{"title": "Inferring synaptic conductances from spike trains with a biophysically inspired point process model", "book": "Advances in Neural Information Processing Systems", "page_first": 954, "page_last": 962, "abstract": "A popular approach to neural characterization describes neural responses in terms of a cascade of linear and nonlinear stages: a linear filter to describe stimulus integration, followed by a nonlinear function to convert the filter output to spike rate. However, real neurons respond to stimuli in a manner that depends on the nonlinear integration of excitatory and inhibitory synaptic inputs. Here we introduce a biophysically inspired point process model that explicitly incorporates stimulus-induced changes in synaptic conductance in a dynamical model of neuronal membrane potential. Our work makes two important contributions. First, on a theoretical level, it offers a novel interpretation of the popular generalized linear model (GLM) for neural spike trains. We show that the classic GLM is a special case of our conductance-based model in which the stimulus linearly modulates excitatory and inhibitory conductances in an equal and opposite \u201cpush-pull\u201d fashion. Our model can therefore be viewed as a direct extension of the GLM in which we relax these constraints; the resulting model can exhibit shunting as well as hyperpolarizing inhibition, and time-varying changes in both gain and membrane time constant. Second, on a practical level, we show that our model provides a tractable model of spike responses in early sensory neurons that is both more accurate and more interpretable than the GLM. Most importantly, we show that we can accurately infer intracellular synaptic conductances from extracellularly recorded spike trains. We validate these estimates using direct intracellular measurements of excitatory and inhibitory conductances in parasol retinal ganglion cells. We show that the model fit to extracellular spike trains can predict excitatory and inhibitory conductances elicited by novel stimuli with nearly the same accuracy as a model trained directly with intracellular conductances.", "full_text": "Inferring synaptic conductances from spike trains\nunder a biophysically inspired point process model\n\nKenneth W. Latimer\n\nThe Institute for Neuroscience\n\nThe University of Texas at Austin\n\nlatimerk@utexas.edu\n\nE. J. Chichilnisky\n\nDepartment of Neurosurgery\n\nHansen Experimental Physics Laboratory\n\nStanford University\nej@stanford.edu\n\nFred Rieke\n\nDepartment of Physiology and Biophysics\n\nHoward Hughes Medical Institute\n\nUniversity of Washington\n\nrieke@u.washington.edu\n\nJonathan W. Pillow\n\nPrinceton Neuroscience Institute\n\nDepartment of Psychology\n\nPrinceton University\n\npillow@princeton.edu\n\nAbstract\n\nA popular approach to neural characterization describes neural responses in terms\nof a cascade of linear and nonlinear stages: a linear \ufb01lter to describe stimulus\nintegration, followed by a nonlinear function to convert the \ufb01lter output to spike\nrate. However, real neurons respond to stimuli in a manner that depends on the\nnonlinear integration of excitatory and inhibitory synaptic inputs. Here we in-\ntroduce a biophysically inspired point process model that explicitly incorporates\nstimulus-induced changes in synaptic conductance in a dynamical model of neu-\nronal membrane potential. Our work makes two important contributions. First, on\na theoretical level, it offers a novel interpretation of the popular generalized linear\nmodel (GLM) for neural spike trains. We show that the classic GLM is a special\ncase of our conductance-based model in which the stimulus linearly modulates ex-\ncitatory and inhibitory conductances in an equal and opposite \u201cpush-pull\u201d fashion.\nOur model can therefore be viewed as a direct extension of the GLM in which we\nrelax these constraints; the resulting model can exhibit shunting as well as hyper-\npolarizing inhibition, and time-varying changes in both gain and membrane time\nconstant. Second, on a practical level, we show that our model provides a tractable\nmodel of spike responses in early sensory neurons that is both more accurate and\nmore interpretable than the GLM. Most importantly, we show that we can ac-\ncurately infer intracellular synaptic conductances from extracellularly recorded\nspike trains. We validate these estimates using direct intracellular measurements\nof excitatory and inhibitory conductances in parasol retinal ganglion cells. The\nstimulus-dependence of both excitatory and inhibitory conductances can be well\ndescribed by a linear-nonlinear cascade, with the \ufb01lter driving inhibition exhibit-\ning opposite sign and a slight delay relative to the \ufb01lter driving excitation. We\nshow that the model \ufb01t to extracellular spike trains can predict excitatory and in-\nhibitory conductances elicited by novel stimuli with nearly the same accuracy as\na model trained directly with intracellular conductances.\n\n1\n\nIntroduction\n\nThe point process generalized linear model (GLM) has provided a useful and highly tractable tool\nfor characterizing neural encoding in a variety of sensory, cognitive, and motor brain areas [1\u20135].\n\n1\n\n\fFigure 1: Schematic of conductance-based spiking model.\n\nHowever, there is a substantial gap between descriptive statistical models like the GLM and more\nrealistic, biophysically interpretable neural models. Cascade-type statistical models describe input\nto a neuron in terms of a set of linear (and sometimes nonlinear) \ufb01ltering steps [6\u201311]. Real neurons,\non the other hand, receive distinct excitatory and inhibitory synaptic inputs, which drive conductance\nchanges that alter the nonlinear dynamics governing membrane potential. Previous work has shown\nthat excitatory and inhibitory conductances in retina and other sensory areas can exhibit substantially\ndifferent tuning. [12, 13].\nHere we introduce a quasi-biophysical interpretation of the generalized linear model. The resulting\ninterpretation reveals that the GLM can be viewed in terms of a highly constrained conductance-\nbased model. We expand on this interpretation to construct a more \ufb02exible and more plausible\nconductance-based spiking model (CBSM), which allows for independent excitatory and inhibitory\nsynaptic inputs. We show that the CBSM captures neural responses more accurately than the stan-\ndard GLM, and allows us to accurately infer excitatory and inhibitory synaptic conductances from\nstimuli and extracellularly recorded spike trains.\n\n2 A biophysical interpretation of the GLM\n\nThe generalized linear model (GLM) describes neural encoding in terms of a cascade of linear,\nnonlinear, and probabilistic spiking stages. A quasi-biological interpretation of GLM is known as\n\u201csoft threshold\u201d integrate-and-\ufb01re [14\u201317]. This interpretation regards the linear \ufb01lter output as a\nmembrane potential, and the nonlinear stage as a \u201csoft threshold\u201d function that governs how the\nprobability of spiking increases with membrane potential, speci\ufb01cally:\n\nVt = k(cid:62)xt\nrt = f (Vt)\n\nyt|rt \u223c Poiss(rt\u2206t),\n\n(1)\n(2)\n(3)\n\nwhere k is a linear \ufb01lter mapping the stimulus xt to the membrane potential Vt at time t, a \ufb01xed\nnonlinear function f maps Vt to the conditional intensity (or spike rate) rt, and spike count yt is a\nPoisson random variable in a time bin of in\ufb01nitesimal width \u2206t. The log likelihood is\n\nT(cid:88)\n\nlog p(y1:T|x1:T , k) =\n\n\u2212rt\u2206t + yt log(rt\u2206t) \u2212 log(yt!).\n\n(4)\n\nt=1\n\nThe stimulus vector xt can be augmented to include arbitrary covariates of the response such as the\nneuron\u2019s own spike history or spikes from other neurons [2, 3]. In such cases, the output does not\nform a Poisson process because spiking is history-dependent.\nThe nonlinearity f is \ufb01xed a priori. Therefore, the only parameters are the coef\ufb01cients of the \ufb01lter\nk. The most common choice is exponential, f (z) = exp(z), corresponding to the canonical \u2018log\u2019\nlink function for Poisson GLMs. Prior work [6] has shown that if f grows at least linearly and at\nmost exponentially, then the log-likelihood is jointly concave in model parameters \u03b8. This ensures\nthat the log-likelihood has no non-global maxima, and gradient ascent methods are guaranteed to\n\ufb01nd the maximum likelihood estimate.\n\n2\n\nstimulusspikesnonlinearityinhibitory filterexcitatory filterPoissonpost-spike filter\f3\n\nInterpreting the GLM as a conductance-based model\n\nA more biophysical interpretation of the GLM can be obtained by considering a single-compartment\nneuron with linear membrane dynamics and conductance-based input:\n\ndV\ndt\n\n= \u2212glV + ge(t)(V \u2212 Ee) \u2212 gi(t)(V \u2212 Ei)\n= \u2212(gl + ge(t) + gi(t))V + ge(t)Ee + gi(t)Ei\n= \u2212gtot(t)V + Is(t),\n\n(5)\nwhere (for simplicity) we have set the leak current reversal potential to zero. The \u201ctotal conductance\u201d\nat time t is gtot(t) = gl +ge(t)+gi(t) and the \u201ceffective input current\u201d is Is(t) = ge(t)Ee +gi(t)Ei.\nSuppose that the stimulus affects the neuron via the synaptic conductances ge and gi. It is then\nnatural to ask under which conditions, if any, the above model can correspond to a GLM. The\nde\ufb01nition of a GLM requires the solution V (t) to be a linear (or af\ufb01ne) function of the stimulus.\nThis arises if the two following conditions are met:\n\n1. Total conductance gtot is constant. Thus, for some constant c:\n\nge(t) + gi(t) = c.\n\n(6)\n\n2. The input Is is linear in x. This holds if we set:\n\nge(xt) = ke\ngi(xt) = ki\n\n(cid:62)xt + be\n(cid:62)xt + bi.\n\n(7)\nWe can satisfy these two conditions by setting ke = \u2212ki, so that the excitatory and inhibitory\nconductances are driven by equal and opposite linear projections of the stimulus. This allows us to\nrewrite the membrane equation (eq. 5):\n\ndV\ndt\n\n= \u2212gtotV + (ke\n= \u2212gtotV + ktot\n\n(cid:62)xt + be)Ee + (ki\n(cid:62)xt + btot,\n\n(cid:62)xt + bi)Ei\n\n(8)\nwhere gtot = gl + be + bi is the (constant) total conductance, ktot = keEe + kiEi, and btot =\nbeEe + biEi. If we take the initial voltage V0 to be btot, the equilibrium voltage in the absence of a\nstimulus, then the solution to this differential equation is\n\nVt =\n\n(cid:90) t\n\n0\n\ne\u2212gtot(t\u2212s)(cid:0)ktot\n\n(cid:62)xs\n(cid:62)xt) + btot\n\n= kleak \u2217 (ktot\n= kglm\n\n(cid:62)xt + btot,\n\n(cid:1) ds + btot\n\n(9)\nwhere kleak \u2217 (ktot\n(cid:62)xt) denotes linear convolution of the exponential decay \u201cleak\u201d \ufb01lter kleak(t) =\ne\u2212gtot t with the linearly projected stimulus train, and kglm = ktot \u2217 kleak is the \u201ctrue\u201d GLM \ufb01lter\n(from eq. 1) that results from temporally convolving the conductance \ufb01lter with the leak \ufb01lter. Since\nthe membrane potential is a linear (af\ufb01ne) function of the stimulus (as in eq. 1), the model is clearly\na GLM.\nThus, to summarize, the GLM can be equated with a synaptic conductance-based dynamical model\nin which the GLM \ufb01lter k results from a common linear \ufb01lter driving excitatory and inhibitory\nsynaptic conductances, blurred by convolution with an exponential leak \ufb01lter determined by the\ntotal conductance.\n\n4 Extending GLM to a nonlinear conductance-based model\n\nFrom the above, it is easy to see how to create a more realistic conductance-based model of neural\nresponses. Such a model would allow the stimulus tuning of excitation and inhibition to differ (i.e.,\nallow ke (cid:54)= \u2212ki), and would include a nonlinear relationship between x and the conductances to\n\n3\n\n\fpreclude negative values (e.g., using a rectifying nonlinearity). As with the GLM, we assume that\nthe only source of stochasticity on the model is in the spiking mechanism: we place no additional\nnoise on the conductances or the voltage. This simplifying assumption allows us to perform ef\ufb01cient\nmaximum likelihood inference using standard gradient ascent methods.\nWe specify the membrane potential of the conductance-based point process model as follows:\n\ndV\ndt\n\n= ge(t)(Ee \u2212 V ) + gi(t)(Ei \u2212 V ) + gl(El \u2212 V ),\nge(t) = fe(ke\n\n(cid:62)xt),\n\n(cid:62)xt),\n\n(11)\nwhere fe and fi are nonlinear functions ensuring positivity of the synaptic conductances. In practice,\nwe evaluate V along a discrete lattice of points (t = 1, 2, 3, . . . T ) of width \u2206t. Assuming ge and gi\nremain constant within each bin, the voltage equation becomes a simple linear differential equation\nwith the solution\n\ngi(t) = fi(ki\n\n(10)\n\n(cid:18)\n\n(cid:19)\n\nV (t + 1) = e\u2212gtot(t)\u2206t\n\nV (t) \u2212 Is(t)\ngtot(t)\n\nV (1) = El\n\ngtot(t) = ge(t) + gi(t) + gl\n\nIs(t) = ge(t)Ee + gi(t)Ei + glEl\n\n+\n\nIs(t)\ngtot(t)\n\n(12)\n\n(13)\n(14)\n(15)\n\nThe mapping from membrane potential to spiking is similar to that in the standard GLM (eq. 3):\n\n(cid:18) (V \u2212 VT )\n\n(cid:19)\n\nrt = f (V (t))\n\nf (V ) = exp\nVS\nyt|rt \u223c Poiss(rt\u2206t).\n\n(16)\n\n(17)\n\n(19)\n\n(18)\nThe voltage-to-spike rate nonlinearity f follows the form proposed by Mensi et al. [17], where VT\nis a soft spiking threshold and VS determines the steepness of the nonlinearity. To account for\nrefractory periods or other spike-dependent behaviors, we simply augment the function to include a\nGLM-like spike history term:\n\n(cid:18) (V \u2212 VT )\n\nVS\n\n(cid:19)\n\nf (V ) = exp\n\n+ h(cid:62)yhist\n\nfe(\u00b7), fi(\u00b7) = log(1 + exp(\u00b7)).\n\nSpiking activity in real neurons in\ufb02uences both the membrane potential and the output nonlinearity.\nWe could include additional conductance terms that depend on either stimuli or spike history, such as\nan after hyper-polarization current; this provides one direction for future work. For spatial stimuli,\nthe model can include a set of spatially distinct recti\ufb01ed inputs (e.g., as employed in [9]).\nTo complete the model, we must select a form for the conductance nonlinearities fe and fi. Although\nwe could attempt to \ufb01t these functions (e.g., as in [9, 18]), we \ufb01xed them to be the soft-rectifying\nfunction:\n\n(20)\nFixing these nonlinearities improved the speed and robustness of maximum likelihood parameter\n\ufb01tting. Moreover, we examined intracellularly recorded conductances and found that the nonlinear\nmapping from linearly projected stimuli to conductance was well described by this function (see\nFig. 4).\nThe model parameters we estimate are {ke, ki, be, bi, h, gl, El}. We set the remaining model param-\neters to biologically plausible values: VT = \u221270mV, VS = 4mV, Ee = 0mV, and Ei = \u221280mV .\nTo limit the total number of parameters, we \ufb01t the linear \ufb01lters ke and ki using a basis consisting of\n12 raised cosine functions, and we used 10 raised cosine functions for the spike history \ufb01lter [3].\nThe log-likelihood function for this model is not concave in the model parameters, which increases\nthe importance to selecting a good initialization point. We initialized the parameters by \ufb01tting a\nsimpli\ufb01ed model which had only one conductance. We initialized the leak terms as El = \u221270mV\nand gl = 200. We assumed a single synaptic conductance with a linear stimulus dependence,\n(cid:62)xt (note that this allows for negative conductance values). We initialized this \ufb01lter\nglin(t) = klin\n\n4\n\n\fFigure 2: Simulation results. (A) Estimates (solid traces) of excitatory (blue) and inhibitory (red) stimulus\n\ufb01lters from 10 minutes of simulated data. (Dashed lines indicate true \ufb01lters). (B) The L2 norm between\nthe estimated input \ufb01lters and the true \ufb01lters (calculated in the low-dimensional basis) as a function of the\namount of training data. (C) The log-likelihood of the \ufb01t CBSM on withheld test data converges to the log\nlikelihood of the true model.\n\nthe GLM \ufb01t, and then numerically maximized the likelihood for klin. We then initialized the pa-\nrameters for the complete model using ke = cklin and ki = \u2212cklin, where 0 < c \u2264 1, thereby\nexploiting the mapping between the GLM and the CBSM. Although this initialization presumes that\nexcitation and inhibition have nearly opposite tuning, we found that standard optimization meth-\nods successfully converged to the true model parameters even when ke and ki had similar tuning\n(simulation results not shown).\n\n5 Results: simulations\n\nTo examine the estimation performance, we \ufb01t spike train data simulated from a CBSM with known\nparameters (see Fig. 2). The simulated data qualitatively mimicked experimental datasets, with input\n\ufb01lters selected to reproduce the stimulus tuning of macaque ON parasol RGCs. The stimulus con-\nsisted of a one dimensional white noise signal, binned at a 0.1ms resolution, and \ufb01ltered with a low\npass \ufb01lter with a 60Hz cutoff frequency. The simulated cell produced a \ufb01ring rate of approximately\n32spikes/s. We validated our maximum likelihood \ufb01tting procedure by examining error in the \ufb01tted\nparameters, and evaluating the log-likelihood on a held out \ufb01ve-minute test set. With increasing\namounts of training data, the parameter estimates converged to the true parameters, despite the fact\nthat the model does not have the concavity guarantees of the standard GLM.\nTo explore the CBSM\u2019s qualitative response properties, we performed simulated experiments using\nstimuli with varying statistics (see Fig. 3). We simulated spike responses from a CBSM with\n\ufb01xed parameters to stimuli with different standard deviations. We then separately \ufb01t responses from\neach simulation with a standard GLM. The \ufb01tted GLM \ufb01lters exhibit shifts in both peak height\nand position for stimuli with different variance. This suggests that the CBSM can exhibit gain\ncontrol effects that cannot be captured by a classic GLM with a spike history \ufb01lter and exponential\nnonlinearity.\n\n6 Results: neural data\n\nWe \ufb01t the CBSM to spike trains recorded from 7 macaque ON parasol RGCs [12]. The spike trains\nwere obtained by cell attached recordings in response to full-\ufb01eld, white noise stimuli (identical to\nthe simulations above). Either 30 or 40 trials were recorded from each cell, using 10 unique 6 second\nstimuli. After the spike trains were recorded, voltage clamp recordings were used to measure the\nexcitatory and inhibitory conductances to the same stimuli. We \ufb01t the model using the spike trains\nfor 9 of the stimuli, and the remaining trials were used to test model \ufb01t. Thus, the models were\neffectively trained using 3 or 4 repeats of 54 seconds of full-\ufb01eld noise stimulus. We compared the\nintracellular recordings to the ge and gi estimated from the CBSM (Fig. 5). Additionally, we \ufb01t the\nmeasured conductances with the linear-nonlinear cascade model from the CBSM (the terms ge and\n\n5\n\n05100102030minutes of training data L2 errorestimated filter errors0510\u22125.33\u22125.31\u22125.29\u22125.27x 104minutes of training datalog likelihoodActualfit to test dataABC50100150200\u22120.8\u22120.400.4filter fitstime (ms)weight\fFigure 3: Qualitative illustration of model\u2019s capacity to exhibit contrast adaptation (or gain control). (A)\nThe GLM \ufb01lters \ufb01t to a \ufb01xed CBSM simulated at various levels of stimulus variance. (B) Filters \ufb01t to two\nreal retinal ganglion cells at two different levels of contrast (from [19]).\n\nFigure 4: Measured conductance vs. output of a \ufb01tted linear stimulus \ufb01lter (gray points), for both the\nexcitatory (left) and inhibitory (right) conductances. The green diamonds correspond to a non-parametric\nestimate of the conductance nonlinearity, given by the mean conductance for each bin of \ufb01lter output. For\nboth conductances, the function is is well described by a soft-rectifying function (black trace).\n\ngi in eq. 11) with a least-squares \ufb01t as an upper bound measure for the best possible conductance\nestimate given our model. The CBSM correctly determined the stimulus tuning for excitation and\ninhibition for these cells: inhibition is oppositely tuned and slightly delayed from excitation.\nFor the side-by-side comparison shown in Fig. 5, we introduced a scaling factor in the estimated\nconductances in order to compare the conductances estimated from spike trains against recorded\nconductances. Real membrane voltage dynamics depend on the capacitance of the membrane, which\nwe do not include because it introduces an arbitrary scaling factor that cannot be estimated by spike\nalone. Therefore, for comparisons we chose a scaling factor for each cell independently. However,\nwe used a single scaling for the inhibitory and excitatory conductances. Additionally, we often had\n2 or 3 repeated trials of the withheld stimulus, and we compared the model prediction to the average\nconductance recorded for the stimulus. The CBSM predicted the synaptic conductances with an\naverage r2 = 0.54 for the excitatory and an r2 = 0.39 for the inhibitory input from spike trains,\ncompared to an average r2 = 0.72 and r2 = 0.59 for the excitatory and inhibitory conductances re-\nspectively from the least-squares \ufb01t directly to the conductances (Fig. 6). To summarize, using only\na few minutes of spiking data, the CBSM could account for 71% of the variance of the excitatory\ninput and 62% of the inhibitory input that can possibly be explained using the LN cascade model of\nthe conductances (eq. 11).\nOne challenge we discovered when \ufb01tting the model to real spike trains was that one \ufb01lter, typically\nki, would often become much larger than the other \ufb01lter. This resulted in one conductance becoming\ndominant, which the intracellular recordings indicated was not the case. This was likely due to the\nfact that we are data-limited when dealing with intracellular recordings: the spike train recordings\ninclude only 1 minute of unique stimulus. To alleviate this problem, we added a penalty term, \u03c6, to\n\n6\n\n050100150200\u22120.0100.010.020.03time (ms)weight\ufb01lters at different contrasts0.25x contrast0.5x 1x2xexperimental data(Chander & Chichilnisky, 2001)ABinhibitory\u221230030\u22121001020304050filter outputmeasured conductance excitatorydata mean\u221240\u22122002040\u2212100102030405015-15\fFigure 5: Two example ON parasol RGC responses to a full-\ufb01eld noise stimulus \ufb01t with the CBSM. The\nmodel parameters were \ufb01t to spike train data, and then used to predict excitatory and inhibitory synaptic\ncurrents recorded separately in response to novel stimuli. For comparison, we show predictions of an LN\nmodel \ufb01t directly to the conductance data. Left: Linear kernels for the excitatory (blue) and inhibitory\n(red) inputs estimated from the conductance-based model (light red, light blue) and estimated by \ufb01tting a\nlinear-nonlinear model directly to the measured conductances (dark red, dark blue). The \ufb01lters represent a\ncombination of events that occur in the retinal circuitry in response to a visual stimulus, and are primarily\nshaped by the cone transduction process. Right: Conductances predicted by our model on a withheld test\nstimulus. Measured conductances (black) are compared to the predictions from the CBSM \ufb01lters (\ufb01t to\nspiking data) and an LN model (\ufb01t to conductance data).\n\nthe log likelihood on the difference of the L2 norms of ke and ki:\n\n\u03c6(ke, ki) = \u03bb(cid:0)||ke||2 \u2212 ||ki||2(cid:1)2\n\n(21)\nThis differentiable penalty ensures that the model will not rely too strongly on one \ufb01lter over the\nother, without imposing any prior on the shape of the \ufb01lters (with \u03bb = 0.05). We note that unlike\nthe a typical situation with statistical models that contain more abstract parameters, the terms we\nwish to regularize can be measured with intracellular recordings. Future work with this model could\ninclude more informative, data-driven priors on ke and ki.\nFinally, we \ufb01t the CBSM and GLM to a population of nine extracellularly recorded macaque RGCs\nin response to a full-\ufb01eld binary noise stimulus [20]. We used a \ufb01ve minute segment for model\n\ufb01tting, and compared predicted spike rate using a 6s test stimulus for which we had repeated trials.\n\n7\n\n250ms10nS050100150200\u22120.3\u22120.2\u22120.100.10.2time (ms)weightestimated filters250ms10nS050100150200\u22120.2\u22120.100.10.2time (ms)weightestimated filtersExample Cell 2Example Cell 1estimated conductancesgegigegifit to conductance:fit to spikes:fit to conductance:fit to spikes:fit to conductance:fit to spikes:fit to conductance:fit to spikes:(spikes)(conductances)(spikes)(conductances)\fFigure 6: Summary of the CBSM \ufb01ts to 7 ON parasol RGCs for which we had both spike train and\nconductance recordings. The axes show model\u2019s ability to predict the excitatory (left) and inhibitory (right)\ninputs to a new stimulus in terms of r2. The CBSM \ufb01t is compared against predictions of an LN model \ufb01t\ndirectly to measured conductances.\n\nFigure 7: (A) Performance on spike rate (PSTH) prediction. The true rate (black) was estimated using 167\nrepeat trials. The GLM prediction is in blue and the CBSM is in red. The PSTHs were smoothed with a\nGaussian kernel with a 1ms standard deviation. (B) Spike rate prediction performance for the population\nof 9 cells. The red circle indicates cell used in left plot.\n\nThe CBSM achieved a 0.08 higher average r2 in PSTH prediction performance compared to the\nGLM. All nine cells showed an improved \ufb01t with the CBSM.\n\n7 Discussion\n\nThe classic GLM is a valuable tool for describing the relationship between stimuli and spike re-\nsponses. However, the GLM describes this map as a mathematically convenient linear-nonlinear\ncascade, which does not take account of the biophysical properties of neural processing. Here we\nhave shown that the GLM may be interpreted as a biophysically inspired, but highly constrained,\nsynaptic conductance-based model. We proposed a more realistic model of the conductance, remov-\ning the arti\ufb01cial constraints present in the GLM interpretation, which results in a new, more accurate\nand more \ufb02exible conductance-based point process model for neural responses. Even without the\nbene\ufb01t of a concave log-likelihood, numerical optimization methods provide accurate estimates of\nmodel parameters.\nQualitatively, the CBSM has a stimulus-dependent time constant, which allows it change gain as a\nfunction of stimulus statistics (e.g., contrast), an effect that cannot be captured by a classic GLM. The\nmodel also allows the excitatory and inhibitory conductances to be distinct functions of the sensory\nstimulus, as is expected in real neurons. We demonstrate that the CBSM not only achieves improved\nperformance as a phenomenological model of neural encoding compared to the GLM, the model\naccurately estimates the tuning of the excitatory and inhibitory synaptic inputs to RGCs purely from\nmeasured spike times. As we move towards more naturalistic stimulus conditions, we believe that\nthe conductance-based approach will become a valuable tool for understanding the neural code in\nsensory systems.\n\n8\n\n00.20.40.60.8100.20.40.60.81spike fit r2fit\u2212to\u2212conductance r2Excitation prediction00.20.40.60.8100.20.40.60.81spike fit r2fit\u2212to\u2212conductance r2Inhibition prediction250ms50 spks/sGLM:CBSM:AB0.40.60.810.40.50.60.70.80.91Conductance Model Rate prediction performanceoff cellon cellGLM \fReferences\n[1] K. Harris, J. Csicsvari, H. Hirase, G. Dragoi, and G. Buzsaki. Organization of cell assemblies\n\nin the hippocampus. Nature, 424:552\u2013556, 2003.\n\n[2] W. Truccolo, U. T. Eden, M. R. Fellows, J. P. Donoghue, and E. N. Brown. A point process\nframework for relating neural spiking activity to spiking history, neural ensemble and extrinsic\ncovariate effects. J. Neurophysiol, 93(2):1074\u20131089, 2005.\n\n[3] J. W. Pillow, J. Shlens, L. Paninski, A. Sher, A. M. Litke, E. J. Chichilnisky, and E. P. Simon-\ncelli. Spatio-temporal correlations and visual signaling in a complete neuronal population.\nNature, 454:995\u2013999, 2008.\n\n[4] S. Gerwinn, J. H. Macke, and M. Bethge. Bayesian inference for generalized linear models for\n\nspiking neurons. Frontiers in Computational Neuroscience, 2010.\n\n[5] I. H. Stevenson, B. M. London, E. R. Oby, N. A. Sachs, J. Reimer, B. Englitz, S. V. David,\nS. A. Shamma, T. J. Blanche, K. Mizuseki, A. Zandvakili, N. G. Hatsopoulos, L. E. Miller,\nand K. P. Kording. Functional connectivity and tuning curves in populations of simultaneously\nrecorded neurons. PLoS Comput Biol, 8(11):e1002775, 2012.\n\n[6] L. Paninski. Maximum likelihood estimation of cascade point-process neural encoding models.\n\nNetwork: Computation in Neural Systems, 15:243\u2013262, 2004.\n\n[7] D. A. Butts, C. Weng, J. Jin, J.M. Alonso, and L. Paninski. Temporal precision in the vi-\nsual pathway through the interplay of excitation and stimulus-driven suppression. J Neurosci,\n31(31):11313\u201311327, Aug 2011.\n\n[8] B Vintch, A Zaharia, J A Movshon, and E P Simoncelli. Ef\ufb01cient and direct estimation of\na neural subunit model for sensory coding. In Adv. Neural Information Processing Systems\n(NIPS*12), volume 25, Cambridge, MA, 2012. MIT Press. To be presented at Neural Infor-\nmation Processing Systems 25, Dec 2012.\n\n[9] J. M. McFarland, Y. Cui, and D. A. Butts. Inferring nonlinear neuronal computation based on\nphysiologically plausible inputs. PLoS computational biology, 9(7):e1003143, January 2013.\n[10] Il M. Park, Evan W. Archer, Nicholas Priebe, and Jonathan W. Pillow. Spectral methods for\nneural characterization using generalized quadratic models. In Advances in Neural Information\nProcessing Systems 26, pages 2454\u20132462, 2013.\n\n[11] L. Theis, A. M. Chagas, D. Arnstein, C. Schwarz, and M. Bethge. Beyond glms: A generative\nmixture modeling approach to neural system identi\ufb01cation. PLoS Computational Biology, Nov\n2013. in press.\n\n[12] P. K. Trong and F. Rieke. Origin of correlated activity between parasol retinal ganglion cells.\n\nNature neuroscience, 11(11):1343\u201351, November 2008.\n\n[13] C. Poo and J. S. Isaacson. Odor representations in olfactory cortex: \u201dsparse\u201d coding, global\n\ninhibition, and oscillations. Neuron, 62(6):850\u201361, June 2009.\n\n[14] H. E. Plesser and W. Gerstner. Noise in integrate-and-\ufb01re neurons: from stochastic input to\n\nescape rates. Neural Comput, 12(2):367\u2013384, Feb 2000.\n\n[15] W. Gerstner. A framework for spiking neuron models: The spike response model. In F. Moss\nand S. Gielen, editors, The Handbook of Biological Physics, volume 4, pages 469\u2013516, 2001.\n[16] L. Paninski, J. W. Pillow, and J. Lewi. Statistical models for neural encoding, decoding, and\n\noptimal stimulus design. Progress in brain research, 165:493\u2013507, January 2007.\n\n[17] S. Mensi, R. Naud, and W. Gerstner. From stochastic nonlinear integrate-and-\ufb01re to general-\n\nized linear models. In NIPS, pages 1377\u20131385, 2011.\n\n[18] M. B. Ahrens, L. Paninski, and M. Sahani. Inferring input nonlinearities in neural encoding\n\nmodels. Network: Computation in Neural Systems, 19(1):35\u201367, January 2008.\n\n[19] D. Chander and E. J. Chichilnisky. Adaptation to Temporal Contrast in Primate and Salamander\n\nRetina. The Journal of Neuroscience, 21(24):9904\u201316, December 2001.\n\n[20] J. W. Pillow, L. Paninski, V. J. Uzzell, E. P. Simoncelli, and E. J. Chichilnisky. Prediction and\ndecoding of retinal ganglion cell responses with a probabilistic spiking model. The Journal of\nneuroscience, 25(47):11003\u201311013, November 2005.\n\n9\n\n\f", "award": [], "sourceid": 592, "authors": [{"given_name": "Kenneth", "family_name": "Latimer", "institution": "UT Austin"}, {"given_name": "E.J.", "family_name": "Chichilnisky", "institution": "Stanford University"}, {"given_name": "Fred", "family_name": "Rieke", "institution": "University of Washington, Seattle"}, {"given_name": "Jonathan", "family_name": "Pillow", "institution": "UT Austin"}]}