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121 nips-2009-Know Thy Neighbour: A Normative Theory of Synaptic Depression

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Author: Jean-pascal Pfister, Peter Dayan, Máté Lengyel

Abstract: Synapses exhibit an extraordinary degree of short-term malleability, with release probabilities and effective synaptic strengths changing markedly over multiple timescales. From the perspective of a ﬁxed computational operation in a network, this seems like a most unacceptable degree of added variability. We suggest an alternative theory according to which short-term synaptic plasticity plays a normatively-justiﬁable role. This theory starts from the commonplace observation that the spiking of a neuron is an incomplete, digital, report of the analog quantity that contains all the critical information, namely its membrane potential. We suggest that a synapse solves the inverse problem of estimating the pre-synaptic membrane potential from the spikes it receives, acting as a recursive ﬁlter. We show that the dynamics of short-term synaptic depression closely resemble those required for optimal ﬁltering, and that they indeed support high quality estimation. Under this account, the local postsynaptic potential and the level of synaptic resources track the (scaled) mean and variance of the estimated presynaptic membrane potential. We make experimentally testable predictions for how the statistics of subthreshold membrane potential ﬂuctuations and the form of spiking non-linearity should be related to the properties of short-term plasticity in any particular cell type. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 uk Abstract Synapses exhibit an extraordinary degree of short-term malleability, with release probabilities and effective synaptic strengths changing markedly over multiple timescales. [sent-12, score-0.295]

2 We suggest an alternative theory according to which short-term synaptic plasticity plays a normatively-justiﬁable role. [sent-14, score-0.38]

3 This theory starts from the commonplace observation that the spiking of a neuron is an incomplete, digital, report of the analog quantity that contains all the critical information, namely its membrane potential. [sent-15, score-0.885]

4 We suggest that a synapse solves the inverse problem of estimating the pre-synaptic membrane potential from the spikes it receives, acting as a recursive ﬁlter. [sent-16, score-1.105]

5 We show that the dynamics of short-term synaptic depression closely resemble those required for optimal ﬁltering, and that they indeed support high quality estimation. [sent-17, score-0.682]

6 Under this account, the local postsynaptic potential and the level of synaptic resources track the (scaled) mean and variance of the estimated presynaptic membrane potential. [sent-18, score-1.575]

7 We make experimentally testable predictions for how the statistics of subthreshold membrane potential ﬂuctuations and the form of spiking non-linearity should be related to the properties of short-term plasticity in any particular cell type. [sent-19, score-1.117]

8 The effect of a spike from a presynaptic neuron on its postsynaptic partner depends on the history of the activity of both pre- and postsynaptic neurons, and thus the efﬁcacy of a synapse undergoes perpetual modiﬁcation. [sent-21, score-0.921]

9 Short-term plasticity typically only depends on the ﬁring pattern 1 of the presynaptic cell [1]; short term depression gradually diminishes the postsynaptic effects of presynaptic spikes that arrive in quick succession (Fig. [sent-23, score-1.29]

10 Given the prominence and ubiquity of synaptic depression in cortical (and subcortical) synapses [2], it is pressing to identify its computational role(s). [sent-25, score-0.633]

11 Here, we propose a theory according which synaptic depression solves a computational problem that is faced by any neural population in which neurons represent and compute with analog quantities, but communicate with discrete spikes. [sent-28, score-0.578]

12 For convenience, we assume this analog quantity to be the membrane potential, but, via a non-linear transformation [7], it could equally well be an analog ﬁring rate. [sent-29, score-0.728]

13 That is, we assume that network computations require the evolution of the membrane potential of a neuron to be a function of the membrane potentials of its presynaptic partners. [sent-30, score-1.903]

14 However, such a neuron does not have (at least not directly, see [8] for an example of indirect interaction) access to these membrane potentials, but rather only to the spikes to which they lead, and so it faces a key estimation problem. [sent-31, score-0.881]

15 It is in the indispensable ﬁrst estimation step that we suggest synaptic depression to be involved. [sent-36, score-0.554]

16 In Section 2 we formalise the problem of estimating presynaptic membrane potentials as an instance of Bayesian inference, and derive an online recursive estimator for it. [sent-37, score-1.191]

17 Given suitable assumptions about presynaptic membrane potential dynamics and spike generation, this optimal estimator can be written in closed form exactly [9, 10]. [sent-38, score-1.535]

18 In Section 3, we introduce a canonical model of postsynaptic membrane potential and synaptic depression dynamics, and show how it relates to the optimal estimator derived earlier. [sent-39, score-1.612]

19 In Section 4, we present results from numerical simulations showing the quality with which synaptic depression can approximate the performance of the optimal estimator, and how much is gained relative to a static synapse without synaptic depression. [sent-40, score-1.021]

20 2 Bayesian estimation of presynaptic membrane potentials The Bayesian estimation problem that needs to be solved by a synapse involves inferring the posterior distribution p (ut |s1. [sent-43, score-1.276]

21 t ) over the presynaptic membrane potential ut at time step t (for discretized time), given the spikes seen from the presynaptic cell up to that time step, s1. [sent-45, score-1.869]

22 We ﬁrst deﬁne a statistical (generative) model of presynaptic membrane potential ﬂuctuations and spiking, and then derive the estimator that is appropriate for it. [sent-48, score-1.244]

23 First we assume that presynaptic membrane potential dynamics are Markovian p(ut |u1. [sent-51, score-1.239]

24 t−1 ) = p(ut |ut−1 ) (1) In particular, we assume that the presynaptic membrane potential evolves as an Ornstein-Uhlenbeck (OU) process, given (again, in discretized time) by √ ut = ut−1 − θ(ut−1 − ur )∆t + Wt ∆t, 2 iid 2 Wt ∼ N (Wt ; 0, σW ) (2) A C 4 3 u [mV] 2 B . [sent-53, score-1.433]

25 Synaptic depression: postsynaptic responses to a train of presynaptic action potentials (not shown) at 40 Hz. [sent-60, score-0.5]

26 Graphical model of the process generating presynaptic subthreshold membrane potential ﬂuctuations, u, and spikes, s. [sent-63, score-1.19]

27 The membrane potential evolves according to a ﬁrst-order Markov process, the Ornstein-Uhlenbeck (OU) process (Eqs. [sent-64, score-0.85]

28 The probability of generating a spike at time t (st = 1) depends only on the current membrane potential, ut , and is determined by a non-linear Poisson (NP) model (Eqs. [sent-66, score-1.004]

29 Sample membrane potential trace (red line) and spike timings (vertical black dotted lines) 2 generated by the OU-NP process; with ur = 0 mV, θ−1 = 100 ms, σW = 0. [sent-69, score-1.092]

30 where 1/θ is the time constant with which the membrane potential decays back to its resting value, ur , and ∆t is the size of the discretized time bins. [sent-71, score-1.055]

31 Because both θ and σW are assumed to be 2 2 constant, the variance of the presynaptic membrane potential, σOU = σW /2θ, is stationary. [sent-72, score-1.0]

32 The second assumption is that spiking activity at any time only depends on the membrane potential at that time: p(st |u1. [sent-73, score-0.952]

33 if the membrane potential, u, is bigger than zero, the neuron emits a spike, and if u < 0, the neuron does not ﬁre. [sent-80, score-0.824]

34 Estimating on-line the membrane potential of the presynaptic cell from its spiking history amounts to computing the posterior probability distribution, p (ut |s1. [sent-81, score-1.336]

35 3 Performing recursive inference (ﬁltering), as described by equation 6, under the generative model described by equations 1-5 results in a posterior distribution that is Gaussian, ut |s1. [sent-98, score-0.244]

36 The mean and variance of this Gaussian evolve (in continuous time, by taking the limit ∆t → 0) as: µ = ˙ σ ˙ 2 −θ(µ − ur ) + βσ 2 (S(t) − γ) 2 = −2θ σ − 2 σOU 2 4 − γβ σ (7) (8) with the normalisation factor given by γ = g0 exp(βu) ut |s1. [sent-101, score-0.336]

37 t = g0 exp βµ + β 2 σ2 2 (9) where S(t) is the spike train of the presynaptic cell (represented as a sum of Dirac delta functions). [sent-103, score-0.515]

38 Equation 7 indicates that each time a spike is observed, the estimated membrane potential should increase proportionally to the uncertainty (variance) about the current estimate. [sent-105, score-0.972]

39 2A shows, the higher the presynaptic membrane potential is, the more spikes are emitted (because the instantaneous ﬁring rate is a monotonic function of membrane potential, see Eq. [sent-109, score-1.889]

40 Therefore the estimation error is smaller for higher membrane potential (see Fig. [sent-111, score-0.862]

41 Conversely, in the absence of spikes, the estimated membrane potential decreases while the variance increases back to its asymptotic value. [sent-113, score-0.88]

42 2C shows that the representation of uncertainty about the membrane potential by σ 2 is self-consistent because it is predictive of the error of the mean estimator, µ. [sent-115, score-0.822]

43 s of equation 7 comes from the prior knowledge about the membrane potential dynamics. [sent-118, score-0.803]

44 Then the dynamics of µ becomes driven only by the prior mean membrane potential ur : µ ˙ −θ (µ − ur ) (10) and so the asymptotic estimated membrane potential will tend to the prior mean membrane potential. [sent-128, score-2.576]

45 This is reasonable since in the small noise limit, the true membrane potential ut will effectively be very close to ur . [sent-129, score-1.07]

46 Furthermore the convergence time constant of the estimated membrane potential should be matched to the time constant θ−1 of the OU process and this is indeed the case in Eq. [sent-130, score-0.933]

47 2 Slow dynamics limit A second interesting limit is where the time constant of the OU process becomes small, i. [sent-133, score-0.267]

48 We can therefore write: √ γ µ S(t) − γ ˙ (11) σW Because the time constant θ−1 of the OU process is slow, the driving force that pulls the membrane potential back to its mean value ur is weak. [sent-145, score-1.014]

49 Therefore the membrane potential estimation dynamics should rely on the observed spikes rather than on the prior information ur . [sent-146, score-1.168]

50 Furthermore, the time constant τ = γ/ /σW0 is not ﬁxed but is a function of the mean estimated membrane potential µ. [sent-149, score-0.854]

51 Red line: presynaptic membrane potential, u, as a function of time, vertical dotted lines: spikes emitted. [sent-156, score-1.125]

52 Estimation error (µ − u)2 as a function of the membrane potential u of the OU process. [sent-161, score-0.822]

53 Black dots: estimation error and true membrane potential in individual time steps, red line: third order polynomial ﬁt. [sent-162, score-0.912]

54 As µ gets closer to the true membrane potential, the time constant increases, leading to an appropriately accurate estimate of the membrane potential. [sent-170, score-1.343]

55 This dynamical time constant therefore helps the estimation avoid the traditional speed vs accuracy trade-off (short time constant are fast but give a noisy estimation; longer time constant are slow but yield a more accurate estimation), by combining the best of the two worlds. [sent-171, score-0.244]

56 3 Depressing synapses as estimators of presynaptic membrane potential In section 2 we have shown that presynaptic spikes have a varying, context-dependent effect on the optimal on-line estimator of presynaptic membrane potential. [sent-172, score-2.833]

57 In this section we will show that the variability that synaptic depression introduces in postsynaptic responses closely resembles the variability of the optimal estimator. [sent-173, score-0.673]

58 A simple way to study the similarity between the optimal estimator and short-term plasticity is to consider their steady state ﬁltering properties. [sent-174, score-0.259]

59 As we saw above, according to the optimal estimator, the higher the input ﬁring rate is, the smaller the posterior variance becomes, and therefore the increment due to subsequent spikes should decrease. [sent-175, score-0.258]

60 This is consistent with depressing synapses for which the amount of excitatory postsynaptic current (EPSC) decreases when the stimulation frequency is increased (see Fig. [sent-176, score-0.4]

61 Steady-state spiking increment βσ 2 of the optimal estimator as a function of r = S (Eq. [sent-184, score-0.31]

62 Synaptic depression in the climbing ﬁbre to Purkinje cell synapse: average (±s. [sent-187, score-0.271]

63 Importantly, the similarity between the optimal membrane potential estimator and short-term plasticity is not limited to stationary properties. [sent-192, score-1.043]

64 Indeed, the actual dynamics of the optimal estimator (Eqs. [sent-193, score-0.269]

65 7-9) can be well approximated by the dynamics of synaptic depression. [sent-194, score-0.409]

66 In a canonical model of short-term depression [14], the postsynaptic membrane potential, v, changes as v − v0 1−x v=− ˙ + J Y x S(t), with x= ˙ − Y x S(t) (12) τ τD where J and Y are constants (synaptic weight and utilisation fraction), and x is a time varying ‘resource’ variable (e. [sent-195, score-1.033]

67 the fraction of presynaptic vesicles ready to fuse to the membrane). [sent-197, score-0.322]

68 Thus, v is increased by each presynaptic spike, and in the absence of spikes it decays to its resting value, v0 , with membrane time constant τ . [sent-198, score-1.174]

69 However, the effect of each spike on v is scaled by x which itself is decreased after each spike and increases between spikes back towards one with time constant τD . [sent-199, score-0.493]

70 Thus, the postsynaptic potential, v, behaves much like the posterior mean of the optimal estimator, µ, while the dynamics of the synaptic resource variable, x, closely resemble that of the posterior variance of the optimal estimator, σ 2 . [sent-200, score-0.762]

71 Indeed, the capacity of a depressing synapse (with appropriate parameters) to estimate the presynaptic membrane potential can be nearly as good as that of the optimal estimator (Fig. [sent-202, score-1.555]

72 Interestingly, although 2 the scaled variance σ 2 /σ∞ does not follow the resource variable dynamics x perfectly just after a spike, these two quantities are virtually identical at the time of the next spike, i. [sent-204, score-0.227]

73 when they are used by the membrane potential estimators (Fig. [sent-206, score-0.829]

74 The second estimator we consider includes synaptic depression, i. [sent-212, score-0.414]

75 Once the parameters of presynaptic membrane potential dynamics (σW , θ, ur ) and spiking (β, g0 ) are ﬁxed, the optimal estimator is entirely determined. [sent-220, score-1.608]

76 For a wide range of parameter values, the depressing synapse performs almost as well as the optimal estimator, and both perform better than the static synapse. [sent-224, score-0.352]

77 2 0 500 1000 time [ms] 1500 2000 Figure 4: Depressing synapses implement near-optimal estimation of presynaptic membrane potentials. [sent-230, score-1.155]

78 Red line, and vertical dotted lines: membrane potential, u, and spikes, S, generated by a simulated presynaptic cell (with parameters as in Fig. [sent-232, score-1.059]

79 Blue line: postsynaptic potential, v, in a depressing synapse (Eq. [sent-234, score-0.398]

80 Black: resource variable, x, in the depressing synapse (Eq. [sent-243, score-0.307]

81 We interpreted short-term synaptic depression, a key feature of synaptic dynamics, as solving the fundamental computational task of estimating the analog membrane potential of the presynaptic cell from observed spikes. [sent-254, score-1.825]

82 Steady-state and dynamical properties of a Bayes-optimal estimator are well-matched by a canonical model of depression; using a ﬁxed synaptic efﬁcacy instead leads to a highly suboptimal estimator. [sent-255, score-0.46]

83 Our theory is readily testable, since it suggests a precise relationship between quantities that have been subject to extensive, separate, empirical study — namely the statistics of a neuron’s membrane potential dynamics (captured by the parameters of Eq. [sent-256, score-0.917]

84 (5)), and the synaptic depression it expresses in its efferent synapses. [sent-258, score-0.556]

85 Accounting for the observation that different efferent synapses of the same cell can express different forms of short-term synaptic plasticity [15] remains a challenge; one obvious possibility is that different synapses are estimating different aspects or functions of the membrane potential. [sent-259, score-1.392]

86 For that model, the spike generation mechanism of the presynaptic neuron was modiﬁed such that even a simple read-out mechanism with ﬁxed efﬁcacies could correctly decode the analogue quantity encoded presynaptically. [sent-261, score-0.557]

87 By contrast, we considered a standard model of spiking [17], and thereby derived an explanation for the evident fact that synapses are not in fact ﬁxed. [sent-262, score-0.24]

88 Comparing the estimation error for different membrane potential estimators as a func2 2 tion of . [sent-274, score-0.888]

89 Blue: depressing synapse with its 5 tuneable parameters (see text) being optimised for each value of . [sent-277, score-0.331]

90 Red: static synapse with its 3 tuneable parameters (see text) being optimised. [sent-278, score-0.232]

91 Analysing the estimation error of the optimal estimator in the slow dynamics limit ( → 0). [sent-282, score-0.387]

92 For example, it would be important to understand in more quantitative detail the mapping between the parameters of the process generating the presynaptic membrane potential and spikes, and the parameters of synaptic depression that will best realize the corresponding optimal estimator. [sent-286, score-1.718]

93 We present some preliminary derivations in the supplementary material that seem to yield at least the right ball-park values for optimal synaptic dynamics. [sent-287, score-0.362]

94 This should also enable us to explore the particular parameter regimes in which depressing synapses have the most (or least) advantage over static synapses in terms of estimation performance, as in Fig. [sent-288, score-0.459]

95 Our assumption about the prior distribution of presynaptic membrane potential dynamics is highly restrictive. [sent-291, score-1.239]

96 One interesting property of smooth trajectories is that a couple of spikes arriving in quick succession may be diagnostic of an upward-going trend in membrane potential which is best decoded with increasing, i. [sent-294, score-0.97]

97 We hope through our theory to be able to provide a teleological account of the rich complexities of real synaptic inconstancy. [sent-301, score-0.295]

98 Interplay between facilitation, depression, and residual calcium at three presynaptic terminals. [sent-316, score-0.322]

99 Modulation of intracortical synaptic potentials by presynaptic somatic membrane potential. [sent-346, score-1.318]

100 Predicting spike timing of neocortical u pyramidal neurons by simple threshold models. [sent-373, score-0.23]

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same-paper 1 0.99999952 121 nips-2009-Know Thy Neighbour: A Normative Theory of Synaptic Depression

Author: Jean-pascal Pfister, Peter Dayan, Máté Lengyel

2 0.29780984 200 nips-2009-Reconstruction of Sparse Circuits Using Multi-neuronal Excitation (RESCUME)

Author: Tao Hu, Anthony Leonardo, Dmitri B. Chklovskii

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. 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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