of the hierarchical ﬁltering stage, and so permits us to model nonlinear V1 responses not captured by the simple and complex cell model components alone [6]. [sent-38, score-0.491]

26 That study also used a ﬁltering model to describe the relationship between natural images and evoked fMRI signals, and used the estimated models in turn to decode (identify) images. [sent-40, score-0.217]

27 However, the earlier study only provided linear pooling of model complex cell ﬁlters. [sent-41, score-0.444]

28 Our results show that the V-SPAM model predicts fMRI responses evoked by natural images better than does the earlier linear pooling model. [sent-42, score-0.373]

29 Furthermore, the spatial receptive ﬁelds, frequency tuning and orientation tuning curves of the V-SPAM model estimated for each voxel appear to be consistent with the known properties of V1, and with the previous results [5]. [sent-43, score-0.613]

30 The sparse additive model (SpAM) framework of Ravikumar et al [7] extends these sparse linear models to the nonparametric domain. [sent-58, score-0.219]

31 A sparse additive model then imposes a sparsity constraint on the set J = {j : fj ≡ 0} of functions fj that are nonzero. [sent-62, score-0.309]

32 j ˆ Soft-threshold: fj = [1 − λ/ˆj ]+ Pj ; s Center: fj ← fj − mean(fj ). [sent-77, score-0.261]

33 Output: Component functions fj and estimator m(Xi ) = j fj (Xij ). [sent-78, score-0.174]

34 ˆ Figure 1: T HE S PAM BACKFITTING A LGORITHM 3 A model for pooled neural activity of voxels Our V-SPAM model combines a biologically-inspired ﬁltering scheme and a novel algorithm that permits nonlinear pooling of the outputs of the ﬁltering stage. [sent-79, score-0.514]

35 The ﬁltering stage itself consists of three distinct layers, arranged hierarchically: simple cells, complex cells, and linear combinations of the complex cells (here called pooled-complex cells). [sent-80, score-0.38]

36 The output of this ﬁltering operation is then fed to an algorithm that estimates a nonlinear pooling function that optimizes predictive power. [sent-81, score-0.237]

37 1 Simple Cell Model The ﬁrst stage of the hierarchical ﬁlter is inspired by simple cells that are known to exist in area V1. [sent-83, score-0.268]

38 The receptive ﬁelds of V1 simple cells are known to be generally consistent with a Gabor wavelet model [6]. [sent-84, score-0.244]

39 Most importantly, they are spatially localized, oriented, spatial frequency band-pass and phase selective. [sent-85, score-0.159]

40 Each row shows a family of Gabor wavelets that share a common spatial location and frequency, but differ in orientation. [sent-88, score-0.151]

41 3 In our model the simple cell ﬁlter bank was implemented as a Gabor wavelet pyramid, as follows. [sent-90, score-0.286]

42 Then our simple cell model, for the activation given the image I as stimulus, is given by, Xj (I) = [ j I ]+ , where is the Euclidean inner product, and [ ]+ is a non-negative rectiﬁcation. [sent-94, score-0.247]

43 image Gabor wavelet output non-negative rectiﬁcation Figure 3: Simple cell model. [sent-97, score-0.299]

44 The activation of a model simple cell given an image is the inner product of the image with a Gabor wavelet, followed by a non-negative rectiﬁcation. [sent-98, score-0.325]

45 2 Complex Cell Model The second stage of the hierarchical ﬁlter is inspired by complex cells that are also known to exist in area V1. [sent-100, score-0.352]

46 Complex cells are similar to simple cells, except they are not sensitive to spatial phase. [sent-101, score-0.209]

47 In our model the complex cell ﬁlter bank was implemented by taking the sum of squares of the outputs of four simple cells (corresponding to the wavelet pairs that are identical up to phase), followed by a ﬁxed output nonlinearity. [sent-102, score-0.608]

48 The activation of the model complex cell given an image I is given by, Xj (I) = log(1 + j j I ]2 + [ + = log(1 + where ure 4). [sent-103, score-0.36]

49 [ [ j I ]2 + [ and j j j I ]2 + [ + j I ]2 + [ + j I ]2 ) + I ]2 ) (1) (2) are Gabor wavelets identical up to phase (also called a quadrature pair; see Fig- + image output Gabor wavelet quadrature pair squaring ﬁxed nonlinearity Figure 4: Complex cell model. [sent-104, score-0.469]

50 The activation of a model complex cell given an image is the sum of squares of the inner products of the image with a quadrature pair of Gabor wavelets followed by a nonlinearity. [sent-105, score-0.498]

51 This is equivalently modeled by summing the squares of 4 simple cell model outputs, followed by a nonlinearity. [sent-106, score-0.247]

52 3 Pooled-complex Cell Model The hierarchical ﬁltering component of our model also includes a third ﬁltering stage, linear pooling of complex cells sharing a common spatial location and frequency. [sent-108, score-0.506]

53 Xjk correspond to complex cells with the same spatial location and frequency, then If Xj1 the corresponding pooled-complex cell (which thus sums over different orientations) is given by, k Zj1 jk = l=1 Xjl . [sent-111, score-0.502]

54 ) 4 + + output image + complex cells Figure 5: Pooled-complex cell model. [sent-113, score-0.461]

55 Subsets of complex cells that share a common spatial location and frequency are summed. [sent-114, score-0.375]

56 4 V-SPAM model Finally, the predicted fMRI response Y is obtained as a sparse additive combination of complex Xp , and the cell and pooled-complex cell outputs. [sent-116, score-0.681]

57 Denote the complex cell outputs by X1 pooled-complex cell outputs by Z1 Zq . [sent-117, score-0.592]

58 Then the fMRI response Y is modeled as a sparse p q additive (nonparametric) model, Y = j=1 fj (Xj ) + l=1 gl (Zl ) + φ. [sent-118, score-0.256]

59 Figure 6 summarizes the entire V-SPAM model, including both ﬁltering and pooling components. [sent-119, score-0.128]

60 + image fMRI voxel response simple cell outputs complex cell outputs pooled-complex cell outputs nonlinearities Figure 6: V-SPAM model. [sent-120, score-1.19]

61 The fMRI voxel response is modeled as the summation of nonlinear functions of complex and pooled-complex cell outputs. [sent-121, score-0.586]

62 1 Experiments Data description The data set analyzed in this paper consists of a total of 1,294 voxels recorded from area V1 of one human observer. [sent-124, score-0.143]

63 A 4T Varian MRI scanner provided voxels of size 2mm x 2mm x 2. [sent-125, score-0.141]

64 2 V-SPAM model ﬁtting The V-SPAM model was ﬁtted separately for each of the 1,294 voxels using the training set of 1,750 images and the evoked fMRI responses. [sent-138, score-0.33]

65 In the ﬁrst stage, the model complex cell outputs are computed according to equation (2) using a pyramid (or family) of Gabor wavelets sampled on a grid of 128 x 128 pixels. [sent-140, score-0.449]

66 At each spatial frequency ω the wavelets are positioned evenly on a ω × ω grid covering the image. [sent-142, score-0.189]

67 In the second stage, the model complex cell outputs are pre-screened in order to eliminate complex cell outputs that are unrelated to a voxel’s response, and to reduce the computational complexity of successive stages of ﬁtting. [sent-145, score-0.743]

68 This is accomplished by considering the squared-correlation of the response of each complex cell with the evoked voxel response, using the 1,750 images in the training set. [sent-146, score-0.672]

69 In the third stage, pooled-complex cells (see Section 3) are formed from the complex cell outputs that passed the pre-screening in ﬁtting stage 2. [sent-149, score-0.55]

70 In the fourth and ﬁnal stage, the complex and pooled-complex cell responses to the images in the training set are used as predictors in the SpAM ﬁtting algorithm (see Figure 1), and this is optimized to ﬁt the voxel responses evoked by the same 1,750 images in the training set. [sent-150, score-0.76]

71 3 Model validation For each voxel, we evaluate the ﬁtted V-SPAM models by computing the predictive R2 (squared correlation) of the predicted and actual fMRI responses evoked by each of the 120 images in the validation set. [sent-153, score-0.246]

72 The sparse linear pooling model aims to predict each voxel’s response as a linear combination of all 10,921 p estimated complex cell outputs. [sent-155, score-0.561]

73 This model has the form, Y (I) = β0 + j=1 βj Xj (I) + , where the Xj (I) are the complex cell outputs estimated according to (2), with the p = 10, 921 Gabor wavelets described in Section 4. [sent-156, score-0.44]

74 5 Results Figure 7 (left) shows a scatterplot comparing the performance of the V-SPAM model with that of the sparse linear pooling model for all 1,294 voxels. [sent-163, score-0.256]

75 The vertical axis gives performance of the V-SPAM model, and the horizontal axis the sparse linear pooling model. [sent-164, score-0.218]

76 The inset region contains 429 voxels for which both models had some predictive power (R2 ≥ 0. [sent-166, score-0.19]

77 For these voxels, the relative improvement of the V-SPAM model over the sparse linear pooling model is shown in the histogram to the right. [sent-168, score-0.236]

78 The predictions of the V-SPAM model were on average 14% better than those of the sparse linear pooling model (standard deviation 17%). [sent-169, score-0.236]

79 1 Estimated receptive ﬁelds and tuning curves Figure 8 shows the spatial receptive-ﬁelds (RF’s) and joint frequency and orientation tuning curves estimated using the V-SPAM model for 3 voxels. [sent-171, score-0.436]

80 These voxels were chosen because they had high predictive power (R2 ’s of 0. [sent-172, score-0.176]

81 6 −20 sparse linear pooling model 0 20 40 60 80 100 relative improvement (%) (mean = 14, SD = 17, median = 12, IQR = 17) Figure 7: Predictive R2 of the ﬁtted V-SPAM model compared against the ﬁtted sparse linear pooling model. [sent-192, score-0.414]

82 (Right) Relative performance for the 429 voxels contained in the inset region on the left. [sent-194, score-0.135]

83 location in the spatial RF represents the standardized predicted response of the voxel to an image stimulus consisting of a single pixel at that location. [sent-195, score-0.445]

84 The spatial RF’s of these voxels are clearly localized in space, consistent with the known retinotopic organization of V1 and previous fMRI results [9]. [sent-196, score-0.198]

85 The lower row of Figure 8 shows the joint frequency and orientation tuning properties of these same 3 voxels. [sent-197, score-0.171]

86 Here the tuning curves were estimated by computing the predicted response of the ﬁtted voxel model to cosine gratings of varying orientation (degrees) and spatial frequency (cycles/ﬁeld of view). [sent-198, score-0.634]

87 All of the voxels are tuned to spatial frequencies above about 8 cycles/ﬁeld of view, while orientation tuning varies from voxel to voxel. [sent-199, score-0.512]

88 The joint spatial frequency and orientation tuning of all 3 voxels appears to be non-separable (i. [sent-200, score-0.369]

89 Each location in the spatial RF shows the standardized predicted response of the voxel to an image consisting of a single pixel at that location. [sent-207, score-0.445]

90 The tuning curves show the standardized predicted response of the voxel to cosine gratings of varying orientation (degrees) and spatial frequency (cycles/ﬁeld of view). [sent-208, score-0.606]

91 2 Nonlinearities One of the potential advantages of the V-SPAM model over other approaches is that it can reveal novel nonlinear tuning and pooling properties, as revealed by the nonlinear summation occurring in the ﬁnal stage of the V-SPAM model. [sent-210, score-0.392]

92 Figure 9 illustrates some of these functions estimated for a typical voxel with high predictive power (R2 of 0. [sent-211, score-0.279]

93 These correspond to the nonlinearities appearing in the ﬁnal stage of the V-SPAM model (see Figure 6). [sent-213, score-0.169]

94 Here the horizontal axis is the input in standard units of the corresponding model complex or pooled-complex cell outputs, and the vertical axis is the output in standard units of predicted responses. [sent-214, score-0.39]

95 0 1 input 2 −1 0 1 2 3 input Figure 9: Nonlinearities estimated in the V-SPAM model for a voxel with high predictive power (R2 : 0. [sent-239, score-0.308]

96 The other 75 nonlinearities for this voxel (overlaid in the right panel) are smaller and contribute less to the predicted response. [sent-242, score-0.289]

97 6 Discussion and conclusions Our V-SPAM model provides better predictions of fMRI activity evoked by natural images than does a sparse linear model similar to that used in an earlier study of this data set [5]. [sent-243, score-0.333]

98 This increased predictive power of the V-SPAM model reﬂects the fact that it can describe explicitly the nonlinear pooling that likely occurs among the many neurons whose pooled activity contributes to measured fMRI signals. [sent-244, score-0.375]

99 These pooled output nonlinearities are likely a critical component of nonlinear computation across the visual hierarchy. [sent-245, score-0.214]

100 Therefore, the SpAM framework may be particularly useful for modeling neurons or fMRI signals recorded in higher and more nonlinear stages of visual processing beyond V1. [sent-246, score-0.143]

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