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205 nips-2013-Multisensory Encoding, Decoding, and Identification

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Author: Aurel A. Lazar, Yevgeniy Slutskiy

Abstract: We investigate a spiking neuron model of multisensory integration. Multiple stimuli from different sensory modalities are encoded by a single neural circuit comprised of a multisensory bank of receptive ﬁelds in cascade with a population of biophysical spike generators. We demonstrate that stimuli of different dimensions can be faithfully multiplexed and encoded in the spike domain and derive tractable algorithms for decoding each stimulus from the common pool of spikes. We also show that the identiﬁcation of multisensory processing in a single neuron is dual to the recovery of stimuli encoded with a population of multisensory neurons, and prove that only a projection of the circuit onto input stimuli can be identiﬁed. We provide an example of multisensory integration using natural audio and video and discuss the performance of the proposed decoding and identiﬁcation algorithms. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We investigate a spiking neuron model of multisensory integration. [sent-6, score-0.648]

2 Multiple stimuli from different sensory modalities are encoded by a single neural circuit comprised of a multisensory bank of receptive ﬁelds in cascade with a population of biophysical spike generators. [sent-7, score-1.121]

3 We demonstrate that stimuli of different dimensions can be faithfully multiplexed and encoded in the spike domain and derive tractable algorithms for decoding each stimulus from the common pool of spikes. [sent-8, score-0.392]

4 We also show that the identiﬁcation of multisensory processing in a single neuron is dual to the recovery of stimuli encoded with a population of multisensory neurons, and prove that only a projection of the circuit onto input stimuli can be identiﬁed. [sent-9, score-1.606]

5 We provide an example of multisensory integration using natural audio and video and discuss the performance of the proposed decoding and identiﬁcation algorithms. [sent-10, score-0.776]

6 Interestingly, recent studies demonstrated that multisensory integration takes place in brain areas that were traditionally considered to be unisensory [2, 6, 7]. [sent-15, score-0.589]

7 This is in contrast to classical brain models in which multisensory integration is relegated to anatomically established sensory convergence regions, after extensive unisensory processing has already taken place [4]. [sent-16, score-0.659]

8 Moreover, multisensory effects were shown to arise not solely due to feedback from higher cortical areas. [sent-17, score-0.476]

9 The computational principles of multisensory integration are still poorly understood. [sent-19, score-0.537]

10 Moreover, although multisensory neuron responses depend on several concurrently received stimuli, existing identiﬁcation methods typically require separate experimental trials for each of the sensory modalities involved [4, 12, 13, 14]. [sent-21, score-0.743]

11 Doing so creates major challenges, especially when unisensory responses are weak or together do not account for the multisensory response. [sent-22, score-0.507]

12 Here we present a biophysically-grounded spiking neural circuit and a tractable mathematical methodology that together allow one to study the problems of multisensory encoding, decoding, and identiﬁcation within a uniﬁed theoretical framework. [sent-23, score-0.584]

13 of multisensory receptive ﬁelds in cascade with a population of neurons that implement stimulus multiplexing in the spike domain. [sent-42, score-0.814]

14 , during attention) (c) joint processing and storage of multisensory signals/stimuli (e. [sent-50, score-0.476]

15 First we show that, under appropriate conditions, each of the stimuli processed by a multisensory circuit can be decoded loss-free from a common, unlabeled set of spikes. [sent-53, score-0.745]

16 These conditions provide clear lower bounds on the size of the population of multisensory neurons and the total number of spikes generated by the entire circuit. [sent-54, score-0.663]

17 We then discuss the open problem of identifying multisensory processing using concurrently presented sensory stimuli. [sent-55, score-0.567]

18 We show that the identiﬁcation of multisensory processing in a single neuron is elegantly related to the recovery of stimuli encoded with a population of multisensory neurons. [sent-56, score-1.376]

19 2 Modeling Sensory Stimuli, their Processing and Encoding Our formal model of multisensory encoding, called the multisensory Time Encoding Machine (mTEM) is closely related to traditional TEMs [18]. [sent-59, score-0.952]

20 However, in contrast to traditional TEMs that encode one or more stimuli of the same dimension n, a general mTEM receives M input stimuli u1 1 , . [sent-62, score-0.316]

21 , N , the results of this processing are aggregated into the dendritic current v i ﬂowing into the spike initiation zone, where it is encoded into a time sequence (ti )k∈Z , with ti denoting the timing of the k th spike of neuron i. [sent-74, score-0.496]

22 1b), the mapping of the current v i into spikes is described by a set of equations formerly known as the t-transform [18]: ti k+1 ti k i v i (s)ds = qk , k ∈ Z, (1) i i i where qk = C i δ i − bi (ti k+1 − tk ). [sent-79, score-0.455]

23 Intuitively, at every spike time tk+1 , the ideal IAF neuron is i i providing a measurement qk of the current v (t) on the time interval [ti , ti ). [sent-80, score-0.426]

24 For practical and computational reasons we choose to work with the space of trigonometric polynomials Hnm deﬁned below, where each element of the space is a function in nm variables (nm ∈ N, m = 1, 2, . [sent-84, score-0.43]

25 , xnm ) = nm l1 =−L1 over the domain Dnm = exp nm n=1 nm n=1 [0, Tn ], jln Ωn xn /Ln / um. [sent-95, score-1.56]

26 , xnm ) = l1 T1 · · · Tnm , with j denoting the imaginary number. [sent-113, score-0.27]

27 Hnm is endowed with the inner product ·, · : Hnm × Hnm → C, where m um , wnm = nm m um (x1 , . [sent-115, score-0.839]

28 nm Dnm (2) Given the inner product in (2), the set of elements el1 . [sent-125, score-0.43]

29 In what follows, we will primarily be concerned with time-varying stimuli, and the dimension xnm will denote the temporal dimension t of the stimulus um , i. [sent-155, score-0.536]

30 We model audio stimuli um = um (t) as elements of the RKHS H1 over the domain 1 1 D1 = [0, T1 ]. [sent-161, score-0.669]

31 An audio signal um ∈ H1 can√ written be 1 L m m m as u1 (t) = l=−L ul el (t), where the coefﬁcients ul ∈ C and el (t) = exp (jlΩt/L) / T . [sent-163, score-0.357]

32 A video signal um ∈ H3 can be written as um (x, y, t) = 3 3 L1 L2 L3 uml2 l3 el1 l2 l3 (x, y, t), where the coefﬁcients uml2 l3 ∈ C and the funcl1 =−L1 l2 =−L2 l3 =−L3 l1 √ l1 tions el1 l2 l3 (x, y, t) = exp (jl1 Ω1 x/L1 + jl2 Ω2 y/L2 + jl3 Ω3 t/L3 ) / T1 T2 T3 . [sent-166, score-0.458]

33 Without loss of generality, we assume that such transformations involve convolution in the time domain (temporal dimension xnm ) and integration in dimensions x1 , . [sent-174, score-0.331]

34 3 Multisensory Decoding Consider an mTEM comprised of a population of N ideal IAF neurons receiving M input signals um of dimensions nm , m = 1, . [sent-215, score-0.841]

35 Assuming that the multisensory processing is given by nm kernels him , m = 1, . [sent-219, score-0.935]

36 , N , the t-transform in (1) can be rewritten as nm i Tki1 [u1 1 ] + Tki2 [u2 2 ] + . [sent-225, score-0.43]

37 + TkiM [uM ] = qk , n n nM k ∈ Z, (4) where Tkim : Hnm → R are linear functionals deﬁned by Tkim [um ] nm ti k+1 = ti k Dnm him (x1 , . [sent-228, score-0.723]

38 nm nm i We observe that each qk in (4) is a real number representing a quantal measurement of all M stimuli, taken by the neuron i on the interval [ti , ti ). [sent-238, score-1.177]

39 We now k demonstrate that it is possible to reconstruct stimuli um , m = 1, . [sent-240, score-0.352]

40 (Multisensory Time Decoding Machine (mTDM)) Let M signals um ∈ Hnm be encoded by a multisensory TEM comprised of N ideal IAF neurons nm and N × M receptive ﬁelds with full spectral support. [sent-248, score-1.405]

41 Then given the ﬁlter kernel coefﬁcients him nm , i = 1, . [sent-250, score-0.465]

42 , N , all inputs um can be perfectly recovered as nm l1 . [sent-253, score-0.624]

43 lnm are elements of u = Φ+ q, and Φ+ denotes the pseudoinverse of Φ. [sent-274, score-0.29]

44 ,−ln −1 ,ln (ti − ti ), lnm = 0  −l k+1 k m m im im i i [Φ ]kl = h−l1 ,−l2 ,. [sent-288, score-0.602]

45 ,−lnm −1 ,lnm Lnm Tnm elnm (tk+1 ) − elnm (tk ) , (6)  , lnm = 0  jlnm Ωnm where the column index l traverses all possible subscript combinations of l1 , l2 , . [sent-291, score-0.415]

46 A necessary condition for recovery is that the total number of spikes generated by all neurons is larger than M nm m=1 n=1 (2Ln +1)+N . [sent-295, score-0.587]

47 nm n=1 (2Ln + 1)/ min(ν − 1, 2Lnm + 1) , i Proof: Substituting (5) into (4), qk = Tki1 [u1 1 ]+. [sent-297, score-0.499]

48 In matrix form the above equality can be written as qi = Φi u, with nm i [qi ]k = qk , Φi = [Φi1 , Φi2 , . [sent-322, score-0.499]

49 To ﬁnd the coefﬁcients im φim n k , we note that φim nm k = Tnm k (el1 . [sent-332, score-0.53]

50 ; um ] with the vector um containing n=1 (2Ln + 1) entries corresponding to coefﬁcients uml2 . [sent-351, score-0.388]

51 This system of linear equations can be solved nm for u, provided that the rank r(Φ) of matrix Φ satisﬁes r(Φ) = m n=1 (2Ln + 1). [sent-365, score-0.43]

52 A necessary condition for the latter is that the total number of measurements generated by all N neurons is nm greater or equal to n=1 (2Ln + 1). [sent-366, score-0.506]

53 Equivalently, the total number of spikes produced by all N nm neurons should be greater than n=1 (2Ln + 1) + N . [sent-367, score-0.588]

54 , u = Φ+ q t u1 L + h21 (t) 1 v 21 (t) Convex Optimization Problem t u1 (t) 1 (tN )k∈Z k (tN )k∈Z k e L1 ,L2 ,L3 (t) (a) (b) Figure 2: Multimodal TEM & TDM for audio and video integration (a) Block diagram of the multimodal TEM. [sent-373, score-0.315]

55 Thus each neuron can produce a maximum of only 2P Lnm + 1 informative measurements, or equivalently, 2P Lnm + 2 informative spikes on a time nm interval [0, Tnm ]. [sent-376, score-0.626]

56 It follows that for each modality, we require at least n=1 (2Ln + 1)/(2Lnm + 1) nm neurons if ν ≥ (2Lnm + 2) and at least n=1 (2Ln + 1)/(ν − 1) neurons if ν < (2Lnm + 2). [sent-377, score-0.582]

57 4 Multisensory Identiﬁcation We now investigate the following nonlinear neural identiﬁcation problem: given stimuli um , nm m = 1, . [sent-378, score-0.782]

58 , M , at the input to a multisensory neuron i and spikes at its output, ﬁnd the multisensory receptive ﬁeld kernels him , m = 1, . [sent-381, score-1.258]

59 We will show that this problem is mathematically dual nm to the decoding problem discussed above. [sent-385, score-0.476]

60 We consider identifying kernels for only one multisensory neuron (identiﬁcation for multiple neurons can be performed in a serial fashion) and drop the superscript i in him nm throughout this section. [sent-390, score-1.147]

61 Instead, we introduce the natural notion of performing multiple experimental trials and use the same superscript i to index stimuli uim on different trials i = 1, . [sent-391, score-0.286]

62 Since for every trial i, an input signal uim , nm m = 1, . [sent-397, score-0.588]

63 , xnm ) by the reproducing property of the RK Knm. [sent-412, score-0.294]

64 dsnm −1 dsnm = nm nm Dnm (a) = (b) = uim (s1 , . [sent-422, score-1.05]

65 dsnm −1 dsnm , nm nm Dnm where (a) follows from the reproducing property and symmetry of Knm and Deﬁnition 2, and (b) from the deﬁnition of Phm in (3). [sent-446, score-0.946]

66 1 can then be written as nm i Li1 [Ph1 1 ] + Li2 [Ph2 2 ] + . [sent-448, score-0.43]

67 , h = Φ+ q t 21 (Ph1 )(t) 1 + (Ph1 )(t) 1 Convex Optimization Problem Σ (tN )k∈Z k (tN )k∈Z k e L1 ,L2 ,L3 (t) (a) (b) Figure 3: Multimodal CIM for audio and video integration (a) Time encoding interpretation of the multimodal CIM. [sent-453, score-0.344]

68 , M , k ∈ Z, are linear functionals deﬁned by k Lim [Phm ] = k nm ti k+1 ti k Dm uim (s1 , . [sent-458, score-0.782]

69 Intuitively, each inter-spike interval [ti , ti ) produced by the IAF neuron is a time k k+1 i measurement qk of the (weighted) sum of all kernel projections Phm , m = 1, . [sent-469, score-0.396]

70 Each projection Phm is determined by the corresponding stimuli uim , i = 1, . [sent-474, score-0.286]

71 , N , nm nm employed during identiﬁcation and can be substantially different from the underlying kernel hm . [sent-477, score-0.948]

72 nm It follows that we should be able to identify the projections Phm , m = 1, . [sent-478, score-0.452]

73 , M , be a collection of N linearly indepenn nM nm i=1 dent stimuli at the input to an mTEM circuit comprised of receptive ﬁelds with kernels hm ∈ Hnm , nm m = 1, . [sent-490, score-1.287]

74 , M , can be perfectly idennm nm L nm L1 m tiﬁed as (Phm )(x1 , . [sent-506, score-0.86]

75 lnm are elements of h = Φ+ q, and Φ+ denotes the pseudoinverse of Φ. [sent-524, score-0.29]

76 ,−ln −1 ,ln (ti − ti ), lnm = 0  −l k+1 k m m im im i i [Φ ]kl = u−l1 ,−l2 ,. [sent-538, score-0.602]

77 ,−lnm −1 ,lnm Lnm Tnm elnm (tk+1 ) − elnm (tk ) , (8)  , lnm = 0  jlnm Ωnm where l traverses all subscript combinations of l1 , l2 , . [sent-541, score-0.415]

78 A necessary condition for identiﬁcation is that the total number of spikes generated in response to all N trials is larger than M nm m=1 n=1 (2Ln + 1) + N . [sent-545, score-0.49]

79 If the neuron produces ν spikes on each trial, a sufﬁcient condition is that the number of trials N ≥ M m=1 nm n=1 (2Ln + 1)/ min(ν − 1, 2Lnm + 1) . [sent-546, score-0.626]

80 Proof: The equivalent representation of the t-transform in equations (4) and (7) implies that the decoding of the stimulus um (in Theorem 1) and the identiﬁcation of the ﬁlter projections Phm nm nm encountered here are dual problems. [sent-547, score-1.15]

81 , M , are encoded with an mTEM nm comprised of N neurons and receptive ﬁelds uim , i = 1, . [sent-551, score-0.807]

82 An analog audio signal u1 (t) and an analog video signal u2 (x, y, t) appear as inputs to temporal ﬁlters with kernels hi1 (t) 1 3 1 and spatiotemporal ﬁlters with kernels hi2 (x, y, t), i = 1, . [sent-567, score-0.343]

83 In practice, the number of components could be different and would be determined by the bandwidth of input stimuli Ω, or equivalently the order L, and the number of spikes produced (Theorems 1-2). [sent-576, score-0.274]

84 Thus each spike train (ti )k∈Z carries information about two stimuli of completely k different modalities (audio and video) and, under certain conditions, the entire collection of spike trains {ti }N , k ∈ Z, can provide a faithful representation of both signals. [sent-581, score-0.36]

85 k i=1 To demonstrate the performance of the algorithm presented in Theorem 1, we simulated a multisensory TEM with each neuron having a non-separable spatiotemporal receptive ﬁeld for video stimuli and a temporal receptive ﬁeld for audio stimuli. [sent-582, score-1.217]

86 The mTEM produced a total of 360, 000 spikes in response to a 6-second-long grayscale video and mono audio of Albert Einstein explaining the mass-energy equivalence formula E = mc2 : “. [sent-587, score-0.343]

87 ” A multisensory TDM was then used to reconstruct the video and audio stimuli from the produced set of spikes. [sent-591, score-0.849]

88 Furthermore, the stimuli now appear as kernels describing the ﬁlters and the inputs to the circuit are kernel projections Phm , m = 1, . [sent-600, score-0.316]

89 In other words, identiﬁcation of a single neuron nm has been converted into a population encoding problem, where the artiﬁcially constructed population of N neurons is associated with the N spike trains generated in response to N experimental trials. [sent-604, score-0.885]

90 6 Conclusion We presented a spiking neural circuit for multisensory integration that encodes multiple information streams, e. [sent-609, score-0.645]

91 We derived conditions for inverting the nonlinear operator describing the multiplexing and encoding in the spike domain and developed methods for identifying multisensory processing using concurrent stimulus presentations. [sent-612, score-0.666]

92 We provided explicit algorithms for multisensory decoding and identiﬁcation and evaluated their performance using natural audio and video stimuli. [sent-613, score-0.715]

93 Our investigations brought to light a key duality between identiﬁcation of multisensory processing in a single neuron and the recovery of stimuli encoded with a population of multisensory neurons. [sent-614, score-1.376]

94 Spatiotemporal RF t = 45 ms x x A m plit ude y Error A mplit ude y Identiﬁed y Original Temporal RF t = 30 ms A m plit ude t = 15 ms x (a) T ime , [ms ] (b) Figure 5: Multisensory identiﬁcation. [sent-633, score-0.268]

95 On the use of superadditivity as a metric for characterizing multisensory integration in functional neuroimaging studies. [sent-685, score-0.537]

96 Comparing bayesian models for o multisensory cue combination without mandatory integration. [sent-695, score-0.476]

97 The inﬂuence of visual and auditory receptive ﬁeld organization on multisensory integration in the superior colliculus. [sent-702, score-0.658]

98 Linking neurons to behavior in multisensory perception: A computational review. [sent-705, score-0.552]

99 Recovery of stimuli encoded with a Hodgkin-Huxley neuron using conditional PRCs. [sent-742, score-0.352]

100 Reconstruction of sensory stimuli encoded with integrate-and-ﬁre neurons with random thresholds. [sent-763, score-0.362]

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