{"title": "Dependence of Orientation Tuning on Recurrent Excitation and Inhibition in a Network Model of V1", "book": "Advances in Neural Information Processing Systems", "page_first": 1769, "page_last": 1776, "abstract": "One major role of primary visual cortex (V1) in vision is the encoding of the orientation of lines and contours. The role of the local recurrent network in these computations is, however, still a matter of debate. To address this issue, we analyze intracellular recording data of cat V1, which combine measuring the tuning of a range of neuronal properties with a precise localization of the recording sites in the orientation preference map. For the analysis, we consider a network model of Hodgkin-Huxley type neurons arranged according to a biologically plausible two-dimensional topographic orientation preference map. We then systematically vary the strength of the recurrent excitation and inhibition relative to the strength of the afferent input. Each parametrization gives rise to a different model instance for which the tuning of model neurons at different locations of the orientation map is compared to the experimentally measured orientation tuning of membrane potential, spike output, excitatory, and inhibitory conductances. A quantitative analysis shows that the data provides strong evidence for a network model in which the afferent input is dominated by strong, balanced contributions of recurrent excitation and inhibition. This recurrent regime is close to a regime of 'instability', where strong, self-sustained activity of the network occurs. The firing rate of neurons in the best-fitting network is particularly sensitive to small modulations of model parameters, which could be one of the functional benefits of a network operating in this particular regime.", "full_text": "Dependence of Orientation Tuning on Recurrent\n\nExcitation and Inhibition in a Network Model of V1\n\nKlaus Wimmer1*, Marcel Stimberg1*, Robert Martin1, Lars Schwabe2, Jorge Mari\u00f1o3,\n\nJames Schummers4, David C. Lyon5, Mriganka Sur4, and Klaus Obermayer1\n\n1 Bernstein Center for Computational Neuroscience and Technische Universit\u00e4t Berlin, Germany\n\n2 Dept of Computer Science and Electrical Engineering, University of Rostock, Germany\n\n3 Dept of Medicine, Neuroscience, and Motor Control Group, Univ. A Coru\u00f1a, Spain\n\n4 Dept of Brain and Cognitive Sci and Picower Ctr for Learning and Memory, MIT, Cambridge\n\n5 Dept of Anatomy and Neurobiology, University of California, Irvine, USA\n\n[klaus, mst]@cs.tu-berlin.de\n\nAbstract\n\nThe computational role of the local recurrent network in primary visual cortex is\nstill a matter of debate. To address this issue, we analyze intracellular record-\ning data of cat V1, which combine measuring the tuning of a range of neuronal\nproperties with a precise localization of the recording sites in the orientation pref-\nerence map. For the analysis, we consider a network model of Hodgkin-Huxley\ntype neurons arranged according to a biologically plausible two-dimensional to-\npographic orientation preference map. We then systematically vary the strength\nof the recurrent excitation and inhibition relative to the strength of the afferent\ninput. Each parametrization gives rise to a different model instance for which the\ntuning of model neurons at different locations of the orientation map is compared\nto the experimentally measured orientation tuning of membrane potential, spike\noutput, excitatory, and inhibitory conductances. A quantitative analysis shows\nthat the data provides strong evidence for a network model in which the affer-\nent input is dominated by strong, balanced contributions of recurrent excitation\nand inhibition. This recurrent regime is close to a regime of \u201cinstability\u201d, where\nstrong, self-sustained activity of the network occurs. The \ufb01ring rate of neurons\nin the best-\ufb01tting network is particularly sensitive to small modulations of model\nparameters, which could be one of the functional bene\ufb01ts of a network operating\nin this particular regime.\n\n1 Introduction\n\nOne of the major tasks of primary visual cortex (V1) is the computation of a representation of\norientation in the visual \ufb01eld. Early models [1], combining the center-surround receptive \ufb01elds of\nlateral geniculate nucleus to give rise to orientation selectivity, have been shown to be over-simplistic\n[2; 3]. Nonetheless, a debate remains regarding the contribution of afferent and recurrent excitatory\nand inhibitory in\ufb02uences [4; 5]. Information processing in cortex changes dramatically with this\n\u201ccortical operating regime\u201d, i. e. depending on the relative strengths of the afferent and the different\nrecurrent inputs [6; 7]. Recently, experimental and theoretical studies have investigated how a cell\u2019s\norientation tuning depends on its position in the orientation preference map [7\u201310]. However, the\ncomputation of orientation selectivity in primary visual cortex is still a matter of debate.\nThe wide range of models operating in different regimes that are discussed in the literature are an\nindication that models of V1 orientation selectivity are underconstrained. Here, we assess whether\nthe speci\ufb01c location dependence of the tuning of internal neuronal properties can provide suf\ufb01cient\n\n*K. Wimmer and M. Stimberg contributed equally to this work.\n\n1\n\n\fconstraints to determine the corresponding cortical operating regime. The data originates from in-\ntracellular recordings of cat V1 [9], combined with optical imaging. This allowed to measure, in\nvivo, the output (\ufb01ring rate) of neurons, the input (excitatory and inhibitory conductances) and a\nstate variable (membrane potential) as a function of the position in the orientation map. Figure 1\nshows the experimentally observed tuning strength of each of these properties depending on the\ndistribution of orientation selective cells in the neighborhood of each neuron. The x-axis spans the\nrange from pinwheels (0) to iso-orientation domains (1), and each y-axis quanti\ufb01es the sharpness\nof tuning of the individual properties (see section 2.2). The tuning of the membrane potential (Vm)\nas well as the tuning of the total excitatory (ge) and inhibitory (gi) conductances vary strongly with\nmap location, whereas the tuning of the \ufb01ring rate (f) does not. Speci\ufb01cally, the conductances and\nthe membrane potential are sharper tuned for neurons within an iso-orientation domain, where the\nneighboring neurons have very similar orientation preferences, as compared to neurons close to a\npinwheel center, where the neighboring neurons show a broad range of orientation preferences.\n\nFigure 1: Variation of the orientation selectivity indices (OSI, cf. Equation 2) of the \ufb01ring rate (f),\nthe average membrane potential (Vm), and the excitatory (ge) and inhibitory (gi) input conductances\nof neurons in cat V1 with the map OSI (the orientation selectivity index of the orientation map at the\nlocation of the measured neuron). Dots indicate the experimentally measured values from 18 cells\n[9]. Solid lines show the result of a linear regression. The slopes (values \u00b1 95% con\ufb01dence interval)\nare \u22120.02 \u00b1 0.24 (f), 0.27 \u00b1 0.22 (Vm), 0.49 \u00b1 0.20 (ge), 0.44 \u00b1 0.19 (gi).\nThis paper focuses on the constraints that this speci\ufb01c map-location dependence of neuronal prop-\nerties imposes on the operating regime of a generic network composed of Hodgkin-Huxley type\nmodel neurons. The model takes into account that the lateral inputs a cell receives are determined\n(1) by the position in the orientation map and (2) by the way that synaptic inputs are pooled across\nthe map. The synaptic pooling radius has been shown experimentally to be independent of map\nlocation [9], resulting in essentially different local recurrent networks depending on whether the\nneighborhood is made up of neurons with similar preferred orientation, such as in an iso-orientation\ndomain, or is highly non-uniform, such as close to a pinwheel. The strength of lateral connections,\non the other hand, is unknown. Mari\u00f1o et al. [9] have shown that their data is compatible with a\nmodel showing strong recurrent excitation and inhibition. However, this approach cannot rule out\nalternative explanations accounting for the emergence of orientation tuning in V1. Here, we sys-\ntematically explore the model space, varying the strength of the recurrent excitation and inhibition.\nThis, in effect, allows us to test the full range of models, including feed-forward-, inhibition- and\nexcitation-dominated models as well as balanced recurrent models, and to determine those that are\ncompatible with the observed data.\n\n2 Methods\n\n2.1 Simulation: The Hodgkin-Huxley network model\n\nThe network consists of Hodgkin-Huxley type point neurons and includes three voltage dependent\ncurrents (Na+ and K+ for generation of action potentials, and a non-inactivating K+-current that\nis responsible for spike-frequency adaptation). Spike-frequency adaptation was reduced by a factor\n0.1 for inhibitory neurons. For a detailed description of the model neuron and the parameter values,\nsee Destexhe et al.\n[11]. Every neuron receives afferent, recurrent and background input. We\n\n2\n\n\fused exponential models for the synaptic conductances originating from GABAA-like inhibitory\nand AMPA-like excitatory synapses [12]. Slow NMDA-like excitatory synapses are modeled by a\ndifference of two exponentials (parameters are summarized in Table 1). Additional conductances\nrepresent background activity (Ornstein-Uhlenbeck conductance noise, cf. Destexhe et al. [11]).\n\nTable 1: Parameters of the Hodgkin-Huxley type neural network.\nVALUE\n\nPARAMETER DESCRIPTION\n\nNAff\nNE\nNI\n\u03c3E = \u03c3I\nEe\nEi\n\u03c4E\n\u03c4I\n\u03c41\n\u03c42\nE, \u03c3d\n\u00b5d\nE\nI, \u03c3d\n\u00b5d\nI\ngAff\nE\ngAff\nI\ngII\ngEI\n\nNumber of afferent exc. synaptic connections per cell\nNumber of recurrent exc. synaptic connections per cell\nNumber of recurrent inh. synaptic connections per cell\nSpread of recurrent connections (std. dev.)\nReversal potential excitatory synapses\nReversal potential inhibitory synapses\nTime constant of AMPA-like synapses\nTime constant of GABAA-like synapses\nTime constant of NMDA-like synapses\nTime constant of NMDA-like synapses\nMean and standard deviation of excitatory synaptic delay\nMean and standard deviation of inhibitory synaptic delay\nPeak conductance of afferent input to exc. cells\nPeak conductance of afferent input to inh. cells\nPeak conductance from inh. to inh. cells\nPeak conductance from inh. to exc. cells\n\n20\n100\n50\n4 units (125 \u00b5m)\n0 mV\n-80 mV\n5 ms\n5 ms\n80 ms\n2 ms\n4 ms, 2 ms\n1.25 ms, 1 ms\n141 nS\n0.73 gAff\nE\n1.33 gAff\nE\n1.33 gAff\nE\n\nThe network was composed of 2500 excitatory cells arranged on a 50 \u00d7 50 grid and 833 inhibitory\nneurons placed at random grid locations, thus containing 75% excitatory and 25% inhibitory cells.\nThe complete network modeled a patch of cortex 1.56 \u00d7 1.56 mm2 in size. Connection probabilities\nfor all recurrent connections (between the excitatory and inhibitory population and within the popu-\nlations) were determined from a spatially isotropic Gaussian probability distribution (for parameters,\nsee Table 1) with the same spatial extent for excitation and inhibition, consistent with experimental\nmeasurements [9]. In order to avoid boundary effects, we used periodic boundary conditions. Re-\ncurrent excitatory conductances were modeled as arising from 70% fast (AMPA-like) versus 30%\nslow (NMDA-like) receptors. If a presynaptic neuron generated a spike, this spike was transferred\nto the postsynaptic neuron with a certain delay (parameters are summarized in Table 1).\nThe afferent inputs to excitatory and inhibitory cortical cells were modeled as Poisson spike trains\nwith a time-independent \ufb01ring rate fAff given by\n\n(cid:18)\n\n(cid:18)\n\n\u2212(\u03b8stim \u2212 \u03b8)2\n(2\u03c3Aff)2\n\n(cid:19)(cid:19)\n\nfAff(\u03b8stim) = 30 Hz\n\nrbase + (1 \u2212 rbase) exp\n\n,\n\n(1)\n\nwhere \u03b8stim is the orientation of the presented stimulus, \u03b8 is the preferred orientation of the cell,\nrbase = 0.1 is a baseline \ufb01ring rate, and \u03c3Aff = 27.5\u00b0 is the tuning width. These input spike\ntrains exclusively trigger fast, AMPA-like excitatory synapses. The orientation preference for each\nneuron was assigned according to its location in an arti\ufb01cial orientation map (Figure 2A). This map\nwas calibrated such that the pinwheel distance and the spread of recurrent connections matches\nexperimental data [9].\nIn order to measure the orientation tuning curves of f, Vm, ge, and gi, the response of the network\nto inputs with different orientations was computed for 1.5 s with 0.25 ms resolution (usually, the\nnetwork settled into a steady state after a few hundred milliseconds). We then calculated the \ufb01ring\nrate, the average membrane potential, and the average total excitatory and inhibitory conductances\nfor every cell in an interval between 0.5 s and 1.5 s.\n\n2.2 Quantitative evaluation: Orientation selectivity index (OSI) and OSI-OSI slopes\n\nWe analyze orientation tuning using the orientation selectivity index [13], which is given by\n\n(cid:113)(cid:0)(cid:80)N\n\ni=1\n\nOSI =\n\nR(\u03c6i) cos(2\u03c6i)(cid:1)2\n\n+(cid:0)(cid:80)N\n\ni=1\n\nR(\u03c6i) sin(2\u03c6i)(cid:1)2\n/(cid:80)N\n\nR(\u03c6i).\n\n(2)\n\ni=1\n\n3\n\n\fFigure 2: (A) Arti\ufb01cial orientation map with four pinwheels of alternating handedness arranged on\na 2-dimensional grid. The white (black) circle denotes the one-(two-) \u03c3-area corresponding to the\nradial Gaussian synaptic connection pro\ufb01le (\u03c3E = \u03c3I = 125 \u00b5m). (B) Map OSI of the arti\ufb01cial\norientation map. Pinwheel centers appear in black.\n\nR(\u03c6i) is the value of the quantity whose tuning is considered, in response to a stimulus of ori-\nthe spiking activity). For all measurements, eight stimulus orientations \u03c6i \u2208\nentation \u03c6i (e. g.\n{\u221267.5,\u221245,\u221222.5, 0, 22.5, 45, 67.5, 90} were presented. The OSI is then a measure of tuning\nsharpness ranging from 0 (unselective) to 1 (perfectly selective). In addition, the OSI was used to\ncharacterize the sharpness of the recurrent input a cell receives based on the orientation preference\nmap. To calculate this map OSI, we estimate the local orientation preference distribution by binning\nthe orientation preference of all pixels within a radius of 250 \u00b5m around a cell into bins of 10\u00b0 size;\nthe number of cells in each bin replaces R(\u03c6i). Figure 2 shows the arti\ufb01cial orientation map and the\nmap OSI for the cells in our network model. The map OSI ranges from almost 0 for cells close to\npinwheel centers to almost 1 in the linear zones of the iso-orientation domains.\nThe dependence of each tuning property on the local map OSI was then described by a linear re-\ngression line using the least squares method. These linear \ufb01ts provided a good description of the\nrelationship between map OSI and the tuning of the neuronal properties in the simulations (mean\nsquared deviation around the regression lines was typically below 0.0025 and never above a value\nof 0.015) as well as in the experimental data (mean squared deviation was between 0.009 (gi) and\n0.015 (f)). In order to \ufb01nd the regions of parameter space where the linear relationship predicted by\nthe models is compatible with the data, the con\ufb01dence interval for the slope of the linear \ufb01t to the\ndata was used.\n\n3 Results\n\nThe parameter space of the class of network models considered in this paper is spanned by the peak\nconductance of synaptic excitatory connections to excitatory (gEE) and inhibitory (gIE) neurons.\nWe shall \ufb01rst characterize the operating regimes found in this model space, before comparing the\nlocation dependence of tuning observed in the different models with that found experimentally.\n\n3.1 Operating regimes of the network model\n\nThe operating regimes of a \ufb01ring rate model can be de\ufb01ned in terms of the strength and shape of the\neffective recurrent input [7]. The de\ufb01nitions of Kang et al. [7], however, are based on the analytical\nsolution of a linear \ufb01ring rate model where all neurons are above threshold and cannot be applied\nto the non-linear Hodgkin-Huxley network model used here. Therefore, we characterize the param-\neter space explored here using a numerical de\ufb01nition of the operating regimes. This de\ufb01nition is\nbased on the orientation tuning of the input currents to the excitatory model cells in the orientation\ndomain (0.6 < map OSI < 0.9). Speci\ufb01cally, if the sum of input currents is positive (negative) for\nall presented orientations, recurrent excitation (inhibition) is dominant, and the regime thus excita-\ntory (EXC; respective inhibitory, INH). If the sum of input currents has a positive maximum and a\nnegative minimum (i. e. Mexican-hat like), a model receives signi\ufb01cant excitation as well as inhibi-\n\n4\n\n\fFigure 3: (A) Operating regimes of the network model as a function of the peak conductance of\nsynaptic excitatory connections to excitatory (gEE) and inhibitory (gIE) neurons: FF \u2013 feed-forward,\nEXC \u2013 recurrent excitatory dominated, INH \u2013 recurrent inhibitory dominated, REC \u2013 strong re-\ncurrent excitation and inhibition, and unstable. The conductances are given as multiples of the\nafferent peak conductance of excitatory neurons (gAff\nE ). The \ufb01gure summarizes simulation results\nfor 38 \u00d7 28 different values of gEE and gIE. (B) Tuning curves for one example network in the REC\nregime (marked by a cross in A). Mean responses across cells are shown for the \ufb01ring rate (f), the\nmembrane potential (Vm), the total excitatory (ge), and the total inhibitory conductance (gi), sep-\narately for cells in iso-orientation domains (0.6 < map OSI < 0.9, thick lines) and cells close to\npinwheel centers (map OSI < 0.3, thin lines). For each cell, responses were individually aligned\nto its preferred orientation and normalized to its maximum response; for the Vm tuning curve, the\nmean membrane potential without any stimulation (Vm = \u221264.5 mV) was subtracted beforehand.\nTo allow comparison of the magnitude of gi and ge responses, both types of conductances were\nnormalized to the maximum gi response.\n\ntion and we refer to such a model as operating in the recurrent regime (REC). An example for the\norientation tuning properties observed in the recurrent regime is shown in Figure 3B. Finally, if the\nsum of the absolute values of the currents through excitatory and inhibitory recurrent synapses of\nthe model cells (at preferred orientation) is less than 30% of the current through afferent synapses,\nthe afferent drive is dominant and we call such regimes feed-forward (FF).\nThe regions of parameter space corresponding to these operating regimes are depicted in Figure 3A\nas a function of the peak conductance of synaptic excitatory connections to excitatory (gEE) and\ninhibitory (gIE) neurons. We refer to the network as \u201cunstable\u201d if the model neurons show strong\nresponses (average \ufb01ring rate exceeds 100 Hz) and remain at high \ufb01ring rates if the afferent input\nis turned off; i. e. the network shows self-sustained activity. In this regime, the model neurons lose\ntheir orientation tuning.\n\n3.2 Orientation tuning properties in the different operating regimes\n\nWe analyzed the dependence of the orientation tuning properties on the operating regimes and com-\npared them to the experimental data. For every combination of gEE and gIE, we simulated the re-\nsponses of neurons in the network model to oriented stimuli in order to measure the orientation\ntuning of Vm, f, ge and gi (see Methods). The OSI of each of the four quantities can then be plotted\nagainst the map OSI to reveal the dependence of the tuning on the map location (similar to the ex-\nperimental data shown in Figure 1). The slope of the linear regression of this OSI-OSI dependence\nwas used to characterize the different operating points of the network. Figure 4 shows these slopes\nfor the tuning of f, Vm, ge and gi, as a function of gEE and gIE of the respective Hodgkin-Huxley\nnetwork models (gray scale). Model networks with strong recurrent excitation (large values of gEE),\nas in the REC regime, predict steeper slopes than networks with less recurrent excitation. In other\nwords, as the regime becomes increasingly more recurrently dominated, the recurrent contribution\nleads to sharper tuning in neurons within iso-orientation domains as compared to neurons near the\n\n5\n\n\fFigure 4: Location dependence of orientation tuning of the conductances, the membrane potential,\nand the \ufb01ring rate in the network model. The \ufb01gure shows the slope values of the OSI-OSI regres-\nsion lines (in gray values) as a function of the peak conductance of synaptic excitatory connections\nto excitatory (gEE) and inhibitory (gIE) neurons, separately for the spike rate (A), the membrane po-\ntential (B), the total synaptic excitatory (C), and inhibitory conductance (D). The conductances are\ngiven as multiples of the afferent peak conductance of excitatory neurons (gAff\nE ). Thin lines denote\nthe borders of the different operating regimes (cf. Figure 3). The region delimited by the thick\nyellow line corresponds to slope values within the 95% con\ufb01dence interval of the corresponding\nexperimental data. Note that in (A) this region covers the whole range of operating regimes except\nthe unstable regime.\n\npinwheel centers. However, yet closer to the line of instability the map-dependence of the tuning\nalmost vanishes (slope approaching zero). This re\ufb02ects the strong excitatory recurrent input in the\nEXC regime which leads to an overall increase in the network activity that is almost untuned and\ntherefore provides very similar input to all neurons, regardless of map location. Also, the strongly\ninhibitory-dominated regimes (large values of gIE) at the bottom right corner of Figure 4 are of in-\nterest. Here, the slope of the location dependence becomes negative for the tuning of \ufb01ring rate f\nand membrane potential Vm. Such a sharpening of the tuning close to pinwheels in an inhibition\ndominated regime has been observed elsewhere [8].\nComparing the slope of the OSI-OSI regression lines to the 95% con\ufb01dence interval of the slopes\nestimated from the experimental data (Figure 1) allows us to determine those regions in parame-\nter space that are compatible with the data (yellow contours in Figure 4). The observed location-\nindependence of the \ufb01ring rate tuning is compatible with all stable models in the parameter space\n(Figure 4A) and therefore does not constrain the model class.\nIn contrast to this, the observed\nlocation-dependence of the membrane potential tuning (Figure 4B) and the inhibitory conductance\ntuning (Figure 4D) excludes most of the feed-forward and about half of the inhibitory-dominated\nregime. Most information, however, is gained from the observed location-dependence of the ex-\ncitatory conductance tuning (Figure 4C). It constrains the network to operate in either a recurrent\nregime with strong excitation and inhibition or in a slightly excitatory-dominated regime.\n\n6\n\n\f3.3 Only the strongly recurrent regime satis\ufb01es all constraints\n\nCombining the constraints imposed by the OSI-OSI relationship of the four measured quantities (yel-\nlow contour in both panels of Figure 5), we can conclude that the experimental data constrains the\nnetwork to operate in a recurrent operating regime, with recurrent excitation and inhibition strong,\napproximately balanced, and dominating the afferent input. In addition, we calculated the sum of\nsquared differences between the data points (Figure 1) and the OSI-OSI relationship predicted by\nthe model, for each operating regime. The \u201cbest \ufb01tting\u201d operating regime, which had the lowest\nsquared difference, is marked with a cross in Figure 5. The corresponding simulated tuning curves\nfor orientation domain and pinwheel cells are shown in Figure 3B.\n\nFigure 5: Ratio between (A) the excitatory current through the recurrent synapses and the cur-\nrent through afferent synapses of excitatory model cells and between (B) the inhibitory recur-\nrent and the excitatory afferent current (in gray values). Currents were calculated for stimuli\nat the cells\u2019 preferred orientations, and averaged over all model cells within orientation domains\n(0.6 < map OSI < 0.9). The region delimited by the thick yellow line corresponds to slope val-\nues that are in the 95% con\ufb01dence interval for each experimentally measured quantity (spike rate,\nmembrane potential, the total synaptic excitatory, and inhibitory conductance). The white cross at\n(2.0, 1.7) denotes the combination of model parameters that yields the best \ufb01t to the experimental\ndata (see text). Thin lines denote the borders of the different operating regimes (cf. Figure 3).\n\nIn line with the de\ufb01nition of the operating regimes, the excitatory current through the recurrent\nsynapses (gray values in Figure 5A) plays a negligible role in the feed-forward and in most of the\ninhibitory-dominated regimes. Only in the recurrent and the excitatory-dominated regime is the\nrecurrent current stronger than the afferent current. A similar observation holds for the inhibitory\ncurrent (Figure 5B). The strong recurrent currents in the excitatory-dominated regime re\ufb02ect the\nstrong overall activity that reduce the map-location dependence of the total excitatory and inhibitory\nconductances (cf. Figure 4C and D).\n\n4 Discussion\n\nAlthough much is known about the anatomy of lateral connections in the primary visual cortex of\ncat, the strengths of synapses formed by short-range connections are largely unknown. In our study,\nwe use intracellular physiological measurements to constrain the strengths of these connections.\nExtensively exploring the parameter space of a spiking neural network model, we \ufb01nd that neither\nfeed-forward dominated, nor recurrent excitatory- or inhibitory-dominated networks are consistent\nwith the tuning properties observed in vivo. We therefore conclude that the cortical network in cat\nV1 operates in a regime with a dominant recurrent in\ufb02uence that is approximately balanced between\ninhibition and excitation.\n\n7\n\n\fThe analysis presented here focuses on the steady state the network reaches when presented with\none non-changing orientation. In this light, it is very interesting, that a comparable operating regime\nhas been indicated in an analysis of the dynamic properties of orientation tuning in cat V1 [14].\nOur main \ufb01nding \u2013 tuning properties of cat V1 are best explained by a network operating in a regime\nwith strong recurrent excitation and inhibition \u2013 is robust against variation of the values chosen for\nother parameters not varied here, e. g. gII and gEI (data not shown). Nevertheless, the network ar-\nchitecture is based on a range of basic assumptions: e. g. all neurons in the network receive equally\nsharply tuned input. The explicit inclusion of location dependence of the input tuning might well\nlead to tuning properties compatible with the experimental data in different operating regimes. How-\never, there is no evidence supporting such a location dependence of the afferent input and therefore\nassuming location-independent input seemed the most prudent basis for this analysis. Another as-\nsumption is the absence of untuned inhibition, since the inhibitory neurons in the network presented\nhere receive tuned afferent input, too. The existence of an untuned inhibitory subpopulation is still\na matter of debate (compare e. g. [15] and [16]). Naturally, such an untuned component would\nconsiderably reduce the location dependence of the inhibitory conductance gi. Given that in our\nexploration only a small region of parameter space exists where the slope of gi is steeper than in the\nexperiment, a major contribution of such an untuned inhibition seems incompatible with the data.\nOur analysis demonstrates that the network model is compatible with the data only if it operates in a\nregime that \u2013 due to the strong recurrent connections \u2013 is close to instability. Such a network is very\nsensitive to changes in its governing parameters, e. g. small changes in connection strengths lead to\nlarge changes in the overall \ufb01ring rate: In the regimes close to the line of instability, increasing gEE\nby just 5% typically leads to increases in \ufb01ring rate of around 40% (EXC), respectively 20% (REC).\nIn the other regimes (FF and INH) \ufb01ring rate only changes by around 2\u20133%. In the \u201cbest \ufb01tting\u201d\noperating regime, a 10% change in \ufb01ring rate, which is of similar magnitude as observed \ufb01ring rate\nchanges under attention in macaque V1 [17], is easily achieved by increasing gEE by 2%. It therefore\nseems plausible that one bene\ufb01t of being in such a regime is the possibility of signi\ufb01cantly changing\nthe \u201coperating point\u201d of the network through only small adjustments of the underlying parameters.\nCandidates for such an adjustment could be contextual modulations, adaptation or attentional effects.\nThe analysis presented here is based on data for cat V1. However, the ubiquitous nature of some\nof the architectural principles in neocortex suggests that our results may generalize to other cortical\nareas, functions and species.\n\nReferences\n[1] Hubel, D. H & Wiesel, T. N. (1962) J Physiol 160, 106\u2013154.\n[2] Sompolinsky, H & Shapley, R. (1997) Curr Opin Neurobiol 7, 514\u2013522.\n[3] Ferster, D & Miller, K. D. (2000) Annu Rev Neurosci 23, 441\u2013471.\n[4] Martin, K. A. C. (2002) Curr Opin Neurobiol 12, 418\u2013425.\n[5] Teich, A. F & Qian, N. (2006) J Neurophysiol 96, 404\u2013419.\n[6] Ben-Yishai, R, Bar-Or, R. L, & Sompolinsky, H. (1995) Proc Natl Acad Sci U S A 92, 3844\u20133848.\n[7] Kang, K, Shelley, M, & Sompolinsky, H. (2003) Proc Natl Acad Sci U S A 100, 2848\u20132853.\n[8] McLaughlin, D, Shapley, R, Shelley, M, & Wielaard, D. J. (2000) Proc Natl Acad Sci U S A 97, 8087\u201392.\n[9] Mari\u00f1o, J, Schummers, J, Lyon, D. C, Schwabe, L, Beck, O, Wiesing, P, Obermayer, K, & Sur, M. (2005)\n\nNat Neurosci 8, 194\u2013201.\n\n[10] Nauhaus, I, Benucci, A, Carandini, M, & Ringach, D. L. (2008) Neuron 57, 673\u2013679.\n[11] Destexhe, A, Rudolph, M, Fellous, J, & Sejnowski, T. (2001) Neuroscience 107, 13\u201324.\n[12] Destexhe, A, Mainen, Z. F, & Sejnowski, T. J. (1998) in Methods in neuronal modeling, eds. Koch, C &\n\nSegev, I. (MIT Press, Cambridge, Mass), 2nd edition, pp. 1\u201325.\n\n[13] Swindale, N. V. (1998) Biol Cybern 78, 45\u201356.\n[14] Schummers, J, Cronin, B, Wimmer, K, Stimberg, M, Martin, R, Obermayer, K, Koerding, K, & Sur, M.\n\n(2007) Frontiers in Neuroscience 1, 145\u2013159.\n\n[15] Cardin, J. A, Palmer, L. A, & Contreras, D. (2007) J Neurosci 27, 10333\u201310344.\n[16] Nowak, L. G, Sanchez-Vives, M. V, & McCormick, D. A. (2008) Cereb Cortex 18, 1058\u20131078.\n[17] McAdams, C. J & Maunsell, J. H. (1999) J Neurosci 19, 431\u2013441.\n\n8\n\n\f", "award": [], "sourceid": 146, "authors": [{"given_name": "Klaus", "family_name": "Wimmer", "institution": null}, {"given_name": "Marcel", "family_name": "Stimberg", "institution": null}, {"given_name": "Robert", "family_name": "Martin", "institution": null}, {"given_name": "Lars", "family_name": "Schwabe", "institution": null}, {"given_name": "Jorge", "family_name": "Mari\u00f1o", "institution": null}, {"given_name": "James", "family_name": "Schummers", "institution": null}, {"given_name": "David", "family_name": "Lyon", "institution": null}, {"given_name": "Mriganka", "family_name": "Sur", "institution": null}, {"given_name": "Klaus", "family_name": "Obermayer", "institution": null}]}