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162 nips-2009-Neural Implementation of Hierarchical Bayesian Inference by Importance Sampling

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Author: Lei Shi, Thomas L. Griffiths

Abstract: The goal of perception is to infer the hidden states in the hierarchical process by which sensory data are generated. Human behavior is consistent with the optimal statistical solution to this problem in many tasks, including cue combination and orientation detection. Understanding the neural mechanisms underlying this behavior is of particular importance, since probabilistic computations are notoriously challenging. Here we propose a simple mechanism for Bayesian inference which involves averaging over a few feature detection neurons which ﬁre at a rate determined by their similarity to a sensory stimulus. This mechanism is based on a Monte Carlo method known as importance sampling, commonly used in computer science and statistics. Moreover, a simple extension to recursive importance sampling can be used to perform hierarchical Bayesian inference. We identify a scheme for implementing importance sampling with spiking neurons, and show that this scheme can account for human behavior in cue combination and the oblique effect. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract The goal of perception is to infer the hidden states in the hierarchical process by which sensory data are generated. [sent-4, score-0.187]

2 Here we propose a simple mechanism for Bayesian inference which involves averaging over a few feature detection neurons which ﬁre at a rate determined by their similarity to a sensory stimulus. [sent-7, score-0.611]

3 This mechanism is based on a Monte Carlo method known as importance sampling, commonly used in computer science and statistics. [sent-8, score-0.24]

4 Moreover, a simple extension to recursive importance sampling can be used to perform hierarchical Bayesian inference. [sent-9, score-0.462]

5 We identify a scheme for implementing importance sampling with spiking neurons, and show that this scheme can account for human behavior in cue combination and the oblique effect. [sent-10, score-0.8]

6 This task involves combining prior knowledge with current data [1], and integrating cues from multiple sensory modalities [2]. [sent-13, score-0.241]

7 Studies of human psychophysics and animal behavior suggest that the brain is capable of solving these problems in a way that is consistent with optimal Bayesian statistical inference [1, 2, 3, 4]. [sent-14, score-0.192]

8 Moreover, complex brain functions such as visual information processing involves multiple brain areas [5]. [sent-15, score-0.265]

9 Identifying neural mechanisms that could support hierarchical Bayesian inference is important, since probabilistic computations can be extremely challenging. [sent-17, score-0.233]

10 One prominent approach to explaining how the brain uses population activities for probabilistic computations has been done in the “Bayesian decoding” framework [7]. [sent-20, score-0.215]

11 In this framework, it is assumed that the ﬁring rate of a population of neurons, r, can be converted to a probability distribution over stimuli, p(s|r), by applying Bayesian inference, where the likelihood p(r|s) reﬂects the probability of that ﬁring pattern given the stimulus s. [sent-21, score-0.179]

12 In this paper, we take a different approach, allowing a population of neurons to encode a probability distribution directly. [sent-25, score-0.43]

13 Rather than relying on a separate decoding operation, we assume that the activity of each neuron translates directly to the weight given to the optimal stimulus for that neuron in the corresponding probability distribution. [sent-26, score-0.423]

14 In particular, we focus on one Monte Carlo method, namely importance sampling with the prior as a surrogate, and show how recursive importance sampling approximates hierarchical Bayesian inference. [sent-28, score-0.85]

15 The expectation E[f (x∗ )|x] can be approximated using a Monte Carlo method known as importance sampling. [sent-35, score-0.24]

16 In its general form, importance sampling approximates the expectation by using a set of samples from some surrogate distribution q(x∗ ) and assigning those samples weights proportional to the ratio p(x∗ |x)/q(x∗ ). [sent-36, score-0.399]

17 Recent work also has suggested that importance sampling might provide a psychological mechanism for performing probabilistic inference, drawing on its connection to exemplar models [9]. [sent-38, score-0.379]

18 In this section, we ﬁrst describe a radial basis function network implementing importance sampling, then discuss the feasibility of three assumptions mentioned above. [sent-40, score-0.421]

19 1 Radial basis function (RBF) networks Radial basis function (RBF) networks are a multi-layer neural network architecture in which the hidden units are parameterized by locations in a latent space x∗ . [sent-43, score-0.217]

20 Using RBF networks to model the brain is not a new idea – similar models have been proposed for pattern recognition [10] and as psychological accounts of human category learning [11]. [sent-51, score-0.194]

21 A RBF neuron is recruited for a stimulus value x∗ drawn from the prior (Fig. [sent-53, score-0.292]

22 For a Gaussian likelihood, the peak ﬁring rate i would be reached at preferred stimulus x = x∗ and diminishes as ||x − x∗ || increases. [sent-56, score-0.173]

23 The ith RBF i i neuron makes a synaptic connection to output neuron j with strength fj (x∗ ), where fj is a function i of interest. [sent-57, score-0.441]

24 The output units also receive input from an inhibitory neuron that sums over all RBF neurons’ activities. [sent-58, score-0.217]

25 The ﬁrst stage is even easier in our formulation, because RBF neurons simply represent samples from the prior, independent of the second stage later in development. [sent-64, score-0.37]

26 Moreover, the performance of RBF networks is relatively insensitive to the precise form of the radial basis functions [12], providing some robustness to differences between the Bayesian likelihood p(x|x∗ ) and the activation function in the network. [sent-65, score-0.165]

27 RBF networks also produce i sparse coding, because localized radial basis likelihood functions mean only a few units will be signiﬁcantly activated for a given input x. [sent-66, score-0.202]

28 First, responses of cortical neurons to stimuli are often characterized by receptive ﬁelds and tuning curves, where receptive ﬁelds specify the domain within a stimulus feature space that modify neuron’s response and tuning curves detail how neuron’s responses change with different feature values. [sent-69, score-0.786]

29 A typical tuning curve (like orientation tuning in V1 simple cells) has a bell-shape that peaks at the neuron’s preferred stimulus parameter and diminishes as parameter diverges. [sent-70, score-0.417]

30 These neurons are effectively measure the likelihood p(x|x∗ ), where x∗ is the preferred stimulus. [sent-71, score-0.424]

31 i i Second, importance sampling requires neurons with preferred stimuli x∗ to appear with frequency i proportional to the prior distribution p(x∗ ). [sent-72, score-0.909]

32 This can be realized if the number of neurons representing x∗ is roughly proportional to p(x∗ ). [sent-73, score-0.459]

33 While systematic study of distribution of neurons over their preferred stimuli is technically challenging, there are cases where this assumption seems to hold. [sent-74, score-0.47]

34 For example, research on the ”oblique effect” supports the idea that the distribution of orientation tuning curves in V1 is proportional to the prior. [sent-75, score-0.247]

35 Electrophysiology [13], optical imaging [14] and 3 fMRI studies [15] have found that there are more V1 neurons tuned to cardinal orientations than to oblique orientations. [sent-76, score-0.789]

36 These ﬁndings are in agreement with the prior distribution of orientations of lines in the visual environment. [sent-77, score-0.212]

37 Repetitive stimulation of a ﬁnger expands its corresponding cortical representation in somatosensory area [16], suggesting more neurons are recruited to represent this stimulus. [sent-79, score-0.399]

38 Alternatively, recruiting neurons x∗ according i to the prior distribution can be implemented by modulating feature detection neurons’ ﬁring rates. [sent-80, score-0.472]

39 This strategy also seems to be used by the brain: studies in parietal cortex [17] and superior colliculus [18] show that increased prior probability at a particular location results in stronger ﬁring for neurons with receptive ﬁelds at that location. [sent-81, score-0.44]

40 It has been suggested that biophysical mechanisms such as shunting inhibition and synaptic depression might account for normalization and gain control [10, 21, 22]. [sent-83, score-0.276]

41 3 Importance sampling by Poisson spiking neurons Neurons communicate mostly by spikes rather than continuous membrane potential signals. [sent-87, score-0.597]

42 Poisson spiking neurons play an important role in other analyses of systems for representing probabilities [8]. [sent-88, score-0.45]

43 Poisson spiking neurons can also be used to perform importance sampling if we have an ensemble of neurons with ﬁring rates λi proportional to p(x|x∗ ), with the values of x∗ drawn from the prior. [sent-89, score-1.219]

44 Assume a neuron tuned to stimulus x∗ emits spikes i ri ∼ Poisson(c · p(x|x∗ )), where c is any positive constant. [sent-95, score-0.373]

45 Thus, Poisson spiking neurons, if plugged into an RBF network, can perform importance sampling and give similar results to “neurons” with analog output, as we conﬁrm later in the paper through simulations. [sent-98, score-0.428]

46 4 Hierarchical Bayesian inference and multi-layer importance sampling Inference tasks solved by the brain often involve more than one random variable, with complex dependency structures between those variables. [sent-99, score-0.5]

47 For example, visual information process in primates involves dozens of subcortical areas that interconnect in a hierarchical structure containing two major pathways [5]. [sent-100, score-0.23]

48 Hierarchical Bayesian inference has been proposed as a solution to this problem, with particle ﬁltering and belief propagation as possible algorithms implemented by the brain [6]. [sent-101, score-0.152]

49 We show how a multi-layer neural network can perform such computations using importance samplers (Fig. [sent-103, score-0.304]

50 To understand brain function, it is often helpful to identify the generative model that determines how stimuli to the brain Sx are generated. [sent-107, score-0.268]

52 Sx is the stimulus presented to the nervous system, while X, Y , and Z are latent variables at increasing levels of abstraction. [sent-113, score-0.166]

53 2, the quantity of interest is the posterior expectation of some function f (z) of a high-level latent variable Z given stimulus Sx , E[f (z)|Sx ] = f (z)p(z|Sx ) dz. [sent-115, score-0.169]

54 After repeatedly using the importance sampling trick (see Eq. [sent-116, score-0.348]

55 5), this hierarchical Bayesian inference problem can decomposed into three importance samplers with ∗ ∗ values x∗ ,yj and zk drawn from the prior. [sent-117, score-0.516]

56 This recursive importance sampling scheme can be used in a variety of graphical models. [sent-119, score-0.382]

57 For example, tracking a stimulus over time is a natural extension where an additional observation is added at each level of the generative model. [sent-120, score-0.181]

58 2 Neural implementation of the multi-layer importance sampler The decomposition of hierarchical inference into recursive importance sampling (Eq. [sent-123, score-0.814]

59 1, composed of feature detection neurons with output proportional to the likelihood p(Sx |x∗ ). [sent-127, score-0.483]

60 The response of neuron yj , summing over synaptic inputs, apj i j ∗ ∗ ∗ proximates p(yj |Sx ). [sent-129, score-0.339]

61 Similarly, the response of zk ≈ p(zk |Sx ), and the activities of these neurons ∗ ∗ are pooled to compute E[f (z)|Sx ]. [sent-130, score-0.563]

62 Note that, at each level, x∗ ,yj and zk are sampled from prior i distributions. [sent-131, score-0.164]

63 Posterior expectations involving any random variable can be computed because the neuron activities at each level approximate the posterior density. [sent-132, score-0.224]

64 A single pool of neurons can also feed activation to multiple higher levels. [sent-133, score-0.37]

65 3b), such a multi-layer importance sampling scheme could be used to account for hierarchical inference in divergent pathways by projecting a set of V2 cells to both MT and V4 areas with corresponding synaptic weights. [sent-135, score-0.662]

66 b) Possible implementation in dorsal-ventral visual inference pathways, with multiple higher levels receiving input from one lower level. [sent-137, score-0.177]

67 5 Simulations In this section we examine how well the mechanisms introduced in the previous sections account for human behavioral data for two perceptual phenomena: cue combination and the oblique effect. [sent-139, score-0.39]

68 1 Haptic-visual cue combination When sensory cues come from multiple modalities, the nervous system is able to combine those cues optimally in the way dictated by Bayesian statistics [2]. [sent-141, score-0.36]

69 4a shows the setup of an experiment where a subject measures the height of a bar through haptic and visual inputs. [sent-143, score-0.392]

70 The object’s visual input is manipulated so that the visual cues can be inconsistent with haptic cues and visual noise 2 can be adjusted to different levels, i. [sent-144, score-0.694]

71 visual cue follows xV ∼ N (SV , σV ) and haptic cue follows 2 2 xH ∼ N (SH , σH ), where SV , SH , σV are controlled parameters. [sent-146, score-0.44]

72 4d shows the percentage of trials that participants report the comparison stimulus (consistent visual/haptic cues from 45-65mm) is larger than the standard stimulus (inconsistent visual/haptic cues, SV = 60mm and SH = 50mm). [sent-148, score-0.302]

73 With the increase of visual noise, haptic input accounts for larger weights in decision making and the percentage curve is shifted towards SH , consistent with Bayesian statistics. [sent-149, score-0.284]

74 We present an alternative solution based on importance sampling that encodes the probability distribution by a population of neurons directly. [sent-153, score-0.778]

75 The importance sampling solution approximates the posterior expectation of the bar’s height x∗ C given SV and SH . [sent-154, score-0.43]

76 It is straightforward to approximate posterior p(xV |SV ) using importance sampling: p(xV = x∗ |SV ) = E[1(xV = x∗ )|SV ] ≈ V V p(SV |x∗ ) V ≈ ∗ i p(SV |xV,i ) rV i rV,i x∗ ∼ p(xV ) V,i (6) where rV,i ∼ Poisson[c·p(SV |x∗ )] is the number of spikes emitted by neuron x∗ . [sent-158, score-0.462]

77 SV and SH are the sensory stimuli, XV and XH the values along the visual and haptic dimensions, and XC the combined estimate of object height. [sent-176, score-0.391]

78 (c) Illustration of importance sampling using two sensory arrays {x∗ }, {x∗ }. [sent-177, score-0.455]

79 The V,i H,j transparent ellipses indicate the tuning curves of high level neurons centered on values x∗ over C,k xV and xH . [sent-178, score-0.586]

80 The big ellipse represents the manipulated input with inconsistent sensory input and different variance structure. [sent-179, score-0.209]

81 Thus, the values x∗ employed in the computation are sampled from C,k normal perceptual conditions, namely consistent visual and haptic inputs (xV = xH ) and normal variance structure (transparent ellipses in Fig. [sent-184, score-0.379]

83 {x∗ },{x∗ } and {x∗ } consist of 20 independently drawn examples each, and the total ﬁring rate V,i H,j C,k of each set of neurons is limited to 30. [sent-198, score-0.37]

84 2 The oblique effect The oblique effect describes the phenomenon that people show greater sensitivity to bars with horizontal or vertical (0o /90o ) orientations than “oblique” orientations. [sent-201, score-0.646]

85 5a shows an experimental setup where subjects exhibited higher sensitivity in detecting the direction of rotation of a bar when the reference bar to which it was compared was in one of these cardinal orientations. [sent-203, score-0.297]

86 D ∆θ | D ~ NT(D) (0,σ∆θ) o Relative detection sensitivity 180 2 90 reference bar test bar 2 p(D=1) = p(D=-1) = 0. [sent-208, score-0.269]

87 The oblique effect is shown in lower panel, being greater sensitivity to orientation near the cardinal directions. [sent-211, score-0.471]

88 (c) The oblique effect emerges from our model, but depends on having the correct prior p(θ). [sent-213, score-0.302]

89 When combined with the angle of the reference bar r (shaded in the graphical model, since it is known), ∆θ generates the orientation of a test bar θ, and θ further generates the observation Sθ , both with normal distributions with variance σθ and σSθ respectively. [sent-215, score-0.216]

90 The oblique effect has been shown to be closely related to the number of V1 neurons that tuned to different orientations [25]. [sent-216, score-0.736]

91 Many studies have found more V1 neurons tuned to cardinal orientations than other orientations [13, 14, 15]. [sent-217, score-0.631]

92 Moreover, the uneven distribution of feature detection neurons is consistent with the idea that these neurons might be sampled proportional to the prior: more horizontal and vertical segments exist in the natural visual environment of humans. [sent-218, score-0.958]

93 Importance sampling provides a direct test of the hypothesis that preferential distribution of V1 neurons around 0o /90o can cause the oblique effect, which becomes a question of whether the oblique effect depends on the use of a prior p(θ) with this distribution. [sent-219, score-1.005]

94 5c shows that detection sensitivity is uncorrelated with orientations if we take a uniform prior p(θ), but exhibits the oblique effect under a prior that prefers cardinal directions. [sent-224, score-0.616]

95 In both cases, 40 neurons ∗ ∗ are used to represent each of ∆θi and θi , and results are averaged over 100 trials. [sent-225, score-0.37]

96 We have shown that a small number of feature detection neurons whose tuning curves represent a small set of typical examples from sensory experience is sufﬁcient to perform some basic forms of Bayesian inference. [sent-230, score-0.671]

97 Components of the importance sampler, such as the tuning curves and their synaptic strengths, need to be updated to match the distributions in the environment. [sent-236, score-0.489]

98 Humans integrate visual and haptic information in a statistically optimal fashion. [sent-251, score-0.284]

99 The orientation and direction selectivity of cells in macaque visual cortex. [sent-345, score-0.169]

100 Unequal representation of cardinal and oblique contours in ferret visual cortex. [sent-354, score-0.42]

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Abstract: One of the central problems in neuroscience is reconstructing synaptic connectivity in neural circuits. Synapses onto a neuron can be probed by sequentially stimulating potentially pre-synaptic neurons while monitoring the membrane voltage of the post-synaptic neuron. Reconstructing a large neural circuit using such a “brute force” approach is rather time-consuming and inefficient because the connectivity in neural circuits is sparse. Instead, we propose to measure a post-synaptic neuron’s voltage while stimulating sequentially random subsets of multiple potentially pre-synaptic neurons. To reconstruct these synaptic connections from the recorded voltage we apply a decoding algorithm recently developed for compressive sensing. Compared to the brute force approach, our method promises significant time savings that grow with the size of the circuit. 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This method can be scaled to reconstruct multiple synaptic connections onto one neuron by combining intracellular recordings from the postsynaptic neuron with photo-activation of pre-synaptic neurons using glutamate uncaging [913] or channelrhodopsin [14, 15], or with multi-electrode arrays [16, 17]. Neurons are sequentially stimulated to fire action potentials by scanning a laser beam (or electrode voltage) over a brain slice, while synaptic weights are measured by recording post-synaptic voltage. Although sequential excitation of single potentially pre-synaptic neurons could reveal connectivity, such a “brute force” approach is inefficient because the connectivity among neurons is sparse. Even among nearby neurons in the cerebral cortex, the probability of connection is only about ten percent [3-8]. Connection probability decays rapidly with the 1 distance between neurons and falls below one percent on the scale of a cortical column [3, 8]. Thus, most single-neuron stimulation trials would result in zero response making the brute force approach slow, especially for larger circuits. Another drawback of the brute force approach is that single-neuron stimulation cannot be combined efficiently with methods allowing parallel recording of neural activity, such as calcium imaging [18-22], voltage-sensitive dyes [23-25] or multi-electrode arrays [17, 26]. As these techniques do not reliably measure sub-threshold potential but report only spiking activity, they would reveal only the strongest connections that can drive a neuron to fire [2730]. Therefore, such combination would reveal only a small fraction of the circuit. We propose to circumvent the above limitations of the brute force approach by stimulating multiple potentially pre-synaptic neurons simultaneously and reconstructing individual connections by using a recently developed method called compressive sensing (CS) [31-35]. In each trial, we stimulate F neurons randomly chosen out of N potentially pre-synaptic neurons and measure post-synaptic activity. Although each measurement yields only a combined response to stimulated neurons, if synaptic inputs sum linearly in a post-synaptic neuron, one can reconstruct the weights of individual connections by using an optimization algorithm. Moreover, if the synaptic connections are sparse, i.e. only K << N potentially pre-synaptic neurons make synaptic connections onto a post-synaptic neuron, the required number of trials M ~ K log(N/K), which is much less than N [31-35]. The proposed method can be used even if only spiking activity is available. Because multiple neurons are driven to fire simultaneously, if several of them synapse on the post-synaptic neuron, they can induce one or more spikes in that neuron. As quantized spike counts carry less information than analog sub-threshold voltage recordings, reconstruction requires a larger number of trials. Yet, the method can be used to reconstruct a complete feedforward circuit from spike recordings. Reconstructing neural circuit with multi-neuronal excitation may be compared with mapping retinal ganglion cell receptive fields. Typically, photoreceptors are stimulated by white-noise checkerboard stimulus and the receptive field is obtained by Reverse Correlation (RC) in case of sub-threshold measurements or Spike-Triggered Average (STA) of the stimulus [36, 37]. Although CS may use the same stimulation protocol, for a limited number of trials, the reconstruction quality is superior to RC or STA. 2 Ma pp i ng sy na pti c inp ut s o nto o n e ne uro n We start by formalizing the problem of mapping synaptic connections from a population of N potentially pre-synaptic neurons onto a single neuron, as exemplified by granule cells synapsing onto a Purkinje cell (Figure 1a). Our experimental protocol can be illustrated using linear algebra formalism, Figure 1b. We represent synaptic weights as components of a column vector x, where zeros represent non-existing connections. Each row in the stimulation matrix A represents a trial, ones indicating neurons driven to spike once and zeros indicating non-spiking neurons. The number of rows in the stimulation matrix A is equal to the number of trials M. The column vector y represents M measurements of membrane voltage obtained by an intra-cellular recording from the post-synaptic neuron: y = Ax. (1) In order to recover individual synaptic weights, Eq. (1) must be solved for x. RC (or STA) solution to this problem is x = (ATA)-1AT y, which minimizes (y-Ax)2 if M>N. In the case M << N for a sparse circuit. In this section we search computationally for the minimum number of trials required for exact reconstruction as a function of the number of non-zero synaptic weights K out of N potentially pre-synaptic neurons. First, note that the number of trials depends on the number of stimulated neurons F. If F = 1 we revert to the brute force approach and the number of measurements is N, while for F = N, the measurements are redundant and no finite number suffices. As the minimum number of measurements is expected to scale as K logN, there must be an optimal F which makes each measurement most informative about x. To determine the optimal number of stimulated neurons F for given K and N, we search for the minimum number of trials M, which allows a perfect reconstruction of the synaptic connectivity x. For each F, we generate 50 synaptic weight vectors and attempt reconstruction from sequentially increasing numbers of trials. The value of M, at which all 50 recoveries are successful (up to computer round-off error), estimates the number of trial needed for reconstruction with probability higher than 98%. By repeating this procedure 50 times for each F, we estimate the mean and standard deviation of M. We find that, for given N and K, the minimum number of trials, M, as a function of the number of stimulated neurons, F, has a shallow minimum. As K decreases, the minimum shifts towards larger F because more neurons should be stimulated simultaneously for sparser x. For the explored range of simulation parameters, the minimum is located close to 0.1N. Next, we set F = 0.1N and explore how the minimum number of measurements required for exact reconstruction depends on K and N. Results of the simulations following the recipe described above are shown in Figure 3a. As expected, when x is sparse, M grows approximately linearly with K (Figure 3b), and logarithmically with N (Figure 3c). N = 1000 180 25 160 20 140 120 15 100 10 80 5 150 250 400 650 1000 Number of potential connections (N) K = 30 220 200 (a) 200 220 Number of measurements (M) 30 Number of measurements (M) Number of actual connections (K) Number of necessary measurements (M) (b) 180 160 140 120 100 80 5 10 15 20 25 30 Number of actual connections (K) 210 (c) 200 190 180 170 160 150 140 130 120 2 10 10 3 Number of potential connections (N) Figure 3: a) Minimum number of measurements M required for reconstruction as a function of the number of actual synapses, K, and the number of potential synapses, N. b) For given N, we find M ~ K. c) For given K, we find M ~ logN (note semi-logarithmic scale in c). 4 R o b ust nes s o f re con st r uc t io n s t o noi se a n d v io la tio n o f si m pli fy in g a ss umpt io n s To make our simulation more realistic we now take into account three possible sources of noise: 1) In reality, post-synaptic voltage on a given synapse varies from trial to trial [4, 5, 46-52], an effect we call synaptic noise. Such noise detrimentally affects reconstructions because each row of A is multiplied by a different instantiation of vector x. 2) Stimulation of neurons may be imprecise exciting a slightly different subset of neurons than intended and/or firing intended neurons multiple times. We call this effect stimulation noise. Such noise detrimentally affects reconstructions because, in its presence, the actual measurement matrix A is different from the one used for recovery. 3) A synapse may fail to release neurotransmitter with some probability. Naturally, in the presence of noise, reconstructions cannot be exact. We quantify the 4 reconstruction x − xr l2 = error ∑ N i =1 by the normalized x − xr l2–error l2 / xl , where 2 ( xi − xri ) 2 . We plot normalized reconstruction error in brute force approach (M = N = 500 trials) as a function of noise, as well as CS and RC reconstruction errors (M = 200, 600 trials), Figure 4. 2 0.9 Normalized reconstruction error ||x-x|| /||x|| 1 r 2 For each noise source, the reconstruction error of the brute force approach can be achieved with 60% fewer trials by CS method for the above parameters (Figure 4). For the same number of trials, RC method performs worse. Naturally, the reconstruction error decreases with the number of trials. The reconstruction error is most sensitive to stimulation noise and least sensitive to synaptic noise. 1 (a) 1 (b) 0.9 0.8 (c) 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0.7 RC: M=200 RC: M=600 Brute force method: M=500 CS: M=200 CS: M=600 0.6 0.5 0 0.05 0.1 0.15 Synaptic noise level 0.2 0.25 0 0.05 0.1 0.15 Stimulation noise level 0.2 0.25 0 0 0.05 0.1 0.15 0.2 0.25 Synaptic failure probability Figure 4: Impact of noise on the reconstruction quality for N = 500, K = 30, F = 50. a) Recovery error due to trial-to-trial variation in synaptic weight. The response y is calculated using the synaptic connectivity x perturbed by an additive Gaussian noise. The noise level is given by the coefficient of variation of synaptic weight. b) Recovery error due to stimulation noise. The matrix A used for recovery is obtained from the binary matrix used to calculate the measurement vector y by shifting, in each row, a fraction of ones specified by the noise level to random positions. c) Recovery error due to synaptic failures. The detrimental effect of the stimulation noise on the reconstruction can be eliminated by monitoring spiking activity of potentially pre-synaptic neurons. By using calcium imaging [18-22], voltage-sensitive dyes [23] or multi-electrode arrays [17, 26] one could record the actual stimulation matrix. Because most random matrices satisfy the reconstruction requirements [31, 34, 35], the actual stimulation matrix can be used for a successful recovery instead of the intended one. If neuronal activity can be monitored reliably, experiments can be done in a different mode altogether. Instead of stimulating designated neurons with high fidelity by using highly localized and intense light, one could stimulate all neurons with low probability. Random firing events can be detected and used in the recovery process. The light intensity can be tuned to stimulate the optimal number of neurons per trial. Next, we explore the sensitivity of the proposed reconstruction method to the violation of simplifying assumptions. First, whereas our simulation assumes that the actual number of connections, K, is known, in reality, connectivity sparseness is known a priori only approximately. Will this affect reconstruction results? In principle, CS does not require prior knowledge of K for reconstruction [31, 34, 35]. For the CoSaMP algorithm, however, it is important to provide value K larger than the actual value (Figure 5a). Then, the algorithm will find all the actual synaptic weights plus some extra non-zero weights, negligibly small when compared to actual ones. Thus, one can provide the algorithm with the value of K safely larger than the actual one and then threshold the reconstruction result according to the synaptic noise level. Second, whereas we assumed a linear summation of inputs [53], synaptic integration may be 2 non-linear [54]. We model non-linearity by setting y = yl + α yl , where yl represents linearly summed synaptic inputs. Results of simulations (Figure 5b) show that although nonlinearity can significantly degrade CS reconstruction quality, it still performs better than RC. 5 (b) 0.45 0.4 Actual K = 30 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 10 20 30 40 50 60 Normalized reconstruction error ||x-x||2/||x|| r 2 Normalized reconstrcution error ||x-x||2/||x|| r 2 (a) 0.9 0.8 0.7 CS RC 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.15-0.12-0.09-0.06-0.03 0 0.03 0.06 0.09 0.12 0.15 Relative strength of the non-linear term α × mean(y ) K fed to CoSaMP l Figure 5: Sensitivity of reconstruction error to the violation of simplifying assumptions for N = 500, K = 30, M = 200, F = 50. a) The quality of the reconstruction is not affected if the CoSaMP algorithm is fed with the value of K larger than actual. b) Reconstruction error computed in 100 realizations for each value of the quadratic term relative to the linear term. 5 Ma pp i ng sy na pti c inp ut s o nto a n e uro na l po pu la tio n Until now, we considered reconstruction of synaptic inputs onto one neuron using subthreshold measurements of its membrane potential. In this section, we apply CS to reconstructing synaptic connections onto a population of potentially post-synaptic neurons. Because in CS the choice of stimulated neurons is non-adaptive, by recording from all potentially post-synaptic neurons in response to one sequence of trials one can reconstruct a complete feedforward network (Figure 6). (a) x p(y=1) A Normalized reconstruction error ||x/||x||2−xr/||xr||2||2 y (b) (c) 100 (d) 500 700 900 1100 Number of spikes 1 STA CS 0.8 0.6 0.4 0.2 0 1000 0 300 3000 5000 7000 9000 Number of trials (M) Ax Figure 6: Mapping of a complete feedforward network. a) Each post-synaptic neuron (red) receives synapses from a sparse subset of potentially pre-synaptic neurons (blue). b) Linear algebra representation of the experimental protocol. c) Probability of firing as a function of synaptic current. d) Comparison of CS and STA reconstruction error using spike trains for N = 500, K = 30 and F = 50. Although attractive, such parallelization raises several issues. First, patching a large number of neurons is unrealistic and, therefore, monitoring membrane potential requires using different methods, such as calcium imaging [18-22], voltage sensitive dyes [23-25] or multielectrode arrays [17, 26]. As these methods can report reliably only spiking activity, the measurement is not analog but discrete. Depending on the strength of summed synaptic inputs compared to the firing threshold, the postsynaptic neuron may be silent, fire once or multiple times. As a result, the measured response y is quantized by the integer number of spikes. Such quantized measurements are less informative than analog measurements of the sub-threshold membrane potential. In the extreme case of only two quantization levels, spike or no spike, each measurement contains only 1 bit of information. Therefore, to achieve reasonable reconstruction quality using quantized measurements, a larger number of trials M>>N is required. We simulate circuit reconstruction from spike recordings in silico as follows. First, we draw synaptic weights from an experimentally motivated distribution. Second, we generate a 6 random stimulation matrix and calculate the product Ax. Third, we linear half-wave rectify this product and use the result as the instantaneous firing rate for the Poisson spike generator (Figure 6c). We used a rectifying threshold that results in 10% of spiking trials as typically observed in experiments. Fourth, we reconstruct synaptic weights using STA and CS and compare the results with the generated weights. We calculated mean error over 100 realizations of the simulation protocol (Figure 6d). Due to the non-linear spike generating procedure, x can be recovered only up to a scaling factor. We propose to calibrate x with a few brute-force measurements of synaptic weights. Thus, in calculating the reconstruction error using l2 norm, we normalize both the generated and recovered synaptic weights. Such definition is equivalent to the angular error, which is often used to evaluate the performance of STA in mapping receptive field [37, 55]. Why is CS superior to STA for a given number of trials (Figure 6d)? Note that spikeless trials, which typically constitute a majority, also carry information about connectivity. While STA discards these trials, CS takes them into account. In particular, CoSaMP starts with the STA solution as zeroth iteration and improves on it by using the results of all trials and the sparseness prior. 6 D i s c uss ion We have demonstrated that sparse feedforward networks can be reconstructed by stimulating multiple potentially pre-synaptic neurons simultaneously and monitoring either subthreshold or spiking response of potentially post-synaptic neurons. When sub-threshold voltage is recorded, significantly fewer measurements are required than in the brute force approach. Although our method is sensitive to noise (with stimulation noise worse than synapse noise), it is no less robust than the brute force approach or RC. The proposed reconstruction method can also recover inputs onto a neuron from spike counts, albeit with more trials than from sub-threshold potential measurements. This is particularly useful when intra-cellular recordings are not feasible and only spiking can be detected reliably, for example, when mapping synaptic inputs onto multiple neurons in parallel. For a given number of trials, our method yields smaller error than STA. The proposed reconstruction method assumes linear summation of synaptic inputs (both excitatory and inhibitory) and is sensitive to non-linearity of synaptic integration. Therefore, it is most useful for studying connections onto neurons, in which synaptic integration is close to linear. On the other hand, multi-neuron stimulation is closer than single-neuron stimulation to the intrinsic activity in the live brain and can be used to study synaptic integration under realistic conditions. In contrast to circuit reconstruction using intrinsic neuronal activity [56, 57], our method relies on extrinsic stimulation of neurons. Can our method use intrinsic neuronal activity instead? We see two major drawbacks of such approach. First, activity of non-monitored presynaptic neurons may significantly distort reconstruction results. Thus, successful reconstruction would require monitoring all active pre-synaptic neurons, which is rather challenging. Second, reliable reconstruction is possible only when the activity of presynaptic neurons is uncorrelated. Yet, their activity may be correlated, for example, due to common input. 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