{"title": "Reconstructing Stimulus-Driven Neural Networks from Spike Times", "book": "Advances in Neural Information Processing Systems", "page_first": 325, "page_last": 332, "abstract": null, "full_text": "Reconstructing Stimulus-Driven Neural\n\nNetworks from Spike Times\n\nDuane Q. Nykamp\n\nUCLA Mathematics Department\n\nLos Angeles, CA 90095\n\nnykamp@math.ucla.edu\n\nAbstract\n\nWe present a method to distinguish direct connections between two neu-\nrons from common input originating from other, unmeasured neurons.\nThe distinction is computed from the spike times of the two neurons in\nresponse to a white noise stimulus. Although the method is based on a\nhighly idealized linear-nonlinear approximation of neural response, we\ndemonstrate via simulation that the approach can work with a more re-\nalistic, integrate-and-\ufb01re neuron model. We propose that the approach\nexempli\ufb01ed by this analysis may yield viable tools for reconstructing\nstimulus-driven neural networks from data gathered in neurophysiology\nexperiments.\n\n1 Introduction\n\nThe pattern of connectivity between neurons in the brain is fundamental to understanding\nthe function the brain\u2019s neural networks. Related properties of closely connected neurons,\nfor example, may lead to inferences on how the observed properties are built or enhanced by\nthe neural connections. Unfortunately, the complexity of higher organisms makes obtaining\ncombined functional and connectivity data extraordinarily dif\ufb01cult.\n\nThe most common tool for recording in vivo the activity of neurons in higher organisms\nis the extracellular electrode. Typically, one uses this electrode to record only the times of\noutput spikes, or action potentials, of neurons. In such an experiment, the states of the mea-\nsured neurons remain hidden. The ability to infer connectivity patterns from spike times\nalone would greatly expand the attainable connectivity data and provide the opportunity to\nbetter address the link between function and connectivity.\n\nAttempts to infer connectivity from spike time data have focused on second-order statistics\nof the spike times of two simultaneously recorded neurons. In particular, the joint peris-\ntimulus time histogram (JPSTH) and its integral, the shuf\ufb02e-corrected correlogram [1, 2, 3]\nhave become widely used tools to analyze such data.\n\nHowever, the JPSTH and correlogram cannot distinguish correlations induced by connec-\ntions between the two measured neurons (direct connection correlations) from correlations\ninduced by common connections from a third, unmeasured neuron (common input correla-\ntions). Inferences from the JPSTH or correlogram about the connections between the two\nmeasured neurons are ambiguous.\n\n\fAnalysis tools such as partial coherence [4] can distinguish between a direct connection\nand common input when one can also measure neurons inducing the common input effects.\nThe distinction of present approach is that all other neurons are unmeasured.\n\nWe demonstrate that, by characterizing how each neuron responds to the stimulus, one\nmay be able to distinguish between direct connection and common input correlations. In\nthat case, one could determine if a connection existed between two neurons simply by\nmeasuring their spike times in response to a stimulus. Since the properties of the neurons\nwould be determined by the same measurements, such an analysis would be the basis for\ninferring links between connectivity and function.\n\n2 The model\n\nTo make the subtle distinction between direct connection correlations and common input\ncorrelations, one needs to exploit an explicit model. The model must be suf\ufb01ciently simple\nso that all necessary model parameters can be determined from experimental measure-\nments. For this reason, the analysis is limited to phenomenological lumped models. We\npresent analysis based on a linear-nonlinear model of neural response to white noise.\n\nLet the stimulus X be a vector of independent Gaussian random variables with zero mean\nand standard deviation (cid:27) = 1. X is a discrete approximation to temporal or spatio-temporal\nwhite noise. Let Ri\np = 1 if neuron p spiked at the discrete time point i and be zero oth-\nerwise. Let the probability of a spike from a neuron be a linear-nonlinear function of the\nstimulus and the previous spike times of the other neurons,\n\n(cid:22)W j\n\nq (cid:17);\nqpri(cid:0)j\n\n(1)\n\np (cid:1) x +Xq6=pXj>0\n\np = 1(cid:12)(cid:12)X = x; Rq = rq;8q(cid:1) = gp(cid:16)hi\n\nPr(cid:0)Ri\np is the linear kernel of neuron p shifted i units in time (normalized so that\npk = 1), gp((cid:1)) is its output nonlinearity (representing, for example, its spike generat-\nqp is a connectivity term representing how a spike of neuron q at a\n\nwhere hi\nkhi\ning mechanism), and (cid:22)W j\nparticular time modi\ufb01es the response of neuron p after j time steps.\nThe network of Eq. (1) is an extension of the standard linear-nonlinear model of a single\nneuron. The linear-nonlinear model of a single neuron can be completely reconstructed\nfrom measured spike times in response to white noise [5]. We will demonstrate that the\nnetwork of linear-nonlinear neurons can be similarly analyzed to determine the connectivity\nbetween two measured neurons.\n\n3 Analysis of model\n\nLet neurons 1 and 2 be the only two measured neurons. The spike times of all other neurons\nwill remain unmeasured. Given further simplifying assumptions detailed below, we can\nisolate the connectivity terms between neurons 1 and 2 ( (cid:22)W j\n21). We will outline a\nmethod to determined these connectivity terms from a few statistics of the two measured\nspikes trains and the white noise stimulus.\n\n12 and (cid:22)W j\n\n3.1 Assumptions\n\nqp. We will neglect all quadratic and higher order terms in the (cid:22)W j\n\nThe \ufb01rst assumption is that the coupling is suf\ufb01ciently weak to justify a \ufb01rst order approx-\nimation in the (cid:22)W j\nqp with\none important exception. Common input correlations are second order in the (cid:22)W j\nqp because\ncommon input requires two connections. Since our analysis must include common input to\nthe measured neurons, we retain terms containing (cid:22)W j\np1\n\nq2 with p; q > 2.\n\n(cid:22)W k\n\n\fThe second assumption is that the unmeasured neurons do not respond to essentially identi-\ncal stimulus features as the measured neurons (1 & 2) or each other. We quantify similarity\nto stimulus features by the inner product between linear kernels, cos (cid:22)(cid:18)k\nq. We\nrequire each cos (cid:22)(cid:18) to be small so that we can ignore terms of the form (cid:22)W cos (cid:22)(cid:18). We al-\nlow one exception and retain (cid:22)W cos (cid:22)(cid:18)k\n21 terms so that no assumption is made on the two\nmeasured neurons.\n\npq = hi(cid:0)k\n\n(cid:1) hi\n\np\n\nLast, we assume the nonlinearity is an error function of the form\n\ngp(x) =\n\n1\n\n2h1 + erf(cid:16) x (cid:0) (cid:22)Tp\n(cid:15)pp2 (cid:17)i\n0 e(cid:0)t2\n\ndt.\n\nwith parameters (cid:22)Tp and (cid:15)p, where erf(y) = 2p(cid:25) R y\n\n(2)\n\n3.2 Outline of method\n\n1g and EfRi\n\nThe \ufb01rst step in analyzing the network response is to ignore the fact that the neurons are\nembedded in a neural network and analyze the spike trains of neurons 1 and 2 as though\neach were an isolated linear-nonlinear system. Using the procedure outlined in Ref. [5],\none can determine the effective linear-nonlinear parameters from the average \ufb01ring rates\n(EfRi\nThese effective linear-nonlinear parameters clearly will not match the parameters for neu-\nrons 1 and 2 in the complete system (Eq. (1)). The network connections alter the mean rates\nand stimulus-spike correlations of neurons 1 and 2, changing the linear-nonlinear parame-\nters reconstructed from these measurements. Nonetheless, these effective linear-nonlinear\nsystem parameters can be written approximately as combinations of parameters from the\nnetwork in Eq. (1).\n\n2g)1 and the stimulus-spike correlations (EfXRi\n\n1g and EfXRi\n\n2g).\n\n1Ri(cid:0)k\n\nThe connectivity between neurons 1 and 2 can then be determined from the correlation\ng measured at different positive and negative delays k and\nbetween their spikes (EfRi\ng) as follows. Given our\nthe correlation of their spike pairs with the stimulus (EfXRi\n21, and (cid:22)W j\nassumptions, we obtain equations linear in (cid:22)W j\nq2. For each delay k, we\np1\n1Ri(cid:0)k\n1Ri(cid:0)k\nobtain three equations: one from EfRi\n2 g\ng. At \ufb01rst\nonto EfXRi\nglance, it appears that the unknowns greatly outnumber the equations.\n\n12, (cid:22)W j\ng, one from the projection of EfXRi\n\n1g, and one from the projection of EfXRi\n\ng onto EfXRi(cid:0)k\n\n1Ri(cid:0)k\n(cid:22)W ~|\n\n1Ri(cid:0)k\n\n2\n\n2\n\n2\n\n2\n\n2\n\nHowever, the system of equations is well-posed because the (cid:22)W j\nq2 appear in the same\np1\ncombination for each of the three equations at a given delay. In fact, we have only two sets\nof unknowns, which can be written as\n\n(cid:22)W ~|\n\n(cid:22)W k = (cid:26) (cid:22)W (cid:0)k\n\n(cid:22)W k\n\n12\n\n21\n\nfor k < 0;\nfor k > 0;\n\nand\n\n(cid:22)U k = Xp>2Xj;~|\n\nckj ~|\np\n\n(cid:22)W j\np1\n\n(cid:22)W ~|\np2:\n\n(3)\n\n(4)\n\nAll other parameters in the equations were already determined in the \ufb01rst stage. If N is the\nnumber of delays considered, then we have 3N linear equations and only 2N unknowns.\nThe factor (cid:22)W k is the direct connection between neurons 1 and 2 (the direction of the con-\nnection depends on the sign of the delay k). The factor (cid:22)U k is the common input to neuron 2\nand neuron 1 (k times steps delayed) from all other neurons in the network. The weighting\n\n1Ef(cid:1)g denotes expected value.\n\n\fp ) of its terms depends on the properties of the unmeasured neurons. Fortunately, since\n\n(ckj ~|\nwe can treat (cid:22)U k as a unit, we don\u2019t need to determine the weighting.\nTo analyze spike train data, we approximate the statistics EfRi\n2g, EfXRi\n1g,\ng by averages over an experiment. We then\nEfXRi\ncompute the least-squares \ufb01t to solve for approximations of (cid:22)W and (cid:22)U. We denote these\napproximations (or correlation measures) as W and U, respectively.\n4 Demonstration\n\ng, and EfXRi\n\n1g, EfRi\n\n2g, EfRi\n\n1Ri(cid:0)k\n\n1Ri(cid:0)k\n\n2\n\n2\n\nWe demonstrate the ability of the measures W and U to distinguish direct connection cor-\nrelations from common input correlations with three example simulations. In the \ufb01rst two\nexamples, we simulated a network of three coupled linear-nonlinear neurons (Eqs. (1) and\n(2)). In the third example, we simulated a pair of integrate-and-\ufb01re neurons driven by the\nstimulus in a manner similar to the linear-nonlinear neurons. In each example, we measured\nonly the spike times of neuron 1 and neuron 2.\n\nSince the white noise stimulus does not repeat, one cannot calculate a JPSTH or shuf\ufb02e-\ncorrected correlogram. Instead, for comparison we calculated the covariance between the\ni, and a stimulus independent correlation mea-\nspike times, Ck = hRi\ni (cid:0) hRi\n21, where hi represents averaging over the\nsure introduced in Ref. [6], S k = hRi\ni (cid:0) (cid:23)k\nentire stimulus. The quantity (cid:23) k\ni if neurons 1 and 2\nwere independent linear-nonlinear systems responding to the same stimulus.\n\n21 is the expected value of hRi\n\n1ihRi(cid:0)k\n1Ri(cid:0)k\n\n1Ri(cid:0)k\n\n1Ri(cid:0)k\n\n2\n\n2\n\n2\n\n2\n\nWe used spatio-temporal linear kernels of the form\n\n(cid:28)h e(cid:0) jjj2\n\np\n\nhp(j; t) = te(cid:0) t\n\nq.\n(cid:1) hi\n\n21 = (cid:22)W 6\n\npq = hi(cid:0)k\n\n40 sin((j1 cos (cid:30)p + j2 sin (cid:30)p)fp + kp)\n\n(5)\nfor t > 0 (hp = 0 otherwise), where j = (j1; j2) denotes a discrete space point. For the\nlinear-nonlinear simulations, we sampled this function on a 20(cid:2) 20(cid:2) 20 grid in space and\ntime, normalizing the resulting vector to obtain the unit vector hi\np. The kernels were chosen\nto be caricatures of receptive \ufb01elds of simple cells in visual cortex. The only geometry of\nthe kernels that appears in the equations is their inner products cos (cid:22)(cid:18)k\nFor the \ufb01rst example, we simulated a network of three linear-nonlinear neurons. Neuron\n2 had an excitatory connection onto neuron 1 with a delay of 5\u20136 units of time (a positive\ndelay for our sign convention): (cid:22)W 5\n21 = 0:6. Neuron 3 had one excitatory connec-\ntion onto neuron 1 and second excitatory connection onto neuron 2 that was delayed by\n6\u20138 units of time (a negative delay): (cid:22)W 1\n32 = 1:5. In this way, the\nspike times from neuron 1 and 2 had positive correlations due to both a direct connection\nand common input. Fig. 1 shows the results after simulating for 600,000 units of time,\nobtaining 16,000\u201322,000 spikes per neuron.\nThe covariance C has peaks at both positive and negative delays, corresponding to the direct\nconnection and common input, respectively, as well as a small peak around zero due to the\nshared stimulus (see Ref. [6]). The measure S eliminates the stimulus-induced correlation,\nbut still cannot distinguish the direct connection from the common input. The proposed\nmeasures W and U, however, do separate the two sources of correlation. W contains a\npeak only at the positive delay corresponding to the direct connection from neuron 2 to\nneuron 1; U contains a peak only at the negative delay corresponding to the common input\nfrom the (unmeasured) third neuron. This distinction was made at the cost of a dramatic\nincrease in the noise. On the order of 20,000 spikes were needed to get clean results even in\nthis idealized simulation, a long experiment given the typically low \ufb01ring rates in response\nto white noise stimuli.\n\n31 = (cid:22)W 2\n\n32 = (cid:22)W 9\n\n31 = (cid:22)W 8\n\nTheoretically, the method should handle inhibitory connections just as well as excitatory\n\n\fa\n\nx 10\u22123\n\n4\n\nC\n\n2\n\n0\n\u221230\n\nx 10\u22123\n\nb\n\nS\n\n3\n2\n1\n0\n\u221230\n\nc\n\nW\n\n1\n0.5\n0\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n\u221230\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n1d\n\n0.5\n\nU\n\n0\n\n\u221230\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\nFigure 1: Results from the spike times of two neurons in a simulation of three linear-\nnonlinear neurons. Delay is in units of time and is the spike time of neuron 1 minus the\nspike time of neuron 2. The correlations at a positive delay are due to a direct connection,\nwhile those a negative delay are due to common input. (a) The covariance C between the\nspike times of neuron 1 and neuron 2 re\ufb02ects both connections. The third peak around zero\n2, is induced by the common stimulus. (b)\ndelay, due to similarity in the kernels hi\nThe correlation measure S removes the correlation induced by the common stimulus, but\ncannot distinguish between the two connectivity induced correlations. (c\u2013d) The measures\nW and U do distinguish the connectivity induced correlations. W re\ufb02ects only the direct\nconnection (c); U re\ufb02ects only the common input (d). Parameters for g((cid:1)): (cid:22)T1 = 2:5,\n(cid:22)T2 = 3:0, (cid:22)T3 = 2:2, (cid:15)1 = 0:5, (cid:15)2 = 1:0, (cid:15)3 = 0:7. Parameters for h: (cid:28)h = 1, (cid:30)1 = 0,\n(cid:30)2 = (cid:25)=8, (cid:30)3 = (cid:25)=4, f1 = 0:5, f2 = 0:8, f3 = 1:0, k1 = 0, k2 = (cid:0)1, k3 = 1.\n\n1 and hi\n\nconnections. To test the inhibitory case, we modi\ufb01ed the connections so that neuron 1\n21 = (cid:22)W 6\nreceived an inhibitory connection from neuron 2 ( (cid:22)W 5\n21 = (cid:0)0:3), and neuron 1\nreceived an inhibitory connection from neuron 3 ( (cid:22)W 1\n31 = (cid:22)W 2\n31 = (cid:0)1:0). Neuron 2 con-\ntinued to receive an excitatory connection from neuron 3 ( (cid:22)W 8\n32 = (cid:22)W 9\n32 = 1:0). The low\n\ufb01ring rates of neurons, however, makes inhibition more dif\ufb01cult to detect via correlations\n[3]. Similarly, the measures W and U performed less well with inhibition. To demonstrate\nthat they could, at least theoretically, work for inhibition, we increased the \ufb01ring rates, used\n(cid:22)W s with smaller magnitudes, and increased the simulation length. Fig. 2 shows the results\nafter simulating for 1,200,000 units of time, obtaining 130,000\u2013140,000 spikes per neuron.\nWith this extraordinarily large number of spikes, W and U successfully distinguish the\ndirect connection correlations from the common input correlations.\n\nTo test the robustness of the method to deviations from the linear-nonlinear model, we\nsimulated a system of two integrate-and-\ufb01re neurons whose input was a threshold-linear\nfunction of the stimulus. The neurons received common input from a threshold-linear unit,\n\n\fa\n\nC\n\nx 10\u22123\n\n5\n\n0\n\n\u22125\n\u221230\n\nb\n\nx 10\u22123\n\n0\n\nS\n\n\u22122\n\n\u22124\n\u221230\n\nc\n\nW\n\n0\n\u22120.1\n\u22120.2\n\u22120.3\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n\u221230\n\n\u221220\n\n\u221210\n\nd\n\nU\n\n0\n\n\u22120.1\n\n\u221230\n\n\u221220\n\n\u221210\n\n0\n\nDelay\n\n0\n\nDelay\n\n10\n\n20\n\n30\n\n10\n\n20\n\n30\n\nFigure 2: Results from the simulation of the same linear-nonlinear network as in Fig. 1,\nexcept that the connections from both neuron 2 and neuron 3 onto neuron 1 were made\ninhibitory. Panels are as in Fig. 1. Again, S eliminates the stimulus-induced peak in C.\nW re\ufb02ects only the direct connection correlations, and U re\ufb02ects only the common input\ncorrelations. This inhibitory example, however, required a long simulation for accurate\nresults (see text). Parameters for g((cid:1)): (cid:22)T1 = 1:2, (cid:22)T2 = 2:0, (cid:22)T3 = 1:5, (cid:15)1 = 0:5, (cid:15)2 = 1:0,\n(cid:15)3 = 0:7. Parameters for h are the same as in Fig. 1.\n\nand neuron 1 received a direct connection from neuron 2 (see Fig. 3).\nWe let t be given in milliseconds, sampled Eq. (5) on a 20(cid:2) 20(cid:2) 30 grid in space and time,\np.\nusing a 2 ms grid in time, and normalized the resulting vector to obtain the unit vector hi\nA two millisecond sample rate of discrete white noise is unrealistic in many experiments,\nso we departed further from the assumptions of the derivation and let the stimulus be white\nnoise sampled at 10 ms. We let the stimulus standard deviation be (cid:27) = 1=p5 so that it had\nthe same power as discrete white noise sampled at 2 ms with (cid:27) = 1.\nAfter one hour of simulated time (360,000 frames), we collected approximately 23,000\u2013\n25,000 spikes per neuron. Fig. 4 shows that the method still effectively distinguishes direct\nconnection correlations from common input correlations. The separation isn\u2019t perfect as\nW becomes negative where the common input correlation is positive and U becomes neg-\native where the direct input correlation is positive. To determine whether a combination\nof positive W and negative U, for example, indicates positive direct connection correlation\nor negative common input correlation, one simply needs to look to see if S is positive or\nnegative.\nFig. 4 dramatically illustrates the increased noise in W and U. For this reason, the mea-\nsures are useful only when one can run a relatively long experiment to get an acceptable\nsignal-to-noise ratio. The noise is due to the conditioning of the (non-square) matrix in the\n\n\fX\n\nh1\n\n3h\n\n2h\n\nT j\n1\n\nT j\n3\n\nT j\n2\n\n1\n\n2\n\nT j\nsp,1\n\nT j\nsp,1\n\nFigure 3: Diagram of two integrate-and-\ufb01re neurons (circles) receiving threshold-linear\ninput from the stimulus. The neurons received common input from threshold-linear unit\n3, and neuron 1 received a direct connection from neuron 2. The evolution of the voltage\nof neuron p in response to input gp(t) was given by (cid:28)m\ndt + Vp + gp(t)(Vp (cid:0) Es) = 0.\nWhen Vp(t) reached 1, a spike was recorded, and the voltage was reset to 0 and held\nthere for an absolute refractory period of length (cid:28)ref . We let gp(t) = gext\np (t),\np (t) + gint\np ) + 0:05Pj G(t (cid:0) T j\nwhere the external input was gext\np (t) = 0:05Pj G(t (cid:0) T j\n3 (cid:0) (cid:14)p)\nwith G(t) = e2\np were drawn\np (cid:1) X(cid:3)+ where [x]+ = x if\nfrom a modulated Poisson process with rate given by (cid:11)p(cid:2)hi\n1 (t) = 0:05Pj G(t (cid:0) T j\nsp;2 are the spike times of neuron 2.\n\nx > 0 and is zero otherwise. The internal input gint\n2 (t) to neuron 2 was set to zero, and\nthe internal input to neuron 1 was set to re\ufb02ect an excitatory connection from neuron 2,\ngint\n\nsp;2 (cid:0) (cid:14)21), where the T j\n\ndVp\n\ne(cid:0)t=(cid:28)s for t > 0 and G(t) = 0 otherwise. The T j\n\n(cid:28)s(cid:1)2\n4 (cid:0) t\n\nleast-square calculation of W and U. The condition numbers in the three examples were\napproximately 70, 50, and 110, respectively. Measurement errors or noise could be mag-\nni\ufb01ed by as much as these factors. The high condition numbers re\ufb02ect the subtlety of the\ndistinction we are making.\nObtaining values of W and U signi\ufb01cantly beyond the noise level in real experiments may\nprove a formidable challenge. However, the utility of W and U with noisy data greatly\nimproves when they are used in conjunction with other measures. One can use a less noisy\nmeasure such as S to \ufb01nd signi\ufb01cant stimulus-independent correlations and determine their\nmagnitudes. Then, assuming one can rule out causes like covariation in latency or excitabil-\nity [7], one simply needs to determine if the correlations were caused by a direct connection\nor by common input. One does not need to use W and U to reject the null hypothesis of\nno connectivity-induced correlations; they are needed only to make the remaining binary\ndistinction.\n\nThe proposed method should be viewed simply as an example of a new framework for\nreconstructing stimulus-driven neural networks. Clearly, extensions beyond the presented\nmodel will be necessary since the linear-nonlinear model can adequately describe the be-\nhavior of only a small subset of neurons in primary sensory areas. Furthermore, methods to\nvalidate the assumed model will be required before results of this approach can be trusted.\n\nThough limited in scope and model-dependent, we have demonstrated what appears to\nbe the \ufb01rst example of a de\ufb01nitive dissociation between direct connection and common\ninput correlations from spike time data. At least in the case of excitatory connections,\nthis distinction can be made with a realistic, albeit large, amount of data. With further\nre\ufb01nements, this approach may yield viable tools for reconstructing stimulus-driven neural\nnetworks.\n\n\fx 10\u22125\n\na\n\nC\n\n6\n4\n2\n0\n\n\u2212150\n\n\u2212100\n\n\u221250\n\nb\n\nx 10\u22125\n\nS\n\n4\n2\n0\n\n\u2212150\n\n\u2212100\n\n\u221250\n\n1c\n\nW\n\n0.5\n0\n\u22120.5\n\n\u2212150\n\n\u2212100\n\n\u221250\n\n0.6d\n\n0.4\n0.2\n0\n\u22120.2\n\nU\n\n0\n\nDelay (ms)\n\n50\n\n100\n\n150\n\n0\n\nDelay (ms)\n\n50\n\n100\n\n150\n\n0\n\nDelay (ms)\n\n50\n\n100\n\n150\n\n\u2212150\n\n\u2212100\n\n\u221250\n\n0\n\nDelay (ms)\n\n50\n\n100\n\n150\n\nFigure 4: Results from the simulation of two integrate-and-\ufb01re neurons, where neuron 2 had\nan excitatory connection onto neuron 1 with a delay (cid:14)21 = 50 ms. Both neurons received\ncommon input, but the common input to neuron 2 was delayed ((cid:14) 1 = 0 ms, (cid:14)2 = 60 ms).\nPanels are as in Fig. 1. S greatly reduces the central, stimulus-induced correlation from\nC. W and U successfully distinguish the direct connection correlations from the common\ninput correlations, but also negatively re\ufb02ect each other. Ambiguity in interpretation of W\nand U can be eliminated by comparison with S. Integrate-and-\ufb01re parameters: (cid:28)m = 5\nms, Es = 6:5, (cid:28)2 = 2 ms, (cid:28)ref = 2 ms, (cid:11)1 = (cid:11)2 = 0:25 ms(cid:0)1, and (cid:11)3 = 0:1 ms(cid:0)1.\nParameters for h are the same as in Fig. 1 except that (cid:28)h = 10 ms.\n\nReferences\n\n[1] D. H. Perkel, G. L. Gerstein, and G. P. Moore. Neuronal spike trains and stochastic point pro-\n\ncesses. II. Simultaneous spike trains. Biophys. J., 7:419\u201340, 1967.\n\n[2] A. M. H. J. Aertsen, G. L. Gerstein, M. K. Habib, and G. Palm. Dynamics of neuronal \ufb01ring\n\ncorrelation: Modulation of \u201ceffective connectivity\u201d. J. Neurophysiol., 61:900\u2013917, 1989.\n\n[3] G. Palm, A. M. H. J. Aertsen, and G. L. Gerstein. On the signi\ufb01cance of correlations among\n\nneuronal spike trains. Biol. Cybern., 59:1\u201311, 1988.\n\n[4] J. R. Rosenberg, A. M. Amjad, P. Breeze, D. R. Brillinger, and D. M. Halliday. The Fourier ap-\nproach to the identi\ufb01cation of functional coupling between neuronal spike trains. Prog. Biophys.\nMol. Biol., 53:1\u201331, 1989.\n\n[5] D. Q. Nykamp and Dario L. Ringach. Full identi\ufb01cation of a linear-nonlinear system via cross-\n\ncorrelation analysis. J. Vision, 2:1\u201311, 2002.\n\n[6] D. Q. Nykamp. A spike correlation measure that eliminates stimulus effects in response to white\n\nnoise. J. Comp. Neurosci., 14:193\u2013209, 2003.\n\n[7] C. D. Brody. Correlations without synchrony. Neural. Comput., 11:1537\u201351, 1999.\n\n\f", "award": [], "sourceid": 2237, "authors": [{"given_name": "Duane", "family_name": "Nykamp", "institution": null}]}