{"title": "Scalable Spike Source Localization in Extracellular Recordings using Amortized Variational Inference", "book": "Advances in Neural Information Processing Systems", "page_first": 4724, "page_last": 4736, "abstract": "Determining the positions of neurons in an extracellular recording is useful for investigating the functional properties of the underlying neural circuitry. In this work, we present a Bayesian modelling approach for localizing the source of individual spikes on high-density, microelectrode arrays. To allow for scalable inference, we implement our model as a variational autoencoder and perform amortized variational inference. We evaluate our method on both biophysically realistic simulated and real extracellular datasets, demonstrating that it is more accurate than and can improve spike sorting performance over heuristic localization methods such as center of mass.", "full_text": "Scalable Spike Source Localization in Extracellular\nRecordings using Amortized Variational Inference\n\nUniversity of Edinburgh, United Kingdom\n\nUniversity of Edinburgh, United Kingdom\n\nKai Xu\n\nSchool of Informatics\n\nkai.xu@ed.ac.uk\n\nCole L. Hurwitz\n\nSchool of Informatics\n\ncole.hurwitz@ed.ac.uk\n\nAkash Srivastava\n\nMIT\u2013IBM Watson AI Lab\nCambridge, United States\n\nAkash.Srivastava@ibm.com\n\nAlessio P. Buccino\n\nDepartment of Informatics\n\nUniversity of Oslo, Oslo, Norway\n\nalessiob@ifi.uio.no\n\nMatthias H. Hennig\nSchool of Informatics\n\nUniversity of Edinburgh, United Kingdom\n\nm.hennig@ed.ac.uk\n\nAbstract\n\nDetermining the positions of neurons in an extracellular recording is useful for\ninvestigating functional properties of the underlying neural circuitry. In this work,\nwe present a Bayesian modelling approach for localizing the source of individual\nspikes on high-density, microelectrode arrays. To allow for scalable inference,\nwe implement our model as a variational autoencoder and perform amortized\nvariational inference. We evaluate our method on both biophysically realistic\nsimulated and real extracellular datasets, demonstrating that it is more accurate\nthan and can improve spike sorting performance over heuristic localization methods\nsuch as center of mass.\n\n1\n\nIntroduction\n\nExtracellular recordings, which measure local potential changes due to ionic currents \ufb02owing through\ncell membranes, are an essential source of data in experimental and clinical neuroscience. The\nmost prominent signals in these recordings originate from action potentials (spikes), the all or none\nevents neurons produce in response to inputs and transmit as outputs to other neurons. Traditionally,\na small number of electrodes (channels) are used to monitor spiking activity from a few neurons\nsimultaneously. Recent progress in microfabrication now allows for extracellular recordings from\nthousands of neurons using microelectrode arrays (MEAs), which have thousands of closely spaced\nelectrodes [13, 2, 14, 1, 36, 55, 32, 25, 12]. These recordings provide insights that cannot be obtained\nby pooling multiple single-electrode recordings [27]. This is a signi\ufb01cant development as it enables\nsystematic investigations of large circuits of neurons to better understand their function and structure,\nas well as how they are affected by injury, disease, and pharmacological interventions [20].\nOn dense MEAs, each recording channel may record spikes from multiple, nearby neurons, while\neach neuron may leave an extracellular footprint on multiple channels. Inferring the spiking activity\nof individual neurons, a task called spike sorting, is therefore a challenging blind source separation\nproblem, complicated by the large volume of recorded data [46]. Despite the challenges presented by\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fspike sorting large-scale recordings, its importance cannot be overstated as it has been shown that\nisolating the activity of individual neurons is essential to understanding brain function [35]. Recent\nefforts have concentrated on providing scalable spike sorting algorithms for large scale MEAs and\nalready several methods can be used for recordings taken from hundreds to thousands of channels\n[42, 31, 10, 54, 22, 26]. However, scalability, and in particular automation, of spike sorting pipelines\nremains challenging [8].\nOne strategy for spike sorting on dense MEAs is to spatially localize detected spikes before clustering.\nIn theory, spikes from the same neuron should be localized to the same region of the recording area\n(near the cell body of the \ufb01ring neuron), providing discriminatory, low-dimensional features for each\nspike that can be utilized with ef\ufb01cient density-based clustering algorithms to sort large data sets\nwith tens of millions of detected spikes [22, 26]. These location estimates, while useful for spike\nsorting, can also be exploited in downstream analyses, for instance to register recorded neurons with\nanatomical information or to identify the same units from trial to trial [9, 22, 41].\nDespite the potential bene\ufb01ts of localization, preexisting methods have a number of limitations. First,\nmost methods are designed for low-channel count recording devices, making them dif\ufb01cult to use with\ndense MEAs [9, 51, 3, 30, 29, 34, 33, 50]. Second, current methods for dense MEAs utilize cleaned\nextracellular action potentials (through spike-triggered averaging), disallowing their use before spike\nsorting [48, 6]. Third, all current model-based methods, to our knowledge, are non-Bayesian, relying\nprimarily on numerical optimization methods to infer the underlying parameters. Given these current\nlimitations, the only localization methods used consistently before spike sorting are simple heuristics\nsuch as a center of mass calculation [38, 44, 22, 26].\nIn this paper, we present a scalable Bayesian modelling approach for spike localization on dense\nMEAs (less than \u223c 50\u00b5m between channels) that can be performed before spike sorting. Our method\nconsists of a generative model, a data augmentation scheme, and an amortized variational inference\nmethod implemented with a variational autoencoder (VAE) [11, 28, 47]. Amortized variational\ninference has been used in neuroscience for applications such as predicting action potentials from\ncalcium imaging data [52] and recovering latent dynamics from single-trial neural spiking data [43],\nhowever, to our knowledge, it has not been used in applications to extracellular recordings.\nAfter training, our method allows for localization of one million spikes (from high-density MEAs) in\napproximately 37 seconds on a TITAN X GPU, enabling real-time analysis of massive extracellular\ndatasets. To evaluate our method, we use biophysically realistic simulated data, demonstrating that\nour localization performance is signi\ufb01cantly better than the center of mass baseline and can lead to\nhigher-accuracy spike sorting results across multiple probe geometries and noise levels. We also\nshow that our trained VAE can generalize to recordings on which it was not trained. To demonstrate\nthe applicability of our method to real data, we assess our method qualitatively on real extracellular\ndatasets from a Neuropixels [25] probe and from a BioCam4096 recording platform.\nTo clarify, our contribution is not full spike sorting solution. Although we envision that our method\ncan be used to improve spike sorting algorithms that currently rely center of mass location estimates,\ninterfacing with and evaluating these algorithms was beyond the scope of our paper.\n\n2 Background\n\n2.1 Spike localization\n\ni=1, be the set of M neurons in the recording and c := {cj}N\n\nWe start with introducing relevant notation. First, we de\ufb01ne the identities and positions of neurons and\nchannels. Let n := {ni}M\nj=1, the set of N\nchannels on the MEA. The position of a neuron, ni, can be de\ufb01ned as pni := (xniyni , zni ) \u2208 R3 and\nsimilarly the position of a channel, cj, pcj := (xcj , ycj , zcj ) \u2208 R3. We further denote pc := {pcj}N\nj=1\nto be the position of all N channels on the MEA. In our treatment of this problem, the neuron and\nchannel positions are single points that represent the centers of the somas and the centers of the\nchannels, respectively. These positions are relative to the origin, which we set to be the center of\nthe MEA. For the neuron, ni, let si := {si,k}Ki\nk=1, be the set of spikes detected during the recording\nwhere Ki is the total number of spikes \ufb01red by ni. The recorded extracellular waveform of si,k on a\nchannel, cj, can then be de\ufb01ned as wi,k,j := {r(0)\ni,k,j \u2208 R and\nt = 0, . . . , T . The set of waveforms recorded by each of the N channels of the MEA during the\n\ni,k,j} where r(t)\n\ni,k,j, r(1)\n\ni,k,j, ..., r(t)\n\ni,k,j, ..., r(T )\n\n2\n\n\fj=1. Finally, for the spike, si,k, the point source location\n\nspike, si,k, is de\ufb01ned as wi,k := {wi,k,j}N\ncan be de\ufb01ned as psi,k := (xsi,k , ysi,k , zsi,k ) \u2208 R3.\nThe problem we attempt to solve can now be stated as follows: Localizing a spike, si,k, is the task of\n\ufb01nding the corresponding point source location, psi,k, given the observed waveforms wi,k and the\nchannel positions, pc.\nWe make the assumption that the point source location, psi,k is actually the location of the \ufb01ring\nneuron\u2019s soma, pni. Given the complex morphological structure of many neurons, this assumption\nmay not always be correct, but it provides a simple way to assess localization performance and\nevaluate future models.\n\n2.2 Center of mass\n\nMany modern spike sorting algorithms localize spikes on MEAs using the center of mass or barycenter\nmethod [44, 22, 26]. We summarize the traditional steps for localizing a spike, si,k using this method.\nFirst, let us de\ufb01ne \u03b1i,k,j := mint wi,k,j to be the negative amplitude peak of the waveform, wi,k,j,\ngenerated by si,k and recorded on channel, cj. We consider the negative peak amplitude as a matter\nof convention since spikes are de\ufb01ned as inward currents. Then, let \u03b1i,k := (\u03b1i,k,j)N\nj=1 be the vector\nof all amplitudes generated by si,k and recorded by all N channels on the MEA.\nTo \ufb01nd the center of mass of a spike, si,k, the \ufb01rst step is to determine the central channel for the\n(cid:80)L+1\ncalculation. This central channel is set to be the channel which records the minimum amplitude\nfor the spike, cjmin := cargminj \u03b1i,k,j The second and \ufb01nal step is to take the L closest channels to\n(cid:80)L+1\nj=1 (ycj )|\u03b1i,k,j|\nj=1 |\u03b1i,k,j| where all of the L + 1\n\n(cid:80)L+1\n(cid:80)L+1\nj=1 (xcj )|\u03b1i,k,j|\nj=1 |\u03b1i,k,j|\n\nchannels\u2019 positions and recorded amplitudes contribute to the center of mass calculation.\nThe center of mass method is inexpensive to compute and has been shown to give informative location\nestimates for spikes in both real and synthetic data [44, 37, 22, 26]. Center of mass, however, suffers\nfrom two main drawbacks: First, since the chosen channels will form a convex hull, the center of mass\nlocation estimates must lie inside the channels\u2019 locations, negatively impacting location estimates for\nneurons outside of the MEA. Second, center of mass is biased towards the chosen central channel,\npotentially leading to arti\ufb01cial separation of location estimates for spikes from the same neuron [44].\n\ncjmin and compute, \u02c6xsi,k =\n\n, \u02c6ysi,k =\n\n3 Method\n\nIn this section, we introduce our scalable, model-based approach to spike localization. We describe\nthe generative model, the data augmentation procedure, and the inference methods.\n\n3.1 Model\n\nOur model uses the recorded amplitudes on each channel to determine the most likely source location\nof si,k. We assume that the peak signal from a spike decays exponentially with the distance from the\nsource, r: a exp(br) where a, b \u2208 R, r \u2208 R+. This assumption is well-motivated by experimentally\nrecorded extracellular potential decay in both a salamander and mouse retina [49, 22], as well as a\ncat cortex [16]. It has also been further corroborated using realistic biophysical simulations [18].\nWe utilize this exponential assumption to infer the source location of a spike, si,k, since localization\nis then equivalent to solving for si,k\u2019s unknown parameters, \u03b8si,k := {ai,k, bi,k, xsi,k , ysi,k , zsi,k}\ngiven the observed amplitudes, \u03b1i,k. To allow for localization without knowing the identity of the\n\ufb01ring neuron, we assume that each spike has individual exponential decay parameters, ai,k, bi,k, and\nindividual source locations, psi,k. We \ufb01nd, however, that \ufb01xing bi,k for all spikes to a constant that is\nequal to an empirical estimate from literature (decay length of \u223c 28\u00b5m) works best across multiple\nprobe geometries and noise levels, so we did not infer the value for bi,k in our \ufb01nal method. We will\nrefer to the \ufb01xed decay rate as b and exclude it from the unknown parameters moving forward.\n\n3\n\n\fThe generative process of our exponential model is as follows,\n\nai,k \u223c N (\u00b5ai,k , \u03c3a), xsi,k \u223c N (\u00b5xsi,k\n\u02c6ri,k = (cid:107)(xsi,k , ysi,k , zsi,k ) \u2212 pc(cid:107)2, \u03b1i,k \u223c N (ai,k exp(b\u02c6ri,k), I)\n\n, \u03c3x), ysi,k \u223c N (\u00b5ysi,k\n\n, \u03c3y), zsi,k \u223c N (\u00b5zsi,k\n\n, \u03c3z)\n\n(1)\n\nIn our observation model, the amplitudes are drawn from an isotropic Gaussian distribution with a\nvariance of one. We chose this Gaussian observation model for computational simplicity and since it\nis convenient to work with when using VAEs. We discuss the limitations of our modeling assumptions\nin Section 5 and propose several extensions for future works.\nFor our prior distributions, we were careful to set sensible parameter values. We found that inference,\nespecially for a spike detected near the edge of the MEA, is sensitive to the mean of the prior\ndistribution of ai,k, therefore, we set \u00b5ai,k = \u03bb\u03b1i,k,jmin where \u03b1i,k,jmin is the smallest negative\namplitude peak of si,k. We choose this heuristic because the absolute value of \u03b1i,k,jmin will always\nbe smaller than the absolute value of the amplitude of the spike at the source location, due to potential\ndecay. Therefore, scaling \u03b1i,k,jmin by \u03bb gives a sensible value for \u00b5ai,k. We empirically choose\n\u03bb = 2 for the \ufb01nal method after performing a grid search over \u03bb = {1, 2, 3}. The parameter, \u03c3a, does\nnot have a large affect on the inferred location so we set it to be approximately the standard deviation\nof the \u03b1i,k,jmin (50). The location prior means, \u00b5xsi,k\n, are set to the location of the\n, for the given spike. The location prior standard deviations,\nminimum amplitude channel, pcjmin\n\u03c3x, \u03c3y, \u03c3z, are set to large constant values to \ufb02atten out the distributions since we do not want the\nlocation estimate to be overly biased towards pcjmin\n\n, \u00b5ysi,k\n\n, \u00b5zsi,k\n\n.\n\n3.2 Data Augmentation\n\nFor localization to work well, the input channels should be centered around the peak spike, which\nis hard for spikes near the edges (edge spikes). To address this issue, we employ a two-step data\naugmentation. First, inputs for edge spikes are padded such that the channel with the largest amplitude\nis at the center of the inputs. Second, all channels are augmented with an indicating variable which\nprovides signal to distinguish them for the inference network. To be more speci\ufb01c, we introduce\nvirtual channels outside of the MEA which have the same layout as the real, recording channels (see\nappendix C). We refer to a virtual channel as an \"unobserved\" channel, cju, and to a real channel on\nthe MEA as an \"observed\" channel, cjo. We de\ufb01ne the amplitude on an unobserved channel, \u03b1i,k,ju,\nto be zero since unobserved channels do not actually record any signals. We let the amplitude for an\nobserved channel, \u03b1i,k,jo, be equal to mint wi,k,jo, as before.\nBefore de\ufb01ning the augmented dataset, we must \ufb01rst introduce an indicator function, 1o : \u03b1 \u2192 {0, 1}:\n\n(cid:26)1,\n\n0,\n\n1o(\u03b1) =\n\nif \u03b1 is from an observed channel,\nif \u03b1 is from an unobserved channel.\n\nwhere \u03b1 is an amplitude from any channel, observed or unobserved.\nTo construct the augmented dataset for a spike, si,k, we take the set of L channels that lie within a\nbounding box of width W centered on the observed channel with the minimum recorded amplitude,\ncjomin\n\n. We de\ufb01ne our newly augmented observed data for si,k as,\n\u03b2i,k := {(\u03b1i,k,j, 1o(\u03b1i,k,j)}L\n\n(2)\nSo, for a single spike, we construct a L \u00d7 2 dimensional vector that contains amplitudes from L\nchannels and indices indicating whether the amplitudes came from observed or unobserved channels.\nSince the prior location for each spike is at the center of the subset of channels used for the observed\ndata, for edge spikes, the data augmentation puts the prior closer to the edge and is, therefore, more\ninformative for localizing spikes near/off the edge of the array. Also, since edge spikes are typically\nseen on less channels, the data augmentation serves to ignore channels which are away from the\nspike, which would otherwise be used if the augmentation is not employed.\n\nj=1\n\n3.3\n\nInference\n\nNow that we have de\ufb01ned the generative process and data augmentation procedure, we would like to\ncompute the posterior distribution for the unknown parameters of a spike, si,k,\n\np(ai,k, xsi,k , ysi,k , zsi,k|\u03b2i,k)\n\n(3)\n\n4\n\n\fFigure 1: Estimated spike locations for the different methods on a 10\u00b5V recording. Center of mass\nestimates (left) are calculated using 16 observed amplitudes. The MCMC estimated locations (middle)\nused 9-25 observed amplitudes for inference, and the VAE model (right) was trained on 9-25 observed\namplitudes and a 10 amplitude jitter (amplitude jitter is described in 3.3.3).\n\ngiven the augmented dataset, \u03b2i,k. To infer the posterior distribution for each spike, we utilize two\nmethods of Bayesian inference: MCMC sampling and amortized variational inference.\n\n3.3.1 MCMC sampling\n\nWe use MCMC to assess the validity and applicability of our model to extracellular data. We\nimplement our model in Turing [15], a probabilistic modeling language in Julia. We run Hamiltonian\nMonte Carlo (HMC) [39] for 10,000 iterations with a step size of 0.01 and a step number of 10. We\nuse the posterior means of the location distributions as the estimated location.1\nDespite the ease use of probabilistic programming and asymptotically guaranteed inference quality of\nMCMC methods, the scalability of MCMC methods to large-scale datasets is limited. This leads us to\nimplement our model as a VAE and to perform amortized variational inference for our \ufb01nal method.\n\n3.3.2 Amortized variational inference\n\nTo speed up inference of the spike parameters, we construct a VAE and use amortized variational in-\nference to estimate posterior distributions for each spike. In variational inference, instead of sampling\nfrom the target intractable posterior distribution of interest, we construct a variational distribution that\nis tractable and minimize the Kullback\u2013Leibler (KL) divergence between the variational posterior\nand the true posterior. Minimizing the KL divergence is equivalent to maximizing the evidence lower\nbound (ELBO) for the log marginal likelihood of the data. In VAEs, the parameters of the variational\nposterior are not optimized directly, but are, instead, computed by an inference network.\nWe de\ufb01ne our variational posterior for x, y, z as a multivariate Normal with diagonal covariance\nwhere the mean and diagonal of the covariance matrix are computed by an inference network\n\nq\u03a6(x, y, z) = N (\u00b5\u00b5\u00b5\u03c61(f\u03c60(\u03c5i,k)), \u03c3\u03c3\u03c32\n\n\u03c62\n\n(f\u03c60 (\u03c5i,k)))\n\n(4)\n\nThe inference network is implemented as a feed-forward, deep neural network parameterized by\n\u03a6 = {\u03c60, \u03c61, \u03c62}. As one can see, the variational parameters are a function of the input \u03c5\u03c5\u03c5.\nWhen using an inference network, the input can be any part of the dataset so for our method, we use,\n\u03c5i,k, as the input for each spike, si,k, which is de\ufb01ned as follows:\n\n\u03c5i,k := {(wi,k,j, 1o(\u03b1i,k,j)}L\n\n(5)\nwhere wi,k,j is the waveform detected on the jth channel (de\ufb01ned in Section 2.1). Similar to our\nprevious augmentation, the waveform for an unobserved channel is set to be all zeros. We choose\nto input the waveforms rather than the amplitudes because, empirically, it encourages the inferred\nlocation estimates for spikes from the same neuron to be better localized to the same region of the\nMEA. For both the real and simulated datasets, we used \u223c2 ms of readings for each waveform.\n\nj=1\n\n1The code for our MCMC implementation is provided in Appendix H.\n\n5\n\n\f10 \u00b5V\n\n20 \u00b5V\n\nCOM\nCOM\nCOM\nCOM\nMCMC\n\nMethod Observed Channels 2D Avg. Spike Distance from Soma (\u00b5m)\n15.84\u00b110.08 16.46\u00b110.39 17.18\u00b110.97\n18.05\u00b111.42 18.59\u00b111.67\n19.22\u00b112.1\n20.94\u00b113.09 21.54\u00b113.46 22.17\u00b113.94\n23.44\u00b114.81 24.31\u00b115.43 25.18\u00b115.98\n13.31\u00b111.67\n11.30\u00b19.85\n9.87\u00b18.64\n9.21\u00b18.00\n12.05\u00b110.35\n10.40\u00b18.97\n8.79\u00b17.49\n11.18\u00b19.56\n9.79\u00b18.31\n8.94\u00b17.91\n10.48\u00b19.334 12.43\u00b111.223\n12.27\u00b110.78\n10.41\u00b19.07\n9.12\u00b17.83\n\n4\n9\n16\n25\n9-25\n4-9\n4-9\n9-25\n9-25\n\nVAE - 0\u00b5V\nVAE - 10\u00b5V\nVAE - 0\u00b5V\nVAE - 10\u00b5V\n\n30 \u00b5V\n\nTable 1: Results for the 2D location estimates. These results are for three simulated, square MEA\ndatasets with noise levels ranging from 10\u00b5V-30\u00b5V. For the VAE methods in the \ufb01rst column, the\namount of amplitude jitter used is displayed to the right (amplitude jitter is described in 3.3.3).\n\n(0)\n\nThe decoder for our method reconstructs the amplitudes from the observed data rather than the\nwaveforms. Since we assume an exponential decay for the amplitudes, the decoder is a simple\nGaussian likelihood function, where given the Euclidean distance vector \u02c6ri,k, computed by samples\nfrom the variational posterior, the decoder reconstructs the mean value of the observed amplitudes\nwith a \ufb01xed variance. The decoder is parameterized by the exponential parameters of the given spike,\nsi,k, so it reconstructs the amplitudes of the augmented data, \u03b2(0)\ni,k , with the following expression:\ni,k := ai,k exp(b\u02c6ri,k) \u00d7 \u03b21\n(0)\n\u02c6\u03b2\ni,k is the reconstructed observed amplitudes. By multiplying\nthe reconstructed amplitude vector by \u03b21\ni,k which consists of either zeros or ones (see Eq. 5), the\nunobserved channels will be reconstructed with amplitudes of zero and the observed channels will be\nreconstructed with the exponential function. For our VAE, instead of estimating the distribution of\nai,k, we directly optimize ai,k when maximizing the lower bound. We set the initial value of ai,k to\nthe mean of the prior. Thus, ai,k can be read as a parameter of the decoder.\nGiven our inference network and decoder, the ELBO we maximize for each spike, si,k, is given by,\n\ni,k where \u02c6\u03b2\n\n(cid:35)\n\n(cid:34) L(cid:88)\n\nlog p(\u03b2i,k; ai,k) \u2265 \u2212KL [q\u03a6(x, y, z)(cid:107) pxpypz] + Eq\u03a6\n\nN (\u03b20\n\ni,k,l|ai,k exp(b\u02c6ri,k), I)\u03b21\n\ni,k,l\n\nwhere KL is the KL-divergence. The location priors, px, py, pz, are normally distributed as described\nin 3.1, with means of zero (the position of the maximum amplitude channel in the observed data) and\nvariances of 80. For more information about the architecture and training, see Appendix F.\n\nl=1\n\n3.3.3 Stabilized Location Estimation\n\nIn this model, the channel on which the input is centered can bias the estimate of the spike location,\nin particular when amplitudes are small. To reduce this bias, we can create multiple inputs for the\nsame spike where each input is centered on a different channel. During inference, we can average the\ninferred locations for each of these inputs, thus lowering the central channel bias. To this end, we\nintroduce a hyperparameter, amplitude jitter, where for each spike, si,k, we create multiple inputs\ncentered on channels with peak amplitudes within a small voltage of the maximum amplitude, \u03b1i,k,j.\nWe use two values for the amplitude jitter in our experiments: 0\u00b5V and 10\u00b5V . When amplitude jitter\nis set to 0\u00b5V , no averaging is performed; when amplitude jitter is set to 10\u00b5V , all channels that have\npeak amplitudes within 10\u00b5V of \u03b1i,k,j are used as inputs to the VAE and averaged during inference.\n\n4 Experiments\n\n4.1 Datasets\n\nWe simulate biophysically realistic ground-truth extracellular recordings to test our model against\na variety of real-life complexities. The simulations are generated using the MEArec [4] package\nwhich includes 13 layer 5 juvenile rat somatosensory cortex neuron models from the neocortical\nmicrocircuit collaboration portal [45]. We simulate three recordings with increasing noise levels\n\n6\n\n\f(ranging from 10\u00b5V to 30\u00b5V ) for two probe geometries, a 10x10 channel square MEA with a 15 \u00b5m\ninter-channel distance and 64 channels from a Neuropixels probe (\u223c25-40 \u00b5m inter-channel distance).\nOur simulations contain 40 excitatory cells and 10 inhibitory cells with random morphological\nsubtypes, randomly distributed and rotated in 3D space around the probe (with a 20 \u00b5m minimum\ndistance between somas). Each dataset has about 20,000 spikes in total (60 second duration). For\nmore details on the simulation and noise model, see Appendix G.\nFor the real datasets, we use public data from a Neuropixels probe [32] and from a mouse retina\nrecorded with the BioCam4096 platform [24]. The two datasets have 6 million and 2.2 million\nspikes, respectively. Spike detection and sorting (with our location estimates) are done using the\nHerdingSpikes2 software [22].\n\n4.2 Evaluation\n\nBefore evaluating the localization methods, we must detect the spikes from each neuron in the\nsimulated recordings. To avoid biasing our results by our choice of detection algorithm, we assume\nperfect detection, extracting waveforms from channels near each spiking neuron. Once the waveforms\nare extracted from the recordings, we perform the data augmentation. For the square MEA we use\nW = 20, 40, which gives L = 4-9, 9-25 real channels in the observed data, respectively. For the\nsimulated Neuropixels, we use W = 35, 45, which gives L = 3-6, 8-14 real channels in the observed\ndata, respectively. Once we have the augmented dataset, we generate location estimates for all the\ndatasets using each localization method. For straightforward comparison with center of mass, we\nonly evaluate the 2D location estimates (in the plane of the recording device).\nIn the \ufb01rst evaluation, we assess the accuracy of each method by computing the Euclidean distance\nbetween the estimated spike locations and the associated \ufb01ring neurons. We report the mean and\nstandard deviation of the localization error for all spikes in each recording.\nIn the second evaluation, we cluster the location estimates of each method using Gaussian mixture\nmodels (GMMs). The GMMs are \ufb01t with spherical covariances ranging from 45 to 75 mixture\ncomponents (with a step size of 5). We report the true positive rate and accuracy for each number\nof mixture components when matched back to ground truth. To be clear, our use of GMMs is not\na proposed spike sorting method for real data (the number of clusters is never known apriori), but\nrather a systematic way to evaluate whether our location estimates are more discriminable features\nthan those of center of mass.\nIn the third evaluation, we again use GMMs to cluster the location estimates, however, this time\ncombined with two principal components from each spike. We report the true positive rate and\naccuracy for each number of mixture components as before. Combining location estimates and\nprincipal components explicitly, to create a new, low-dimensional feature set, is introduced in Hilgen\n(2017). In this work, the principal components are whitened and then scaled with a hyperparameter,\n\u03b1. To remove any bias from choosing an \u03b1 value in our evaluation, we conduct a grid search over\n\u03b1 = {4, 6, 8, 10} and report the best metric scores for each method.\nIn the fourth evaluation, we assess the generalization performance of the method by training a VAE\non an extracellular dataset and then trying to infer the spike locations in another dataset where the\nneuron locations are different, but all other aspects are kept the same (10\u00b5V noise level, square MEA).\nThe localization and sorting performance is then compared to that of a VAE trained directly on the\nsecond dataset and to center of mass.\nTaken together, the \ufb01rst evaluation demonstrates how useful each method is purely as a localization\ntool, the second evaluation demonstrates how useful the location estimates are for spike sorting\nimmediately after localizing, the third evaluation demonstrates how much the performance can\nimprove given extra waveform information, and the fourth evaluation demonstrates how our method\ncan be used across similar datasets without retraining. For all of our sorting analysis, we use\nSpikeInterface version 0.9.1 [5].\n\n4.3 Results\n\nTable 1 reports the localization accuracy of the different localization methods for the square MEA\nwith three different noise levels. Our model-based methods far outperform center of mass with any\nnumber of observed channels. As expected, introducing amplitude jitter helps lower the mean and\n\n7\n\n\fFigure 2: Spike Sorting Performance on square MEA. We compare the sorting performance of the\nVAE localization method and the COM localization method with and without principal components\nacross all noise levels. For the VAE, we include the results with 0\u00b5V and 10\u00b5V amplitude jitter and\nwith different amounts of observed channels (4-9 and 9-25). For COM, we plot the highest sorting\nperformance (25 observed channels). The test data set has 50 neurons.\n\nstandard deviation of the location spike distance. Using a small width of 20\u00b5m when constructing\nthe augmented data (4-9 observed channels) has the highest performance for the square MEA.\nThe location estimates for the square MEA are visualized in Figure 1. Recording channels are plotted\nas grey squares and the true soma locations are plotted as black stars. The estimated individual\nspike locations are colored according to their associated \ufb01ring neuron identity. As can be seen in the\nplot, center of mass suffers both from arti\ufb01cial splitting of location estimates and poor performance\non neurons outside the array, two areas in which the model-based approaches excel. The MCMC\nand VAE methods have very similar location estimates, highlighting the success of our variational\ninference in approximating the true posterior. See Appendix A for a location estimate plot when the\nVAE is trained and tested on simulated Neuropixels recordings.\nIn Figure 2, spike sorting performance on the square MEA is visualized for all localization methods\n(with and without waveform information). Here, we only show the sorting results for center of mass\non 25 observed channels, where it performs at its best. Overall, the model-based approaches have\nsigni\ufb01cantly higher precision, recall, and accuracy than center of mass across all noise levels and\nall different numbers of mixtures. This illustrates how model-based location estimates provide a\nmuch more discriminatory feature set than the location estimates from the center of mass approaches.\nWe also \ufb01nd that the addition of waveform information (in the form of principal components)\nimproves spike sorting performance for all localization methods. See Appendix A for a spike sorting\nperformance plot when the VAE is trained and tested on simulated Neuropixels recordings.\nAs shown in Appendix D, when our method is trained on one simulated recording, it can generalize\nwell to another simulated recording with different neuron locations. The localization accuracy and\nsorting performance are only slightly lower than the VAE that is trained directly on the new recording.\nOur method also still outperforms center of mass on the new dataset even without training on it.\nFigure 3 shows our localization method as applied to two real, large-scale extracellular datasets.\nIn these plots, we color the location estimates based on their unit identity after spike sorting with\nHerdingSpikes2. These extracellular recordings do not have ground truth information as current,\nground-truth recordings are limited to a few labeled neurons [56, 19, 21, 40, 54]. Therefore, to\ndemonstrate that the units we \ufb01nd likely correspond to individual neurons, we visualize waveforms\nfrom a local grouping of sorted units on the Neuropixels probe. This analysis illustrates that are\nmethod can already be applied to large-scale, real extracellular recordings.\nIn Appendix E, we demonstrate that the inference time for the VAE is much faster than that of\nMCMC, highlighting the excellent scalability of our method. The inference speed of the VAE allows\nfor localization of one million spikes in approximately 37 seconds on a TITAN X GPU, enabling\nreal-time analysis of large-scale extracellular datasets.\n\n8\n\n5060700.30.40.50.60.70.80.91.05060705060700.30.40.50.60.70.80.91.0VAE-0VAE-0-PCsVAE-10VAE-10-PCsCOMCOM-PCsVAE9-25VAE4-9Number of Mixtures10\u03bcV20\u03bcV30\u03bcVPrecisionRecallAccuracyPrecisionRecallAccuracyPrecisionRecallAccuracy506070506070506070506070506070506070\fFigure 3: Estimated spike locations for two real recordings. A, Analysis of a one hour recording\nfrom an awake, head-\ufb01xed mouse with a Neuropixels probe. Spikes were detected using the HS2\npackage [22], their locations estimated using the VAE model, and clustered with mean shift, together\nwith the \ufb01rst two principal components obtained from the waveforms. Shown are a large section of\nthe probe, a magni\ufb01cation and corresponding spike waveforms from the clustered units. B, The same\nanalysis performed on a recording from a mouse retina with a BioCam array from ref [24].\n\n5 Discussion\n\nHere, we introduce a Bayesian approach to spike localization using amortized variational inference.\nOur method signi\ufb01cantly improves localization accuracy and spike sorting performance over the\npreexisting baseline while remaining scalable to the large volumes of data generated by MEAs. Scal-\nability is particularly relevant for recordings from thousands of channels, where a single experiment\nmay yield in the order of 100 million spikes.\nWe validate the accuracy of our model assumptions and inference scheme using biophysically realistic\nground truth simulated recordings that capture much of the variability seen in real recordings. Despite\nthe realism of our simulated recordings, there are some factors that we did not account for, including:\nbursting cells with event amplitude \ufb02uctuations, electrode drift, and realistic intrinsic variability of\nrecorded spike waveforms. As these factors are dif\ufb01cult to model, future analysis of real recordings\nor advances in modeling software will help to understand possible limitations of the method.\nAlong with limitations of the simulated data, there are also limitations of our model. Although\nwe assume a monopole current-source, every part of the neuronal membrane can produce action\npotentials [7]. This means that a more complicated model, such as a dipole current [50], line current-\nsource [50], or modi\ufb01ed ball-and-stick [48], might be a better \ufb01t to the data. Since these models have\nonly ever been used after spike sorting, however, the extent at which they can improve localization\nperformance before spike sorting is unclear and is something we would like to explore in future work.\nAlso, our model utilizes a Gaussian observation model for the spike amplitudes. In real recordings,\nthe true noise distribution is often non-Gaussian and is better approximated by pink noise models ( 1\nf\nnoise) [53]. We plan to explore more realistic observation models in future works.\nSince our method is Bayesian, we hope to better utilize the uncertainty of the location estimates in\nfuture works. Also, as our inference network is fully differentiable, we imagine that our method can\nbe used as a submodule in a more complex, end-to-end method. Other work indicates there is scope\nfor constructing more complicated models to perform event detection and classi\ufb01cation [31], and\nto distinguish between different morphological neuron types based on their activity footprint on the\narray [6]. Our work is thus a \ufb01rst step towards using amortized variational inference methods for the\nunsupervised analysis of complex electrophysiological recordings.\n\n9\n\n\fReferences\n[1] Ballini Marco, Muller Jan, Livi Paolo, Chen Yihui, Frey Urs, Stettler Alexander, Shadmani\nAmir, Viswam Vijay, Jones Ian Lloyd, Jackel David, Radivojevic Milos, Lewandowska Marta K.,\nGong Wei, Fiscella Michele, Bakkum Douglas J., Heer Flavio, Hierlemann Andreas. 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