{"title": "Objective and efficient inference for couplings in neuronal networks", "book": "Advances in Neural Information Processing Systems", "page_first": 4971, "page_last": 4980, "abstract": "Inferring directional couplings from the spike data of networks is desired in various scientific fields such as neuroscience. Here, we apply a recently proposed objective procedure to the spike data obtained from the Hodgkin-Huxley type models and in vitro neuronal networks cultured in a circular structure. As a result, we succeed in reconstructing synaptic connections accurately from the evoked activity as well as the spontaneous one. To obtain the results, we invent an analytic formula approximately implementing a method of screening relevant couplings. This significantly reduces the computational cost of the screening method employed in the proposed objective procedure, making it possible to treat large-size systems as in this study.", "full_text": "Objective and ef\ufb01cient inference for couplings in\n\nneuronal networks\n\nYu Terada1;2, Tomoyuki Obuchi2, Takuya Isomura1, Yoshiyuki Kabashima2\n\n1Laboratory for Neural Computation and Adaptation,\n\nRIKEN Center for Brain Science,\n\n2-1 Hirosawa, Wako, Saitama 351-0198, Japan\n\n2Department of Mathematical and Computer Science\n\nTokyo Institute of Technology\n\nTokyo 152-8550, Japan\n\nyu.terada@riken.jp, obuchi@c.titech.ac.jp,\ntakuya.isomura@riken.jp, kaba@c.titech.ac.jp\n\nAbstract\n\nInferring directional couplings from the spike data of networks is desired in var-\nious scienti\ufb01c \ufb01elds such as neuroscience. Here, we apply a recently proposed\nobjective procedure to the spike data obtained from the Hodgkin\u2013Huxley type\nmodels and in vitro neuronal networks cultured in a circular structure. As a result,\nwe succeed in reconstructing synaptic connections accurately from the evoked ac-\ntivity as well as the spontaneous one. To obtain the results, we invent an analytic\nformula approximately implementing a method of screening relevant couplings.\nThis signi\ufb01cantly reduces the computational cost of the screening method em-\nployed in the proposed objective procedure, making it possible to treat large-size\nsystems as in this study.\n\n1\n\nIntroduction\n\nRecent advances in experimental techniques make it possible to simultaneously record the activity\nof multiple units. In neuroscience, multi-electrodes and optical imaging techniques capture large-\nscale behaviors of neuronal networks, which facilitate a deeper understanding of the information\nprocessing mechanism of nervous systems beyond the single neuron level [1-6]. This preferable\nsituation, however, involves technical issues in dealing with such datasets because they usually con-\nsist of a large amount of high-dimensional data which are dif\ufb01cult to be handled by naive usages of\nconventional statistical methods.\nA statistical-physics-based approach for tackling these issues was presented using the Ising model\n[7]. Although the justi\ufb01cation to use the Ising model for analyzing neuronal systems is not com-\npletely clear [8,9,10], its performance was empirically demonstrated [7], which triggered further\napplications [11-22]. An advantage of using the Ising model is that several analytical techniques for\ninverse problems are available [23-29], which allows us to infer couplings between neurons with a\nfeasible computational cost. Another advantage is that it is straightforward to introduce variants of\nthe model. Beyond the conventional data analysis, an important variant is the kinetic Ising model,\nwhich is more suitable to take into account the correlations in time, since this extended model re-\nmoves the symmetric-coupling constraint of the Ising model. A useful mean-\ufb01eld (MF) inverse\nformula for the kinetic Ising model has been presented in [25,26].\nTwo problems arise when treating neuronal systems\u2019 data in the framework of the Ising models.\nThe \ufb01rst problem is how to determine an appropriate size of time bins when discretizing original\nsignals in time; the appropriate size differs from the intrinsic time-scale of the original neuronal sys-\n\n32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montr\u00e9al, Canada.\n\n\ftems because the Ising models are regarded as a coarse-grained description of the original systems.\nHence, the way of the transformation to the models of this type is nontrivial. The second problem\nis extracting relevant couplings from the solution of the inverse problem; unavoidable noises in ex-\nperimental data contaminate the inferred couplings, and hence, we need to screen the relevant ones\namong them.\nIn a previous study [30], an information-theoretic method and a computational-statistical technique\nwere proposed for resolving the aforementioned \ufb01rst and second problems, respectively. Those\nmethods were validated in two cases: in a numerical simulation based on the Izhikevich models and\nin analyzing in vitro neuronal networks. The result is surprisingly good: their synaptic connections\nare reconstructed with fairly high accuracy. This \ufb01nding motivates us to further examine the methods\nproposed in [30].\nBased on this motivation, this study applies these methods to the data from the Hodgkin\u2013Huxley\nmodel, which describes the \ufb01ring dynamics of a biological neuron more accurately than the Izhike-\nvich model. Further, we examine the situation where responses of neuronal networks are evoked by\nexternal stimuli. We implement this situation both in the Hodgkin\u2013Huxley model and in a cultured\nneuronal network of a previously described design [31], and test the methods in both the cases.\nBesides, based on the previously described MF formula of [25,26], we derive an ef\ufb01cient formula\nimplementing the previous method of screening relevant couplings within a signi\ufb01cantly smaller\ncomputational cost. In practice, the naive implementation of the screening method is computation-\nally expensive, and can be a bottleneck when applied to large-scale networks. Hence, we exploit\nthe simplicity of the model, and use the advanced statistical processing with reasonable time in\nthis work. Below, we address those three points by employing the simple kinetic Ising model, to\nef\ufb01ciently infer synaptic couplings in neuronal networks.\n\nInference procedure\n\n2\nThe kinetic Ising model consists of N units, fsigN\n(cid:6)1. Its dynamics is governed by the so-called Glauber dynamics:\nP (s(t + 1)js(t);fJij; (cid:18)i(t)g) =\n\u2211\n\nN\u220f\n\ni=1\n\ni=1, and each unit takes bipolar values as si(t) =\n\nexp [si(t + 1)Hi(t;fJij; (cid:18)i(t)g)]\n\nexp [Hi(t;fJij; (cid:18)i(t)g)] + exp [(cid:0)Hi(t;fJij; (cid:18)i(t)g)]\n\n;\n\n(1)\n\n{\n\nR\u2211\n\nM\u2211\n\nwhere Hi(t) is the effective \ufb01eld, de\ufb01ned as Hi(t) = (cid:18)i(t) +\nj=1 Jijsj(t), (cid:18)i(t) is the external\nforce, and Jij is the coupling strength from j to i. This model also corresponds to a generalized\nMcCulloch\u2013Pitts model in theoretical neuroscience and logistic regression in statistics. When ap-\nplying this to spike train data, we regard the state si(t) = 1 (-1) as the \ufb01ring (non-\ufb01ring) state.\nThe inference framework we adopt here is the standard maximum-likelihood (ML) framework. We\nrepeat R experiments and denote a \ufb01ring pattern fs\nir(t)gN\ni=1 for t = 1; 2;(cid:1)(cid:1)(cid:1) ; M in an experiment\n(cid:3)\nr(= 1; 2;(cid:1)(cid:1)(cid:1) ; R). The ML framework requires us to solve the following maximization problem on\nthe variable set fJij; (cid:18)i(t)g:\n\nN\n\n}\n\nr=1\n\nt=1\n\n:\n\n1\nR\n\nf ^Jij; ^(cid:18)i(t)g = arg max\nfJij ;(cid:18)i(t)g\n\nr(t + 1)j s\nr(t);fJij; (cid:18)i(t)g)\n(cid:3)\n(cid:3)\nlog P ( s\n\n(2)\nThis cost function is concave with respect to fJij; (cid:18)i(t)g, and hence, a number of ef\ufb01cient solvers\nare available [32]. However, we do not directly maximize eq. (2) in this study but instead we\nemploy the MF formula proposed previously [25,26]. The MF formula is reasonable in terms of\nthe computational cost and suf\ufb01ciently accurate when the dataset size R is large. Moreover, the\navailability of an analytic formula enables us to construct an effective approximation to reduce the\ncomputational cost in the post-processing step, as shown in Sec. 2.3.\nUnfortunately, in many experimental settings, it is not easy to conduct a suf\ufb01cient number of inde-\npendent experiments [33,34], as in the case of Sec. 4. Hence, below we assume the stationarity of\nany statistics, and ignore the time dependence of (cid:18)(t). This allows us to identify the average over\ntime as the ensemble average, which signi\ufb01cantly improves statistics. We admit this assumption is\nnot always valid, particularly in the case where time-dependent external forces are present, although\nwe treat such cases in Sec. 3.2 and Sec. 4.2. Despite this limitation, we still stress that the present\napproach can extract synaptic connections among neurons accurately, although the existence of the\n\n2\n\n\ftime-dependent inputs may decrease its performance. Possible directions to overcome this limitation\nare discussed in Sec. 5.\n\n2.1 Pre-processing: Discretization of time and binarization of state\n\nIn the pre-processing step, we have to decide the duration of the interval that should be used to\ntransform the real time to the unit time \u2206(cid:28) in the Ising scheme. We term \u2206(cid:28) the bin size. Once the\nbin size is determined, the whole real time interval [0;T ] is divided into the set of time bins that are\nlabelled as ftgM =T =\u2206(cid:28)\n. Given this set of the time bins, we binarize the neuron states: if there is no\ni (t) = (cid:0)1; otherwise s\n(cid:3)\n(cid:3)\nspike train of the neuron i in the time bin with a label t, then s\ni (t) = 1. This\nis the whole pre-processing step we adopt, and is a commonly used approach [7].\nDetermination of the bin size \u2206(cid:28) can be a crucial issue: different values of \u2206(cid:28) may lead to dif-\nferent results. To determine it in an objective way, we employ an information-theory-based method\nproposed previously [30]. Following this method, we determine the bin size as\n\nt=1\n\n\u2206(cid:28)opt = arg max\n\n\u2206(cid:28)\n\n^I\u2206(cid:28) (si(t + 1); sj(t))\n\n(3)\n\nwhere I\u2206(cid:28) (si(t + 1); sj(t)) denotes the mutual information between si(t + 1) and sj(t) in the\ncoarse-grained series with \u2206(cid:28), and ^I\u2206(cid:28) (si(t + 1); sj(t)) is its plug-in estimator. The explicit for-\nmula is\n\n^I\u2206(cid:28) (si(t + 1); sj(t)) =\n\nr(cid:11)(cid:12)(i; t + 1; j; t) log\n\nr(cid:11)(cid:12)(i; t + 1; j; t)\nr(cid:11)(i; t + 1)r(cid:12)(j; t)\n\n;\n\n(4)\n\n((cid:11);(cid:12))2f+;(cid:0)g2\n\nwhere r++(i; t + 1; j; t) denotes the realized ratio of the pattern (si(t + 1); sj(t)) = (+1; +1),\nr++(i; t + 1; j; t) (cid:17) (1=(M (cid:0) 1))#f(si(t + 1); sj(t)) = (+1; +1)g, and the other double-subscript\nquantities fr+(cid:0); r(cid:0)+; r(cid:0)(cid:0)g are de\ufb01ned similarly. Single-subscript quantities are also the realized\nratios of the corresponding state, for example, r+(j; t) (cid:17) (1=M )#fsj(t) = +1g.\nThe meaning of eq. (3) is clear: the formula inside the brace brackets of the right-hand side, hereafter\ntermed gross mutual information, is merely the likelihood of a (null) hypothesis that si(t + 1) and\nsj(t) are \ufb01ring without any correlation. The optimal value \u2206(cid:28)opt is chosen to reject this hypothesis\nmost strongly. This can also be regarded as a generalization of the chi-square test.\n\n2.2\n\nInference algorithm: The MF formula\n\n( T\n\n8<:\n\n\u2206(cid:28)\n\n)\u2211\n\ni\u0338=j\n\n(cid:0) 1\n\n\u2211\n\n9=; ;\n\nThe previously derived MF formula [25,26] is given by\n(cid:0)1DC\n(cid:0)1;\n\n^J MF = A\n\nwhere\n\n8><>: (cid:22)i(t) = \u27e8si(t)\u27e9 ;\n(\n\n1 (cid:0) (cid:22)2\n\n)\n\n(cid:14)ij;\n\ni (t)\n\nAij(t) =\nCij(t) = \u27e8si(t)sj(t)\u27e9 (cid:0) (cid:22)i(t)(cid:22)j(t);\nDij(t) = \u27e8si(t + 1)sj(t)\u27e9 (cid:0) (cid:22)i(t + 1)(cid:22)j(t):\n\u2211\n\n^(cid:18)MF\ni\n\n(t) = tanh\n\n(cid:0)1 ((cid:22)i(t + 1)) (cid:0)\n\n^J MF\nij (cid:22)j(t);\n\nj\n\nNote that the estimate ^J MF seemingly depends on time, but it is known that the time dependence is\nvery weak and ignorable. Once given ^J MF, the MF estimate of the external \ufb01eld is given as\n\nalthough we focus on the couplings between neurons and do not estimate the external force in this\nstudy. The literal meaning of the brackets is the ensemble average corresponding to (1=R)\nr=1 in\neq. (2), but here we identify it as the average over time. Here, we use the time-averaged statistics of\nf(cid:22); C; D; (cid:18)g, as declared above.\n\nR\n\n(5)\n\n(6)\n\n(7)\n\n\u2211\n\n3\n\n\f2.3 Post-processing: Screening relevant couplings and its fast approximation\n\nThe basic idea of our screening method is to compare the coupling estimated from the original\ndata with the one estimated from randomized data in which the time series of \ufb01ring patterns of\neach neuron is randomly independently permuted. We do not explain the detailed procedures here\nbecause similar methods have been described previously [7,30]. Instead, here we state the essential\npoint of the method and derive an approximate formula implementing the screening method in a\ncomputationally ef\ufb01cient manner.\nThe key of the method is to compute the probability distribution of ^Jij, P ( ^Jij), when applying our\ninference algorithm to the randomized data. Once we obtain the probability distribution, we can\njudge how unlikely our original estimate is as compared to the estimates from the randomized data.\nIf the original estimate is suf\ufb01ciently unlikely, we accept it as a relevant coupling; otherwise, we\nreject it.\nEvaluation of the above probability distribution is not easy in general, and hence, it is common to\nhave recourse to numerical sampling, which can be a computational burden. Here, we avoid this\nproblem by computing it in an analytical manner under a reasonable approximation.\nFor the randomized data, we may assume that two neurons si and sj \ufb01re independently with \ufb01xed\nmeans (cid:22)i and (cid:22)j, respectively. Under this assumption, by the central limit theorem, each diagonal\ncomponent of C converges to Cii = 1 (cid:0) (cid:22)2\ni = Aii, while its non-diagonal component becomes a\nzero-mean Gaussian variable whose variance is proportional to 1=(M (cid:0) 1), and is thus, small. All\nthe components of D behave similarly to the non-diagonal ones of C. This consideration leads to\nthe expression\n\n(cid:0)1)jj =\n\n1\n\n(1 (cid:0) (cid:22)2\n\ni )(1 (cid:0) (cid:22)2\nj )\n\n(8)\nDij:\ni )(1 (cid:0) (cid:22)2\nj )=(M (cid:0) 1).\n\n^J ran\nij =\n\n(A\n\n\u2211\n\nk\n\n(\n\n(cid:0)1)iiDik(C\n(\n\n(cid:0)1)kj (cid:25) (A\n)\n\nij\n\nj (cid:21) (cid:8)th\n)\n\nj ^J ran\n\nij\n\nj (cid:21) (cid:8)th\n\nP\n\n(cid:25) 1 (cid:0) erf\n\n(cid:0)1)iiDij(A\n\u221a\n\n0@(cid:8)th\n\u222b\n\nBy the independence between si and sj, the variance of Dij becomes (1 (cid:0) (cid:22)2\nHence the probability P\n\nis obtained as\n\nj ^J ran\n\n(1 (cid:0) (cid:22)2\n\ni )(1 (cid:0) (cid:22)2\n\nj )(M (cid:0) 1)\n\nwhere erf(x) is the error function de\ufb01ned as\nerf(x) (cid:17) 2p\n(cid:25)\n\u221a\n\nInserting the absolute value of the original estimate of ^Jij in (cid:8)th, we obtain its likelihood, and can\njudge whether it should be accepted. Below, we set the signi\ufb01cance level pth associated with ((cid:8)th)ij\nas\n\n2\n\n(cid:0)y2\n\n:\n\ndy e\n\nx\n\n0\n\n(9)\n\n(10)\n\n1A ;\n\n((cid:8)th)ij =\n\n2\n\n(1 (cid:0) (cid:22)2\n\ni )(1 (cid:0) (cid:22)2\n\nj )(M (cid:0) 1)\n\nand accept only ^Jij such that j ^Jijj > ((cid:8)th)ij.\n\n(cid:0)1 (1 (cid:0) pth)\n\nerf\n\n(11)\n\n3 Hodgkin\u2013Huxley networks\n\nWe \ufb01rst evaluate the accuracy of our methods using synthetic systems consisting of the Hodgkin\u2013\nHuxley neurons. The dynamics of the neurons are given by\n\nC\n\ndVi\nd(cid:28)\ndni\nd(cid:28)\ndmi\nd(cid:28)\ndhi\nd(cid:28)\n\ni hi (Vi (cid:0) ENa) (cid:0) (cid:22)gL (Vi (cid:0) EL) + Iex\ni ;\n\n= (cid:0)(cid:22)gKn4\ni (Vi (cid:0) EK) (cid:0) (cid:22)gNam3\n= (cid:11)n (Vi) (1 (cid:0) ni) (cid:0) (cid:12)n (Vi) ni;\n= (cid:11)m (Vi) (1 (cid:0) mi) (cid:0) (cid:12)m (Vi) mi;\n= (cid:11)h (Vi) (1 (cid:0) hi) (cid:0) (cid:12)h (Vi) hi;\n\n(12)\n\n(13)\n\n(14)\n\n(15)\n\n4\n\n\fwhere Vi is the membrane potential of ith neuron, ni is the activation variable that represents the\nratio of the open channels for K+ ion, and mi and hi are the activation and inactivation variables\nfor Na+ ion, respectively. All parameters, except the external input term Iex\ni , are set as described in\n[35]. The input forces are given by\n\nIex\ni = ci((cid:28) ) +\n\nKijVj(cid:2) (Vj (cid:0) Vth) + a\n\n;\n\n(16)\n\nN\u2211\n\nj=1\n\n)\n\n\u2211\n\n(\n\n(cid:28) (cid:0) (cid:28) k\n\ni\n\n(cid:14)\n\nk\n\nwhere ci(t) represents the environmental noise with a Poisson process, the second term represents\nthe couplings with the threshold voltage Vth = 30 mV and the Heaviside step function (cid:2)((cid:1)), and\nthe last term denotes the impulse stimulations with the delta function. Here, we consider no-delay\nsimple couplings, which we term the synaptic connections, and aim to reconstruct their structure\nwith the excitatory/inhibitory signs using our methods. We use N = 100 neuron networks, where\nthe 90 neurons are excitatory and have positive outgoing couplings while the others are inhibitory.\nThe rate and strength of the Poisson process are set as (cid:21) = 180 Hz and b = 2 mV, respectively,\nfor all neurons. We generate their time series, integrating (12)-(15) by the Euler method with d(cid:28) =\n0:01 ms, where we suppose a neuron is \ufb01ring when its voltage exceeds Vth, and use the spike train\ndata with the whole period T = 106 ms for our inference.\n\n3.1 Spontaneous activity case\n\n\u221a\n\n(1 (cid:0) (cid:22)2\n\ni )(1 (cid:0) (cid:22)2\n\nAt \ufb01rst, we consider a system on a chain network in which each neuron has three synaptic connec-\ntions to adjoint neurons in one direction. The connection strength Kij is drawn from the uniform\ndistributions in [0:015; 0:03] for the excitatory and in [(cid:0)0:06;(cid:0)0:03] for the inhibitory neurons, re-\nspectively. Here, we set a = 0 mV to study the spontaneous activity. An example of the spike\ntrains generated during 3 seconds is shown in Fig. 1 (a), where the spike times and correspond-\ning neuronal indices are plotted. Subsequently, using the whole spike train data, we calculate the\ngross mutual information for different \u2206(cid:28), and the result is indicated by the red curve in Fig. 1\n(b). The curve has the unimodal feature, which implies the existence of the optimal time bin size of\napproximately \u2206(cid:28) = 3 ms, although the original system does not have the delay. We suppose that\ninputs must accumulate suf\ufb01ciently to generate a spike, which costs some time scale, and this is a\npossible reason for the emergence of the nontrivial time-scale. To validate our approximation (8),\nwe randomize the coarse-grained series with \u2206(cid:28) = 3 ms in the time direction independently, rescale\nj )(M (cid:0) 1), and compare the results of 1000 randomized data\n^J ran\nij by multiplying\nwith the standard Gauss distribution in Fig. 1 (c), which shows their good correspondence. Using\n\u2206(cid:28) = 3 ms to make the spike trains coarse-grained, we apply the inverse formula to the series and\n(cid:0)3, which leads to the estimated coupling matrix shown in\nscreen relevant couplings with pth = 10\nFig. 1 (e), while the one used to generate the data is shown in Fig. 1 (d). The asymmetric network\nstructure is recovered suf\ufb01ciently with the discrimination of the signs of the couplings. The con-\nditional ratios of the correctness are shown in Fig. 1 (f), where the inference results obtained with\ndifferent values of \u2206(cid:28) are also shown. This demonstrates the fairly accurate reconstruction result\nobtained using our inference procedure. We also show the receiver operating characteristic (ROC)\ncurves obtained by gradually changing the value pth in Fig. 1 (g), with the different values of \u2206(cid:28).\nWe conclude that using non-optimal time bins drastically decreases the accuracy of the inference\nresults.\nTo illustrate the robustness of the optimality of the time bin, in Fig. 1 (i) we plot the means and\nstandard deviations of the gross mutual information through the 10 different simulations, showing\nthat the variance is small enough and the result is well robust.\nTo consider a more general situation, we also employ a Hodgkin\u2013Huxley system on a random net-\nwork. The directional synaptic connection between every pair of neurons is generated with the\nprobability 0:1, and the excitatory and inhibitory couplings are drawn from the uniform distribu-\ntions within [0:01; 0:02] and [(cid:0)0:04;(cid:0)0:02], respectively. The corresponding inference results for\nits spontaneous activity are shown by green curves in Figs. 1 (b) and (f). The ROC curves for the\nthree different three values of \u2206(cid:28) are also shown in (h). We con\ufb01rm that the inference is suf\ufb01ciently\neffective in the random-network system as well as in the chain system.\n\n5\n\n\fFigure 1: Application of the proposed approach to the Hodgkin\u2013Huxley models. (a) Spontaneous\nspike trains during 3 seconds. (b) Gross mutual information v.s. time bin size \u2206(cid:28). The red curve\nshows the chain network while the green curve shows the random network. (c) Histogram of rescaled\n^J ran\nij obtained by randomizing the original series, and the standard Gauss distribution. (d) An exam-\nple of the chain networks that we used, where the red and blue elements indicate the excitatory and\ninhibitory couplings, respectively. (e) Corresponding inferred coupling network with \u2206(cid:28) = 3 ms.\n(f) Conditional correctness ratios for the existence, absence, excitatory coupling, and inhibitory cou-\npling, where the standard deviations of 10 different simulations are shown with the error bars. (g,h)\nReceiver operating characteristic curves for different coarse-grained series in the systems (g) on the\nchain and (h) on the random network, where the error bars indicate the standard deviations of 10\n(cid:0)3 used in (e) and (f). (i) The mean\ndifferent simulations. The marked points indicate pth = 10\nand standard deviation of the gross mutual information for 10 independent simulations of the chain\nsystems. The result is shown to be robust.\n\n3.2 Evoked activity case\n\nWe next investigate performance in systems where responses are evoked by impulse stimuli. The\nmodel parameters, except for a, are the same as those in the chain model in Sec. 3.1. The strength of\nthe external force is set as a = 5:3 mV, and the stimulations are injected to all neurons with interval\n1 s. In Fig. 2 (a) we show the spike trains, where we observe that most of the neurons \ufb01re at the\ninjection times (cid:28) = 0:5; 1:5; 2:5 s. The gross mutual information against \u2206(cid:28) is shown in Fig. 2 (b).\nAlthough the curve feature is modi\ufb01ed due to the existence of the impulse inputs, we observe that\nits peak is located at a similar value of \u2206(cid:28). Therefore, we use the same value \u2206(cid:28) = 3 ms. Applying\n(cid:0)3, we obtain the inferred couplings which\nour inference procedure with \u2206(cid:28) = 3 ms and pth = 10\nare shown in Fig. 2 (c), where the original network is in Fig. 1 (d). On comparing Fig. 2 (c) with\nFig. 1 (e), while the inference detects the existence of the synaptic connections, we observe more\nfalse couplings in the evoked case. The conditional ratios in Fig. 2 (d) indicate that the existence\nof the external inputs may increase the false positive rate with the same pth. The ROC curves are\nshown in Fig. 2 (f).\n\n6\n\n(a)(b)(c)(d)(e)(f)0100020003000102030405060708090100[ms]1011001021030[ms]24681014122040608010020406080100204060801002040608010010.50ExistenceAbsenceExcitatoryInhibitory\u0394\u03c4 = 3 ms (chain)\u0394\u03c4 = 1 ms (chain)\u0394\u03c4 = 10 ms (chain)\u0394\u03c4 = 3 ms (random network)chainrandom network420-2-400.20.40.10.3StandardGaussdistributionRescaled(g)\u0394\u03c4 = 3 ms \u0394\u03c4 = 1 ms \u0394\u03c4 = 10 ms 00110.50.5(h)00110.50.5\u0394\u03c4 = 3 ms \u0394\u03c4 = 1 ms \u0394\u03c4 = 10 ms timeneuron\u0394\u03c4jijiFalse Positive RatioTrue Positive RatioFalse Positive RatioTrue Positive Ratio(i)101100102103[ms]\u0394\u03c402468101412\fFigure 2: Application of the proposed approach to the evoked activity in the Hodgkin\u2013Huxley\nmodels. (a) Evoked spike trains during 3 seconds, where the red line expresses the injection times\nof the stimuli. (b) Gross mutual information v.s. time bin size. (c) Inferred coupling matrix with\nthe red excitatory and blue inhibitory elements using \u2206(cid:28) = 3 ms, where the generative network is\nthe one shown in Fig. 1 (b). (d) Conditional correctness ratios. (e) Receiver operating characteristic\n(cid:0)3 are marked. (f)\ncurves for different coarse-grained series, where the points denoting pth = 10\nThe mean and standard deviation of the gross mutual information for 10 independent simulations.\nThe result is shown to be robust.\n\n4 Cultured neuronal networks\n\nWe apply our inference methods to the real neuronal systems introduced in a previous study [31],\nwhere rat cortical neurons were cultured in micro wells. The wells had a circular structure, and con-\nsequently the synapses of the neurons were likely to form a physically asymmetric chain network,\nwhich is similar to the situation in the Hodgkin\u2013Huxley models we used in Sec. 3. The activity of\nthe neurons was recorded by the multi-electrode array with 40 (cid:22)s time resolution, and the Ef\ufb01cient\nTechnology of Spike sorting method [36] was used to identify the spike events of individual neurons.\nWe study the spontaneous and evoked activities here.\n\n4.1 Spontaneous activity case\n\nWe \ufb01rst use the spontaneous activity data recorded during 120 s. The spike sorting identi\ufb01ed 100\nneurons which generated the spikes. The spike raster plot during 3 seconds is displayed in Fig.\n3 (a). We calculate the gross mutual information as in case of the Hodgkin\u2013Huxley models, and\nthe obtained optimal bin size is approximately \u2206(cid:28) = 5 ms. We also con\ufb01rm that the inferred\ncouplings are similar to the results described previously [30], and this supports the validity of our\nnovel approximation method introduced in Sec. 2.3. We show the inferred network in Figs. 3 (b-\n(cid:0)9, where we locate the nodes denoting the neurons\nd) with different values pth = 10\non a circle following the experimental design [31]. A more strict threshold provides us with clear\ndemonstration of the relevant couplings here.\n\n(cid:0)6; 10\n\n(cid:0)3; 10\n\n4.2 Evoked activity case\n\nWe next study an evoked neuronal system, where an electrical pulse stimulation is injected from\nan electrode after every 3 seconds, and the other experimental settings are similar to those of the\nspontaneous case. In this case the activity of 149 neurons were identi\ufb01ed by the spike sorting. The\nexample of the spike trains is shown in Fig. 4 (a). The gross mutual information is shown in Fig. 4\n(cid:0)6,\n(b), where we can see the peak around \u2206(cid:28) = 10 ms. Setting \u2206(cid:28) = 10 ms and pth = 10\nwe obtain the estimated coupling matrices in Figs. 4 (c,d). In these cases, we can also observe\nthe bold diagonal elements representing the asymmetric chain structure, although with the lower\n\n(cid:0)3; 10\n\n7\n\n(a)(b)(c)(d)(e)0100020003000102030405060708090100[ms]1011001021030[ms]5102015204060801002040608010010.50ExistenceAbsenceExcitatoryInhibitory25\u0394\u03c4 = 3 ms\u0394\u03c4 = 1 ms\u0394\u03c4 = 10 ms00110.50.5False Positive RatioTrue Positive Ratiotimeneuron\u0394\u03c4ji\u0394\u03c4 = 3 ms\u0394\u03c4 = 1 ms\u0394\u03c4 = 10 ms1011001021030[ms]510201525\u0394\u03c4(f)\fFigure 3: Application of the proposed approach to a cultured-neuronal system. (a) Spike trains\nduring 3 seconds. (b-d) Inferred networks, where the nodes are located on the circle corresponding to\n(cid:0)9.\nthe experimental design. The different signi\ufb01cant levels used are: (b) 10\nThe red and blue directional arrows represent the excitatory and inhibitory couplings, respectively.\n(e) The gross mutual information for the 1st and 2nd halves of the data. The \ufb01gure shows the\nrobustness of the result.\n\n(cid:0)6, and (d) 10\n\n(cid:0)3, (c) 10\n\nsigni\ufb01cant level some far-diagonal elements emerge due to the existence of the external inputs,\nwhich is a situation similar to that in the Hodgkin\u2013Huxley simulation in Sec. 3.2. The inferred\n(cid:0)9 is displayed in Fig. 4 (e), where some long-range\nnetwork with the strict threshold pth = 10\ncouplings are still estimated while physical connections corresponding to them do not exist because\nof the experimental design.\n\n5 Conclusion and discussion\n\nWe propose a systematic inference procedure for extracting couplings from point-process data. The\ncontribution of this study is three-fold: (i) invention of an analytic formula to screen relevant cou-\nplings in a computationally ef\ufb01cient manner; (ii) examination in the Hodgkin\u2013Huxley model, with\nand without impulse stimuli; (iii) examination in an evoked cultured neuronal network.\nThe applications to the synthetic data, with and without the impulse stimuli, demonstrate the fairly\naccurate reconstructions of synaptic connections by our inference methods. The application to the\nreal data of the spontaneous activity in the cultured neuronal system also highlights the effectiveness\nof the proposed methods in detecting the synaptic connections.\nFrom the comparison between the analyses of the spontaneous and evoked activities, we found that\nthe inference accuracy becomes degraded by the external stimuli. One of the potential origins is\nthe breaking of our stationary assumption of the statistics f(cid:22); C; Dg because of the time-varying\nexternal force (cid:18). To overcome this, certain techniques resolving the insuf\ufb01ciency of samples, such\nas regularization, will be helpful. A promising approach might be the introduction of an \u21131 regular-\nization into eq. (2), which enables us to automatically screen out irrelevant couplings. Comparing it\nwith the present approach based on computational statistics will be an interesting future work.\n\nAcknowledgments\n\nThis work was supported by MEXT KAKENHI Grant Numbers 17H00764 (YT, TO, and YK) and\n18K11463 (TO), and RIKEN Center for Brain Science (YT and TI).\n\n8\n\n(b)(c)(d)(d)(a)20406080100100020003000[ms]timeneuron1st half2nd half10110010210-1[ms]\u0394\u03c401234\fFigure 4: Application of the proposed approach to an evoked cultured-neuronal system. 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