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13 nips-2009-A Neural Implementation of the Kalman Filter

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Author: Robert Wilson, Leif Finkel

Abstract: Recent experimental evidence suggests that the brain is capable of approximating Bayesian inference in the face of noisy input stimuli. Despite this progress, the neural underpinnings of this computation are still poorly understood. In this paper we focus on the Bayesian ﬁltering of stochastic time series and introduce a novel neural network, derived from a line attractor architecture, whose dynamics map directly onto those of the Kalman ﬁlter in the limit of small prediction error. When the prediction error is large we show that the network responds robustly to changepoints in a way that is qualitatively compatible with the optimal Bayesian model. The model suggests ways in which probability distributions are encoded in the brain and makes a number of testable experimental predictions. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Recent experimental evidence suggests that the brain is capable of approximating Bayesian inference in the face of noisy input stimuli. [sent-4, score-0.15]

2 In this paper we focus on the Bayesian ﬁltering of stochastic time series and introduce a novel neural network, derived from a line attractor architecture, whose dynamics map directly onto those of the Kalman ﬁlter in the limit of small prediction error. [sent-6, score-0.603]

3 When the prediction error is large we show that the network responds robustly to changepoints in a way that is qualitatively compatible with the optimal Bayesian model. [sent-7, score-0.433]

4 We develop a neural network whose dynamics can be shown to approximate those of a one-dimensional Kalman ﬁlter, the Bayesian model when all the distributions are Gaussian. [sent-14, score-0.424]

5 Where the approximation breaks down, for large prediction errors, the network performs something akin to outlier or change detection and this ‘failure’ suggests ways in which the network can be extended to deal with more complex, non-Gaussian distributions over multiple dimensions. [sent-15, score-0.682]

6 Our approach rests on the modiﬁcation of the line attractor network of Zhang [26]. [sent-16, score-0.612]

7 In particular, we make three changes to Zhang’s network, modifying the activation rule, the weights and the inputs in such a way that the network’s dynamics map exactly onto those of the Kalman ﬁlter when the prediction error is small. [sent-17, score-0.256]

8 Crucially, these modiﬁcations result in a network that is no longer a line attractor and thus no longer suffers from many of the limitations of these networks. [sent-18, score-0.612]

9 ˆ ¯ 3 The neural network The network is a modiﬁcation of Zhang’s line attractor model of head direction cells [26]. [sent-22, score-1.088]

10 We use rate neurons and describe the state of the network at time t with the membrane potential vector, u(t), where each component of u(t) denotes the membrane potential of a single neuron. [sent-23, score-0.623]

11 In discrete time, the update equation is then u(t + 1) = wJf [u(t)] + I(t + 1) (4) where w scales the strength of the weights, J is the connectivity matrix, f [·] is the activation rule that maps membrane potential onto ﬁring rate, and I(t + 1) is the input at time t + 1. [sent-24, score-0.396]

12 When w = 1, γ(t) = 0 and I(t) = 0, the network is a line attractor over a wide range of Kw , σw , c, S and µ, having a continuum of ﬁxed points (as N → ∞). [sent-28, score-0.612]

13 Each ﬁxed point has the same shape, taking the form of a smooth membrane potential proﬁle, U(x) = Jsym f [U(x)], centered at location, x, in the network. [sent-29, score-0.146]

14 When γ(t) = 0, the bump of activity can be made to move over time (without losing its shape) [26] and hence, so long as γ(t) = v(t), implement the prediction step of the Kalman ﬁlter (equation 1). [sent-30, score-0.73]

15 That is, if the bump at time t is centered at x(t), i. [sent-31, score-0.48]

16 Thus, in ¯ ˆ x x this conﬁguration, the network can already implement the ﬁrst step of the Kalman ﬁlter through its recurrent connectivity. [sent-36, score-0.419]

17 The next two steps, equations 2 and 3, however, remain inaccessible as the network has no way of encoding uncertainty and it is unclear how it will deal with external inputs. [sent-37, score-0.432]

18 4 Relation to Kalman ﬁlter - small prediction error case In this section we outline how the neural network dynamics can be mapped onto those of a Kalman ﬁlter. [sent-38, score-0.533]

19 2 Our approach is to analyze the network in terms of U, which, for clarity, we deﬁne here to be the ﬁxed point membrane potential proﬁle of the network when w = 1, γ(t) = 0, I(t) = 0, S = S0 and µ = µ0 . [sent-40, score-0.693]

20 We begin by making the assumption that both the input, I(t), and the network membrane potential, u(t), take the form of scaled versions U, with the former encoding the noisy observations, z(t), and the latter encoding the network’s estimate of position, x(t), i. [sent-42, score-0.487]

21 w= S S0 + µ0 I (15) are identical to equations 2 and 3 so long as (a) α(t) ∝ 1 σx (t)2 ˆ (b) A(t) ∝ 1 σz (t)2 (c) µI ∝ σv (t)2 S (16) Thus the network dynamics, when the prediction error is small, map directly onto the Kalman ﬁlter equations. [sent-50, score-0.429]

22 1 Implications Reciprocal code for uncertainty in input and estimate Equation 16a provides a link between the strength of activity in the network and the overall uncertainty in the estimate of the Kalman ﬁlter, σx (t), with uncertainty decreasing as the activity increases. [sent-54, score-0.844]

23 A similar relation is also implied for the ˆ uncertainty in the observations, σz (t), where equation 16b suggests that this should be reﬂected in the magnitude of the input, A(t). [sent-55, score-0.13]

24 Code for velocity signal As with Zhang’s line attractor network [26], the mean of the velocity signal, v(t) is encoded into the recurrent connections of the network, with the degree of asymmetry in the weights, γ(t), proportional to the speed. [sent-57, score-0.902]

25 Such hard coding of the velocity signal represents a limitation of the model, as we would like to be able to deal with arbitrary, time varying speeds. [sent-58, score-0.178]

26 However, this kind of change could be implemented by pre-synaptic inhibition [24] or by using a ‘double-ring’ network similar to [25]. [sent-59, score-0.384]

27 Equation 16c implies that the variance of the velocity signal, σv (t), is encoded in the strength of the divisive feedback, µ (assuming constant S). [sent-60, score-0.255]

28 This is very different from Zhang’s model, that has no concept of uncertainty and is also very different from the traditional view of divisive inhibition that sees it as a mechanism for gain control [14, 13]. [sent-61, score-0.232]

29 The network is no longer a line attractor This can be seen by considering the ﬁxed point values of the scale factor, α(t), when the input current, I(t) = 0. [sent-62, score-0.68]

30 This is a key result as it removes all of the constraints required for line attractor dynamics such as inﬁnite precision in the weights and lack of noise in the network and thus the network is much more biologically plausible. [sent-64, score-1.118]

31 2 An example In ﬁgure 1 we demonstrate the ability of the network to approximate the dynamics of a onedimensional Kalman ﬁlter. [sent-66, score-0.429]

32 The input, shown in ﬁgure 1A, is a noiseless bump of current centered 4 B 100 80 80 neuron # 100 neuron # A 60 40 20 40 20 0 C 60 20 40 60 time step 80 100 0 D 60 20 40 60 time step 80 100 0 20 40 60 time step 80 100 2 σx(t) position 50 40 30 1. [sent-67, score-0.903]

33 5 20 0 0 20 40 60 time step 80 100 Figure 2: Response of the network when presented with a noisy moving bump input. [sent-69, score-0.856]

34 The observation noise has standard deviation σz (t) = 5, the speed v(t) = 0. [sent-71, score-0.136]

35 In accordance with equation 16b, the height of each bump is scaled by 1/σz (t)2 . [sent-75, score-0.508]

36 In ﬁgure 1B we plot the output activity of the network over time. [sent-76, score-0.454]

37 We assume that the network gets the correct velocity signal, i. [sent-78, score-0.362]

38 As can be seen from the plot, the amount of activity in the network steadily grows from zero over time to an asymptotic value, corresponding to the network’s increasing certainty in its predictions. [sent-85, score-0.456]

39 The position of the bump of activity in the network is also much less jittery than the input bump, reﬂecting a certain amount of smoothing. [sent-86, score-1.053]

40 In ﬁgure 1C we compare the positions of the input bumps (gray dots) with the position of the network bump (black line) and the output of the equivalent Kalman ﬁlter (red line). [sent-87, score-0.969]

41 The network clearly tracks the Kalman ﬁlter estimate extremely well. [sent-88, score-0.312]

42 The same is true for the network’s estimate of the uncertainty, computed as 1/ α(t) and shown as the black line in ﬁgure 1D, which tracks the Kalman ﬁlter uncertainty (red line) almost exactly. [sent-89, score-0.218]

43 5 Effect of input noise We now consider the effect of noise on the ability of the network to implement a Kalman ﬁlter. [sent-90, score-0.608]

44 In particular we consider noise in the input signal, which for this simple one layer network is equivalent to having noise in the update equation. [sent-91, score-0.569]

45 Speciﬁcally, we consider input signals where the only source of noise is in the input current i. [sent-93, score-0.243]

46 there is no additional jitter in the position of the bump as there was in the noiseless case, thus we write I(t) = A(t)U (x(t)) + (t) (18) where (t) is some noise vector. [sent-95, score-0.74]

47 The main effect of the noise is that it perturbs the effective position of the input bump. [sent-96, score-0.315]

48 This can be modeled by extracting the maximum likelihood estimate of the input position given the noisy input and then using this position as the input to the equivalent Kalman ﬁlter. [sent-97, score-0.535]

49 Because of the noise, this extracted position is not, in general, the same as the noiseless input position and for zero mean Gaussian noise with covariance Σ, the variance of the perturbation, σz (t), 5 B 10 6 ERMS vs KF A α 4 2 0 8 6 4 2 0 0 0. [sent-98, score-0.497]

50 is approximately given by σz (t) ≈ 1 A(t) 2 U T Σ−1 U (19) Now, for the network to approximate a Kalman ﬁlter, equation 16b must hold which means that we require the magnitude of the covariance matrix to scale in proportion to the strength of the input signal, A(t), i. [sent-105, score-0.464]

51 Interestingly this relation is true for Poisson noise, the type of noise that is found all over the brain. [sent-108, score-0.131]

52 In ﬁgure 2 we demonstrate the ability of the network to approximate a Kalman ﬁlter. [sent-109, score-0.287]

53 In panel A we show the input current which is a moving bump of activity corrupted by independent Gaussian noise of standard deviation σnoise = 0. [sent-110, score-0.802]

54 This is a high noise setting and it is hard to see the bump location by eye. [sent-112, score-0.528]

55 The network dramatically cleans up this input signal (ﬁgure 2B) and the output activity, although still noisy, reﬂects the position of the underlying stimulus much more faithfully than the input. [sent-113, score-0.562]

56 In panel C we compare the position of the output bump in the network (black line) with that of the equivalent Kalman ﬁlter. [sent-115, score-0.848]

57 To do this we ﬁrst ﬁt the noisy input bump at each time to obtain input positions z(t) shown as gray dots. [sent-116, score-0.64]

58 which closely match those of the network (black line). [sent-119, score-0.287]

59 Similarly, there is good agreement between the two estimates of the uncertainty, σx (t), panel D (black line - network, red line - Kalman ﬁlter). [sent-120, score-0.209]

60 1 Performance of the network as a function of noise magnitude The noise not only affects the position of the input bump but also, in a slightly more subtle manner, causes a gradual decline in the ability of the network to emulate a Kalman ﬁlter. [sent-122, score-1.446]

61 The reason for this (outlined in more detail in the supplementary material) is that the output bump scale factor, α, decreases as a function of the noise variance, σnoise . [sent-123, score-0.587]

62 The average results of simulations on 100 neurons are shown as the red dots, while the black line represents the results of the theory in the supplementary material. [sent-125, score-0.331]

63 The reason for the decline in α as σnoise goes up is that, because of the rectifying non-linearity in the activation rule, increasing σnoise increases the amount of noisy activity in the network. [sent-126, score-0.252]

64 Because of inhibition (both divisive and subtractive) in the network, this ‘noisy activity’ competes with the bump activity and decreases it - thus reducing α. [sent-127, score-0.713]

65 We quantify this difference in ﬁgure 3B where we plot the root mean squared error (in units of neural position) between the network and the equivalent Kalman ﬁlter as a function of σnoise . [sent-129, score-0.342]

66 As before, the results of simulations are shown as red dots and the theory (outlined in the supplementary material) is the black line. [sent-130, score-0.229]

67 To give some sense for the scale on this plot, the horizontal blue line corresponds to the maximum height of the (noise free) input bump. [sent-131, score-0.186]

68 Thus we may conclude that the performance of the network and the theory are robust up to fairly large values of σnoise . [sent-132, score-0.287]

69 6 Response to changepoints (and outliers) - large prediction error case We now consider the dynamics of the network when the prediction error is large. [sent-134, score-0.621]

70 By large we mean that the prediction error is greater than the width of the bump of activity in the network. [sent-135, score-0.634]

71 a sustained large and abrupt change in the input position at a random time. [sent-138, score-0.24]

72 In the interests of space we focus only on the latter case and such an input, with a changepoint at t = 50, is shown in ﬁgure 4A. [sent-139, score-0.167]

73 As before, prior to the change, there is a single bump of activity whose position approximates that of a Kalman ﬁlter. [sent-141, score-0.698]

74 However, after the changepoint, the network maintains two bumps of activity for several time steps. [sent-142, score-0.509]

75 One at the original position, that shrinks over time and essentially predicts where the input would be if the change had not occurred, and a second, that grows over time, at the location of the input after the changepoint. [sent-143, score-0.2]

76 Thus in the period immediately after the changepoint, the network can be thought of as encoding two separate and competing hypotheses about the position of the stimulus, one corresponding to the case where no change has occurred, and the other, the case where a change occurred at t = 50. [sent-144, score-0.548]

77 In ﬁgure 4C we compare the position of the bump(s) in the network (black dots whose size reﬂects the size of each bump) to the output from the Kalman ﬁlter (red line). [sent-145, score-0.475]

78 Finally, in ﬁgure 4D we plot the scale factor, αi (t), of each bump as computed from the simulations (black dots) and from the approximate analytic solution described in the supplementary material (red line for bump at 30, blue line for bump at 80). [sent-148, score-1.579]

79 Thus, when confronted with a changepoint, the network no longer approximates a Kalman ﬁlter and instead maintains two competing hypotheses in a way that is qualitatively similar to that of the run-length distribution in [1]. [sent-150, score-0.287]

80 Although this work, at ﬁrst glance, seems similar to what is presented here, there are several major differences, the main one being that our network is not a line attractor at all, while the results in [8] rely on this property. [sent-155, score-0.612]

81 Combined with divisive normalization, these networks can implement a Kalman ﬁlter exactly, while the model presented here can ‘only’ approximate one. [sent-158, score-0.129]

82 While this may seem like a limitation of our network, we see it as an advantage as the breakdown of the approximation leads to a more robust response to outliers and changepoints than a pure Kalman ﬁlter. [sent-159, score-0.156]

83 One possible way around this limitation is hinted at by the response of the network in the changepoint case where we saw two, largely independent bumps of activity in the network. [sent-162, score-0.703]

84 This ability to encode multiple ‘particles’ in the network may allow networks of this kind to implement something like the dynamics of a particle ﬁlter [12] that can approximate the inference process for non-linear and non-Gaussian systems. [sent-163, score-0.438]

85 3 Experimental predictions The model makes at least two easily testable predictions about the response of head direction cells [21, 22, 23] in rats. [sent-166, score-0.265]

86 The ﬁrst comes by considering the response of the neurons in the ‘dark’. [sent-167, score-0.119]

87 Assuming that all bearing cues can indeed be eliminated, by setting A(t) = 0 in equation 13, we expect the activity of the neurons to fall off as 1/t and that the shape of the tuning curves will remain approximately constant. [sent-168, score-0.256]

88 Note that this prediction is vastly different from the behaviour of a line attractor, where we would not expect the level of activity to fall off at all in the dark. [sent-169, score-0.297]

89 In particular, one could imagine a training phase, where the position of one landmark is jittered over time, such that each time the rat encounters it it is at a slightly different heading. [sent-171, score-0.247]

90 In the test case, all other, reliable, landmark cues would be removed and the response of head direction cells measured in response to presentation of the unreliable cue alone. [sent-172, score-0.375]

91 The prediction of the model is that this would reduce the strength of the input, A, which in turn reduces the level of activity in the head direction cells, α. [sent-173, score-0.372]

92 This prediction is very different from that of a line attractor which would predict a constant level of activity regardless of the reliability of the landmark cues. [sent-175, score-0.638]

93 8 Conclusions In this paper we have introduced a novel neural network model whose dynamics map directly onto those of a one-dimensional Kalman ﬁlter when the prediction error is small. [sent-176, score-0.533]

94 This property is robust to noise and when the prediction error is large, such as for changepoints, the output of the network diverges from that of the Kalman ﬁlter, but in a way that is both interesting and useful. [sent-177, score-0.47]

95 Finally, the model makes two easily testable experimental predictions about head direction cells. [sent-178, score-0.151]

96 The contribution of noise to contrast invariance of orientation tuning in cat visual cortex. [sent-195, score-0.138]

97 Optimal sensorimotor integration in recurrent cortical networks: a neural implementation of kalman ﬁlters. [sent-239, score-0.641]

98 Head-direction cells recorded from postsubiculum in freely moving rats. [sent-329, score-0.167]

99 Head-direction cells recorded from postsubiculum in freely moving rats. [sent-340, score-0.167]

100 Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. [sent-367, score-0.137]

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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. We use computer simulations to find optimal stimulation parameters and explore the feasibility of our reconstruction method under realistic experimental conditions including noise and non-linear synaptic integration. Multineuronal stimulation allows reconstructing synaptic connectivity just from the spiking activity of post-synaptic neurons, even when sub-threshold voltage is unavailable. By using calcium indicators, voltage-sensitive dyes, or multi-electrode arrays one could monitor activity of multiple postsynaptic neurons simultaneously, thus mapping their synaptic inputs in parallel, potentially reconstructing a complete neural circuit. 1 In tro d uc tio n Understanding information processing in neural circuits requires systematic characterization of synaptic connectivity [1, 2]. The most direct way to measure synapses between a pair of neurons is to stimulate potentially pre-synaptic neuron while recording intra-cellularly from the potentially post-synaptic neuron [3-8]. 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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