{"title": "Scalable Inference for Neuronal Connectivity from Calcium Imaging", "book": "Advances in Neural Information Processing Systems", "page_first": 2843, "page_last": 2851, "abstract": "Fluorescent calcium imaging provides a potentially powerful tool for inferring connectivity in neural circuits with up to thousands of neurons. However, a key challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can be orders of magnitude slower than the underlying neural spiking process. Bayesian inference based on expectation-maximization (EM) have been proposed to overcome these limitations, but they are often computationally demanding since the E-step in the EM procedure typically involves state estimation in a high-dimensional nonlinear dynamical system. In this work, we propose a computationally fast method for the state estimation based on a hybrid of loopy belief propagation and approximate message passing (AMP). The key insight is that a neural system as viewed through calcium imaging can be factorized into simple scalar dynamical systems for each neuron with linear interconnections between the neurons. Using the structure, the updates in the proposed hybrid AMP methodology can be computed by a set of one-dimensional state estimation procedures and linear transforms with the connectivity matrix. This yields a computationally scalable method for inferring connectivity of large neural circuits. Simulations of the method on realistic neural networks demonstrate good accuracy with computation times that are potentially significantly faster than current approaches based on Markov Chain Monte Carlo methods.", "full_text": "Scalable Inference for Neuronal Connectivity from\n\nCalcium Imaging\n\nAlyson K. Fletcher\n\nSundeep Rangan\n\nAbstract\n\nFluorescent calcium imaging provides a potentially powerful tool for inferring\nconnectivity in neural circuits with up to thousands of neurons. However, a key\nchallenge in using calcium imaging for connectivity detection is that current sys-\ntems often have a temporal response and frame rate that can be orders of magni-\ntude slower than the underlying neural spiking process. Bayesian inference meth-\nods based on expectation-maximization (EM) have been proposed to overcome\nthese limitations, but are often computationally demanding since the E-step in the\nEM procedure typically involves state estimation for a high-dimensional nonlin-\near dynamical system. In this work, we propose a computationally fast method\nfor the state estimation based on a hybrid of loopy belief propagation and approx-\nimate message passing (AMP). The key insight is that a neural system as viewed\nthrough calcium imaging can be factorized into simple scalar dynamical systems\nfor each neuron with linear interconnections between the neurons. Using the struc-\nture, the updates in the proposed hybrid AMP methodology can be computed by a\nset of one-dimensional state estimation procedures and linear transforms with the\nconnectivity matrix. This yields a computationally scalable method for inferring\nconnectivity of large neural circuits. Simulations of the method on realistic neural\nnetworks demonstrate good accuracy with computation times that are potentially\nsigni\ufb01cantly faster than current approaches based on Markov Chain Monte Carlo\nmethods.\n\n1\n\nIntroduction\n\nDetermining connectivity in populations of neurons is fundamental to understanding neural com-\nputation and function. In recent years, calcium imaging has emerged as a promising technique for\nmeasuring synaptic activity and mapping neural micro-circuits [1\u20134]. Fluorescent calcium-sensitive\ndyes and genetically-encoded calcium indicators can be loaded into neurons, which can then be im-\naged for spiking activity either in vivo or in vitro. Current methods enable imaging populations of\nhundreds to thousands of neurons with very high spatial resolution. Using two-photon microscopy,\nimaging can also be localized to speci\ufb01c depths and cortical layers [5]. Calcium imaging also has\nthe potential to be combined with optogenetic stimulation techniques such as in [6].\nHowever, inferring neural connectivity from calcium imaging remains a mathematically and com-\nputationally challenging problem. Unlike anatomical methods, calcium imaging does not directly\nmeasure connections. Instead, connections must be inferred indirectly from statistical relationships\nbetween spike activities of different neurons. In addition, the measurements of the spikes from cal-\ncium imaging are indirect and noisy. Most importantly, the imaging introduces signi\ufb01cant temporal\nblurring of the spike times: the typical time constants for the decay of the \ufb02uorescent calcium con-\ncentration, [Ca2+], can be on the order of a second \u2013 orders of magnitude slower than the spike rates\nand inter-neuron dynamics. Moreover, the calcium imaging frame rate remains relatively slow \u2013\noften less than 100 Hz. Hence, determining connectivity typically requires super-resolution of spike\ntimes within the frame period.\n\n1\n\n\fTo overcome these challenges, the recent work [7] proposed a Bayesian inference method to esti-\nmate functional connectivity from calcium imaging in a systematic manner. Unlike \u201cmodel-free\u201d\napproaches such as in [8], the method in [7] assumed a detailed functional model of the neural dy-\nnamics with unknown parameters including a connectivity weight matrix W. The model parameters\nincluding the connectivity matrix can then be estimated via a standard EM procedure [9]. While the\nmethod is general, one of the challenges in implementing it is the computational complexity. As\nwe discuss below, the E-step in the EM procedure essentially requires estimating the distributions\nof hidden states in a nonlinear dynamical system whose state dimension grows linearly with the\nnumber of neurons. Since exact computation of these densities grows exponentially in the state di-\nmension, [7] uses an approximate method based on blockwise Gibbs sampling where each block of\nvariables consists of the hidden states associated with one neuron. Since the variables within a block\nare described as a low-dimensional dynamical system, the updates of the densities for the Gibbs\nsampling can be computed ef\ufb01ciently via a standard particle \ufb01lter [10, 11]. However, simulations of\nthe method show that the mixing between blocks can still take considerable time to converge.\nThis paper provides a novel method that can potentially signi\ufb01cantly improve the computation time\nof the state estimation. The key insight is to recognize that a high-dimensional neural system can be\n\u201cfactorized\u201d into simple, scalar dynamical systems for each neuron with linear interactions between\nthe neurons. As described below, we assume a standard leaky integrate-and-\ufb01re model for each\nneuron [12] and a \ufb01rst-order AR process for the calcium imaging [13]. Under this model, the\ndynamics of N neurons can be described by 2N systems, each with a scalar (i.e. one-dimensional)\nstate. The coupling between the systems will be linear as described by the connectivity matrix\nW. Using this factorization, approximate state estimation can then be ef\ufb01ciently performed via\napproximations of loopy belief propagation (BP) [14]. Speci\ufb01cally, we show that the loopy BP\nupdates at each of the factor nodes associated with the integrate-and-\ufb01re and calcium imaging can\nbe performed via a scalar standard forward\u2013backward \ufb01lter. For the updates associated with the\nlinear transform W, we use recently-developed approximate message passing (AMP) methods.\nAMP was originally proposed in [15] for problems in compressed sensing. Similar to expectation\npropagation [16], AMP methods use Gaussian and quadratic approximations of loopy BP but with\nfurther simpli\ufb01cations that leverage the linear interactions. AMP was used for neural mapping from\nmulti-neuron excitation and neural receptive \ufb01eld estimation in [17, 18]. Here, we use a so-called\nhybrid AMP technique proposed in [19] that combines AMP updates across the linear coupling\nterms with standard loopy BP updates on the remainder of the system. When applied to the neural\nsystem, we show that the estimation updates become remarkably simple: For a system with N\nneurons, each iteration involves running 2N forward\u2013backward scalar state estimation algorithms,\nalong with multiplications by W and WT at each time step. The practical complexity scales as\nO(N T ) where T is the number of time steps. We demonstrate that the method can be signi\ufb01cantly\nfaster than the blockwise Gibbs sampling proposed in [7], with similar accuracy.\n\n2 System Model\n\nWe consider a recurrent network of N spontaneously \ufb01ring neurons. All dynamics are approximated\nin discrete time with some time step \u2206, with a typical value \u2206 = 1 ms. Importantly, this time step\nis typically smaller than the calcium imaging period, so the model captures the dynamics between\nobservations. Time bins are indexed by k = 0, . . . , T \u22121, where T is the number of time bins so that\nT \u2206 is the total observation time in seconds. Each neuron i generates a sequence of spikes (action\npotentials) indicated by random variables sk\ni taking values 0 or 1 to represent whether there was a\nspike in time bin k or not. It is assumed that the discretization step \u2206 is suf\ufb01ciently small such that\nthere is at most one action potential from a neuron in any one time bin. The spikes are generated\nvia a standard leaky integrate-and-\ufb01re (LIF) model [12] where the (single compartment) membrane\nvoltage vk\n\ni evolve as\n\ni of each neuron i and its corresponding spike output sequence sk\ni = (1 \u2212 \u03b1IF )vk\n\u02dcvk+1\n\nWijsk\u2212\u03b4\n\nj + bIF,i,\n\nqk\ni =\n\ni + qk\n\ni + dk\nvi\n\n,\n\nN(cid:88)\n(cid:26)(\u02dcvk\n\nj=1\n\n2\n\nand\n\n(vk+1\n\ni\n\n, sk+1\n\ni\n\n) =\n\ni , 0)\n(0, 1)\n\nif vk\nif \u02dcvk\n\ni < \u00b5,\ni \u2265 \u00b5,\n\ndk\nvi\n\n\u223c N (0, \u03c4IF ),\n\n(1)\n\n(2)\n\n\fwhere \u03b1IF is a time constant for the integration leakage; \u00b5 is the threshold potential at which the\ni is the increase in the membrane potential from the\nneurons spikes; bIF,i is a constant bias term; qk\npre-synaptic spikes from other neurons and dk\nvi is a noise term including both thermal noise and\ncurrents from other neurons that are outside the observation window. The voltage has been scaled\nso that the reset voltage is zero. The parameter \u03b4 is the integer delay (in units of the time step\n\u2206) between the spike in one neuron and the increase in the membrane voltage in the post-synaptic\nneuron. An implicit assumption in this model is the post-synaptic current arrives in a single time bin\nwith a \ufb01xed delay.\nTo determine functional connectivity, the key parameter to estimate will be the matrix W of the\nweighting terms Wij in (1). Each parameter Wij represents the increase in the membrane voltage in\nneuron i due to the current triggered from a spike in neuron j. The connectivity weight Wij will be\nzero whenever neuron j has no connection to neuron i. Thus, determining W will determine which\nneurons are connected to one another and the strengths of those connections.\nFor the calcium imaging, we use a standard model [7], where the concentration of \ufb02uorescent Cal-\ncium has a fast initial rise upon an action potential followed by a slow exponential decay. Speci\ufb01-\ni = [Ca2+]k be the concentration of \ufb02uorescent Calcium in neuron i in time bin k and\ncally, we let zk\nassume it evolves as \ufb01rst-order auto-regressive AR(1) model,\ni + sk\ni ,\n\n(3)\nwhere \u03b1CA is the Calcium time constant. The observed net \ufb02uorescence level is then given by a\nnoisy version of zk\ni ,\n\ni = (1 \u2212 \u03b1CA,i)zk\nzk+1\n\ni = aCA,izk\nyk\n\ni + bCA,i + dk\nyi\n\n(4)\nwhere aCA,i and bCA,i are constants and dyi is white Gaussian noise with variance \u03c4y. Nonlinearities\nsuch as saturation described in [13] can also be modeled.\nAs mentioned in the Introduction, a key challenge in calcium imaging is the relatively slow frame\nrate which has the effect of subsampling of the \ufb02uorescence. To model the subsampling, we\nlet IF denote the set of time indices k on which we observe F k\ni . We will assume that \ufb02u-\norescence values are observed once every TF time steps for some integer period TF so that\nIF = {0, TF , 2TF , . . . , KTF} where K is the number of Calcium image frames.\n\n\u223c N (0, \u03c4y),\n\n,\n\ndk\nyi\n\n3 Parameter Estimation via Message Passing\n\n3.1 Problem Formulation\n\nLet \u03b8 be set of all the unknown parameters,\n\n\u03b8 = {W, \u03c4IF , \u03c4CA, \u03b1IF , bIF,i, \u03b1CA, aCA,i, bCA,i, i = 1, . . . , N},\n\n(5)\nwhich includes the connectivity matrix, time constants and various variances and bias terms. Esti-\nmating the parameter set \u03b8 will provide an estimate of the connectivity matrix W, which is our main\ngoal.\nTo estimate \u03b8, we consider a regularized maximum likelihood (ML) estimate\nL(y|\u03b8) + \u03c6(\u03b8), L(y|\u03b8) = \u2212 log p(y|\u03b8),\n\n(cid:98)\u03b8 = arg max\ny = {y1, . . . , yN} , yi =(cid:8)yk\n\n(6)\nwhere y is the set of observed values; L(y|\u03b8) is the negative log likelihood of y given the parameters\n\u03b8 and \u03c6(\u03b8) is some regularization function. For the calcium imaging problem, the observations y\nare the observed \ufb02uorescence values across all the neurons,\ni ,\n\n(7)\nwhere yi is the set of \ufb02uorescence values from neuron i, and, as mentioned above, IF is the set of\ntime indices k on which the \ufb02uorescence is sampled.\nThe regularization function \u03c6(\u03b8) can be used to impose constraints or priors on the parameters. In\nthis work, we will assume a simple regularizer that only constrains the connectivity matrix W,\n\nk \u2208 IF\n\n(cid:9) ,\n\n\u03b8\n\n\u03c6(\u03b8) = \u03bb(cid:107)W(cid:107)1,\n\n(cid:107)W(cid:107)1 :=\n\n|Wij|,\n\n(8)\n\n(cid:88)\n\nij\n\n3\n\n\fwhere \u03bb is a positive constant. The (cid:96)1 regularizer is a standard convex function used to encourage\nsparsity [20], which we know in this case must be valid since most neurons are not connected to one\nanother.\n\n3.2 EM Estimation\n\nExact computation of (cid:98)\u03b8 in (6) is generally intractable, since the observed \ufb02uorescence values y\n\ndepend on the unknown parameters \u03b8 through a large set of hidden variables. Similar to [7], we thus\nuse a standard EM procedure [9]. To apply the EM procedure to the calcium imaging problem, let\nx be the set of hidden variables,\n\n(9)\nwhere v are the membrane voltages of the neurons, z the calcium concentrations, s the spike outputs\nand q the linearly combined spike inputs. For any of these variables, we will use the subscript i (e.g.\nvi) to denote the values of the variables of a particular neuron i across all time steps and superscript\nk (e.g. vk) to denote the values across all neurons at a particular time step k. Thus, for the membrane\nvoltage\n\nx = {v, z, q, s} ,\n\nv =(cid:8)vk\n\n(cid:9) , vk =(cid:0)vk\n\ni\n\n1 , . . . , vk\nN\n\n(cid:1) , vi =(cid:0)v0\n\ni , . . . , vT\u22121\n\ni\n\n(cid:1) .\n\nThe EM procedure alternately estimates distributions on the hidden variables x given the current\nparameter estimate for \u03b8 (the E-step); and then updates the estimates for parameter vector \u03b8 given\nthe current distribution on the hidden variables x (the M-step).\n\n(10)\nwhich is the posterior distribution of the hidden variables x given the observations y and\n\n\u2022 E-Step: Given parameter estimates(cid:98)\u03b8(cid:96), estimate\nP (x|y,(cid:98)\u03b8(cid:96)),\ncurrent parameter estimate(cid:98)\u03b8(cid:96).\nE(cid:104)\nL(x, y|\u03b8)|(cid:98)\u03b8(cid:96)(cid:105)\n\n\u2022 M-step Update the parameter estimate via the minimization,\n\n(cid:98)\u03b8(cid:96)+1 = arg min\n\nwhere L(x, y|\u03b8) is the joint negative log likelihood,\n\n\u03b8\n\n+ \u03c6(\u03b8),\n\n(11)\n\n(12)\nIn (11) the expectation is with respect to the distribution found in (10) and \u03c6(\u03b8) is the\nparameter regularization function.\n\nL(x, y|\u03b8) = \u2212 log p(x, y|\u03b8).\n\nThe next two sections will describe how we approximately perform each of these steps.\n\n3.3 E-Step estimation via Approximate Message Passing\n\ni and N states for the bound Ca concentration levels zk\n\nFor the calcium imaging problem, the challenging step of the EM procedure is the E-step, since\nthe hidden variables x to be estimated are the states and outputs of a high-dimensional nonlinear\ndynamical system. Under the model in Section 2, a system with N neurons will require N states\ni , resulting in\nfor the membrane voltages vk\na total state dimension of 2N. The E-step for this system is essentially a state estimation problem,\nand exact inference of the states of a general nonlinear dynamical system grows exponentially in the\nstate dimension. Hence, exact computation of the posterior distribution (10) for the system will be\nintractable even for a moderately sized network.\nAs described in the Introduction, we thus use an approximate messaging passing method that ex-\nploits the separable structure of the system. For the remainder of this section, we will assume the\n\nparameters \u03b8 in (5) are \ufb01xed to the current parameter estimate(cid:98)\u03b8(cid:96). Then, under the assumptions of\n\nSection 2, the joint probability distribution function of the variables can be written in a factorized\nform,\n\nP (x, y) = P (q, v, s, z, y) =\n\n1\nZ\n\n1{qk=Wsk}\n\n\u03c8IF\n\ni\n\n(qi, vi, si)\u03c8CA\n\ni\n\n(si, zi, yi),\n\n(13)\n\nT\u22121(cid:89)\n\nN(cid:89)\n\nk=0\n\ni=1\n\n4\n\n\fmembrane voltage\n\nvi\n\nCa2+ concentration\n\nzi\n\ninput currents\n\nqi\n\n\u03c8IF\n(qi, vi, si)\nIntegrate-and-\ufb01re\n\ni\n\ndynamics\n\nspike outputs\n\nsi\n\n\u03c8CA\n\n(si, zi, yi)\n\ni\nCa imaging\ndynamics\n\nobserved\n\n\ufb02uorescence\n\nyi\n\nNeuron i, i = 1, . . . , N\n\nqk = Wsk\nConnectivity\n\nbetween neurons\n\nTime step k, k = 0, . . . , T \u22121\n\nFigure 1: Factor graph plate representation of the system where the spike dynamics are described\nby the factor node \u03c8IF\n(qi, vi, si) and the calcium image dynamics are represented via the factor\nnode \u03c8CA\n(si, zi, yi). The high-dimensional dynamical system is described as 2N scalar dynamical\nsystems (2 for each neuron) with linear interconnections, qk = Wsk between the neurons. A\ncomputational ef\ufb01cient approximation of loopy BP [19] is applied to this graph for approximate\nBayesian inference required in the E-step of the EM algorithm.\n\ni\n\ni\n\ni\n\ni\n\n(qi, vi, si) is the potential function relating the summed\nwhere Z is a normalization constant; \u03c8IF\nspike inputs qi to the membrane voltages vi and spike outputs si; \u03c8CA\n(si, zi, yi) relates the spike\noutputs si to the bound calcium concentrations zi and observed \ufb02uorescence values yi; and the term\n1{qk=Wsk} indicates that the distribution is to be restricted to the set satisfying the linear constraints\nqk = Wsk across all time steps k.\nAs in standard loopy BP [14], we represent the distribution (13) in a factor graph as shown in\nFig. 1. Now, for the E-step, we need to compute the marginals of the posterior distribution p(x|y)\nfrom the joint distribution (13). Using the factor graph representation, loopy BP iteratively updates\nestimates of these marginal posterior distributions using a message passing procedure, where the\nestimates of the distributions (called beliefs) are passed between the variable and factor nodes in\nthe graph. In general, the computationally challenging component of loopy BP is the updates on\nthe belief messages at the factor nodes. However, using the factorized structure in Fig. 1 along\nwith approximate message passing (AMP) simpli\ufb01cations as described in [19], these updates can be\ncomputed easily.\nDetails are given in the full paper [21], but the basic procedure for the factor node updates and the\nreasons why these computations are simple can be summarized as follows. At a high level, the factor\ngraph structure in Fig. 1 partitions the 2N-dimensional nonlinear dynamical system into N scalar\nsystems associated with each membrane voltage vk\ni and an additional N scalar systems associated\ni . The only coupling between these systems is through the\nwith each calcium concentration level zk\nlinear relationships qk = Wsk. As shown in Appendix ??, on each of the scalar systems, the factor\nnode updates required by loopy BP essentially reduces to a state estimation problem for this system.\nSince the state space of this system is scalar (i.e. one-dimensional), we can discretize the state space\nwell with a small number of points \u2013 in the experiments below we use L = 20 points per dimension.\nOnce discretized, the state estimation can be performed via a standard forward\u2013backward algorithm.\nIf there are T time steps, the algorithm will have a computational cost of O(T L2) per scalar system.\nHence, all the factor node updates across all the 2N scalar systems has total complexity O(N T L2).\nFor the factor nodes associated with the linear constraints qk = Wsk, we use the AMP approxi-\nmations [19]. In this approximation, the messages for the transform outputs qk\ni are approximated as\nGaussians which is, at least heuristically, justi\ufb01ed since the they are outputs of a linear transform of\na large number of variables, sk\ni . In the AMP algorithm, the belief updates for the variables qk and\nsk can then be computed simply by linear transformations of W and WT . Since W represents a\nconnectivity matrix, it is generally sparse. If each row of W has d non-zero values, multiplication\n\n5\n\n\fby W and WT will be O(N d). Performing the multiplications across all time steps results in a total\ncomplexity of O(N T d).\nThus, the total complexity of the proposed E-step estimation method is O(N T L2 + N T d) per loopy\nBP iteration. We typically use a small number of loopy BP iterations per EM update (in fact, in the\nexperiments below, we found reasonable performance with one loopy BP update per EM update).\nIn summary, we see that while the overall neural system is high-dimensional, it has a linear + scalar\nstructure. Under the assumption of the bounded connectivity d, this structure enables an approximate\ninference strategy that scales linearly with the number of neurons N and time steps T . Moreover,\nthe updates in different scalar systems can be computed separately allowing a readily parallelizable\nimplementation.\n\n3.4 Approximate M-step Optimization\n\nThe M-step (11) is computationally relatively simple. All the parameters in \u03b8 in (5) have a linear\nrelationship between the components of the variables in the vector x in (9). For example, the pa-\nrameters aCA,i and bCA,i appear in the \ufb02uorescence output equation (4). Since the noise dk\nyi in this\nequation is Gaussian, the negative log likelihood (12) is given by\n\nL(x, y|\u03b8) =\n\n1\n2\u03c4yi\n\ni \u2212 aCA,izk\n(yk\n\ni \u2212 bCA,i)2 +\n\nT\n2\n\nlog(\u03c4yi) + other terms,\n\nwhere \u201cother terms\u201d depend on parameters other than aCA,i and bCA,i.\n\nE(L(x, y|\u03b8)|(cid:98)\u03b8(cid:96)) will then depend only on the mean and variance of the variables yk\n\nThe expectation\ni , which\ni and zk\nare provided by the E-step estimation. Thus, the M-step optimization in (11) can be computed via a\nsimple least-squares problem. Using the linear relation (1), a similar method can be used for \u03b1IF,i\nand bIF,i, and the linear relation (3) can be used to estimate the calcium time constant \u03b1CA.\nTo estimate the connectivity matrix W, let rk = qk \u2212 Wsk so that the constraints in (13) is equiva-\nlent to the condition that rk = 0. Thus, the term containing W in the expectation of the negative log\n\nlikelihood E(L(x, y|\u03b8)|(cid:98)\u03b8(cid:96)) is given by the negative log probability density of rk evaluated at zero.\nimate the density as follows: Let(cid:98)q and(cid:98)s be the expectation of the variables q and s given by the\nE-step. Hence, the expectation of rk is(cid:98)qk \u2212 W(cid:98)sk. As a simple approximation, we will then assume\n\nIn general, this density will be a complex function of W and dif\ufb01cult to minimize. So, we approx-\n\nthat the variables rk\nsimplifying assumption, the M-step optimization of W with the (cid:96)1 regularizer (8) reduces to\n\ni are Gaussian, independent and having some constant variance \u03c32. Under this\n\n(cid:99)W = arg min\n\nW\n\nT\u22121(cid:88)\n\nk=0\n\n1\n2\n\n(cid:107)(cid:98)qk \u2212 W(cid:98)sk(cid:107)2 + \u03c32\u03bb(cid:107)W(cid:107)1,\n\n(cid:88)\n\nk\u2208IF\n\n(14)\n\nFor a given value of \u03c32\u03bb, the optimization (14) is a standard LASSO optimization [22] which can be\nevaluated ef\ufb01ciently via a number of convex programming methods. In this work, in each M-step,\nwe adjust the regularization parameter \u03c32\u03bb to obtain a desired \ufb01xed sparsity level in the solution W.\n\n3.5\n\nInitial Estimation via Sparse Regression\n\nSince the EM algorithm cannot be guaranteed to converge a global maxima, it is important to pick\nthe initial parameter estimates carefully. The time constants and noise levels for the calcium image\ncan be extracted from the second-order statistics of \ufb02uorescence values and simple thresholding can\nprovide a coarse estimate of the spike rate.\nThe key challenge is to obtain a good estimate for the connectivity matrix W. For each neuron i, we\ni = 1|yi) from the observed \ufb02uorescence\n\ufb01rst make an initial estimate of the spike probabilities P (sk\nvalues yi, assuming some i.i.d. prior of the form P (st\ni) = \u03bb\u2206, where \u03bb is the estimated average spike\nrate per second. This estimation can be solved with the \ufb01ltering method in [13] and is also equivalent\nto the method we use for the factor node updates. We can then threshold these probabilities to make\na hard initial decision on each spike: sk\ni = 0 or 1. We then propose to estimate W from the spikes\nas follows. Fix a neuron i and let wi be the vector of weights Wij, j = 1, . . . , N. Under the\nassumption that the initial spike sequence sk\ni is exactly correct, it is shown in the full paper [21], that\n\n6\n\n\fParameter\nNumber of neurons, N\nConnection sparsity\n\nMean \ufb01ring rate per neuron\nSimulation time step, \u2206\nTotal simulation time, T \u2206\nIntegration time constant, \u03b1IF\nConduction delay, \u03b4\nIntegration noise, dk\nvi\nCa time constant, \u03b1CA\nFluorescence noise, \u03c4CA\nCa frame rate , 1/TF\n\nValue\n100\n10% with random connections. All connections are excitatory\nwith the non-zero weights Wij being exponentially distributed.\n10 Hz\n1 ms\n10 sec (10,000 time steps)\n20 ms\n2 time steps = 2 ms\nProduced from two unobserved neurons.\n500 ms\nSet to 20 dB SNR\n100 Hz\n\nTable 1: Parameters for the Ca image simulation.\n\nFigure 2: Typical network simulation\ntrace. Top panel: Spike traces for\nthe 100 neuron simulated network.\nBottom panel: Calcium image \ufb02u-\norescence levels. Due to the ran-\ndom network topology, neurons often\n\ufb01re together, signi\ufb01cantly complicat-\ning connectivity detection. Also, as\nseen in the lower panel, the slow de-\ncay of the \ufb02uorescent calcium blurs\nthe spikes in the calcium image.\n\na regularized maximum likelihood estimate of wi and bias term bIF,i is given by\n\n((cid:98)wi,(cid:98)bIF,i) = arg min\n\nwi,bIF,i\n\nT\u22121(cid:88)\n\nLik(uT\n\nk wi + cikbIF,i \u2212 \u00b5, sk\n\ni ) + \u03bb\n\n|Wij|,\n\n(15)\n\nk=0\n\nj=1\n\nN(cid:88)\n\nwhere Lik is a probit loss function and the vector uk and scalar cik can be determined from the\nspike estimates. The optimization (15) is precisely a standard probit regression used in sparse linear\nclassi\ufb01cation [23]. This form arises due to the nature of the leaky integrate-and-\ufb01re model (1) and\n(2). Thus, assuming the initial spike sequences are estimated reasonably accurately, one can obtain\ngood initial estimates for the weights Wij and bias terms bIF,i by solving a standard classi\ufb01cation\nproblem.\n\n4 Numerical Example\n\nThe method was tested using realistic network parameters, as shown in Table 1, similar to those\nfound in neurons networks within a cortical column [24]. Similar parameters are used in [7]. The\nnetwork consisted of 100 neurons with each neuron randomly connected to 10% of the other neu-\nrons. The non-zero weights Wij were drawn from an exponential distribution. As a simpli\ufb01cation,\nall weights were positive (i.e. the neurons were excitatory \u2013 there were no inhibitory neurons in the\nsimulation). A typical random matrix W generated in this manner would not in general result in a\nstable system. To stabilize the system, we followed the procedure in [8] where the system is simu-\nlated multiple times. After each simulation, the rows of the matrix W were adjusted up or down to\nincrease or decrease the spike rate until all neurons spiked at a desired target rate. In this case, we\nassumed a desired average spike rate of 10 Hz.\n\n7\n\n\fFigure 3: Weight estimation accuracy. Left: Normalized mean-squared error as a function of the\niteration number. Right: Scatter plot of the true and estimated weights.\n\nFrom the parameters in Table 1, we can immediately see the challenges in the estimation. Most\nimportantly, the calcium imaging time constant \u03b1CA is set for 500 ms. Since the average neurons\nspike rate is assumed to be 10 Hz, several spikes will typically appear within a single time constant.\nMoreover, both the integration time constant and inter-neuron conduction time are much smaller\nthan the\nA typical simulation of the network after the stabilization is shown in Fig. 2. Observe that due to\nthe random connectivity, spiking in one neuron can rapidly cause the entire network to \ufb01re. This\nappears as the vertical bright stripes in the lower panel of Fig. 2. This synchronization makes the\nconnectivity detection dif\ufb01cult to detect under temporal blurring of Ca imaging since it is hard to\ndetermine which neuron is causing which neuron to \ufb01re. Thus, the random matrix is a particularly\nchallenging test case.\nThe results of the estimation are shown in Fig. 3. The left panel shows the relative mean squared\nerror de\ufb01ned as\n\n(cid:80)\nij |Wij \u2212 \u03b1(cid:99)Wij|2\n(cid:80)\nij |Wij|2\n\n,\n\nmin\u03b1\n\nwhere(cid:99)Wij is the estimate for the weight Wij. The minimization over all \u03b1 is performed since the\n\nrelative MSE =\n\nmethod can only estimate the weights up to a constant scaling. The relative MSE is plotted as a\nfunction of the EM iteration, where we have performed only a single loopy BP iteration for each\nEM iteration. We see that after only 30 iterations we obtain a relative MSE of 7% \u2013 a number at\nleast comparable to earlier results in [7], but with signi\ufb01cantly less computation. The right panel\n\nshows a scatter plot of the estimated weights(cid:99)Wij against the true weights Wij.\n\n(16)\n\n5 Conclusions\n\nWe have presented a scalable method for inferring connectivity in neural systems from calcium\nimaging. The method is based on factorizing the systems into scalar dynamical systems with linear\nconnections. Once in this form, state estimation \u2013 the key computationally challenging component\nof the EM estimation \u2013 is tractable via approximating message passing methods. 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