{"title": "On the relations of LFPs & Neural Spike Trains", "book": "Advances in Neural Information Processing Systems", "page_first": 2060, "page_last": 2068, "abstract": "One of the goals of neuroscience is to identify neural networks that correlate with important behaviors, environments, or genotypes. This work proposes a strategy for identifying neural networks characterized by time- and frequency-dependent connectivity patterns, using convolutional dictionary learning that links spike-train data to local field potentials (LFPs) across multiple areas of the brain. Analytical contributions are: (i) modeling dynamic relationships between LFPs and spikes; (ii) describing the relationships between spikes and LFPs, by analyzing the ability to predict LFP data from one region based on spiking information from across the brain; and (iii) development of a clustering methodology that allows inference of similarities in neurons from multiple regions. Results are based on data sets in which spike and LFP data are recorded simultaneously from up to 16 brain regions in a mouse.", "full_text": "On the Relationship Between LFP & Spiking Data\n\nDavid E. Carlson1, Jana Schaich Borg2, Kafui Dzirasa2, and Lawrence Carin1\n\n1Department of Electrical and Computer Engineering\n2Department of Psychiatry and Behavioral Sciences\n\n{david.carlson, jana.borg, kafui.dzirasa, lcarin}@duke.edu\n\nDuke University\nDuham, NC 27701\n\nAbstract\n\nOne of the goals of neuroscience is to identify neural networks that correlate with\nimportant behaviors, environments, or genotypes. This work proposes a strategy\nfor identifying neural networks characterized by time- and frequency-dependent\nconnectivity patterns, using convolutional dictionary learning that links spike-train\ndata to local \ufb01eld potentials (LFPs) across multiple areas of the brain. Analytical\ncontributions are: (i) modeling dynamic relationships between LFPs and spikes;\n(ii) describing the relationships between spikes and LFPs, by analyzing the ability\nto predict LFP data from one region based on spiking information from across the\nbrain; and (iii) development of a clustering methodology that allows inference\nof similarities in neurons from multiple regions. Results are based on data sets in\nwhich spike and LFP data are recorded simultaneously from up to 16 brain regions\nin a mouse.\n\nIntroduction\n\n1\nOne of the most fundamental challenges in neuroscience is the \u201clarge-scale integration problem\u201d:\nhow does distributed neural activity lead to precise, uni\ufb01ed cognitive moments [1]. This paper seeks\nto examine this challenge from the perspective of extracellular electrodes inserted into the brain. An\nextracellular electrode inserted into the brain picks up two types of signals: (1) the local \ufb01eld poten-\ntial (LFP), which represents local oscillations in frequencies below 200 Hz; and (2) single neuron\naction potentials (also known as \u201cspikes\u201d), which typically occur in frequencies of 0.5 kHz. LFPs\nrepresent network activity summed over long distances, whereas action potentials represent the pre-\ncise activity of cells near the tip of an electrode. Although action potentials are often treated as the\n\u201ccurrency\u201d of information transfer in the brain, relationships between behaviors and LFP activity\ncan be equally precise, and sometimes even more precise, than those with the activity of individual\nneurons [2,3]. Further, LFP network disruptions are highly implicated in many forms of psychiatric\ndisease [4]. This has led to much interest in understanding the mechanisms of how LFPs and action\npotentials interact to create speci\ufb01c types of behaviors. New multisite recording techniques that\nallow simultaneous recordings from a large number of brain regions provide unprecedented oppor-\ntunities to study these interactions. However, this type of multi-dimensional data poses signi\ufb01cant\nchallenges that require new analysis techniques.\nThree of the most challenging characteristics of multisite recordings are that: 1) the networks they\nrepresent are dynamic in space and time, 2) subpopulations of neurons within a local area can have\ndifferent functions and may therefore relate to LFP oscillations in speci\ufb01c ways, and 3) different fre-\nquencies of LFP oscillations often relate to single neurons in speci\ufb01c ways [5]. Here new models are\nproposed to examine the relationship between neurons and neural networks that accommodate these\ncharacteristics. First, each LFP in a brain region is modeled as convolutions between a bounded-time\ndictionary element and the observed spike trains. Critically, the convolutional factors are allowed\nto be dynamic, by binning the LFP and spike time series, and modeling the dictionary element for\n\n1\n\n\feach bin of the time series. Next, a clustering model is proposed making each neuron\u2019s dictionary\nelement a scaled version of an autoregressive template shared among all neurons in a cluster. This\nallows one to identify sub-populations of neurons that have similar dynamics over their functional\nconnectivity to a brain region. Finally, we provide a strategy for exploring which frequency bands\ncharacterize spike-to-LFP functional connectivity. We show, using two novel multi-region electro-\nphysiology datasets from mice, how these models can be used to identify coordinated interactions\nwithin and between different neuronal subsystems, de\ufb01ned jointly by the activity of single cells and\nLFPs. These methods may lead to better understanding of the relationship between brain activity\nand behavior, as well as the pathology underlying brain diseases.\n2 Model\n2.1 Data and notation\nThe data used here consists of multiple LFP and spike-train time series, measured simultaneously\nfrom M regions of a mouse brain. Spike sorting is performed on the spiking data by a VB imple-\nmentation of [6], from which J single units are assumed detected from across the multiple regions\n(henceforth we refer to single units as \u201cneurons\u201d); the number of observed neurons J depends on\nthe data considered, and is inferred as discussed in [6]. Since multiple microwires are inserted into\nsingle brain regions in our experiments (described in [7]), we typically detect between 4-50 neurons\nfor each of the M regions in which the microwires are inserted (discussed further when presenting\nresults). The analysis objective is to examine the degree to which one may relate (predict) the LFP\ndata from one brain region using the J-neuron spiking data from all brain regions. This analysis al-\nlows the identi\ufb01cation of multi-site neural networks through the examination of the degree to which\nneurons in one region are predictive of LFPs in another.\nLet x \u2208 RT represent a time series of LFP data measured from a particular brain region. The T\nsamples are recorded on a regular grid, with temporal interval \u2206. The spike trains from J differ-\nent neurons (after sorting) are represented by the set of vectors {y1, . . . , yJ}, binned in the same\nmanner temporally as the LFP data. Each yj \u2208 ZT\n+ is re\ufb02ective of the number of times neuron\nj \u2208 {1, . . . , J} \ufb01red within each of the T time bins, where Z+ represents nonnegative integers.\nIn the proposed model LFP data x are represented as a superposition of signals associated with each\nneuron yj, plus a residual that captures LFP signal unrelated to the spiking data. The contribution\nto x from information in yj is assumed generated by the convolution of yj with a bounded-time\ndictionary element dj (residing within the interval -L to L, with L (cid:28) T ). This model is related to\nconvolutional dictionary learning [8], where the observed (after spike sorting) signal yj represents\nthe signal we convolve the learned dictionary dj against.\nWe model dj as time evolving, motivated by the expectation that neuron j may contribute differently\nto speci\ufb01ed LFP data, based upon the latent state of the brain (which will be related to observed\nanimal activity). The time series x is binned into a set of B equal-size contiguous windows, where\nx = vec([x1, . . . , xB]), and likewise y = vec([yj1, . . . , yjB]). The dictionary element for neuron\nj is similarly binned as {dj1, . . . , djB}, and the contribution of neuron j to xb is represented as a\nconvolution of djb and yjb. This bin size is a trade-off between how \ufb01nely time is discretized and\nthe computational costs.\nIn the experiments, in one example the bins are chosen to be 30 seconds wide (novel-environment\ndata) and in the other 1 minute (sleep-cycle data), and these are principally chosen for computational\nconvenience (the second data set is nine times longer). Similar results were found with windows as\nnarrow as 10 second, or as wide as 2 minutes.\n2.2 Modeling the LFP contribution of multiple neurons jointly\nGiven {y1, . . . , yJ}, the LFP voltage at time window b is represented as\n\nxb =\n\nyjb \u2217 djb + \u0001b\n\n(1)\n\nJ(cid:88)\n\nj=1\n\nwhere \u2217 represents the convolution operator. Let Dj = [dj1, . . . , djB] \u2208 R(2L+1)\u00d7B represent\nthe sequence of dictionary elements used to represent the LFP data over the B windows, from the\nperspective of neuron j. We impose the clustering prior\n\nDj = \u03b6jAj, Aj \u223c G, G \u223c DP(\u03b2, G0)\n\n(2)\n\n2\n\n\fwhere G is a draw from a Dirichlet process (DP) [9, 10], with scale parameter \u03b2 > 0 and base\nprobability measure G0. Note that we cluster the shape of the dictionary elements, and each neuron\nhas its own scaling \u03b6 \u2208 R. Concerning the base measure, we impose an autoregressive prior on the\ntemporal dynamics, and therefore G0 is de\ufb01ned by an AR(\u03b1, \u03b3) process\n\nk=1 \u03c0k\u03b4A\u2217\n\nk\n\nh1, . . . , a\u2217\n\nab = \u03b1ab\u22121 + \u03bdt, \u03bdt \u223c N (0, \u03b3\u22121I)\n\nhB), with G =(cid:80)\u221e\n\n(3)\n(cid:81)\nwhere I is the identity matrix. This AR prior is used to constitute the B columns of the DP \u201catoms\u201d\nA\u2217\nh = (a\u2217\n. The elements of the vector \u03c0 = (\u03c01, \u03c02, . . . ) are\ni