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210 nips-2009-STDP enables spiking neurons to detect hidden causes of their inputs

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Author: Bernhard Nessler, Michael Pfeiffer, Wolfgang Maass

Abstract: The principles by which spiking neurons contribute to the astounding computational power of generic cortical microcircuits, and how spike-timing-dependent plasticity (STDP) of synaptic weights could generate and maintain this computational function, are unknown. We show here that STDP, in conjunction with a stochastic soft winner-take-all (WTA) circuit, induces spiking neurons to generate through their synaptic weights implicit internal models for subclasses (or “causes”) of the high-dimensional spike patterns of hundreds of pre-synaptic neurons. Hence these neurons will ﬁre after learning whenever the current input best matches their internal model. The resulting computational function of soft WTA circuits, a common network motif of cortical microcircuits, could therefore be a drastic dimensionality reduction of information streams, together with the autonomous creation of internal models for the probability distributions of their input patterns. We show that the autonomous generation and maintenance of this computational function can be explained on the basis of rigorous mathematical principles. In particular, we show that STDP is able to approximate a stochastic online Expectation-Maximization (EM) algorithm for modeling the input data. A corresponding result is shown for Hebbian learning in artiﬁcial neural networks. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 STDP enables spiking neurons to detect hidden causes of their inputs Bernhard Nessler, Michael Pfeiffer, and Wolfgang Maass Institute for Theoretical Computer Science, Graz University of Technology A-8010 Graz, Austria {nessler,pfeiffer,maass}@igi. [sent-1, score-0.537]

2 at Abstract The principles by which spiking neurons contribute to the astounding computational power of generic cortical microcircuits, and how spike-timing-dependent plasticity (STDP) of synaptic weights could generate and maintain this computational function, are unknown. [sent-3, score-0.591]

3 We show here that STDP, in conjunction with a stochastic soft winner-take-all (WTA) circuit, induces spiking neurons to generate through their synaptic weights implicit internal models for subclasses (or “causes”) of the high-dimensional spike patterns of hundreds of pre-synaptic neurons. [sent-4, score-0.72]

4 Hence these neurons will ﬁre after learning whenever the current input best matches their internal model. [sent-5, score-0.309]

5 1 Introduction It is well-known that synapses change their synaptic efﬁcacy (“weight”) w in dependence of the difference tpost − tpre of the ﬁring times of the post- and presynaptic neuron according to variations of a generic STDP rule (see [1] for a recent review). [sent-10, score-0.421]

6 However, the computational beneﬁt of this learning rule is largely unknown [2, 3]. [sent-11, score-0.125]

7 Finally, it has been conjectured for quite some while, on the basis of theoretical considerations, that the discovery and representation of hidden causes of their high-dimensional afferent spike inputs is a generic computational operation of cortical networks of neurons [5]. [sent-14, score-0.412]

8 We show in this extended abstract that STDP in WTA-circuits approximates EM for discovering hidden causes of large numbers of input spike trains. [sent-17, score-0.222]

9 In section 3 we show that the functioning of this demonstration can be explained on the basis of EM for simpler non-spiking approximations to the spiking network considered in section 2. [sent-19, score-0.337]

10 1 2 Discovery of hidden causes for a benchmark dataset We applied the network architecture shown in Fig. [sent-20, score-0.229]

11 1A to handwritten digits from the MNIST dataset [6]. [sent-21, score-0.188]

12 1 This dataset consists of 70, 000 28 × 28-pixel images of handwritten digits2 , from which we picked the subset of 20, 868 images containing only the digits 0, 3 and 4. [sent-22, score-0.188]

13 Simple STDP curve Complex STDP curve ∆w ki c · e−wki -1 σ 0 -1 0 tpost − tpre A B Figure 1: A) Architecture for learning with STDP in a WTA-network of spiking neurons. [sent-24, score-0.46]

14 The synaptic weight wki is changed in dependence of the ﬁring times tpre of the presynaptic neuron yi and tpost of the postsynaptic neuron zk . [sent-26, score-1.496]

15 If zk ﬁres at time t without a ﬁring of yi in the interval [t − σ, t + 2σ], wki is reduced by 1. [sent-27, score-1.103]

16 Pixel values xj were encoded through population coding by binary variables yi (spikes were produced for each variable yi by a Poisson process with a rate of 40 Hz for yi = 1, and 0 Hz for yi = 0, at a simulation time step of 1ms, see Fig. [sent-29, score-0.811]

17 Every neuron yi was connected to all K = 10 output neurons z1 , . [sent-32, score-0.478]

18 A Poisson process caused ﬁring of one of the neurons zk on average every 5ms (see [8] for a more realistic ﬁring mechanism). [sent-36, score-0.619]

19 The WTA-mechanism ensured that only one of the output neurons could ﬁre at any time step. [sent-37, score-0.227]

20 1B was applied to all synapses wki for an input consisting of a continuous sequence of spike encodings of handwritten digits, each presented for 50ms (see 1 A similar network of spiking neurons had been applied successfully in [7] to learn with STDP the classiﬁcation of symbolic (i. [sent-40, score-1.27]

21 All pixels that were black in less than 5% of the training examples were removed, leaving m = 429 external variables xj , that were encoded by n = 858 spiking neurons yi . [sent-45, score-0.779]

22 Our approach works just as well for external variables xj that assume any ﬁnite number of values, provided that they are presented to the network through population coding with one variable yi for every possible value of xj . [sent-46, score-0.524]

23 In fact, the approach appears to work also for the commonly considered population coding of continuous external variables. [sent-47, score-0.13]

24 3 This amounts to a representation of the EPSP caused by a ﬁring of neuron yi by a step function, which facilitates the theoretical analysis in section 3. [sent-48, score-0.251]

25 Learning with the spiking network works just as well for biologically realistic EPSP forms. [sent-49, score-0.354]

26 For testing we presented three examples from an independent test set of handwritten digits 0, 3, 4 from the MNIST dataset, and compared the ﬁring of the output-neurons before and after learning. [sent-51, score-0.208]

27 A) Representation of the three handwritten digits 0, 3, 4 for 50ms each by 858 spiking neurons yi . [sent-52, score-0.753]

28 C) Response of the output neurons after STDP (according to Fig. [sent-54, score-0.227]

29 1B) was applied to their weights wki for a continuous sequence of spike encodings of 4000 randomly drawn examples of handwritten digits 0, 3, 4, each represented for 50ms (like in panel A). [sent-55, score-0.842]

30 The three output neurons z4 , z9 , z6 that respond have generated internal models for the three shown handwritten digits according to Fig. [sent-56, score-0.516]

31 4 The learning rate η was chosen locally according to the variance tracking rule 7 . [sent-60, score-0.188]

32 2C shows that for subsequent representations of new handwritten samples of the same digits only one neuron responds during each of the 50ms while a handwritten digit is shown. [sent-62, score-0.461]

33 The implicit internal models which the output neurons z1 , . [sent-63, score-0.335]

34 Since there were more output neurons than digits, several output neurons created internal models for different ways of writing the same digit. [sent-68, score-0.58]

35 The adaptation of the spiking network to the examples shown so far is measured in Fig. [sent-70, score-0.357]

36 3A by the normalized conditional entropy H(L|Z)/H(L, Z), where L denotes the correct classiﬁcation of each handwritten digit y, and Z is the random variable which denotes the cluster assignment with p(Z = k|y) = p(zk = 1|y), the ﬁring probabilities at the presentation of digit y, see (1). [sent-71, score-0.267]

37 Since after training by STDP each of the output neurons ﬁre preferentially for one digit, we can measure the emergent classiﬁcation capability of the network. [sent-72, score-0.227]

38 3 Underlying theoretical principles We show in this section that one can analyze the learning dynamics of the spiking network considered in the preceding section (with the simple STDP curve of Fig. [sent-76, score-0.395]

39 1B with the help of Hebbian learning (using rule (12)) in a corresponding non-spiking neural network Nw . [sent-77, score-0.232]

40 Nw is a stochastic artiﬁcial neural network with the architecture shown in Fig. [sent-78, score-0.174]

41 , zK and weights wki for the connection from the ith input node yi (i = 1, . [sent-85, score-0.73]

42 05 0 0 500 1000 1500 2000 2500 3000 3500 4000 Training Examples A B C Figure 3: Analysis of the learning progress of the spiking network for the MNIST dataset. [sent-102, score-0.355]

43 A) Normalized conditional entropy (see text) for the spiking network with the two variants of STDP learning rules illustrated in Fig. [sent-103, score-0.375]

44 1B (red solid and blue dashed lines), as well as two non-spiking approximations of the network with learning rule (12) that are analyzed in section 3. [sent-104, score-0.232]

45 According to this analysis the non-spiking network with 35% missing attributes (dash-dotted line) is expected to have a very similar learning behavior to the spiking network. [sent-105, score-0.468]

46 2000 random examples of handwritten digits 0 and 3 were presented (for 50ms each) to the spiking network as the ﬁrst 2000 examples. [sent-106, score-0.545]

47 Then for the next 2000 examples also samples of handwritten digit 4 were included. [sent-107, score-0.194]

48 B) The implicit internal models created by the neurons after 2000 training examples are made explicit by drawing for each pixel the difference wki − wk(i+1) of the weights for input yi and yi+1 that encode the two possible values (black/white) of the variable xj that encodes this pixel value. [sent-108, score-1.122]

49 One can clearly see that neurons created separate internal models for different ways of writing the two digits 0 and 3. [sent-109, score-0.396]

50 C) Re-organized internal models after 2000 further training examples that included digit 4. [sent-110, score-0.171]

51 Two output neurons had created internal models for the newly introduced digit 4. [sent-111, score-0.402]

52 , K} deﬁned by p(zk = 1|y, w) = n euk wki yi + wk0 . [sent-115, score-0.755]

53 with uk = K eul (2) i=1 l=1 We consider the case where there are arbitrary discrete external variables x1 , . [sent-116, score-0.164]

54 , yn for n = m · M with i=1 yi = m according to the rule y(j−1)·M +r = 1 ⇐⇒ xj = r , for j = 1, . [sent-125, score-0.339]

55 In other K words: there exists some distribution over hidden binary variables zk with k=1 zk = 1, where the k with zk = 1 is usually referred to as a hidden “cause” in the generation of x, such that K p(zk = 1) · pk (x). [sent-149, score-1.461]

56 The problem of learning a generative model for some arbitrarily given input distribution p∗ (x) (or p∗ (y) after recoding according to (3)), by the neural network Nw is to ﬁnd a weight vector w such that p(y|w) deﬁned by (5) models p∗ (y) as accurately as possible. [sent-160, score-0.252]

57 We will not enforce the normalization (7) explicitly during the subsequently considered learning process, but rather use a learning rule (12) that turns out to automatically approximate such normalization in the limit. [sent-163, score-0.143]

58 1B for the spiking network of section 2 can be viewed as an approximation to an online version of this EM method. [sent-166, score-0.371]

59 1 the standard EM-approach, and show that the Hebbian learning rule (12) provides a stochastic approximation to the maximization step. [sent-168, score-0.168]

60 In the E-step one sets q(z|y) = p(z|y, wold ) for the current parameter values w = wold , thereby achieving Ep∗ [KL(q(z|y)||p(z|y, wold ))] = 0. [sent-172, score-0.222]

61 One can easily show that this is achieved by setting ∗ ∗ wki = log p∗ (yi = 1|zk = 1), and wk0 = log p∗ (zk = 1), (11) with values for the variables zk generated by q(z|y) = p(z|y, wold ), while the values for the variables y are generated by the external distribution p∗ . [sent-174, score-1.22]

62 Note that this distribution of z is exactly the distribution (2) of the output of the neural network Nw for inputs y generated by p∗ . [sent-175, score-0.177]

63 5 In the following section we will show that this M -step can be approximated by applying iteratively a simple Hebbian learning rule to the weights w of the neural network Nw . [sent-176, score-0.265]

64 With similar elementary calculations one can show that E[∆wki ] has for any w a value that moves wki in the direction of ∗ wki (in fact, exponentially fast). [sent-181, score-1.03]

65 One can actually show that one single step of (12) is a linear approximation of the ideal incremental update of wki = log aki , with aki and Nk representing the values of the corresponding sufﬁcient Nk +1 1 statistics, as log aki+1 = wki + log(1 + ηe−wk i ) − log(1 + η) for η = Nk . [sent-182, score-1.208]

66 In order to guarantee the stochastic convergence (see [12]) of the learning rule one has to use a ∞ ∞ decaying learning rate η (t) such that t=1 η (t) = ∞ and t=1 (η (t) )2 = 0. [sent-184, score-0.226]

67 7 The learning rule (12) is similar to a rule that had been introduced in [13] in the context of supervised learning and reinforcement learning. [sent-185, score-0.25]

68 That rule had satisﬁed an equilibrium condition similar to (13). [sent-186, score-0.144]

69 One can easily see the correspondence between the update of wki in (12) and in the simple STDP rule of Fig. [sent-188, score-0.622]

70 In fact, if each time where neuron zk ﬁres in the spiking network, each presynaptic neuron yi that currently has a high ﬁring rate has ﬁred within the last σ = 10ms before the ﬁring of zk , the two learning rules become equivalent. [sent-190, score-1.536]

71 4 the case for the non-spiking network where some attributes are missing (i. [sent-192, score-0.22]

72 , yi = 0 for all i ∈ Gj ; for some group Gj that encodes an external variable xj via population coding). [sent-194, score-0.314]

73 We ﬁrst show that the Hebbian learning rule (12) is also meaningful in the case of online learning of Nw , which better matches the online learning process for the spiking network. [sent-195, score-0.459]

74 3 Stochastic online EM The preceding arguments justify an application of learning rule (12) for a number of steps within each M-step of a batch EM approach for maximizing E∗ [log p(y|w)]. [sent-197, score-0.159]

75 If we assume that the consecutive values of the weights represent independent samples of their true stochastic distribution at the current learning rate, then this observed distribution is the log of a beta-distribution of the above mentioned parameters of the sufﬁcient statistics. [sent-201, score-0.131]

76 Analytically this distribution has the ﬁrst and second moments E[wki ] ≈ E[w2 ]−E[w ]2 2 new 1 1 ki ki log aki and E[wki ] ≈ E[wki ]2 + a1 + Ni , leading to the estimate ηki = Ni = e−E[wki ] +1 . [sent-202, score-0.193]

77 The Ni ki empirical estimates of these ﬁrst two moments can be gathered online by exponentially decaying averages using the same learning rate ηki . [sent-203, score-0.152]

78 6 is applied just once (for zk resulting from p(z|y, w) for the current weights w, or simpler: for the zk that is output by Nw for the current input y). [sent-204, score-0.979]

79 Our strategy for showing that a single application of learning rule (12) is expected to provide progress in an online EM-setting is the following. [sent-205, score-0.159]

80 Hence even a single application of learning rule (12) to a single new example y, drawn according to p∗ , is expected to increase Ep∗ [log p(y|w)] under the constraints (7). [sent-211, score-0.148]

81 2 that learning in the spiking network corresponds to learning in the non-spiking network Nw with missing attributes. [sent-214, score-0.53]

82 Their analysis implies that the correct learning action is then not to change the weights wki for i ∈ Gj . [sent-216, score-0.566]

83 1B, as well as (12), reduce also these weights by η if zk ﬁres. [sent-218, score-0.469]

84 This yields a modiﬁcation of the equilibrium analysis (13): E[∆wki ] = 0 ⇔ (1 − r) p∗ (yi =1|zk =1)η(e−wki − 1) − p∗ (yi =0|zk =1)η − rη = 0 ⇔ wki = log p∗ (yi =1|zk =1) + log(1 − r) , (17) where r is the probability that i belongs to a group Gj where the value of xj is missing. [sent-219, score-0.646]

85 Since this probability r is independent of the neuron zk and also independent of the current value of the external variable xi , this offset of log(1 − r) is expected to be the same for all weights. [sent-220, score-0.614]

86 5 Relationship between the spiking and the non-spiking network As indicated at the end of section 3. [sent-223, score-0.337]

87 2, the learning process for the spiking network from section 2 with the simple STDP curve from Fig. [sent-224, score-0.395]

88 1B (and external variables xj encoded by input spike trains from neurons yi ) is equivalent to a somewhat modiﬁed learning process of the non-spiking network Nw with the Hebbian learning rule (12) and external variables xj encoded by binary variables yi . [sent-225, score-1.258]

89 Each ﬁring of a neuron zk at some time t corresponds to a discrete time step in Nw with an application of the Hebbian learning rule (12). [sent-226, score-0.66]

90 Each neuron yi that had ﬁred during the time interval [t − 10ms, t] contributes a value yi (t) = 1 to the membrane potential uk (t) of the neuron zk at time ˜ t, and a value yi (0) = 0 if it did not ﬁre during [t − 10ms, t]. [sent-227, score-1.16]

91 , m} one has yi = 0 for all i ∈ Gj (since none of the neurons yi with i ∈ Gj ﬁred in the spiking network during [t − 10ms, t]). [sent-232, score-0.824]

92 This corresponds to several applications of rule (12) to the same input (but with different choices of missing attributes) in the non-spiking network. [sent-236, score-0.187]

93 In the experiments in section 2, every example in the non-spiking network with missing attributes was therefore presented for 10 steps, such that the average number of learning steps is the same as in the spiking case. [sent-237, score-0.468]

94 The learning process of the spiking network corresponds to a slight variation of the stochastic online EM algorithm that is implemented through (12) according to the analysis of section 3. [sent-238, score-0.455]

95 4 Discussion The model for discovering hidden causes of inputs that is proposed in this extended abstract presents an interesting shortcut for implementing and learning generative models for input data in networks of neurons. [sent-240, score-0.226]

96 3B) that is encoded in the weights of neurons in a simple WTA-circuit. [sent-242, score-0.253]

97 One might call it a Vapnik-style [16] approach towards generative modeling, since it focuses directly on the task to represent the most likely hidden causes of the inputs through neuronal ﬁring. [sent-243, score-0.148]

98 One just needs to add the stochastic and non-feedforward parts required for implementing stochastic WTA circuits to a 1-layer feedforward network, and apply the Hebbian learning rule (12) to the feedforward weights. [sent-245, score-0.281]

99 The simulation results of section 2 show that a simpliﬁed STDP rule that can be understood clearly from the perspective of stochastic online EM with a suitable Hebbian learning rule, provides good performance in discovering hidden causes for a standard benchmark dataset. [sent-248, score-0.33]

100 Spike-timing dependent plasticity performs stochastic expectation maximization to reveal the hidden causes of complex spike inputs. [sent-308, score-0.25]

similar papers computed by tfidf model

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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. We thank Ashok Veeraraghavan for introducing us to CS, Anthony Leonardo for making a retina dataset available for the analysis, Lou Scheffer and Hong Young Noh for commenting on the manuscript and anonymous reviewers for helpful suggestions. References [1] Luo, L., Callaway, E.M. & Svoboda, K. (2008) Genetic dissection of neural circuits. Neuron 57(5):634-660. [2] Helmstaedter, M., Briggman, K.L. & Denk, W. (2008) 3D structural imaging of the brain with photons and electrons. Current opinion in neurobiology 18(6):633-641. [3] Holmgren, C., Harkany, T., Svennenfors, B. & Zilberter, Y. (2003) Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. Journal of Physiology 551:139-153. [4] Markram, H. (1997) A network of tufted layer 5 pyramidal neurons. Cerebral Cortex 7(6):523-533. 7 [5] Markram, H., Lubke, J., Frotscher, M., Roth, A. & Sakmann, B. (1997) Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. Journal of Physiology 500(2):409-440. [6] Thomson, A.M. & Bannister, A.P. (2003) Interlaminar connections in the neocortex. Cerebral Cortex 13(1):5-14. [7] Thomson, A.M., West, D.C., Wang, Y. & Bannister, A.P. (2002) Synaptic connections and small circuits involving excitatory and inhibitory neurons in layers 2-5 of adult rat and cat neocortex: triple intracellular recordings and biocytin labelling in vitro. Cerebral Cortex 12(9):936-953. [8] Song, S., Sjostrom, P.J., Reigl, M., Nelson, S. & Chklovskii, D.B. (2005) Highly nonrandom features of synaptic connectivity in local cortical circuits. Plos Biology 3(3):e68. [9] Callaway, E.M. & Katz, L.C. (1993) Photostimulation using caged glutamate reveals functional circuitry in living brain slices. Proceedings of the National Academy of Sciences of the United States of America 90(16):7661-7665. [10] Dantzker, J.L. & Callaway, E.M. (2000) Laminar sources of synaptic input to cortical inhibitory interneurons and pyramidal neurons. Nature Neuroscience 3(7):701-707. [11] Shepherd, G.M. & Svoboda, K. (2005) Laminar and columnar organization of ascending excitatory projections to layer 2/3 pyramidal neurons in rat barrel cortex. Journal of Neuroscience 25(24):5670-5679. [12] Nikolenko, V., Poskanzer, K.E. & Yuste, R. (2007) Two-photon photostimulation and imaging of neural circuits. Nature Methods 4(11):943-950. [13] Shoham, S., O'connor, D.H., Sarkisov, D.V. & Wang, S.S. (2005) Rapid neurotransmitter uncaging in spatially defined patterns. Nature Methods 2(11):837-843. [14] Gradinaru, V., Thompson, K.R., Zhang, F., Mogri, M., Kay, K., Schneider, M.B. & Deisseroth, K. (2007) Targeting and readout strategies for fast optical neural control in vitro and in vivo. Journal of Neuroscience 27(52):14231-14238. [15] Petreanu, L., Huber, D., Sobczyk, A. & Svoboda, K. (2007) Channelrhodopsin-2-assisted circuit mapping of long-range callosal projections. Nature Neuroscience 10(5):663-668. [16] Na, L., Watson, B.O., Maclean, J.N., Yuste, R. & Shepard, K.L. (2008) A 256×256 CMOS Microelectrode Array for Extracellular Neural Stimulation of Acute Brain Slices. Solid-State Circuits Conference, 2008. ISSCC 2008. Digest of Technical Papers. IEEE International. [17] Fujisawa, S., Amarasingham, A., Harrison, M.T. & Buzsaki, G. (2008) Behavior-dependent shortterm assembly dynamics in the medial prefrontal cortex. Nature Neuroscience 11(7):823-833. [18] Ikegaya, Y., Aaron, G., Cossart, R., Aronov, D., Lampl, I., Ferster, D. & Yuste, R. (2004) Synfire chains and cortical songs: temporal modules of cortical activity. Science 304(5670):559-564. [19] Ohki, K., Chung, S., Ch'ng, Y.H., Kara, P. & Reid, R.C. (2005) Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature 433(7026):597-603. [20] Stosiek, C., Garaschuk, O., Holthoff, K. & Konnerth, A. (2003) In vivo two-photon calcium imaging of neuronal networks. Proceedings of the National Academy of Sciences of the United States of America 100(12):7319-7324. [21] Svoboda, K., Denk, W., Kleinfeld, D. & Tank, D.W. (1997) In vivo dendritic calcium dynamics in neocortical pyramidal neurons. Nature 385(6612):161-165. [22] Sasaki, T., Minamisawa, G., Takahashi, N., Matsuki, N. & Ikegaya, Y. (2009) Reverse optical trawling for synaptic connections in situ. Journal of Neurophysiology 102(1):636-643. [23] Zecevic, D., Djurisic, M., Cohen, L.B., Antic, S., Wachowiak, M., Falk, C.X. & Zochowski, M.R. (2003) Imaging nervous system activity with voltage-sensitive dyes. Current Protocols in Neuroscience Chapter 6:Unit 6.17. [24] Cacciatore, T.W., Brodfuehrer, P.D., Gonzalez, J.E., Jiang, T., Adams, S.R., Tsien, R.Y., Kristan, W.B., Jr. & Kleinfeld, D. (1999) Identification of neural circuits by imaging coherent electrical activity with FRET-based dyes. Neuron 23(3):449-459. [25] Taylor, A.L., Cottrell, G.W., Kleinfeld, D. & Kristan, W.B., Jr. (2003) Imaging reveals synaptic targets of a swim-terminating neuron in the leech CNS. Journal of Neuroscience 23(36):11402-11410. [26] Hutzler, M., Lambacher, A., Eversmann, B., Jenkner, M., Thewes, R. & Fromherz, P. (2006) High-resolution multitransistor array recording of electrical field potentials in cultured brain slices. Journal of Neurophysiology 96(3):1638-1645. [27] Egger, V., Feldmeyer, D. & Sakmann, B. (1999) Coincidence detection and changes of synaptic efficacy in spiny stellate neurons in rat barrel cortex. Nature Neuroscience 2(12):1098-1105. [28] Feldmeyer, D., Egger, V., Lubke, J. & Sakmann, B. (1999) Reliable synaptic connections between pairs of excitatory layer 4 neurones within a single 'barrel' of developing rat somatosensory cortex. Journal of Physiology 521:169-190. [29] Peterlin, Z.A., Kozloski, J., Mao, B.Q., Tsiola, A. & Yuste, R. (2000) Optical probing of neuronal circuits with calcium indicators. Proceedings of the National Academy of Sciences of the United States of America 97(7):3619-3624. 8 [30] Thomson, A.M., Deuchars, J. & West, D.C. (1993) Large, deep layer pyramid-pyramid single axon EPSPs in slices of rat motor cortex display paired pulse and frequency-dependent depression, mediated presynaptically and self-facilitation, mediated postsynaptically. Journal of Neurophysiology 70(6):2354-2369. [31] Baraniuk, R.G. (2007) Compressive sensing. Ieee Signal Processing Magazine 24(4):118-120. [32] Candes, E.J. (2008) Compressed Sensing. Twenty-Second Annual Conference on Neural Information Processing Systems, Tutorials. [33] Candes, E.J., Romberg, J.K. & Tao, T. (2006) Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics 59(8):1207-1223. [34] Candes, E.J. & Tao, T. (2006) Near-optimal signal recovery from random projections: Universal encoding strategies? Ieee Transactions on Information Theory 52(12):5406-5425. [35] Donoho, D.L. (2006) Compressed sensing. Ieee Transactions on Information Theory 52(4):12891306. [36] Ringach, D. & Shapley, R. (2004) Reverse Correlation in Neurophysiology. Cognitive Science 28:147-166. [37] Schwartz, O., Pillow, J.W., Rust, N.C. & Simoncelli, E.P. (2006) Spike-triggered neural characterization. Journal of Vision 6(4):484-507. [38] Candes, E.J. & Tao, T. (2005) Decoding by linear programming. Ieee Transactions on Information Theory 51(12):4203-4215. [39] Needell, D. & Vershynin, R. (2009) Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit. Foundations of Computational Mathematics 9(3):317-334. [40] Tropp, J.A. & Gilbert, A.C. (2007) Signal recovery from random measurements via orthogonal matching pursuit. Ieee Transactions on Information Theory 53(12):4655-4666. [41] Dai, W. & Milenkovic, O. (2009) Subspace Pursuit for Compressive Sensing Signal Reconstruction. Ieee Transactions on Information Theory 55(5):2230-2249. [42] Needell, D. & Tropp, J.A. (2009) CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis 26(3):301-321. [43] Varshney, L.R., Sjostrom, P.J. & Chklovskii, D.B. (2006) Optimal information storage in noisy synapses under resource constraints. Neuron 52(3):409-423. [44] Brunel, N., Hakim, V., Isope, P., Nadal, J.P. & Barbour, B. (2004) Optimal information storage and the distribution of synaptic weights: perceptron versus Purkinje cell. Neuron 43(5):745-757. [45] Napoletani, D. & Sauer, T.D. (2008) Reconstructing the topology of sparsely connected dynamical networks. Physical Review E 77(2):026103. [46] Allen, C. & Stevens, C.F. (1994) An evaluation of causes for unreliability of synaptic transmission. Proceedings of the National Academy of Sciences of the United States of America 91(22):10380-10383. [47] Hessler, N.A., Shirke, A.M. & Malinow, R. (1993) The probability of transmitter release at a mammalian central synapse. Nature 366(6455):569-572. [48] Isope, P. & Barbour, B. (2002) Properties of unitary granule cell-->Purkinje cell synapses in adult rat cerebellar slices. Journal of Neuroscience 22(22):9668-9678. [49] Mason, A., Nicoll, A. & Stratford, K. (1991) Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro. Journal of Neuroscience 11(1):72-84. [50] Raastad, M., Storm, J.F. & Andersen, P. (1992) Putative Single Quantum and Single Fibre Excitatory Postsynaptic Currents Show Similar Amplitude Range and Variability in Rat Hippocampal Slices. European Journal of Neuroscience 4(1):113-117. [51] Rosenmund, C., Clements, J.D. & Westbrook, G.L. (1993) Nonuniform probability of glutamate release at a hippocampal synapse. Science 262(5134):754-757. [52] Sayer, R.J., Friedlander, M.J. & Redman, S.J. (1990) The time course and amplitude of EPSPs evoked at synapses between pairs of CA3/CA1 neurons in the hippocampal slice. Journal of Neuroscience 10(3):826-836. [53] Cash, S. & Yuste, R. (1999) Linear summation of excitatory inputs by CA1 pyramidal neurons. Neuron 22(2):383-394. [54] Polsky, A., Mel, B.W. & Schiller, J. (2004) Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience 7(6):621-627. [55] Paninski, L. (2003) Convergence properties of three spike-triggered analysis techniques. Network: Computation in Neural Systems 14(3):437-464. [56] Okatan, M., Wilson, M.A. & Brown, E.N. (2005) Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity. Neural Computation 17(9):1927-1961. [57] Timme, M. (2007) Revealing network connectivity from response dynamics. Physical Review Letters 98(22):224101. 9

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