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49 nips-2013-Bayesian Inference and Online Experimental Design for Mapping Neural Microcircuits

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Author: Ben Shababo, Brooks Paige, Ari Pakman, Liam Paninski

Abstract: With the advent of modern stimulation techniques in neuroscience, the opportunity arises to map neuron to neuron connectivity. In this work, we develop a method for efﬁciently inferring posterior distributions over synaptic strengths in neural microcircuits. The input to our algorithm is data from experiments in which action potentials from putative presynaptic neurons can be evoked while a subthreshold recording is made from a single postsynaptic neuron. We present a realistic statistical model which accounts for the main sources of variability in this experiment and allows for signiﬁcant prior information about the connectivity and neuronal cell types to be incorporated if available. Due to the technical challenges and sparsity of these systems, it is important to focus experimental time stimulating the neurons whose synaptic strength is most ambiguous, therefore we also develop an online optimal design algorithm for choosing which neurons to stimulate at each trial. 1

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Summary: the most important sentenses genereted by tfidf model

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1 edu Abstract With the advent of modern stimulation techniques in neuroscience, the opportunity arises to map neuron to neuron connectivity. [sent-8, score-0.59]

2 In this work, we develop a method for efﬁciently inferring posterior distributions over synaptic strengths in neural microcircuits. [sent-9, score-0.382]

3 The input to our algorithm is data from experiments in which action potentials from putative presynaptic neurons can be evoked while a subthreshold recording is made from a single postsynaptic neuron. [sent-10, score-0.801]

4 We present a realistic statistical model which accounts for the main sources of variability in this experiment and allows for signiﬁcant prior information about the connectivity and neuronal cell types to be incorporated if available. [sent-11, score-0.292]

5 Due to the technical challenges and sparsity of these systems, it is important to focus experimental time stimulating the neurons whose synaptic strength is most ambiguous, therefore we also develop an online optimal design algorithm for choosing which neurons to stimulate at each trial. [sent-12, score-1.371]

6 1 Introduction A major goal of neuroscience is the mapping of neural microcircuits at the scale of hundreds to thousands of neurons [1]. [sent-13, score-0.448]

7 By mapping, we speciﬁcally mean determining which neurons synapse onto each other and with what weight. [sent-14, score-0.284]

8 In this paper, we speciﬁcally address the mapping experiment in which a set of putative presynaptic neurons are optically stimulated while an electrophysiological trace is recorded from a designated postsynaptic neuron. [sent-16, score-0.91]

9 For example, while it has been shown that multiple neurons can be stimulated simultaneously [4, 5], successful mapping experiments have thus far only stimulated a single neuron per trial which increases experimental time [2, 3, 6]. [sent-19, score-0.966]

10 Stimulating multiple neurons simultaneously and with high accuracy requires well-tuned hardware, and even then some level of stimulus uncertainty may remain. [sent-20, score-0.421]

11 In this paper, we address these issues by developing a procedure for sparse Bayesian inference and information-based experimental design which can reconstruct neural microcircuits accurately and quickly despite the issues listed above. [sent-24, score-0.372]

12 , N , the experimenter stimulates R of K possible presynaptic neurons. [sent-30, score-0.239]

13 We represent the chosen set of neurons for each trial with the binary vector zn ∈ {0, 1}K , which has a one in each of the the R entries corresponding to the stimulated neurons on that trial. [sent-31, score-1.11]

14 One of the difﬁculties of optical stimulation lies in the experimenter’s inability to stimulate a speciﬁc neuron without possibly failing to stimulate the target neuron or engaging other nearby neurons. [sent-32, score-0.936]

15 In general, this is a result of the fact that optical excitation does not stimulate a single point in space but rather has a point spread function that is dependent on the hardware and the biological tissue. [sent-33, score-0.374]

16 To complicate matters further, each neuron has a different rheobase (a measure of how much current is needed to generate an action potential) and expression level of the optogenetic protein. [sent-34, score-0.299]

17 While some work has shown that it may be possible to stimulate exact sets of neurons, this setup requires very speciﬁc hardware and ﬁne tuning [4, 5]. [sent-35, score-0.276]

18 In addition, even if a neuron ﬁres, there is some probability that synaptic transmission will not occur. [sent-36, score-0.524]

19 Because these events are difﬁcult or impossible to observe, we model this uncertainty by introducing a second binary vector xn ∈ {0, 1}K denoting the neurons that actually release neurotransmitter in trial n. [sent-37, score-0.564]

20 The conditional distribution of xn given zn can be chosen by the experimenter to match their hardware settings and understanding of synaptic transmission rates in their preparation. [sent-38, score-0.753]

21 2 Sparse connectivity Numerous studies have collected data to estimate both connection probabilities and synaptic weight distributions as a function of distance and cell identity [2, 3, 6, 7, 8, 9, 10, 11, 12]. [sent-40, score-0.597]

22 Generally, the data show that connectivity is sparse and that most synaptic weights are small with a heavy tail of strong connections. [sent-41, score-0.45]

23 To capture the sparsity of neural connectivity, we place a “spike-and-slab” prior on the synaptic weights wk [13, 14, 15], for each presynaptic neuron k = 1, . [sent-42, score-0.843]

24 Note that we do not need to restrict the “slab” distributions (the conditional distributions of wk given that wk is nonzero) to the traditional Gaussian choice, and in fact each weight can have its own parameters. [sent-46, score-0.239]

25 3 Postsynaptic response In our model a subthreshold response is measured from a designated postsynaptic neuron. [sent-49, score-0.419]

26 The postsynaptic response for each synaptic event in a given trial can be modeled using an appropriate template function fk (·) for each presynaptic neuron k. [sent-51, score-1.26]

27 For this paper we use an alpha function to model the shape of each neuron’s contribution to the postsynaptic current, parameterized by time constants τk which deﬁne the rise and decay time. [sent-52, score-0.257]

28 As with the synaptic weight priors, the template functions could be designed based on the cells’ identities. [sent-53, score-0.309]

29 The onset of each postsynaptic 1 A cell’s identity can be general such as excitatory or inhibitory, or more speciﬁc such as VIP- or PVinterneurons. [sent-54, score-0.379]

30 These identities can be identiﬁed by driving the optogenetic channel with a particular promotor unique to that cell type or by coexpressing markers for various cell types along with the optogenetic channel. [sent-55, score-0.337]

31 2 Presynaptic weights Location of presynaptic neurons and stimuli Weight 1 0 − 1 0 20 40 10 Current [pA] Neuron k 60 80 100 Postsynaptic current trace 0 − 10 − 20 − 30 0 50 100 Time [samples] 150 200 Figure 1: A schematic of the model experiment. [sent-56, score-0.655]

32 The left ﬁgure shows the relative location of 100 presynaptic neurons; inhibitory neurons are shown in yellow, and excitatory neurons in purple. [sent-57, score-0.903]

33 Neurons marked with a black outline have a nonzero connectivity to the postsynaptic neuron (shown as a blue star, in the center). [sent-58, score-0.653]

34 The true connectivity weights are shown on the upper right, with blue vertical lines marking the ﬁve neurons which were actually ﬁred as a result of this stimulus. [sent-60, score-0.506]

35 The resulting time series postsynaptic current trace is shown in the bottom right. [sent-61, score-0.299]

36 The connected neurons which ﬁred are circled in red, the triangle and star marking their weights and corresponding postsynaptic events in the plots at right. [sent-62, score-0.627]

37 response may be jittered such that each event starts at some time dnk after t = 0, where the delays could be conditionally distributed on the parameters of the stimulation and cells. [sent-63, score-0.363]

38 To infer the marginal distribution of the synaptic weights, one can use standard Bayesian methods such as Gibbs sampling or variational inference, both of which are discussed below. [sent-69, score-0.502]

39 An example set of neurons and connectivity weights, along with the set of stimuli and postsynaptic current trace for a single trial, is shown in Figure 1. [sent-70, score-0.838]

40 1 Charge as synaptic strength To reduce the space over which we perform inference, we collapse the variables wk and τk into a single variable ck = t wk fk (t − dnk , τk ) which quantiﬁes the charge transfer during the synaptic event and can be used to deﬁne the strength of a connection. [sent-77, score-1.108]

41 p(y|X, c) = (3) n We found that na¨ve MCMC sampling over the posterior of w, τ , γ, X, and D insufﬁciently exı plored the support and inference was unsuccessful. [sent-80, score-0.227]

42 We approximate the prior over c as a spike-and-slab with Gaussian slabs where the slabs could be truncated if the cells’ excitatory or inhibitory identity is known. [sent-87, score-0.394]

43 Each xnk can be sampled by computing the odds ratio, and following [15] we draw each ck , γk from the joint distribution p(ck , γk |Z, y, X, {cj , γj |j = k}) by sampling ﬁrst γk from p(γk |Z, y, X, {cj |j = k}), then p(ck |Z, y, X, {cj , |j = k}, γk ). [sent-88, score-0.3]

44 This means that we must be able to perform inference of the posterior as well as choose the next stimulus extremely quickly. [sent-92, score-0.326]

45 To achieve this decrease in runtime, we approximate the posterior distribution of c and γ using a variational approach [16]. [sent-94, score-0.314]

46 The use of variational inference for spike-and-slab regression models has been explored in [17, 18], and we follow their methods with some minor changes. [sent-95, score-0.271]

47 As is the case with fully-factorized variational distributions, updating the posterior involves an iterative algorithm which cycles through the parameters for each factor. [sent-102, score-0.314]

48 4 Therefore, since the product of a spike-and-slab and a Gaussian is still a spike-and-slab, if we stimulate only one neuron at each trial then this posterior is also spike-and-slab, and the variational approximation becomes exact in this limit. [sent-105, score-0.957]

49 We Monte Carlo approximate this integral in a manner similar to the approach used for integrating over the hyperparameters in [17]; however, here we further approximate by sampling over potential stimuli xnk from p(xnk = 1|zn ). [sent-107, score-0.24]

50 In practice we will see this approximation sufﬁces for experimental design, with the overall variational approach performing nearly as well for posterior weight reconstruction as Gibbs sampling from the true posterior. [sent-108, score-0.488]

51 4 Optimal experimental design The preparations needed to perform these type of experiments tend to be short-lived, and indeed, the very act of collecting data — that is, stimulating and probing cells — can compromise the health of the preparation further. [sent-109, score-0.338]

52 We are thus strongly motivated to optimize the experimental design: to choose the optimal subset of neurons zn to stimulate at each trial to minimize N , the overall number of trials required for good inference. [sent-112, score-1.075]

53 , (zn−1 , yn−1 )} are ﬁxed and yn is dependent on the stimulus zn , our problem is reduced to choosing the optimal next stimulus, denoted zn , in expectation over yn , (7) zn = arg max Eyn |zn [I(θ; D)] = arg min Eyn |zn [H(θ|D)] . [sent-120, score-1.121]

54 zn zn 5 Experimental design procedure The optimization described in Section 4 entails performing a combinatorial optimization over zn , where for each zn we consider an expectation over all possible yn . [sent-121, score-1.16]

55 1 Computing the objective function The variational posterior distribution of ck , γk can be used to characterize our general objective function described in Section 4. [sent-125, score-0.493]

56 We deﬁne the cost function J to be the right-hand side of Equation 7, J ≡ Eyn |zn [H(c, γ|D)] (8) such that the optimal next stimulus zn can be found by minimizing J. [sent-126, score-0.408]

57 (10) k,n 2 k 5 Here, we have introduced additional notation, using αk,n , µk,n , and sk,n to refer to the parameters of the variational posterior distribution given the data through trial n. [sent-129, score-0.556]

58 Intuitively, we see that equation 10 represents a balance between minimizing the sparsity pattern entropy H[γk ] of each neuron and minimizing the weight entropy H[ck |γk = 1] proportional to the probability αk that the presynaptic neuron is connected. [sent-130, score-0.847]

59 In algorithm behavior, we see when the probability that a neuron is connected increases, we spend time stimulating it to reduce the uncertainty in the corresponding nonzero slab distribution. [sent-132, score-0.514]

60 For any particular candidate zn , this can be Monte Carlo approximated by ﬁrst sampling yn from the posterior distribution p(yn |zn , c, Dn−1 ), where c is drawn from the variational posterior inferred at trial n − 1. [sent-134, score-1.051]

61 Each sampled yn may be used to estimate the variational parameters αk,n and sk,n with which we evaluate H[ck , γk ]; we average over these evaluations of the entropy from each sample to compute an estimate of J in Eq. [sent-135, score-0.393]

62 Once we have chosen zn , we execute the actual trial and run the variational inference procedure on the full data to obtain the updated variational posterior parameters αk,n , µk,n , and sk,n which are needed for optimization. [sent-137, score-1.066]

63 Once the experiment has concluded, Gibbs sampling can be run, though we found only a limited gain when comparing Gibbs sampling to variational inference. [sent-138, score-0.274]

64 It is not feasible to evaluate the right-hand side of equation 10 for every zn because as K grows there is a combinatorial explosion of possible stimuli. [sent-141, score-0.239]

65 To avoid an exhaustive search over possible zn , we adopt a greedy approach for choosing which R of the K locations to stimulate. [sent-142, score-0.239]

66 First we rank the K neurons based on an ˜n approximation of the objective function. [sent-143, score-0.284]

67 To do this, we propose K hypothetical stimuli, zk , each all zeros except the k th entry equal to 1 — that is, we examine only the K stimuli which represent ∗ ˜n stimulating a single location. [sent-144, score-0.315]

68 We then set znk = 1 for the R neurons corresponding to the zk which give the smallest values for the objective function and all other entries of z∗ to zero. [sent-145, score-0.334]

69 We found that n the neurons selected by a brute force approach are most likely to be the neurons that the greedy selection process chooses (see Figure 1 in the Appendix). [sent-146, score-0.598]

70 For each of ˜n the K proposed stimuli zk , to approximate the expected entropy we must compute the variational ˜ ˜ posterior for M samples of [X1:n−1 xn ] and L samples of yn (where xn is the random variable corresponding to p(˜ n |˜n )). [sent-148, score-0.754]

71 Therefore we run the variational inference procedure on the full data x z on the order of O(M KL) times at each trial. [sent-149, score-0.271]

72 As the system size grows, running the variational inference procedure this many times becomes intractable because the number of iterations needed to converge the coordinate ascent algorithm is dependent on the correlations between the rows of X. [sent-150, score-0.338]

73 Note that the stronger dependence here is on R; when R = 1 the variational parameter updates become exact and independent across the neurons, and therefore no coordinate ascent is necessary and the runtime becomes linear in K. [sent-152, score-0.234]

74 We therefore take one last measure to speed up the optimization process by implementing an online Bayesian approach to updating the variational posterior (in the stimulus selection phase only). [sent-153, score-0.511]

75 Since the variational posterior of ck and γk takes the same form as the prior distribution, we can use the posterior from trial n − 1 as the prior at trial n, allowing us to effectively summarize the previous data. [sent-154, score-1.175]

76 In this online setting, when we stimulate only one neuron, only the parameters of that speciﬁc ˜ n ˜n neuron change. [sent-155, score-0.431]

77 If during optimization we temporarily assume that xk = zk , this results in explicit updates for each variational parameter, with no coordinate ascent iterations required. [sent-156, score-0.284]

78 The combined accelerations described in this section result in a speed up of several orders of magnitude which allows the full inference and optimization procedure to be run in real time, running at approximately one second per trial in our computing environment for K = 500, R = 8. [sent-158, score-0.315]

79 We chose to parallelize over M which distributes the sampling of X and the running of variational inference for each sample. [sent-160, score-0.309]

80 The heavy red and blue lines indicate the results when running the Gibbs sampler at that point in the experiment, and the thinner magenta and cyan lines indicate the results from variational inference. [sent-182, score-0.23]

81 6 Experiments and results We ran our inference and optimal experimental design algorithm on data sets generated from the model described in Section 2. [sent-187, score-0.261]

82 Baseline results are shown in Figure 2, over a range of values for stimulations per trial R and baseline postsynaptic noise levels ν. [sent-189, score-0.499]

83 The results here use an informative prior, where we assume the excitatory or inhibitory identity is known, and we set individual prior connectivity probabilities for each neuron based on that neuron’s identity and distance from the postsynaptic cell. [sent-190, score-0.911]

84 We choose to let X be unobserved and let the stimuli Z produce Gaussian ellipsoids which excite neurons that are located nearby. [sent-191, score-0.403]

85 The optimal procedure was able to achieve equivalent reconstruction quality as a random stimulation paradigm in signiﬁcantly fewer trials when the number of stimuli per trial and response noise were in an experimentally realistic range (R = 4 and ν = 2. [sent-194, score-0.685]

86 As the the number of stimuli per trial R increases, we start to see improved weight estimates and faster convergence but a decrease in the relative beneﬁt of optimal design; the random approach “catches up” to the optimal approach as R becomes large. [sent-197, score-0.468]

87 4 0 200 400 trial, n 600 800 0 200 400 trial, n 600 800 Figure 3: The results of inference and optimal design (A) with a single spike-andslab prior for all connections (prior connection probability of . [sent-209, score-0.281]

88 ) Finally, we see that we are still able to recover the synaptic strengths when we use a more general prior as in Figure 3A where we placed a single spike-and-slab prior across all the connections. [sent-216, score-0.348]

89 Since we assumed the cells’ identities were unknown, we used a zero-centered Gaussian for the slab and a prior connection probability of . [sent-217, score-0.215]

90 While we allow for stimulus uncertainty, it will likely soon be possible to stimulate multiple neurons with high accuracy. [sent-219, score-0.594]

91 The algorithms proposed by [23] are based on computing a maximum a posteriori (MAP) estimate of the weights w; note that to pursue the optimal Bayesian experimental design methods proposed here, it is necessary to compute (or approximate) the full posterior distribution, not just the MAP estimate. [sent-222, score-0.352]

92 ) In the simulated experiments of [23], stimulating roughly 30 of 500 neurons per trial is found to be optimal; extrapolating from Fig. [sent-226, score-0.672]

93 First, the implementation of an inference algorithm which performs well on the full model such that we can recover the synaptic weights, the time constants, and the delays would allow us to avoid compressing the responses to scalar values and recover more information about the system. [sent-231, score-0.408]

94 Also, it may be necessary to improve the noise model as we currently assume that there are no spontaneous synaptic events which will confound the determination of each connection’s strength. [sent-232, score-0.266]

95 Nadal, “What can we learn from synaptic weight distributions? [sent-291, score-0.309]

96 Yuste, “Stereotyped position of local synaptic targets in neocortex,” Science, vol. [sent-307, score-0.266]

97 Reyes, “Spatial proﬁle of excitatory and inhibitory synaptic connectivity in mouse primary auditory cortex,” The Journal of Neuroscience, vol. [sent-315, score-0.575]

98 Markram, “A synaptic organizing principle for cortical neuronal groups,” Proceedings of the National Academy of Sciences, vol. [sent-323, score-0.266]

99 Chklovskii, “Highly nonrandom features of o o synaptic connectivity in local cortical circuits. [sent-334, score-0.402]

100 Stephens, “Scalable variational inference for bayesian variable selection in regression, and its accuracy in genetic association studies,” Bayesian Analysis, vol. [sent-368, score-0.341]

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