nips nips2013 nips2013-262 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: David Carlson, Vinayak Rao, Joshua T. Vogelstein, Lawrence Carin
Abstract: With simultaneous measurements from ever increasing populations of neurons, there is a growing need for sophisticated tools to recover signals from individual neurons. In electrophysiology experiments, this classically proceeds in a two-step process: (i) threshold the waveforms to detect putative spikes and (ii) cluster the waveforms into single units (neurons). We extend previous Bayesian nonparametric models of neural spiking to jointly detect and cluster neurons using a Gamma process model. Importantly, we develop an online approximate inference scheme enabling real-time analysis, with performance exceeding the previous state-of-theart. Via exploratory data analysis—using data with partial ground truth as well as two novel data sets—we find several features of our model collectively contribute to our improved performance including: (i) accounting for colored noise, (ii) detecting overlapping spikes, (iii) tracking waveform dynamics, and (iv) using multiple channels. We hope to enable novel experiments simultaneously measuring many thousands of neurons and possibly adapting stimuli dynamically to probe ever deeper into the mysteries of the brain. 1
Reference: text
sentIndex sentText sentNum sentScore
1 In electrophysiology experiments, this classically proceeds in a two-step process: (i) threshold the waveforms to detect putative spikes and (ii) cluster the waveforms into single units (neurons). [sent-5, score-0.869]
2 We extend previous Bayesian nonparametric models of neural spiking to jointly detect and cluster neurons using a Gamma process model. [sent-6, score-0.468]
3 Crucial for this endeavor is the advancement of our ability to understand the dynamics of the brain, via the measurement of large populations of neural activity at the single neuron level. [sent-11, score-0.347]
4 , electrodes or calcium imaging), real-time decoding of individual neuron responses requires identifying and labeling individual spikes from recordings from large populations. [sent-15, score-0.664]
5 In other words, real-time decoding requires real-time spike sorting. [sent-16, score-0.504]
6 Automatic spike sorting methods are continually evolving to deal with more sophisticated experiments. [sent-17, score-0.557]
7 Most recently, several methods have been proposed to (i) learn the number of separable neurons on each electrode or “multi-trode” [1, 2], or (ii) operate online to resolve overlapping spikes from multiple neurons [3]. [sent-18, score-0.773]
8 Our model explains the continuous output of each neuron by a latent marked Poisson process, with the “marks” characterizing the shape of each spike. [sent-21, score-0.405]
9 Previous efforts to address overlapping spiking often assume a fixed kernel for each waveform, but joint intracellular and extracellular recording clearly indicate that this assumption is false (see Figure 3c). [sent-22, score-0.541]
10 Our work therefore suggests that further improvements in real-time decoding of activity may be most effective if directed at simultaneous real-time spike sorting and decoding. [sent-30, score-0.641]
11 2 Model Our data is a time-series of multielectrode recordings X ⌘ (x1 , · · · , xT ), and consists of T recordings from M channels. [sent-32, score-0.244]
12 Each neuron generates a continuous-time voltage trace, and the outputs of all neurons are superimposed and discretely sampled to produce the recordings X. [sent-35, score-0.616]
13 1 we model the continuous-time output of each neuron as a series of idealized Poisson events smoothed with appropriate kernels, while §2. [sent-37, score-0.266]
14 1 Modeling the continuous-time output of a single neuron There is a rich literature characterizing the spiking activity of a single neuron [4] accounting in detail for factors like non-stationarity, refractoriness and spike waveform. [sent-44, score-1.196]
15 First, we model the spiking activity of each neuron are stationary and memoryless, so that its set of spike times are distributed as a homogeneous Poisson process (PP). [sent-46, score-0.939]
16 We model the neurons themselves are heterogeneous, with the ith neuron having an (unknown) firing rate i . [sent-47, score-0.425]
17 Call the ordered set of spike times of the ith neuron Ti = (⌧i1 , ⌧i2 , . [sent-48, score-0.74]
18 The actual electrical output of a neuron is not binary; instead each spiking event is a smooth perturbation in voltage about a resting state. [sent-53, score-0.463]
19 This perturbation forms the shape of the spike, with the spike shapes varying across neurons as well as across different spikes of the same neuron. [sent-54, score-0.919]
20 However, each neuron has its own characteristic distribution over shapes, and we let ✓ ⇤ 2 ⇥ parametrize this i distribution for neuron i. [sent-55, score-0.532]
21 Whenever this neuron emits a spike, a new shape is drawn independently from the corresponding distribution. [sent-56, score-0.33]
22 This waveform is then offset to the time of the spike, and contributes to the voltage trace associated with that spike. [sent-57, score-0.494]
23 The complete recording from the neuron is the superposition of all these spike waveforms plus noise. [sent-58, score-1.088]
24 We model each spike shape as weighted superpositions of a dictionary of K basis functions d(t) ⌘ (d1 (t), · · · , dK (t))T . [sent-62, score-0.593]
25 Each spike time ⌧ij is associated with a ran⇤ ⇤ ⇤ dom K-dimensional weight vector yij ⌘ (yij1 , . [sent-66, score-0.474]
26 yijK )T , and the shape of this spike at time t PK ⇤ ⇤ is given by the weighted sum k=1 yijk dk (t ⌧ij ). [sent-69, score-0.642]
27 Then, at any time t, the output of neuron i is xi (t) = j=1 k=1 yijk dk (t ⌧ij ). [sent-71, score-0.37]
28 Assume for the moment there are N neurons, and define T ⌘ [i2[N ] Ti as the (ordered) union of the spike times of all neurons. [sent-73, score-0.474]
29 Let ⌧l 2 T indicate the time of the lth overall spike, whereas ⌧ij 2 Ti is the time of the j th spike of neuron i. [sent-74, score-0.777]
30 In words, ⌫l 2 N is the neuron to which the lth element of T belongs, while pl indexes this spike in the spike train T⌫l . [sent-78, score-1.279]
31 Let ✓ l ⌘ (µl , ⌃l ) be the neuron parameter ⇤ associated with spike l, so that ✓ l = ✓ ⇤l . [sent-79, score-0.74]
32 , ylK )T ⌘ y⌫j pj as the weight ⌫ 2 vector of spike ⌧l . [sent-83, score-0.474]
33 Each event ⌧l 2 T has a pair of labels, its neuron parameter ✓ l ⌘ (µl , ⌃l ), and yl , the weight-vector characterizing the spike shape. [sent-86, score-0.926]
34 i With probability one, the neurons have distinct parameters, so that the mark ✓ l identifies the neuron which produced spike l: G(✓ l = ✓ ⇤ ) = P(⌫l = i) = i /⇤. [sent-92, score-0.899]
35 The output waveform x(t) is then a linear functional of this marked Poisson process. [sent-95, score-0.447]
36 2 A nonparametric model of population activity In practice, the number of neurons driving the recorded activity is unknown. [sent-97, score-0.314]
37 Recalling that each neuron is characterized by a pair of parameters ( i , ✓ ⇤ ), we map i P1 the infinite collection of pairs {( i , ✓ ⇤ )} to an random measure ⇤(·) on ⇥: ⇤(d✓) = i=1 i ✓⇤ . [sent-106, score-0.266]
38 Crucially, a normalized Gamma process is the Dirichlet process (DP) [15], so that the spike parameters ✓ are i. [sent-127, score-0.616]
39 For spike l, the shape vector is drawn from a normal with parameters (µl , ⌃l ): these are thus draws from a DP mixture (DPM) of Gaussians [16]. [sent-131, score-0.58]
40 We can exploit the connection with the DP to integrate out the infinite-dimensional measure G(·) (and thus ⇤(·)), and assign spikes to neurons via the so-called Chinese restaurant process (CRP) [17]. [sent-132, score-0.452]
41 Under this scheme, the lth spike is assigned the same parameter as an earlier spike with probability proportional to the number of earlier spikes having that parameter. [sent-133, score-1.235]
42 It is assigned a new parameter (and thus, a new neuron is observed) with probability proportional to ↵. [sent-134, score-0.294]
43 The shape of the j th spike is now a vector of length L, and for matrix D e a weight vector y, is given by Dy. [sent-154, score-0.538]
44 e e Thus, ⌫t and ✓t are the neuron and neuron parameter associated with time bin t, and yt is its weighte vector. [sent-158, score-0.564]
45 Let the binary variable zt indicate whether or not a spike is present in time bin t (recall that e e zt ⇠ Bernoulli(⇤ )). [sent-159, score-0.604]
46 If there is no spike associated with bin t, then we ignore the marks µ and y. [sent-160, score-0.574]
47 We let every spike affect the recordings at all channels, with the spike shape varying across channels. [sent-167, score-1.109]
48 For spike l in m channel m, call the weight-vector yl . [sent-168, score-0.74]
49 All these vectors must be correlated as they correspond to the same spike; we do this simply by concatenating the set of vectors into a single M K-element vector 1 M yl = (yl ; · · · ; yl ), and modeling this as a multivariate normal. [sent-169, score-0.258]
50 We also relax the requirement that the parameters ✓ ⇤ of each neuron remain constant, and instead allow µ⇤ , the mean of the weight-vector distribution, to evolve with time (we keep the covariance parameter ⌃⇤ fixed, however). [sent-171, score-0.266]
51 Each neuron is now associated with a vector-valued function ✓ ⇤ (·), rather than a constant. [sent-176, score-0.266]
52 When a spike at time ⌧l is assigned to neuron i, it is assigned a weight-vector yl drawn from a Gaussian with mean µ⇤ (⌧l ). [sent-177, score-0.925]
53 It also maintains the identities of the neurons that it assigned each of these spikes to, as well as the weight vectors determining the shapes of the associated spike waveforms. [sent-196, score-0.883]
54 We indicate these point estimates with the hat operator, so, for b example Tit is the set of estimated spike times before time t assigned to neuron i. [sent-197, score-0.768]
55 i 2 [C i ⇤ over the parameters ✓i ⌘ (µ⇤ , ⌃⇤ ) of neuron i given the observations until time t. [sent-201, score-0.266]
56 i i Having identified the time and shape of spikes from earlier times, we can calculate their contribution to the recordings xL ⌘ (xt , · · · , xt+L 1 )T . [sent-202, score-0.383]
57 Recalling that the basis functions D, t and thus all spike waveforms, span L time bins, the residual at time t + t1 is then given by P xt+t1 = xt b y b h2[L t1 ] zt h Dbt h (at time t, for t1 > 0, we define zt+t1 = 0). [sent-203, score-0.637]
58 The latter is used to calculate qi,t+1 (✓i ), the new distribution over neuron parameters at time t + 1. [sent-205, score-0.266]
59 5 we decide that there is a spike z present at time t, otherwise, we set zt = 0. [sent-211, score-0.523]
60 In the event of a spike (bt = 1), we use these point estimates to update the posterior distribution z over parameters of cluster ⌫t , to obtain qi,t+1 (·) from qi,t (·); this is straightforward because of b conjugacy. [sent-214, score-0.627]
61 The recording was made simultaneously on all electrodes and was set up such that the cell with the intracellular electrode was also recorded on the extracellular array implanted in the hippocampus of an anesthetized rat. [sent-223, score-0.499]
62 The intracellular recording is relatively noiseless and gives nearly certain firing times of the intracellular neuron. [sent-224, score-0.409]
63 The extracellular recording contains the spike waveforms from the intracellular neuron as well as an unknown number of additional neurons. [sent-225, score-1.294]
64 The neighboring electrode sites in these devices have 30 µm between electrode edges and 60 µm between electrode centers. [sent-232, score-0.331]
65 These devices are close enough that a locallyfiring neuron could appear on multiple electrode sites [2], so neighboring channels warrant joint processing. [sent-233, score-0.497]
66 To define D, we used the first five principle components of all spikes detected with a threshold (three times the standard deviation of the noise above the mean) in the first five seconds. [sent-238, score-0.279]
67 In this dataset, we only have a partial ground truth, so we can only verify accuracy for the neuron with the intracellular (IC) recording. [sent-244, score-0.462]
68 We define a detected spike to be an IC spike if the IC recording has a spike within 0. [sent-245, score-1.533]
69 5 milliseconds (ms) of the detected spike in the extracellular recording. [sent-246, score-0.581]
70 We define the cluster with the greatest number of intracellular spikes as a the “IC cluster”. [sent-247, score-0.456]
71 We refer to these data as “partial ground truth data”, because we know the ground truth spike times for one of the neurons, but not all the others. [sent-248, score-0.608]
72 The spike detections for the offline methods used a threshold of three times the noise standard deviation [5] (unless stated otherwise), and windowed at a size L = 30. [sent-259, score-0.536]
73 For multichannel data, we concatenated the M channels for each waveform to obtain a M ⇥ L-dimensional vector. [sent-260, score-0.591]
74 Running time in unoptimized MATLAB code for 4 minutes of data was 31 seconds for a single channel and 3 minutes for all 4 channels on a 3. [sent-269, score-0.273]
75 Performance on partial ground truth data The main empirical result of our contribution is that all variants of O P A S S detect more true positives with fewer false positives than any of the other algorithms on the partial ground truth data (see Fig. [sent-272, score-0.349]
76 Figure 2: O P A S S detects multiple overlapping waveforms (Top Left) The observed voltage (solid black), MAP waveform 1 (red), MAP waveform 2 (blue), and waveform from the sum (dashed-black). [sent-300, score-1.619]
77 When spikes overlap, although the result can accurately be modeled as a linear sum in voltage space, the resulting waveform often does not appear in any cluster in PC space (see [1]). [sent-311, score-0.786]
78 Note that even though the waveform peaks are approximately 1 ms from one another, thresholding algorithms do not pick up these spikes, because they look different in PC space. [sent-315, score-0.438]
79 Indeed, by virtue of estimating the presence of multiple spikes, the residual squared error between the expected voltage and observed voltage shrinks for this snippet (bottom left). [sent-316, score-0.259]
80 Of the 135 pairs of overlapping spikes, 37 of those spikes came from the intracellular neuron. [sent-320, score-0.479]
81 Thus, while it seems detecting overlapping spikes helps, it does not fully explain the improvements over the competitor algorithms. [sent-321, score-0.346]
82 Time-Varying Waveform Adaptation As has been demonstrated previously [26], the waveform shape of a neuron may change over time. [sent-322, score-0.73]
83 The mean waveform over time for the intracellular neuron is shown in Fig. [sent-323, score-0.83]
84 3c shows that the auto-regressive model for the mean dictionary weights yields a timevarying posterior (top), whereas the static prior yields a constant posterior mean with increasing posterior marginal variances (bottom). [sent-330, score-0.247]
85 5 PCA Component 1 (b) 2 0 1 ms 2 3 (c) Figure 3: The IC waveform changes over time, which our posterior parameters track. [sent-341, score-0.492]
86 Each colored line represents the mean of the waveform averaged over 24 seconds with color denoting the time interval. [sent-343, score-0.434]
87 This neuron decreases in amplitude over the period of the recording. [sent-344, score-0.311]
88 (c) The mean and standard deviation of the waveforms at three time points for the auto-regressive prior on the mean waveform (top) and static prior (bottom). [sent-346, score-0.662]
89 Note that the left panel has a waveform that appears on both channel 2 and channel 3, whereas the waveform in the right panel only appears in channel 3. [sent-354, score-1.289]
90 Had only the third electrode been used, these two waveforms would not be distinct (as evidenced by their substantial overlap in PC space upon using only the third channel in Fig. [sent-361, score-0.47]
91 5 Discussion Our improved sensitivity and specificity seem to arise from multiple sources including (i) improved detection, (ii) accounting for correlated noise, (iii) capturing overlapping spikes, (iv) tracking waveform dynamics, and (v) utilizing multiple channels. [sent-366, score-0.558]
92 While others have developed closely related Bayesian models for clustering [8, 27], deconvolution based techniques [1], time-varying waveforms [26], or online methods [25, 3], we are the first to our knowledge to incorporate all of these. [sent-367, score-0.271]
93 An interesting implication of our work is that it seems that our errors may be irreconcilable using merely first order methods (that only consider the mean waveform to detect and cluster). [sent-368, score-0.447]
94 8a shows the mean waveform of the true and false positives are essentially identical, suggesting that even in the full 30-dimensional space excluding those waveforms from intracellular cluster would be difficult. [sent-371, score-0.976]
95 Projecting each waveform into the first two PCs is similarly suggestive, as the missed positives do not seem to be in the cluster of the true positives (Supp. [sent-372, score-0.586]
96 A model-based spike sorting algorithm for removing correlation artifacts in multi-neuron recordings. [sent-379, score-0.557]
97 Fast, scalable, c Bayesian spike identification for multi-electrode arrays. [sent-382, score-0.474]
98 An online spike detection and spike classification algorithm capable of instantaneous resolution of overlapping spikes. [sent-385, score-1.114]
99 A review of methods for spike sorting: the detection and classification of neural action potentials. [sent-391, score-0.508]
100 Kalman filter mixture model for spike sorting of non-stationary data. [sent-465, score-0.599]
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