{"title": "Inferring spike-timing-dependent plasticity from spike train data", "book": "Advances in Neural Information Processing Systems", "page_first": 2582, "page_last": 2590, "abstract": "Synaptic plasticity underlies learning and is thus central for development, memory, and recovery from injury. However, it is often difficult to detect changes in synaptic strength in vivo, since intracellular recordings are experimentally challenging. Here we present two methods aimed at inferring changes in the coupling between pairs of neurons from extracellularly recorded spike trains. First, using a generalized bilinear model with Poisson output we estimate time-varying coupling assuming that all changes are spike-timing-dependent. This approach allows model-based estimation of STDP modification functions from pairs of spike trains. Then, using recursive point-process adaptive filtering methods we estimate more general variation in coupling strength over time. Using simulations of neurons undergoing spike-timing dependent modification, we show that the true modification function can be recovered. Using multi-electrode data from motor cortex we then illustrate the use of this technique on in vivo data.", "full_text": "Inferring spike-timing-dependent plasticity from\n\nspike train data\n\nIan H. Stevenson and Konrad P. Kording\n\nDepartment of Physical Medicine and Rehabilitation\n{i-stevenson, kk}@northwestern.edu\n\nNorthwestern University\n\nAbstract\n\nSynaptic plasticity underlies learning and is thus central for development, mem-\nory, and recovery from injury. However, it is often dif\ufb01cult to detect changes in\nsynaptic strength in vivo, since intracellular recordings are experimentally chal-\nlenging. Here we present two methods aimed at inferring changes in the coupling\nbetween pairs of neurons from extracellularly recorded spike trains. First, using\na generalized bilinear model with Poisson output we estimate time-varying cou-\npling assuming that all changes are spike-timing-dependent. This approach allows\nmodel-based estimation of STDP modi\ufb01cation functions from pairs of spike trains.\nThen, using recursive point-process adaptive \ufb01ltering methods we estimate more\ngeneral variation in coupling strength over time. Using simulations of neurons un-\ndergoing spike-timing dependent modi\ufb01cation, we show that the true modi\ufb01cation\nfunction can be recovered. Using multi-electrode data from motor cortex we then\nillustrate the use of this technique on in vivo data.\n\n1\n\nIntroduction\n\nOne of the fundamental questions in computational neuroscience is how synapses are modi\ufb01ed by\nneural activity [1, 2]. A number of experimental results, using intracellular recordings in vitro, have\nshown that synaptic plasticity depends on the precise pairing of pre- and post-synaptic spiking [3].\nWhile such spike-timing-dependent plasticity (STDP) is thought to serve as a powerful regulatory\nmechanism [4], measuring STDP in vivo using intracellular recordings is experimentally dif\ufb01cult\n[5]. Here we instead attempt to estimate STDP in vivo by using simultaneously recorded extracel-\nlular spike trains and develop methods to estimate the time-varying strength of synapses.\nIn the past few years model-based methods have been developed that allow the estimation of cou-\npling between pairs of neurons from spike train data [6, 7, 8, 9, 10, 11]. These methods have been\nsuccessfully applied to data from a variety of brain areas including retina [10], hippocampus [8], as\nwell as cortex [12]. While anatomical connections between pairs of extracellularly recorded neu-\nrons are generally not guaranteed, these phenomenological methods regularly improve encoding\naccuracy and provide a statistical description of the functional coupling between neurons.\nHere we present two techniques that extend these statistical methods to time-varying coupling be-\ntween neurons and allow the estimation of spike-timing-dependent plasticity from spike trains. First\nwe introduce a generative model for time-varying coupling between neurons where the changes\nin coupling strength depend on the relative timing of pre- and post-synaptic spikes: a bilinear-\nnonlinear-Poisson model. We then present two approaches for inferring STDP modi\ufb01cation func-\ntions from spike data. We test these methods on both simulated data and data recorded from the\nmotor cortex of a sleeping macaque monkey.\n\n1\n\n\fFigure 1: Generative model. A) A generative model of spikes where the coupling between neu-\nrons undergoes spike-timing dependent modi\ufb01cation. Post-synaptic spiking is modeled as a doubly\nstochastic Poisson process with a conditional intensity that depends on the neuron\u2019s own history and\ncoupling to a pre-synaptic neuron. We consider the case where the strength of the coupling changes\nover time, depending on the relative timing of pre- and post-synaptic spikes through a modi\ufb01cation\nfunction. B) As the synaptic strength changes over time, the in\ufb02uence of the pre-synaptic neuron on\nthe post-synaptic neuron changes. Insets illustrate two points in time where synaptic strength is low\n(left) and high (right), respectively. Red lines illustrate the time-varying in\ufb02uence of the pre-synaptic\nneuron, while the black lines denote the static in\ufb02uence.\n\n2 Methods\n\nMany studies have examined nonstationarity in neural systems, including for decoding [13], unitary\nevent detection [14], and assessing statistical dependencies between neurons [15]. Here we focus\nspeci\ufb01cally on non-stationarity in coupling between neurons due to spike-timing dependent modi\ufb01-\ncation of synapses. Our aim is to provide a framework for inferring spike-timing dependent mod-\ni\ufb01cation functions from spike train data alone. We \ufb01rst present a generative model for spike trains\nwhere neurons are undergoing STDP. We then present two methods for estimating spike-timing\ndependent modi\ufb01cation functions from spike train data: a direct method based on a time-varying\ngeneralized linear model (GLM) and an indirect method based on point-process adaptive \ufb01ltering.\n\n2.1 A generative model for coupling with spike-timing dependent modi\ufb01cation\n\nWhile STDP has traditionally been modeled using integrate-and-\ufb01re neurons [4, 16], here we\nmodel neurons undergoing STDP using a simple rate model of coupling between neurons, a linear-\nnonlinear-Poisson (LNP) model. In our LNP model, the conditional intensity (instantaneous \ufb01ring\nrate) of a neuron is given by a linear combination of covariates passed through a nonlinearity. Here,\nwe assume that this nonlinearity is exponential, and the LNP reduces to generalized linear model\n(GLM) with a canonical log link function.\nThe covariates driving variations in the neuron\u2019s \ufb01ring rate can depend on the past spiking history of\nthe neuron, the past spiking history of other neurons (coupling), as well as any external covariates\nsuch as visual stimuli [10] or hand movement [12]. To model coupling from a pre-synaptic neuron\nto a post-synaptic neuron, here we assume that the post-synaptic neuron\u2019s \ufb01ring is generated by\n\n\u03bb(t | H t, \u03b1, \u03b2) = exp\n\nfi (npost(t \u2212 \u03c4 : t)) \u03b1i +\n\nfj(npre(t \u2212 \u03c4 : t))\u03b2j\n\nnpost(t) \u223c P oisson(\u03bb(t | H t, \u03b1, \u03b2)\u2206t)\n\n(1)\nwhere \u03bb(t | H t, \u03b1, \u03b2) is the conditional intensity of the post-synaptic neuron at time t, given a short\nhistory of past spikes from the two neurons Ht and the model parameters. \u03b10 de\ufb01nes a baseline\n\ufb01ring rate, which is modulated by both the neuron\u2019s own spike history from t\u2212\u03c4 to t, npost(t\u2212\u03c4 : t),\nand the history of the pre-synaptic neuron npre(t \u2212 \u03c4 : t) (together abbreviated as Ht). Here we\nhave assumed that the post-spike history and coupling effects are mapped into a smooth basis by\na set of functions fi and then weighted by a set of post-spike coef\ufb01cients \u03b1 and a set of coupling\n\n\uf8eb\uf8ed\u03b10 +\n\n(cid:88)\n\ni\n\n(cid:88)\n\nj\n\n\uf8f6\uf8f8\n\n2\n\nNonlinearityPredictedSpikingCoupling to Pre-Synaptic NeuronPost-Spike HistoryModification Functiontpre - tpostPost-SynapticSpikesPre-SynapticSpikes+0.51.51 minSynaptic Strength200 mslog(\u03bb)PrePostABSynapticStrengthx\fcoef\ufb01cients \u03b2. Finally, we assume that spikes npost(t) are generated by a Poisson random variable\nwith rate \u03bb(t | H t, \u03b1, \u03b2)\u2206t.\nThis model has been used extensively over the past few years to model coupling between neurons\n[10, 12]. Details and extensions of this basic form have been previously published [6]. It is important\nto note, however, that the parameters \u03b1 and \u03b2 can be easily estimated by maximizing the log-\nlikelihood. Since the likelihood is log-concave [9], there is a single, global solution which can be\nfound quickly by a number of methods, such as iterative reweighted least squares (IRLS, used here).\nHere we consider the case where the coupling strength can vary over time, and particularly as a\nfunction of precise timing between pre- and post-synaptic spikes. To incorporate these spike-timing\ndependent changes in coupling into the generative model we introduce a time-varying coupling\nstrength or \u201dsynaptic weight\u201d w(t)\n\n\u03bb(t | X, \u03b1, \u03b2) = exp (\u03b10 + X s(t)\u03b1 + w(t)X c(t)\u03b2)\n\nnpost(t) \u223c P oisson(\u03bb(t | X, \u03b1, \u03b2)\u2206t)\n\n(2)\n\nwhere w(t) changes based on the relative timing of pre- and post-synaptic spikes. Here, for sim-\nplicity, we have re-written the stable post-spike history and coupling terms in matrix form. The\nvector X s(t) summarizes the post-spike history covariates at time t while X c(t) summarizes the\ncovariates related to the history of the pre-synaptic neuron. In this model, the synaptic weight w(t)\nsimply acts to scale the stable coupling de\ufb01ned by \u03b2, and we update w(t) such that every pre-post\nspike pair alters the synaptic weight independently following the second spike.\nUnder this model, the \ufb01ring rate of the post-synaptic neuron is in\ufb02uenced by it\u2019s own past spiking, as\nwell as the activity of a pre-synaptic neuron. A synaptic weight determining the strength of coupling\nbetween the two neurons changes over time depending on the relative spike-timing (Fig 1A).\nIn the simulations that follow we consider three types of modi\ufb01cation functions: 1) a traditional\ndouble-exponential function that accurately models STDP found in cortical and hippocampal slices,\n2) a mexican-hat type function that qualitatively matches STDP found in GABA-ergic neurons in\nhippocampal cultures, and 3) a smoothed double-exponential function that has recently been demon-\nstrated to stabilize weight distributions [17].\nThe double-exponential modi\ufb01cation function is consistent with original STDP observations [2, 3]\nand has been used extensively in simulated populations of integrate-and-\ufb01re neurons [4, 16]. In this\ncase each pair of pre- and post-synaptic spikes modi\ufb01es the synapse by\n\n\u2206w(tpre \u2212 tpost) =\n\n\uf8f1\uf8f2\uf8f3A+ exp\n\nA\u2212 exp\n\n(cid:16) tpre\u2212tpost\n(cid:17)\n(cid:16)\u2212 tpre\u2212tpost\n\n\u03c4+\n\n\u03c4\u2212\n\n(cid:17)\n\nif tpre < tpost\nif tpre \u2265 tpost\n\n(3)\n\nwhere tpre and tpost denote the relative spike times, and the parameters A+, A\u2212, \u03c4+, and \u03c4\u2212 de-\ntermine the magnitude and drop-off of each side of the double-exponential. This creates a sharp\nboundary where the synapse is strengthened whenever pre-synaptic spikes appear to cause post-\nsynaptic spikes and weakened when post-synaptic spikes do not immediately proceed pre-synaptic\nspikes.\nSimilarly, mexican-hat type functions qualitatively match observations of STDP in GABA-ergic\nneurons in hippocampal cultures [18] where\n\n(cid:18)\u2212(tpre \u2212 tpost)2\n\n(cid:19)\n\n2\u03c4 2\n+\n\n(cid:18)\u2212(tpre \u2212 tpost)2\n\n(cid:19)\n\n2\u03c4 2\u2212\n\n\u2206w(tpre \u2212 tpost) = A+ exp\n\n+ A\u2212 exp\n\n(4)\n\nFor \u03c4\u2212 > \u03c4+ this corresponds to a more general Hebbian rule, where synapses are strengthened\nwhenever pre- and post-synaptic spikes occur in close proximity. When spikes do not occur in close\nproximity the synapse is weakened. In this case, the parameters A+, A\u2212, \u03c4+, and \u03c4\u2212 determine\nthe magnitude and standard deviation of the positive and negative components of the modi\ufb01cation\nfunction.\nFinally, we consider a smoothed double-exponential modi\ufb01cation function that has recently been\nshown to stabilize weight distributions. The sharp causal boundary in the classical double-\nexponential tends to drive synaptic weights either towards a maximum or to zero. By adding noise\n\n3\n\n\fto tpre \u2212 tpost, this causal boundary can be smoothed and weight distributions become stable [17].\nHere we add Gaussian noise to (3) such that (tpre \u2212 tpost)(cid:48) = (tpre \u2212 tpost) + \u0001, \u0001 \u223c N (0, \u03c32).\nIt is important to note that, unlike more common integrate-and-\ufb01re models of STDP, these modi\ufb01ca-\ntion function do not describe a change in the magnitude of post synaptic potentials (PSPs). Rather,\n\u2206w de\ufb01nes a change in the statistical in\ufb02uence of the pre-synaptic neuron on the post-synaptic\nneuron. When w(t)X c(t)\u03b2 is large, the post-synaptic neuron is more likely to \ufb01re following a pre-\nsynaptic spike. However, in this bilinear form, w(t) is only uniquely de\ufb01ned up to a multiplicative\nconstant.\nThis generative model includes two distinct components: a GLM that de\ufb01nes the stationary \ufb01ring\nproperties of the post-synaptic neuron and a modi\ufb01cation function that de\ufb01nes how the coupling\nbetween the pre- and post-synaptic neuron changes over time as a function of relative spike timing.\nIn simulating isolated pairs of neurons, each of the modi\ufb01cation functions described above induces\nlarge variations in the synaptic weight. For the sake of stable simulation we add an additional long-\ntimescale forgetting factor that pushes the synaptic weights back to 1. Namely,\n\nw(t + \u2206t) =\n\nw(t) \u2212 \u2206t\nw(t) \u2212 \u2206t\n\n\u03c4f\n\n(w(t) \u2212 1) + \u2206w(tpre \u2212 tpost)\n(w(t) \u2212 1)\n\n\u03c4f\n\nif npre or npost = 1\notherwise\n\n(5)\n\n(cid:40)\n\nwhere, here, we use \u03c4f = 60s. The next sections describe two methods for estimating time-varying\nsynaptic strength as well as STDP modi\ufb01cation functions from spike train data.\n\n2.2 Point-process adaptive \ufb01ltering of coupling strength\n\nSeveral recent studies have examined the possibility that the tuning properties of neurons may drift\nover time. In this context, techniques for estimating arbitrary changes in the parameters of LNP mod-\nels have been especially useful. Point-process adaptive \ufb01ltering is one such method which allows\naccurate estimation of arbitrary time-varying parameters within LNP models and GLMs [19, 20].\nThe goal of this \ufb01ltering approach is to update the model parameters at each time step, follow-\ning spike observations, based on the instantaneous likelihood. Here we use this approach to track\nvariations in coupling strength between two neurons over time.\nDetails and a complete derivation of this model have been previously presented [20]. Brie\ufb02y, the\nbasic recursive point-process adaptive \ufb01lter follows a standard state-space modeling approach and\nassumes that the model parameters in a GLM, such as (1), vary according to a random walk\n\n(6)\nwhere Ft denotes the transition matrix from one timestep to the next and \u03b7t \u223c N (0, Qt) denotes\nGaussian noise with covariance Qt. Given this state-space assumption, we can update the model\nparameters \u03b2 given incoming spike observations. The prediction density at each timestep is given\nby\n\n\u03b2t+1 = F t\u03b2t + \u03b7t\n\n\u03b2t|t\u22121 = F t\u03b2t\u22121|t\u22121\nW t|t\u22121 = F tW t\u22121|t\u22121F T\n\nt + Qt\n\n(7)\n\nwhere \u03b2t\u22121|t\u22121 and W t\u22121|t\u22121 denote the estimated mean and covariance from the previous\ntimestep. Given a new spike count observation nt, we then integrate this prior information with\nthe likelihood to obtain the posterior. Here, for simplicity, we use a quadratic expansion of the\nlog-posterior (a Laplace approximation). When log \u03bb is linear in the parameters, the conditional\nintensity and posterior are given by\n\n(cid:17)\n\n(cid:16)\n\u03bbt = exp\nt|t = W \u22121\nt|t\u22121 + X T\n\u03b2t|t = \u03b2t|t\u22121 + W t|t\n\nW \u22121\n\n(cid:104)\n\nX t\u03b2t|t\u22121 + ct\n\nt [\u03bbt\u2206t]X t\n\n(cid:105)\nt (nt \u2212 \u03bbt\u2206t)\n\nX T\n\n(8)\n\nwhere X t denotes the covariates corresponding to the state-space variable, and ct describes variation\nin log \u03bb that is assumed to be stable over time. Here, the state-space variable is coupling strength,\n\n4\n\n\fand stable components of the model, such as post-spike history effects, are summarized with ct. The\ninitial values of \u03b2 and W can be estimated using a short training period before \ufb01ltering. The only\nfree parameters are those describing the state-space: F and Q. In the analysis that follows we will\nreduce the problem to a single dimension, where the shape of coupling is \ufb01xed during training, and\nwe apply the point-process adaptive \ufb01lter to a single coef\ufb01cient for the covariate X(cid:48)(t) = Xc(t)\u03b2.\nTogether, (7) and (8) allow us to track changes in the model parameters over time. Given an estimate\nof the time-varying synaptic weight \u02c6w(t), we can then estimate the modi\ufb01cation function \u2206 \u02c6w(tpre\u2212\ntpost) by correlating the estimated changes in \u02c6w(t) with the relative spike timings that we observe.\n\n2.3\n\nInferring STDP with a nonparametric, generalized bilinear model\n\nPoint-process adaptive \ufb01ltering allows us to track noisy changes in coupling strength over time.\nHowever, it does not explicitly model the fact that these changes may be spike-timing dependent.\nIn this section we introduce a method to directly infer modi\ufb01cation functions from spike train data.\nSpeci\ufb01cally, we model the modi\ufb01cation function non-parametrically by generating covariates W\nthat depend on the relative spike timing. This non-parametric approximation to the modi\ufb01cation\ngives a generalized bilinear model (GBLM).\n\u03bb(t | X, W , \u03b1, \u03b2, \u03b2w) = exp\n\n\u03b10 + X s(t)\u03b1 + \u03b2T\n\nwW T (t)X c(t)\u03b2\n\n(cid:16)\n\n(cid:17)\n\nnpost(t) \u223c P oisson(\u03bb(t | X, W , \u03b1, \u03b2, \u03b2w)\u2206t)\n\n(9)\n\nwhere \u03b2w describes the modi\ufb01cation function and W (t)\u03b2w approximates w(t). Each of the K\nSTDP covariates, W k, describes the cumulative effect of spike pairs tpre \u2212 tpost within a speci\ufb01c\nrange [T \u2212\n\nk , T +\nk ],\n\nWk(t + \u2206t) = Wk(t) \u2212 \u2206t\n\u03c4f\n\n(Wk(t) \u2212 1) + 1(tpre \u2212 tpost \u2208 [T \u2212\n\nk , T +\n\nk ])\n\n(10)\n\nsuch that, together, W (t)\u03b2w captures the time-varying coupling due to pre-post spike pairs within a\ngiven window (i.e. -100 to 100ms). To model any decay in STDP over time, we, again, allow these\ncovariates to decay exponentially with \u03c4f .\nIn this form, maximum likelihood estimation along each axis is a log-concave optimization problem\n[21]. The parameters describing the modi\ufb01cation function \u03b2w and the parameters describing the\nstable parts of the model \u03b1 and \u03b2 can be estimated by holding one set of parameters \ufb01xed while\nupdating the other and alternating between the two optimizations. In practice, convergence is rel-\natively fast, with the deviance changing by < 0.1% within 3 iterations (Fig 3A), and, empirically,\nusing random restarts, we \ufb01nd that the solutions tend to be stable. In addition to estimates of the\npost-spike history and coupling \ufb01lters, the GBLM thus provides a non-parametric approximation\nto the modi\ufb01cation function and explicitly accounts for spike-timing dependent modi\ufb01cation of the\ncoupling strength.\n\n3 Results\n\nTo examine the accuracy and convergence properties of the two inference methods presented above,\nwe sampled spike trains from the generative model with various parameters. We simulated a pre-\nsynaptic neuron as a homogeneous Poisson process with a \ufb01ring rate of 5Hz, and the post-synaptic\nneuron as a conditionally Poisson process with a baseline \ufb01ring rate of 5Hz. Through the GBLM,\nthe post-synaptic neuron\u2019s \ufb01ring rate is affected by its own post-spike history as well as the activity\nof the pre-synaptic neuron (modeled using 5 raised cosine basis functions [10]). However, as STDP\noccurs the strength of coupling between the neurons changes according to one of three modi\ufb01cation\nfunctions: a double-exponential, a mexican-hat, or a smoothed double-exponential (Fig 2).\nWe \ufb01nd that both point-process adaptive \ufb01ltering and the generalized bilinear model are able to ac-\ncurately reconstruct the time-varying synaptic weight for each type of modi\ufb01cation function (Fig 2,\nleft). However, adaptive \ufb01ltering generally provides a much less accurate estimate of the underlying\nmodi\ufb01cation function than the GBLM (Fig 2, center). Since the adaptive \ufb01lter only updates the\n\n5\n\n\fFigure 2: Recovering simulated STDP. Spikes were simulated from two neurons whose coupling\nvaried over time, depending on the relative timing of pre- and post-synaptic spikes. Using two\ndistinct methods (point-process adaptive \ufb01ltering and the GBLM) we estimated the time-varying\ncoupling strength and modi\ufb01cation function from simulated spike train data. Results are shown\nfor three different modi\ufb01cation functions A) double-exponential, B) Mexican-hat, and C) smoothed\ndouble-exponential. Black lines denote true values, red lines denote estimates from adaptive \ufb01lter-\ning, and blue lines denote estimates from the GBLM. The post-spike history and coupling terms are\nshown at left for the GBLM as multiplicative gains exp(\u03b2). Error bars denote standard errors for the\npost-spike and coupling \ufb01lters and 95% con\ufb01dence intervals for the modi\ufb01cation function estimates.\n\nsynaptic weight following the observations nt, this is not entirely unsurprising. Changes in coupling\nstrength are only detected by the \ufb01lter after they have occurred and become evident in the spiking\nof the post-synaptic neuron. In contrast to the GBLM, there is a substantial delay between changes\nin the true synaptic weight and those estimated by the adaptive \ufb01lter. In this case, we \ufb01nd that the\naccuracy of the adaptive \ufb01lter follows changes in the synaptic weight approximately exponentially\nwith \u03c4 \u223c 25ms (Fig 3B).\nAn important question for the practical application of these methods is how much data is necessary to\ndetect and accurately estimate modi\ufb01cation functions for various effect sizes. Since the size of spike-\ntiming dependent changes may be small in vivo, it is essential that we know under which conditions\nmodi\ufb01cation functions can be recovered. Here we simulated the standard double-exponential STDP\nmodel with several different effect-sizes, modifying A+ and A\u2212 and examining the estimation error\nin both \u02c6w(t) and \u2206 \u02c6w(tpre \u2212 tpost) (Fig 3). The three different effect-sizes simulated here used\ncoupling kernels similar to Fig 2A and began with w(t) = 1. After spike simulation the standard\ndeviation in w(t) was 0.060\u00b10.002 for the small effect size, 0.13\u00b10.01 for the medium effect size,\nand 0.27\u00b10.01 for the large effect size. For all effect sizes, we found that with small amounts of data\n(< 1000 s), the GBLM tends to over-\ufb01t the data. In these situations Adaptive Filtering reconstructs\nboth the synaptic weight (Fig 3E) and modi\ufb01cation function (Fig 3F) more accurately than the\nGBLM (Fig 3C,E). However, once enough data is available maximum likelihood estimation of the\nGBLM out-performs both the stable coupling model and adaptive \ufb01ltering. The extent of over-\ufb01tting\ncan be assessed by the cross-validated log likelihood ratio relative to the homogeneous Poisson\nprocess (Fig 3G, shown in log2 for 2-fold cross-validation). Here, the stable coupling model has an\naverage cross-validated log likelihood ratio relative to a homogeneous Poisson process of 0.185\u00b1\n0.004 bits/spike across all effect sizes. Even in this controlled simulation the contribution of time-\nvarying coupling is relatively small. Both the GBLM and Adaptive Filtering only increase the log\nlikelihood relative to a homogeneous Poisson process by 3-4% for the parameters used here at the\nlargest recording length.\n\n6\n\n012\u2212505x 10\u22123\u2212505x 10\u22123012\u22122024x 10\u22123\u22122024x 10\u22123060012-1000100\u2212505x 10\u22123-1000100\u2212505x 10\u221230101011212010012BACSynaptic StrengthGBLMAdaptive FilterTime [min]tpre-tpost [ms]tpre-tpost [ms]Time [ms]30SimulatedAFGBLM\fFigure 3: Estimation errors for simulated STDP. A) Convergence of the joint optimization problem\nfor three different effect sizes. Filled circles denote updates of the stable coupling terms. Open\ncircles denote updates of the modi\ufb01cation function terms. Note that after 3 iterations the deviance\nis changing by < 0.1% and the model has (essentially) converged. B) Cross-correlation between\nchanges in the true synaptic weight and estimated weight for the GBLM and Adaptive Filter. Note\nthat Adaptive Filtering fails to predict weight changes as they occur. Error bars denote SEM across\nN=10 simulations at the largest effect size. C,D) Correlation between the simulated and estimated\nsynaptic weight (C) and modi\ufb01cation function (D) for the GBLM as a function of the recording\nlength. E,F) Correlation between the simulated and estimated synaptic weight and modi\ufb01cation\nfunction for Adaptive Filtering. Error bars denote SEM across N=40 simulations for each effect\nsize. G) Cross-validated (2-fold) log likelihood relative to a homogeneous Poisson process for the\nGBLM and Adaptive Filtering models. The GBLM (blue) over-\ufb01ts for small amounts of data, but\neventually out-performs both the stable coupling model (gray) and Adaptive Filtering (red). Error\nbands denote SEM across N=120 simulations, all effect sizes.\n\nFigure 4: Results for data from monkey motor cortex. A) Log likelihood relative to a homogeneous\nPoisson process for each of four models: a stable GLM with only post-spike history (PSH), a stable\nGLM with PSH and coupling, the GBLM, and the Adaptive Filter. Bars and error bars denote\nmedian and inter-quartile range. * denotes signi\ufb01cance under a paired t-test, p<0.05. B) The average\nmodi\ufb01cation function estimated under the GBLM for N=75 pairs of neurons. C) The modi\ufb01cation\nfunction estimated from adaptive \ufb01ltering for the same data. In both cases there does not appear to be\na strong, stereotypically shaped modi\ufb01cation function. D) The degree to which adding nonstationary\ncoupling improves model accuracy does not appear to be related to coupling strength as measured\nby how much the PSH+Coupling model improves model accuracy over the PSH model.\n\n7\n\n1030.10.2Recording Length [s]1234510\u2212610\u2212410\u22122100102ConvergenceIterations10301GBLM10301Adaptive Filter10301Recording Length [s]10301Recording Length [s]Log Likelihood [bits/s]Relative Deviance\u03c3w=0.27\u03c3w=0.13\u03c3w=0.06PSH+CouplingPost-Spike HistoryAGCDEFCorrelationCorrelationCorrelationOver-FittingB\u2206t [ms]Cross-CorrelationEstimation DelayGBLMAdaptive FilterEffect SizesModelsCorrelation\u221250050100150-10-6-22x 10\u22124\u03c4~25 msSynaptic WeightSynaptic WeightModificationFunctionModificationFunction-1000100\u22120.020.020.06-1000100\u22120.020.020.0600.10.20.3\u22120.0100.010.0200.040.080.12Model ComparisonALog-Likelihood [bits/s]PSHPSH+CoupingGBLMAdaptive FilterBCDGBLMAdaptive FilterAveverage Modification Function [AU]tpre-tpost [ms]tpre-tpost [ms]LGBLM- LPSH+Coup [bits/spike]LPSH+Coup - LPSH [bits/spike]000.040.04*\fFinally, to test these methods on actual neural recordings, we examined multi-electrode recordings\nfrom the motor cortex of a sleeping macaque monkey. The experimental details of this task have\nbeen previously published [22]. Approximately 180 minutes of data from 83 neurons were collected\n(after spike sorting) during REM and NREM sleep.\nIn the simulations above we assumed that the forgetting factor \u03c4f was known. For the GBLM \u03c4f\ndetermines the timescale of the spike-timing dependent covariates X w, while for adaptive \ufb01ltering\n\u03c4f de\ufb01nes the transition matrix F . In the analysis that follows we make the simplifying assumption\nthat the forgetting factor is \ufb01xed at \u03c4f = 60s. Additionally, during adaptive \ufb01ltering we \ufb01t the\nvariance of the process noise Q by maximizing the cross-validated log-likelihood.\nAnalyzing the most strongly correlated 75 pairs of neurons during the 180 minute recording (2-fold\ncross-validation) we \ufb01nd that the GBLM and Adaptive Filtering both increase model accuracy (Fig\n4A). However, the resulting modi\ufb01cation functions do not show any of the structure previously seen\nin intracellular experiments. In both individual pairs and the average across pairs (Fig 4B,C) the\nmodi\ufb01cation functions are noisy and generally not signi\ufb01cantly different from zero. Additionally,\nwe \ufb01nd that the increase in model accuracy provided by adding non-stationary coupling to the tra-\nditional, stable coupling GLM does not appear to be correlated with the strength of coupling itself.\nThese results suggest that STDP may be dif\ufb01cult to detect in vivo, requiring even longer recordings\nor, possibly, different electrode con\ufb01gurations. Particularly, with the electrode array used here (Utah\narray, 400 \u00b5m electrode spacing), neurons are unlikely to be mono-synaptically connected.\n\n4 Discussion\n\nHere we have presented two methods for estimating spike-timing dependent modi\ufb01cation functions\nfrom multiple spike train data: an indirect method based on point-process adaptive \ufb01ltering and a\ndirect method using a generalized bilinear model. We have shown that each of these methods is able\nto accurately reconstruct both ongoing \ufb02uctuations in synaptic weight and modi\ufb01cation functions in\nsimulation. However, there are several reasons that detecting similar STDP in vivo may be dif\ufb01cult.\nIn vivo, pairs of neurons do not act in isolation. Rather, each neuron receives input from thousands of\nother neurons, inputs which may confound estimation of the coupling between a given pair. It would\nbe relatively straightforward to include multiple pre-synaptic neurons in the model using either\nstable coupling [6, 10] or time-varying, spike-timing dependent coupling. Additionally, unobserved\ncommon input or external covariates, such as hand position, could also be included in the model.\nThese extra covariates should further improve spike prediction accuracy, and could, potentially,\nresult in better estimation of STDP modi\ufb01cation functions.\nDespite these caveats the statistical description of time-varying coupling presented here shows\npromise. Although the neurons in vivo are not guaranteed to be anatomically connected and es-\ntimated coupling must be always be interpreted cautiously [11], including synaptic modi\ufb01cation\nterms does improve model accuracy on in vivo data. Several experimental studies have even sug-\ngested that understanding plasticity may not require well-isolated pairs of neurons. The effects of\nSTDP may be visible through poly-synaptic potentiation [23, 24, 25]. In analyzing real data our\nability to detect STDP may vary widely across experimental preparations. For instance, recordings\nfrom hippocampal slice or dissociated neuronal cultures may reveal substantially more plasticity\nthan in vivo cortical recordings and are less likely to be confounded by unobserved common-input.\nThere are a number of extensions to the basic Adaptive Filtering and GBLM frameworks that may\nyield more accurate estimation and more biophysically realistic models of STDP. The over-\ufb01tting\nobserved in the GBLM could be reduced by regularizing the modi\ufb01cation function, and Adaptive\nSmoothing (using both forward and backward updates) will likely out-perform Adaptive Filtering\nas used here. By changing the functional form of the covariates included in the GBLM we may be\nable to distinguish between standard models of STDP where spike pairs are treated independently\nand other models such as those with self-normalization [16] or where spike triplets are considered\n[26]. Ultimately, the framework presented here extends recent GLM-based approaches to modeling\ncoupling between neurons to allow for time-varying coupling between neurons and, particularly,\nchanges in coupling related to spike-timing dependent plasticity. Although it may be dif\ufb01cult to\nresolve the small effects of STDP in vivo, both improvements in recording techniques and statistical\nmethods promise to make the observation of these ongoing changes possible.\n\n8\n\n\fReferences\n[1] LF Abbott and SB Nelson. Synaptic plasticity: taming the beast. Nature Neuroscience, 3:1178\u20131183,\n\n2000.\n\n[2] G Bi and M Poo. Synaptic modi\ufb01cation by correlated activity: Hebb\u2019s postulate revisited. 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