nips nips2009 nips2009-165 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Zhi Yang, Qi Zhao, Edward Keefer, Wentai Liu
Abstract: Studying signal and noise properties of recorded neural data is critical in developing more efficient algorithms to recover the encoded information. Important issues exist in this research including the variant spectrum spans of neural spikes that make it difficult to choose a globally optimal bandpass filter. Also, multiple sources produce aggregated noise that deviates from the conventional white Gaussian noise. In this work, the spectrum variability of spikes is addressed, based on which the concept of adaptive bandpass filter that fits the spectrum of individual spikes is proposed. Multiple noise sources have been studied through analytical models as well as empirical measurements. The dominant noise source is identified as neuron noise followed by interface noise of the electrode. This suggests that major efforts to reduce noise from electronics are not well spent. The measured noise from in vivo experiments shows a family of 1/f x spectrum that can be reduced using noise shaping techniques. In summary, the methods of adaptive bandpass filtering and noise shaping together result in several dB signal-to-noise ratio (SNR) enhancement.
Reference: text
sentIndex sentText sentNum sentScore
1 edu Abstract Studying signal and noise properties of recorded neural data is critical in developing more efficient algorithms to recover the encoded information. [sent-3, score-0.52]
2 Important issues exist in this research including the variant spectrum spans of neural spikes that make it difficult to choose a globally optimal bandpass filter. [sent-4, score-0.489]
3 Also, multiple sources produce aggregated noise that deviates from the conventional white Gaussian noise. [sent-5, score-0.462]
4 In this work, the spectrum variability of spikes is addressed, based on which the concept of adaptive bandpass filter that fits the spectrum of individual spikes is proposed. [sent-6, score-0.844]
5 Multiple noise sources have been studied through analytical models as well as empirical measurements. [sent-7, score-0.428]
6 The dominant noise source is identified as neuron noise followed by interface noise of the electrode. [sent-8, score-1.39]
7 This suggests that major efforts to reduce noise from electronics are not well spent. [sent-9, score-0.44]
8 The measured noise from in vivo experiments shows a family of 1/f x spectrum that can be reduced using noise shaping techniques. [sent-10, score-1.475]
9 In summary, the methods of adaptive bandpass filtering and noise shaping together result in several dB signal-to-noise ratio (SNR) enhancement. [sent-11, score-0.949]
10 While single neurons only require a detection algorithm to identify the firings, multiple neurons require the separation of superimposed activities to obtain individual neuron firings. [sent-15, score-0.447]
11 Also, synthesized sequences with benchmarks obtained through neuron models [8], isolated single neuron recordings [2], or simultaneous intra- and extra- cellular recordings [9] lack the in vivo recording environment. [sent-21, score-0.692]
12 Specifically, a procedure of online estimating both individual spike and noise spectrum is first applied. [sent-25, score-0.817]
13 Based on the estimation, a bandpass filter that fits the spectrum of the underlying spike is selected. [sent-26, score-0.616]
14 This maximally reduces the broad band noise without sacrificing the signal integrity. [sent-27, score-0.531]
15 In addition, a comprehensive study of multiple noise sources are performed through lumped circuit model as well as measurements. [sent-28, score-0.563]
16 Experiments suggest that the dominant noise is not from recording electronics, 1 Figure 1: Block diagram of the proposed noise reduction procedures. [sent-29, score-0.909]
17 More importantly, the measured noise generally shows a family of 1/f x spectrum, which can be reduced by using noise shaping techniques [10, 11]. [sent-31, score-1.102]
18 Section 4 describes a noise shaping technique that uses fractional order differentiation. [sent-36, score-0.789]
19 2 Noise Spectrum and Noise Model Recorded neural spikes are superimposed with noise that exhibit non-Gaussian characteristics and can be approximated as 1/f x noise. [sent-39, score-0.549]
20 The frequency dependency of noise is contributed by multiple sources. [sent-40, score-0.495]
21 Except electrolyte bulk noise (4kT Rb in Figure 2) that has a flattened spectrum, the rest show frequency dependency. [sent-42, score-0.627]
22 Specifically, 1/f α −neuron noise is induced from distant neurons [12–14]. [sent-43, score-0.493]
23 Ree generates noise that is attenuated quadratically to frequency in high frequency region by the interface capacitance (Cee ). [sent-46, score-0.78]
24 Electronic noise consists of two major components: thermal noise (∼ kT /gm [16]) and flicker noise (or 1/f noise [16]). [sent-47, score-1.628]
25 Flicker noise dominates at lower frequency range and is fabrication process dependent. [sent-48, score-0.495]
26 Next, we will address the noise model that will later be used to develop noise removal techniques in Section 3 and Section 4, and verified by experiment results in section 5. [sent-49, score-0.8]
27 spike, synaptic release [17–19]) overlap the spectrum of the recorded spike signal. [sent-53, score-0.513]
28 1, the power spectrum of Vneu is P {Vneu } = i k |Xi (f )|2 fi < e2πjf (ti,k1 +k −ti,k1 ) >, 2 (2) where < > represents the average over the ensemble and over k1 , P {} is the spectrum operation, Xi (f ) is the fourier transform of vi. [sent-59, score-0.401]
29 The spectrum of a delta function spike pulse 2 Figure 2: Noise illustration for extracellular spikes. [sent-62, score-0.434]
30 As this term multiplies |Xi (f )|2 , the unresolved spiking activities of distant neurons contribute a spectrum of 1/f x within the signal spectrum. [sent-64, score-0.495]
31 In such a case, the electrode interface can be modeled as a lumped resistor Ree in parallel with a lumped capacitor Cee . [sent-74, score-0.496]
32 (4) 1 Ree Ree 1/Ree + jωCee + 1/(Rb + jωCi ) Referring to the hypothesis that the amplifier input capacitance (Ci ) is sufficiently small, introducing negligible waveform distortion, the integrated noise by electrode interface satisfies fc2 fc2 Ne. [sent-78, score-0.939]
33 tan−1 2πRee Cee f |f =fc2 < f =fc1 2 |1 + 2πjf Ree Cee | πCee Cee (5) Equation 5 suggests reducing electrode interface noise by increasing double layer capacitance (Cee ). [sent-80, score-0.783]
34 Section 5 will compare conventional electrodes and CNT coated electrodes from a noise point of view. [sent-82, score-0.757]
35 In regions away from the interface boundary, ni (x) = 0 results in a flattened noise spectrum. [sent-83, score-0.531]
36 Here we use a lumped bulk resistance Rb in series with the double-layer interface for modeling noise ρtissue Ne. [sent-84, score-0.666]
37 At the frequency of interest, there are two major components: thermal noise of transistors and flicker noise 4kT K 1 Nelectronic = Nc. [sent-90, score-1.006]
38 Given a design schematic, circuit thermal noise can be reduced by increasing transconductance (gm ), which is to the first order linear to bias current thus power consumption. [sent-95, score-0.605]
39 Flicker noise can be reduced using design techniques such as large size input transistors and chopper modulations [22]. [sent-96, score-0.447]
40 By using advanced semiconductor technologies, also, power and area trade off to noise [16], and elegant design techniques like chopper modulation, current feedback [23], the state-of-the-art low noise neural amplifier can provide less than 2µV total noise [24]. [sent-97, score-1.22]
41 Such design costs can be necessary and useful if electronics noise contributes significantly to the total noise. [sent-98, score-0.44]
42 Section 5 will present experiments of evaluating noise contribution from different sources, which show that electronics are not the dominant noise source in our experiments. [sent-100, score-0.854]
43 4 Total Noise The noise sources as shown in Figure 2 include unresolved neuron activities (Nneu ), electrodeelectrolyte interface noise (Ne. [sent-102, score-1.165]
44 The noise spectrum is empirically fitted by N (f ) = Nneu + Ne. [sent-107, score-0.564]
45 Equation 8 describes a combination of both colored noise (1/f x ) and broad band noise, which can be reduced by using noise removal techniques. [sent-112, score-0.89]
46 Section 4 presents a noise shaping technique used to improve the differentiation between signals and noise within the passband. [sent-114, score-1.102]
47 3 Adaptive Bandpass Filtering SNR is calculated by integrating both signal and noise spectrum. [sent-115, score-0.441]
48 While a passband that only fits one spike template may introduce waveform distortion to spikes of other templates, a passband that covers every template will introduce more noise to spikes of every template. [sent-118, score-1.241]
49 A possible solution is to adaptively assign a passband to each spike waveform such that each span will be just wide enough to cover the underlying waveform. [sent-119, score-0.489]
50 This section presents the steps used in order to achieve this solution and includes spike detection, spectrum estimation, and filter generation. [sent-120, score-0.434]
51 1 Spike Detection In this work, spike detection is performed using a nonlinear energy operator (NEO) [25] that captures instantaneous high frequency and high magnitude activities. [sent-122, score-0.405]
52 Because spikes are high frequency activities by definition, NEO outputs a larger score when spikes are present. [sent-134, score-0.444]
53 An example of NEO based spike detection is shown in Figure 4, where NEO improves the separation between spikes and the background activity. [sent-135, score-0.491]
54 2 Corner Frequency Estimation Spectrum estimation of individual spikes is performed to select a corresponding bandpass filter that balances the spectrum distortion and noise. [sent-145, score-0.526]
55 In the frequency domain, denoting PXX and PN N as the signal and noise spectra, Weiner filter is W (f ) = PXX (f ) SN R(f ) = . [sent-147, score-0.553]
56 4 Noise Shaping The adaptive scheme presented in Section 3 tacitly assigns a matched frequency mask to individual spikes and balances noise and spectrum integrity. [sent-153, score-0.85]
57 The remaining noise exhibits 1/f x frequency dependency according to Section 2. [sent-154, score-0.495]
58 In this section, we focus on noise shaping techniques to further distinguish signal from noise. [sent-155, score-0.777]
59 Instead of equally amplifying the spectrum, a noise shaping filter allocates more weight to high SNR regions while reducing weight at low SNR regions. [sent-157, score-0.719]
60 This results in an increased ratio of the integrated signal power over the noise power. [sent-158, score-0.48]
61 In general, there are a variety of noise shaping filters that can improve the integrated SNR [10]. [sent-159, score-0.719]
62 In this work, we use a series of fractional derivative operation for noise shaping dp h(x) , (13) dxp where h(x) is a general function, p is a positive number (can be integer or non-integer) that adjusts the degree of noise shaping; the larger the p, the more emphasis on high frequency spectrum. [sent-160, score-1.316]
63 5 Noise power, unit (uV)2 Noise measured using conventinal electrode Noise measured using CNT coated electrode Recording, unit mV 1. [sent-167, score-0.566]
64 5 2 10 1 0 1 2 3 Time, unit minute 4 10 5 0 10 20 30 (a) 50 60 (b) 25 25 0 minute 15 minute 30 minute 45 minute 0 minute 15 minute 30 minute 45 minute 20 Power/frequency (dB/Hz) 20 Power/frequency (dB/Hz) 40 Time, unit minute 15 10 5 15 10 5 0 0 −5 −5 1 1. [sent-170, score-1.14]
65 5 6 Frequency (kHz) (c) (d) Figure 4: In vivo recording for identifying noise sources. [sent-180, score-0.687]
66 Black curve represents the noise recorded from a custom tungsten electrode; red curve represents the noise recorded from a CNT coated electrodes with the same size. [sent-184, score-1.246]
67 In (d), a CNT coated tungsten electrode of equal size is used for comparison. [sent-187, score-0.435]
68 where Ispike (f ) and Inoise (f ) are power spectrums of spike and noise respectively. [sent-188, score-0.707]
69 5 Experiment To verify the noise analysis presented in Section 2, an in vivo experiment is performed that uses two sharp tungsten electrodes separated by 125 µm to record the hippocampus neuronal activities of a rat. [sent-194, score-0.865]
70 In Figure 4(b), the estimated noise from 600Hz to 6KHz for both recording sites are plotted, where noise dramatically reduces (> 80%) after the drug takes effect. [sent-200, score-0.95]
71 Initially, the CNT electrode records a comparatively larger noise (697µV 2 ) compared with the uncoated electrode (610µV 2 ). [sent-201, score-0.885]
72 After a few minutes, the background noise recorded by the CNT electrode quickly reduces eventually reaching 37µV 2 that is about 1/3 of noise recorded by its counterpart (112µV 2 ), suggesting the noise floor of using the uncoated tungsten electrode (112µV 2 ) is set by the electrode. [sent-202, score-1.937]
73 From these two plots, we can estimate that the neuron noise is around 500 ∼ 600µV 2 , electrode interface noise is ∼ 80µV , while the sum of electronic noise and electrolyte bulk noise is less than 37µV 2 (only ∼ 5% of the total noise). [sent-203, score-2.138]
74 Figure 4(c) displays the 1/f x noise spectrum recorded from the uncoated tungsten electrode (x = 1. [sent-204, score-1.014]
75 Figure 4(d) displays 1/f x noise spectrum recorded from the CNT coated electrode (x = 2. [sent-209, score-0.982]
76 6 Table 1: Statistics of 1/f x noise spectrum from in vivo preparations. [sent-214, score-0.756]
77 2 (d) Figure 5: In vivo experiment of evaluating the proposed adaptive bandpass filter. [sent-267, score-0.422]
78 In the second experiment, 77 recordings of in vivo preparations are used to explore the stochastic distribution of 1/f x noise spectrum. [sent-272, score-0.632]
79 This recording is used to compare the feature extraction results produced by a global bandpass filter (conventional one) and the proposed adaptive bandpass filter, discussed in Section 3. [sent-280, score-0.599]
80 In Figure 5(c), feature extraction results using PCA (a widely used feature extraction algorithm in spike sorting applications) with a global bandpass filter are displayed. [sent-284, score-0.667]
81 In the fourth experiment, earth mover’s distance (EMD), as a cross-bin similarity measure that is robust to waveform misalignment [28], is applied to synthesized data for evaluation of the spike waveform separation before and after noise shaping. [sent-286, score-1.122]
82 to be the spike waveform bundles from candidate neuron A and B. [sent-293, score-0.543]
83 To estimate the spike variation of a candidate neuron, two waveforms are randomly picked from a same waveform bundle, and the distance between them is calculated using EMD. [sent-294, score-0.63]
84 The only difference from plots shown in Figure 6(a)-(d) is that the waveforms after noise shaping are used rather than their original counterparts. [sent-304, score-0.845]
85 In Figure 6(e)-(h), the red curves separate from the black/blue traces, suggesting that the noise shaping filter improves waveform differentiations. [sent-305, score-0.875]
86 In the fifth experiment, we apply different orders of noise shaping filters and the same feature extraction algorithm to evaluate the feature extraction results. [sent-306, score-0.869]
87 The noise shaping technique is developed as a general tool that can be incorporated into an unspecified feature extraction algorithm. [sent-307, score-0.794]
88 Figures from left to right display the feature extraction results with different orders of noise shaping; from 0 (no noise shaping) to 3. [sent-310, score-0.841]
89 (a)-(d) and (e)-(h) are results of 4 different pairs of neurons before and after noise shaping respectively. [sent-316, score-0.792]
90 6 (p) Figure 7: Feature extraction results using PCA with different orders of noise shaping. [sent-527, score-0.458]
91 Each column represents a different order of noise shaping (p in dp f (x) dxp ), sweeping from 0 (without noise shaping) to 3. [sent-529, score-1.134]
92 For both sequences, increased numbers of isolated clusters can be obtained by appropriately choosing the order of the noise shaping filter. [sent-537, score-0.719]
93 6 Conclusion In this paper, a study of multiple noise sources for in vivo neural recording is carried out. [sent-538, score-0.732]
94 The dominant noise source is identified to be neuron noise followed by interface noise of the electrode. [sent-539, score-1.39]
95 The concept of adaptive bandpass filter is proposed to reduce noise because it maintains the signal spectrum integrity while maximally reducing the broad band noise. [sent-541, score-0.942]
96 To reduce the noise within the signal passband and improve waveform separation, a series of fractional order differentiator based noise shaping filters are proposed. [sent-542, score-1.466]
97 The proposed noise removal techniques are generally applicable to an unspecified spike sorting algorithm. [sent-543, score-0.752]
98 Experiment results from in vivo preparations, synthesized sequences, and comparative recordings using both conventional and CNT coated electrodes are reported, which verify the noise model and demonstrate the usefulness of the proposed noise removal techniques. [sent-544, score-1.381]
99 Automated spike sorting using density grid contour clustering and subtractive waveform decomposition. [sent-563, score-0.491]
100 Comprehensive study of noise processes in electrode electrolyte interfaces. [sent-604, score-0.69]
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simIndex simValue paperId paperTitle
same-paper 1 0.99999928 165 nips-2009-Noise Characterization, Modeling, and Reduction for In Vivo Neural Recording
Author: Zhi Yang, Qi Zhao, Edward Keefer, Wentai Liu
Abstract: Studying signal and noise properties of recorded neural data is critical in developing more efficient algorithms to recover the encoded information. Important issues exist in this research including the variant spectrum spans of neural spikes that make it difficult to choose a globally optimal bandpass filter. Also, multiple sources produce aggregated noise that deviates from the conventional white Gaussian noise. In this work, the spectrum variability of spikes is addressed, based on which the concept of adaptive bandpass filter that fits the spectrum of individual spikes is proposed. Multiple noise sources have been studied through analytical models as well as empirical measurements. The dominant noise source is identified as neuron noise followed by interface noise of the electrode. This suggests that major efforts to reduce noise from electronics are not well spent. The measured noise from in vivo experiments shows a family of 1/f x spectrum that can be reduced using noise shaping techniques. In summary, the methods of adaptive bandpass filtering and noise shaping together result in several dB signal-to-noise ratio (SNR) enhancement.
2 0.20480427 52 nips-2009-Code-specific policy gradient rules for spiking neurons
Author: Henning Sprekeler, Guillaume Hennequin, Wulfram Gerstner
Abstract: Although it is widely believed that reinforcement learning is a suitable tool for describing behavioral learning, the mechanisms by which it can be implemented in networks of spiking neurons are not fully understood. Here, we show that different learning rules emerge from a policy gradient approach depending on which features of the spike trains are assumed to influence the reward signals, i.e., depending on which neural code is in effect. We use the framework of Williams (1992) to derive learning rules for arbitrary neural codes. For illustration, we present policy-gradient rules for three different example codes - a spike count code, a spike timing code and the most general “full spike train” code - and test them on simple model problems. In addition to classical synaptic learning, we derive learning rules for intrinsic parameters that control the excitability of the neuron. The spike count learning rule has structural similarities with established Bienenstock-Cooper-Munro rules. If the distribution of the relevant spike train features belongs to the natural exponential family, the learning rules have a characteristic shape that raises interesting prediction problems. 1
3 0.12783226 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. 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