{"title": "A framework for studying synaptic plasticity with neural spike train data", "book": "Advances in Neural Information Processing Systems", "page_first": 2330, "page_last": 2338, "abstract": "Learning and memory in the brain are implemented by complex, time-varying changes in neural circuitry. The computational rules according to which synaptic weights change over time are the subject of much research, and are not precisely understood. Until recently, limitations in experimental methods have made it challenging to test hypotheses about synaptic plasticity on a large scale. However, as such data become available and these barriers are lifted, it becomes necessary to develop analysis techniques to validate plasticity models. Here, we present a highly extensible framework for modeling arbitrary synaptic plasticity rules on spike train data in populations of interconnected neurons. We treat synaptic weights as a (potentially nonlinear) dynamical system embedded in a fully-Bayesian generalized linear model (GLM). In addition, we provide an algorithm for inferring synaptic weight trajectories alongside the parameters of the GLM and of the learning rules. Using this method, we perform model comparison of two proposed variants of the well-known spike-timing-dependent plasticity (STDP) rule, where nonlinear effects play a substantial role. On synthetic data generated from the biophysical simulator NEURON, we show that we can recover the weight trajectories, the pattern of connectivity, and the underlying learning rules.", "full_text": "A framework for studying synaptic plasticity\n\nwith neural spike train data\n\nScott W. Linderman\nHarvard University\n\nCambridge, MA 02138\n\nChristopher H. Stock\n\nHarvard College\n\nCambridge, MA 02138\n\nRyan P. Adams\nHarvard University\n\nCambridge, MA 02138\n\nswl@seas.harvard.edu\n\ncstock@post.harvard.edu\n\nrpa@seas.harvard.edu\n\nAbstract\n\nLearning and memory in the brain are implemented by complex, time-varying\nchanges in neural circuitry. The computational rules according to which synaptic\nweights change over time are the subject of much research, and are not precisely\nunderstood. Until recently, limitations in experimental methods have made it chal-\nlenging to test hypotheses about synaptic plasticity on a large scale. However, as\nsuch data become available and these barriers are lifted, it becomes necessary\nto develop analysis techniques to validate plasticity models. Here, we present\na highly extensible framework for modeling arbitrary synaptic plasticity rules\non spike train data in populations of interconnected neurons. We treat synap-\ntic weights as a (potentially nonlinear) dynamical system embedded in a fully-\nBayesian generalized linear model (GLM). In addition, we provide an algorithm\nfor inferring synaptic weight trajectories alongside the parameters of the GLM and\nof the learning rules. Using this method, we perform model comparison of two\nproposed variants of the well-known spike-timing-dependent plasticity (STDP)\nrule, where nonlinear effects play a substantial role. On synthetic data generated\nfrom the biophysical simulator NEURON, we show that we can recover the weight\ntrajectories, the pattern of connectivity, and the underlying learning rules.\n\nIntroduction\n\n1\nSynaptic plasticity is believed to be the fundamental building block of learning and memory in the\nbrain. Its study is of crucial importance to understanding the activity and function of neural circuits.\nWith innovations in neural recording technology providing access to the simultaneous activity of\nincreasingly large populations of neurons, statistical models are promising tools for formulating and\ntesting hypotheses about the dynamics of synaptic connectivity. Advances in optical techniques [1,\n2], for example, have made it possible to simultaneously record from and stimulate large populations\nof synaptically connected neurons. Armed with statistical tools capable of inferring time-varying\nsynaptic connectivity, neuroscientists could test competing models of synaptic plasticity, discover\nnew learning rules at the monosynaptic and network level, investigate the effects of disease on\nsynaptic plasticity, and potentially design stimuli to modify neural networks.\nDespite the popularity of GLMs for spike data, relatively little work has attempted to model the\ntime-varying nature of neural interactions. Here we model interaction weights as a dynamical system\ngoverned by parametric synaptic plasticity rules. To perform inference in this model, we use particle\nMarkov Chain Monte Carlo (pMCMC) [3], a recently developed inference technique for complex\ntime series. We use this new modeling framework to examine the problem of using recorded data to\ndistinguish between proposed variants of spike-timing-dependent plasticity (STDP) learning rules.\n\n1\n\n\fFigure 1: A simple network of four sparsely connected neurons whose synaptic weights are changing over time.\nHere, the neurons have inhibitory self connections to mimic refractory effects, and are connected via a chain of\nexcitatory synapses, as indicated by the nonzero entries A1\u21922, A2\u21923, and A3\u21924. The corresponding weights\nof these synapses are strengthening over time (darker entries in W ), leading to larger impulse responses in the\n\ufb01ring rates and a greater number of induced post-synaptic spikes (black dots), as shown below.\n\n2 Related Work\nThe GLM is a probabilistic model that considers spike trains to be realizations from a point process\nwith conditional rate \u03bb(t) [4, 5]. From a biophysical perspective, we interpret this rate as a nonlinear\nfunction of the cell\u2019s membrane potential. When the membrane potential exceeds the spiking thresh-\nold potential of the cell, \u03bb(t) rises to re\ufb02ect the rate of the cell\u2019s spiking, and when the membrane\npotential decreases below the spiking threshold, \u03bb(t) decays to zero. The membrane potential is\nmodeled as the sum of three terms: a linear function of the stimulus, I(t), for example a low-pass\n\ufb01ltered input current, the sum of excitatory and inhibitory PSPs induced by presynaptic neurons, and\na constant background rate. In a network of N neurons, let Sn = {sn,m}Mn\nm=1 \u2282 [0, T ] be the set of\nobserved spike times for neuron n, where T is the duration of the recording and Mn is the number\nof spikes. The conditional \ufb01ring rate of a neuron n can be written,\n\n\uf8eb\uf8edbn +\n\n(cid:90) t\n\n0\n\n\u03bbn(t) = g\n\nkn(t \u2212 \u03c4 ) \u00b7 I(\u03c4 ) d\u03c4 +\n\nhn(cid:48)\u2192n(t \u2212 sn(cid:48),m) \u00b7 I[sn(cid:48),m < t]\n\n(1)\n\n\uf8f6\uf8f8 ,\n\nN(cid:88)\n\nMn(cid:48)(cid:88)\n\nn(cid:48)=1\n\nm=1\n\nwhere bn is the background rate, the second term is a convolution of the (potentially vector-valued)\nstimulus with a linear stimulus \ufb01lter, kn(\u2206t), and the third is a linear summation of impulse re-\nsponses, hn(cid:48)\u2192n(\u2206t), which preceding spikes on neuron n(cid:48) induce on the membrane potential of\nneuron n. Finally, the rectifying nonlinearity g : R \u2192 R+ converts this linear function of stimulus\nand spike history into a nonnegative rate. While the spiking threshold potential is not explicitly\nmodeled in this framework, it is implicitly inferred in the amplitude of the impulse responses.\nFrom this semi-biophysical perspective it is clear that one shortcoming of the standard GLM is that it\ndoes not account for time-varying connectivity, despite decades of research showing that changes in\nsynaptic weight occur over a variety of time scales and are the basis of many fundamental cognitive\nprocesses. This absence is due, in part, to the fact that this direct biophysical interpretation is not\nwarranted in most traditional experimental regimes, e.g., in multi-electrode array (MEA) recordings\nwhere electrodes are relatively far apart. However, as high resolution optical recordings grow in\npopularity, this assumption must be revisited; this is a central motivation for the present model.\nThere have been a few efforts to incorporate dynamics into the GLM. Stevenson and Koerding [6]\nextended the GLM to take inter-spike intervals as a covariates and formulated a generalized bilinear\nmodel for weights. Eldawlatly et al. [7] modeled the time-varying parameters of a GLM using a\ndynamic Bayesian network (DBN). However, neither of these approaches accommodate the breadth\nof synaptic plasticity rules present in the literature. For example, parametric STDP models with hard\n\n2\n\ntime\fbounds on the synaptic weight are not congruent with the convex optimization techniques used by\n[6], nor are they naturally expressed in a DBN. Here we model time-varying synaptic weights as a\npotentially nonlinear dynamical system and perform inference using particle MCMC.\nNonstationary, or time-varying, models of synaptic weights have also been studied outside the con-\ntext of GLMs. For example, Petreska et al. [8] applied hidden switching linear dynamical sys-\ntems models to neural recordings. This approach has many merits, especially in traditional MEA\nrecordings where synaptic connections are less likely and nonlinear dynamics are not necessarily\nwarranted. Outside the realm of computational neuroscience and spike train analysis, there exist a\nnumber of dynamic statistical models, such as West et al. [9], which explored dynamic generalized\nlinear models. However, the types of models we are interested in for studying synaptic plasticity\nare characterized by domain-speci\ufb01c transition models and sparsity structure, and until recently, the\ntools for effectively performing inference in these models have been limited.\n3 A Sparse Time-Varying Generalized Linear Model\nIn order to capture the time-varying nature of synaptic weights, we extend the standard GLM by \ufb01rst\nfactoring the impulse responses in the \ufb01ring rate of Equation 1 into a product of three terms:\n\nhn(cid:48)\u2192n(\u2206t, t) \u2261 An(cid:48)\u2192n Wn(cid:48)\u2192n(t) rn(cid:48)\u2192n(\u2206t).\n\ni.e. (cid:82) \u221e\n\n(2)\nHere, An(cid:48)\u2192n \u2208 {0, 1} is a binary random variable indicating the presence of a direct synapse\nfrom neuron n(cid:48) to neuron n, Wn(cid:48)\u2192n(t) : [0, T ] \u2192 R is a non stationary synaptic \u201cweight\u201d tra-\njectory associated with the synapse, and rn(cid:48)\u2192n(\u2206t) is a nonnegative, normalized impulse response,\n0 rn(cid:48)\u2192n(\u03c4 )d\u03c4 = 1. Requiring rn(cid:48)\u2192n(\u2206t) to be normalized gives meaning to the synaptic\nweights: otherwise W would only be de\ufb01ned up to a scaling factor. For simplicity, we assume r(\u2206t)\ndoes not change over time, that is, only the amplitude and not the duration of the PSPs are time-\nvarying. This restriction could be adapted in future work.\nAs is often done in GLMs, we model the normalized impulse responses as a linear combination of\nbasis functions. In order to enforce the normalization of r(\u00b7), however, we use a convex combination\nof normalized, nonnegative basis functions. That is,\n\nrn(cid:48)\u2192n(\u2206t) \u2261 B(cid:88)\n\n\u03b2(n(cid:48)\u2192n)\n\nb\n\nrb(\u2206t),\n\nwhere(cid:82) \u221e\n\n0 rb(\u03c4 ) d\u03c4 = 1, \u2200b and(cid:80)B\n\nb=1\n\nb=1 \u03b2(n(cid:48)\u2192n)\n\nb\n\n= 1, \u2200n, n(cid:48). The same approach is used to model\n\nthe stimulus \ufb01lters, kn(\u2206t), but without the normalization and non-negativity constraints.\nThe binary random variables An(cid:48)\u2192n, which can be collected into an N \u00d7 N binary matrix A,\nmodel the connectivity of the synaptic network. Similarly, the collection of weight trajecto-\nries {{Wn(cid:48)\u2192n(t)}}n(cid:48),n, which we will collectively refer to as W (t), model the time-varying synap-\ntic weights. This factorization is often called a spike-and-slab prior [10], and it allows us to separate\nour prior beliefs about the structure of the synaptic network from those about the evolution of synap-\ntic weights. For example, in the most general case we might leverage a variety of random network\nmodels [11] as prior distributions for A, but here we limit ourselves to the simplest network model,\nthe Erd\u02ddos-Renyi model. Under this model, each An(cid:48)\u2192n is an independent identically distributed\nBernoulli random variable with sparsity parameter \u03c1.\nFigure 1 illustrates how the adjacency matrix and the time-varying weights are integrated into the\nGLM. Here, a four-neuron network is connected via a chain of excitatory synapses, and the synapses\nstrengthen over time due to an STDP rule. This is evidenced by the increasing amplitude of the\nimpulse responses in the \ufb01ring rates. With larger synaptic weights comes an increased probability\nof postsynaptic spikes, shown as black dots in the \ufb01gure. In order to model the dynamics of the\ntime-varying synaptic weights, we turn to a rich literature on synaptic plasticity and learning rules.\n3.1 Learning rules for time-varying synaptic weights\nDecades of research on synapses and learning rules have yielded a plethora of models for the evolu-\ntion of synaptic weights [12]. In most cases, this evolution can be written as a dynamical system,\n\ndW (t)\n\ndt\n\n= (cid:96) (W (t), {sn,m : sn,m < t} ) + \u0001(W (t), t),\n\n3\n\n\ffunctions.\n\nwhere (cid:96) is a potentially nonlinear learning rule that determines how synaptic weights change as a\nfunction of previous spiking. This framework encompasses rate-based rules such as the Oja rule\n[13] and timing-based rules such as STDP and its variants. The additive noise, \u0001(W (t), t), need not\nbe Gaussian, and many models require truncated noise distributions.\nintuition, many common learning rules factor into a product of sim-\nFollowing biological\npler\nFor example, STDP (de\ufb01ned below) updates each synapse indepen-\ndently such that dWn(cid:48)\u2192n(t)/dt only depends on Wn(cid:48)\u2192n(t) and the presynaptic spike his-\ntory Sn