A model for intradendritic computation of binocular ... - Semantic Scholar
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A model for intradendritic computation of binocular disparity Kevin A. Archie1 and Bartlett W. Mel2 1
Neuroscience Program, University of Southern California, Los Angeles, California 90089-2520, USA
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Department of Biomedical Engineering, University of Southern California, Los Angeles, California 90089-1451, USA
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Correspondence should be addressed to K.A.A. ([email protected])
Many complex cells in mammalian primary visual cortex are finely tuned to binocular disparity. In the prevailing model, several binocular simple cells drive each disparity-tuned complex cell. However, some cat complex cells receive direct LGN input, and binocular simple cells are rare in macaque. In our biophysically detailed compartmental model, active dendrites of a single neuron perform the multiple simple-cell-like subunit computations that underlie both orientation and disparity tuning. The responses of our detailed model could be predicted by a simple algebraic formula closely related to an ‘energy’ model. Adding inhibitory synapses led to sharper, more contrast-invariant tuning curves. Thus active dendrites could contribute to binocular-disparity tuning in complex cells.
Neurons in mammalian primary visual cortex are grouped into two broad categories: simple cells, sensitive to stimulus orientation and position (or phase), and complex cells, relatively insensitive to the spatial phase of an oriented stimulus. Many complex cells receive input from both eyes and are tuned to binocular disparity, the horizontal offset of features between the two eyes that conveys stimulus depth1,2. Whereas simple cells are often approximated as linear3, complex cells are fundamentally nonlinear: their responses to combinations of stimuli cannot be predicted by summing the responses to individual stimuli. The form of the response nonlinearity suggests a hierarchy in which outputs of multiple simple-cell-like subunits are pooled by complex cells4,5. In a disparity-tuned complex cell, those subunits seem to be binocular and oriented6,7. An early proposal held that simple cells receive input from unoriented cells in the lateral geniculate nucleus (LGN) in cat or cortical layer 4C in monkey, and complex cells pool over the oriented simple-cell outputs4,8. In similar models for disparity tuning, a set of binocular simple cells performs the requisite subunit computations6,7,9,10. However, evidence rules out such a strict hierarchy. In cat, some ‘first order’ complex cells receive direct LGN input, circumventing the simple-cell stage11,12. In macaque, the binocular simple cells required by conventional models are rare or nonexistent. Although many disparity-tuned neurons are found in macaque V1, a recent study found no binocular simple cells: a few had spatially offset light and dark subregions (accounting for the small proportion of binocular cells classified as simple in earlier studies13,14), but all binocular cells showed position-invariant disparity tuning15. If the simple-cell stage is bypassed, where might the required subunit computations be carried out? It has been suggested that simple-cell-like subunit computations might be performed directly within the dendrites of complex cells8,16,17. In theoretical studies, spatially extended dendrites electrotonically compartmentalize synaptic potentials18,19–21, potentially allowing different dendrites of a single neuron to perform quasi-independent nonlinear subunit 54
computations19,22–27. Also, dendrites of neocortical pyramidal cells contain voltage-dependent channels that could profoundly influence their integrative behavior, including NMDA channels and voltage-dependent Na+ and Ca2+ conductances that can amplify synaptic inputs and generate both fast and slow dendritic spikes28,29. Based on this evidence, two modeling studies show that a single neuron with active dendrites can reproduce the defining spatial nonlinearities of complex cells, including position-invariant orientation tuning17 and binocular-disparity tuning30. The morphologically realistic pyramidal cell and the random subunit allocation used in these studies, however, make it difficult to quantify contributions of active dendritic currents, or to compare the behavior of the biophysical model to existing mathematical models. Here we developed a simplified model, patterned after an existing ‘energy’ model for disparity tuning6,7, in which four binocular simple-cell-like subunits were mapped onto four basal branches of a dendritic tree. In addition, although it was previously shown that excitatory inputs onto active dendrites can produce nonlinear boosting interactions, we asked whether observed suppressive interactions between stimuli5–7,15 could likewise result from intradendritic computations, by assuming more realistic (nonzero) background firing rates of inputs, or by including synaptic inhibition16,19.
RESULTS We used compartmental simulations to study the response of a simplified pyramidal neuron with active dendrites (Fig. 1). Low input resistance at the soma (87 MΩ) meant that voltage signals generated within the thin basal dendrites were strongly attenuated when recorded at the cell body, electrically isolating the branches from one another such that a synaptic input changed voltage within its associated branch much more than in the other three. Nonlinear interactions between synaptic inputs were thus limited primarily to those among synapses on the same branch and mediated by dendritic Na+ and K+ currents and voltage-dependent NMDA-type synaptic currents. nature neuroscience • volume 3 no 1 • january 2000
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Fig. 1. Connectivity between LGN center–surround cells and the model complex cell. The cortical neuron had a simplified pyramidal-cell morphology, with a spherical soma 20 µm in diameter (1 compartment), 4 basal dendrites 1.5 µm in diameter and 200 µm long (16 compartments each), and one apical dendrite 5 µm in diameter and 500 µm long (8 compartments). Visual images were presented to the LGN, which consisted of four arrays of spatial filters (ON- and OFF-center cells in left and right eyes). Each of the 4 basal dendrites received input from 16 excitatory synapses (small black circles on dendrites), each provided by a different LGN cell. The collection of LGN cells providing input to each branch corresponded to one of the simple-cell-like subunits in a binocular energy model7. The type and spatial layout of LGN cells driving each branch were identical for each eye (insets).
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To determine the input–output relation of each branch and the degree of crosstalk between branches, we measured the mean firing rate of the cell as varying numbers of synapses were stimulated at 100 Hz on 2 of the 4 basal branches (Fig. 2). Under the conditions tested, the single-branch input–output relation was expansively nonlinear with a threshold of approximately nine active synapses (Fig. 2a). The cell showed significant functional compartmentalization, as indicated by the drop in the cell’s firing rate when a given number of active synapses on a single branch were dispersed over two branches. The branches were not perfectly independent (compare Fig. 2a with response predicted by the sum of two single-branch responses in Fig. 2b; difference in Fig. 2c). Significant facilitative crosstalk peaked when both branches were driven by 10–12 synapses. The two-branch response plot was compared to that of a hypothetical cell with perfectly isolated squaring subunits—the building blocks of an energy model (Fig. 2d). Whereas the compartmental model responds differently than a quadratic subunit model overall, many iso-synapse-count contours (constant levels of synaptic activity) are similar in the two cases. When the number of activated synapses is held constant across stimulus conditions, as in most of our studies, the biophysical model closely mimics a quadratic subunit model. In contrast, an electrically compact neuron does not support independent subunits and would therefore respond at a constant level along every iso-synapse-count contour (Fig. 2e). To test the orientation tuning of the model cell, we presented sinusoidal gratings at various orientations and phases to both eyes at zero horizontal disparity. A grating at the cell’s optimal (vertical) orientation directed most synaptic input to one branch, giving a high output firing rate (37 Hz, Fig. 3a). A grating at the null (horizontal) orientation led to a more diffuse arrangement of the same synaptic activation values, and a far weaker overall response (15 Hz, Fig. 3b). Based on this spatial concentration, the cell showed clear orientation tuning (Fig. 3c, diamonds). The modest variation in the cell’s response with grating phase (Fig. 3d) was due to subsampling of the LGN array, yielding 25% peak-to-peak modulation in 55
total input to the cell. This modulation was distinct from that seen in responses of typical simple cells in that the peaks and valleys of the modulation occurred at identical locations for light and dark stimuli, confirming precise registration rather than spatial segregation of ON and OFF subfields. To verify the importance of the subunit organization, we randomly scrambled the 64 afferents over the basal dendrites. Orientation tuning was abolished by spatial scrambling (Fig. 3c, squares) or by blocking NMDA and dendritic Na+ channels (Fig. 3c, crosses), replicating findings using a morphologically realistic cell17 and showing that a small number of excitatory afferents projecting to a few dendritic branches could produce clear complex-cell-like responses. Table 1. Biophysical simulation parameters. Parameter Rm Eleak Ra Cm Somatic gNa, gDR Dendritic gNa, gDR gAMPA τAMPA (on, off) gNMDA τNMDA (on, off) Esyn (excitory) gGABA τGABA (on, off) EGABA gGABA τAMPA (on, off) EGABA A
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Value 10 kΩ•cm2 –70 mV 250 Ω•cm2 1.0 µF/cm2 0.02, 0.012 S/cm2 0.006, 0.0012 S/cm2 0.18–0.56 nS 0.1 ms, 2 ms 0.36–1.13 nS 0.1 ms, 80 ms 0 mV 0.33 nS 0.1 ms, 50 ms –65 mV 0.099 nS 62.5 ms, 213 ms –95 mV
Hodgkin-Huxley-style voltage-dependent Na+ and K+ channels were distributed throughout the cell, with higher concentrations in the soma than in the dendrites. The channel implementation27 was adapted from ref. 48. This set of conductances led to a somatic input resistance of 87 MΩ for small perturbations around a steady-state voltage of –70 mV, comparable to the measured input resistance of visual cortical neurons in vivo49. Synapses were modeled according to a described kinetic scheme50, with activation and inactivation time constants as shown. Excitatory synapses had both voltageindependent (AMPA/kainate) and voltage-dependent (NMDA) components; peak conductances were scaled inversely by the local input resistance to reduce the dependence of local EPSP size on synapse location. The local peak of an isolated EPSP varied from 0.94 mV near the soma to 2.15 mV at the distal tips, with the corresponding somatic peak varying from 0.81 mV (proximal synapse) to 0.23 mV (distal synapse).
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two bars were presented at corresponding locations in the two eyes (zero horizontal disparity). 80 80 The peak response region was elongated along the zero-disparity diagonal, indicating that the 60 60 Hz Hz cell’s disparity preference was invariant to the 40 40 absolute positions of the bars within the recep20 20 tive field. When the bars were in either of the 16 16 0 0 12 12 ‘opposite-contrast’ conditions (light and dark, 16 16 12 12 8 8 or dark and light), the response pattern was 8 8 4 n1 4 4 4 n2 n1 complementary, with weak responses along the 0 0 n2 0 0 zero-disparity diagonal, and strong responses in the nonzero-disparity flanks. c Our cell’s measured response differed from a linear prediction based on the sum of the 80 responses to the same stimuli presented monocularly (Fig. 4b). Responses to same-contrast con60 ditions showed a modest nonlinear Hz 16 12 40 enhancement along the zero-disparity diagonal n1 8 and somewhat stronger suppression at nonzero 20 4 disparities, whereas responses to opposite-con0 0 trast cases showed a complementary pattern of 0 4 8 12 16 n2 strong suppression at zero disparity and weak boosting at nonzero disparities. This pattern of boosting and suppression is consistent with e dd e recordings from disparity-tuned complex cells 80 80 in macaque V1 (ref. 15). 60 60 A ‘binocular kernel’ was obtained for the Hz Hz model cell by subtracting the mean opposite40 40 contrast response from the mean same-contrast 20 20 response (Fig. 5a). The diagonally elongated 16 16 0 0 12 12 16 16 structure with alternating excitatory and 12 12 8 8 8 8 inhibitory lobes closely resembled kernels gen4 4 4 4 n1 n1 erated from binocular complex cells in cat visun n 0 0 2 2 0 0 al cortex (Fig. 5b), which are often well fit by an Fig. 2. Interaction between synaptic inputs on two branches. Each frame shows a cell’s mean energy model consisting of four simple-cell-like firing rate (averaged over 64 simulations, 1 s each) as the number of fully active (100 Hz) subunits (negative and positive parts of evensynapses was increased on the 2 branches. Within each branch, synapse locations were ran7 dom. The other 2 branches each contained 16 synapses firing at a background rate of 15 Hz. and odd-symmetric Gabor filters) . We attempted to predict the binocular kernels of our bioCell parameters were as in Table 1. (a) Response of the model cell. Solid and dashed lines show two iso-synapse-count contours (see text; solid, 12 total synapses, dashed, 16 total physically modeled cell using a simple algebraic synapses). Dips in these contours reflect compartmentalization of cell. (b) Response of an formula inspired by the structure of the energy idealized cell with single-branch nonlinearity as in (a), but with perfect independence of the model. To do this, we calculated the linear stage two branches. Each branch was assumed to produce an independent nonlinear response f(n), of each of the four subunits by adding the mean where n was the number of active synapses on that branch. f(n) was taken as the average of firing rates of the 16 afferent inputs to each the values on the branch 1 and branch 2 axes in (a), which differed slightly because of the ran- branch. Each of the subtotals was then squared, domization of spike train inputs used in these simulations. The idealized cell output was the and the four resulting values were added to presum of the individual branch responses, f(n1) + f(n2). (c) Difference between (a) and (b). Raised portions represent cell responses above linear prediction. (d) Model cell whose out- dict the cell’s overall response. The resulting put is proportional to n12 + n22. The result is scaled for comparison with other frames. Iso- binocular disparity kernel (Fig. 5c) resembled synapse-count contours are similar to those shown in (a). (e) Response of a cell with branch corresponding plots generated by the compartnonlinearity f(n) as in (a), but with no subunit isolation, giving a response f(n1 + n2). All iso- mental simulations (Fig. 5a) as well as experisynapse-count contours are flat, indicating that the cell is insensitive to the arrangement of mental data (Fig. 5b). synapses across the two branches. To assess the importance of the squaring nonlinearity in determining the basic form of the binocular kernel, we applied three other expansive nonlinear functions to the branchsubunit outputs in the algebraic model: x1.05 (Fig. 5d), x3(Fig. 5e) We tested the binocular-disparity tuning of the cell using bars at its optimal orientation. Following experimental protocols used and ex/10(Fig. 5f). Binocular kernels produced by all cases were in cat7 and monkey15, we ran four stimulus conditions consistsimilar, and on this basis would be difficult to distinguish experimentally. However, the magnitude of the nonlinear response ing of light and dark bars presented simultaneously to the left components that make up the kernel depended strongly on the and right eyes in each of the four possible combinations (Fig. 4). subunit exponent: the output of the x1.05 model, for instance, For each condition, we presented the two bars at 64 horizontal locations in each of the two eyes to generate a two-dimensional contained a nonlinear component that reached only 3% of the binocular-response plot (Fig. 4a). In the ‘same-contrast’ condipeak cell response, whereas the contribution rose to 38% in the tions (both light or both dark), the cell responded best when the quadratic model. For comparison, the magnitude of the nonlinnature neuroscience • volume 3 no 1 • january 2000
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postsynaptic compartment. As in a previous proposal16, the functional role of the synaptic inhibition in this model was to define a subunit both in terms of a stimulus that should be present (carried by excita25 Hz tion), and a stimulus that should be absent (carried by inhibition). 50 Hz 0 0 We examined two different spatial dis75 Hz mV mV tributions of inhibitory synapses to test a 100 Hz –60 –60 biophysical principle formulated assuming 0 250 250 ms ms 0 passive dendritic integration19: that shunting inhibition is most suppressive—serving as a ‘veto’ mechanism—when physically interposed between a distal site of synaptic excitation and the axosomatic site of spike initiation. In the ‘proximal’ condition, the d 50 c 50 16 inhibitory contacts impinging on each branch were distributed over the first 20 40 40 active, ordered µm of the branch where it emerged from the cell body. In the ‘distal’ condition, 30 30 passive, ordered inhibitory contacts were distributed uni20 20 formly over the distal 180 µm of the branch, avoiding the 20 proximal µm. active, scrambled 10 10 Counter to expectations, proximal inhibition weakened orientation tuning relative 0 0 –90° –45° 0° 45° 90° 0° 120° 240° 360° to that seen in the excitation-only condition (compare Fig. 6a with Fig. 3c). Blunting of Grating orientation Grating phase orientation tuning was particularly evident Fig. 3. Response of the model cell to sinusoidal gratings. (a) Presynaptic firing rates are indicated at higher contrasts, and was due to both an by the size of the synapse marker; synapses with firing rate of 0 Hz are not shown. An input at the elevation of responses at null orientations preferred (vertical) orientation leads to grouping of the most active synapses on a single branch. This grouped input leads to robust firing at the cell body (inset voltage trace). The visual stimulus is and an anomalous suppression of responsshown above the voltage trace, with the grid of LGN cell centers indicated by black dots. One LGN es to optimal stimuli. Binocular-response receptive field is shown for scale. (b) An input at the null orientation leads to dispersal of the most plots (Fig. 6b and c) were qualitatively simactive synapses throughout the basal dendrites; the cell responds only weakly to this diffuse pattern ilar to those in the excitation-only condiof input. (c) Firing rate as a function of grating orientation shows clear orientation tuning (dia- tion, though monocular responses were monds) that is abolished when afferents were randomly reassigned to the four branches (squares) very weak. The binocular interaction plots or when the NMDA and dendritic Na+ channels were blocked (crosses). Higher synaptic conduc- differed from the excitation-only case in that tances were used in the latter case to boost firing rates. Each data point is an average over 1-s runs they consisted almost entirely of boosting, at each of 18 different phases. (d) Firing rate for an optimally oriented grating as a function of gratbut not suppresive nonlinearities (Fig. 6d ing phase (diamonds). Modulation is due to spatial variation in total LGN input to cell (see text). and e). Although suppressive binocular Each data point represents a single 1-s run. Two lower curves are for conditions as in part (c). interactions were almost absent in this cell, the binocular disparity kernel appeared normal in form, containing both positive and negative lobes (Fig. 6f). In contrast to proximal inhibition, distal inhibition produced ear component of the biophysically modeled cell response was both sharper orientation tuning than was seen with excitation in the range of 25–60% of the peak response of the cell (10–15 alone, and approximately multiplicative scaling of the tuning Hz boost over a 40-Hz peak above background in the same-concurve with increased contrast (Fig. 6g). The cell lacked monocutrast conditions; 20-Hz suppression relative to peak of 35 Hz in lar responses almost entirely, but showed strong, disparity-tuned different-contrast conditions), values comparable to those reportbinocular responses (Fig. 6h and i). Binocular interactions were ed for a disparity-tuned complex cell in monkey V1 (ref. 15). again exclusively boosting (Fig. 6j and k), and binocular kernels We added synaptic inhibition to the model to investigate seemed normal for a disparity-tuned complex cell, with prowhether it could also contribute to the dendritic subunit comnounced positive and negative lobes (Fig. 6l). putations underlying disparity tuning. Through an inhibitory To identify the source of differences between cases with and interneuron, each excitatory LGN afferent was assumed to drive without inhibition, we recorded simultaneously from each of the a single GABAA-type synapse placed on one of the four dendritfour dendritic branches and the soma while presenting an optiic compartments. This ‘shunting’ inhibitory synapse was mapped mally oriented grating (Fig. 7a). Somatic action potentials were onto a pyramidal cell dendrite at random and balanced so that most often triggered when two or more dendritic branches fired every branch received an identical number of inhibitory contacts in rapid succession, one spike riding ‘piggyback’ on the other. and direct excitatory and indirect inhibitory inputs driven by any For an optimally oriented stimulus, the initial spike was most given LGN afferent never fell on the same branch. This wiring often generated within the primary subunit, followed quickly by constraint reflected the assumption that a Hebb-type rule gova secondary spike in another branch. Isolated dendritic spikes erning synapse survival would discourage stabilization of correthat did not trigger a somatic spike led to EPSP-like somatic depolated excitatory and inhibitory synapses within the same
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one subunit far more than the other three (Fig. 3a). In this configuration, spikes continued to be generated frequently within the primary subunit, but weak activation of the other branches reduced the incidence of secondary spikes in a these branches and increased spike 45 Hz 2° failures at the cell body. In contrast to proximal inhibi0° tion, an identical set of inhibitory synapses distributed distally led to –2° a substantial reduction in the rate 0 Hz of spike initiation within the branch itself, particularly when +25 Hz 2° excitation to the branch was relab b tively weak. This accounted for the 0° strong suppression of responses to 0 Hz stimuli at non-preferred orientations, which tended to produce a –2° –25 Hz mixture of middling excitation –2° 0° 2° –2° 0° 2° –2° 0° 2° –2° 0° 2° and middling inhibition within Left stimulus position each of the four branches. The Fig. 4. Response of the cell to optimally oriented pairs of bars. The four stimulus conditions are shown at near elimination of dendritic top. Each gray box represents a full array of LGN center locations; stimulus bars are shown to scale. (a) Firing spikes in this condition meant that rate of the cell as a function of bar position in the left and right eyes. The elongated band of strong responses somatic action potentials were along the zero-disparity diagonal in the two same-contrast conditions indicates position-invariant disparity tuning. In opposite-contrast cases, strongest responses are in symmetrical off-diagonal flanks. Responses rarely triggered. Optimally orientwere smoothed by convolution with a Gaussian filter (σ = 0.08°) before contours were generated. ed stimuli continued to drive the Conventional disparity-tuning curve is shown above each frame, derived by integrating the binocular- cell strongly in this case, however, response plot along iso-disparity diagonals7. Monocular responses were computed in separate runs with bars as most of the excitation was conpresented to a single eye, but were nearly identical to responses shown here along the edges of the plot (in centrated within the primary substimulus configurations with a bar at either extreme end of the receptive field in one eye). (b) Nonlinear unit, whereas inhibition was component of the firing rate, computed by subtracting the linear prediction based on the monocular largely relegated to distal portions responses. Positive contours are solid, negative contours are dashed. Both suppressive and boosting interac- of the other three branches. tions are seen in complementary patterns for same-contrast and opposite-contrast conditions. Similar mechanisms accounted for the difference in the pattern of binocular interactions seen in cases with and without inhibition—in particular, the disappearance of suppressive nonlinear interaclarizations of several millivolts. The precise size of these events tions when inhibition was included in the model. As for stimdepended on many factors, including the assumed concentrauli at non-optimal orientations, monocular stimuli were poorly tions of Na+ and K+ channels in the dendritic branches. In gensuited to evoke dendritic spikes because they produced only eral, however, the impedance mismatch between the thin half the excitation of binocular stimuli within their respective dendritic branches and the much larger cell body led to a prosubunits; even at high contrast, monocular responses with disnounced attenuation of dendritic spike height as measured at the tal inhibition peaked at 3 Hz. Weak monocular responses led, in soma. The magnitude of this attenuation was illustrated by plotturn, to a floor effect: binocular responses had little room to ting the voltage along a dendritic branch during a dendritic spike drop below the monocular prediction. For such a purely binocfollowed by a somatic spike (Fig. 7b). ular cell, any nonzero response indicated a boosting interacObserved interactions between dendritic and somatic spikes tion between the two eyes. As a result, the binocular-interaction largely accounted for the differential effects of proximal versus plots were nearly identical to the corresponding binoculardistal inhibition. Whereas we expected proximal shuntingresponse plots under both inhibitory conditions. Because many inhibitory synapses to produce powerful branch-specific supalternative assumptions could produce more powerful monocpression, we found that they did little to impede production of ular responses in this model, the balance of boosting versus dendritic spikes in the ‘unclamped’ distal portions of their suppressive interactions in the cell’s binocular receptive field respective branches. Rather, the girdle of perisomatic inhibishould be viewed as a weakly specified parameter of the model, tion in the proximal case led mainly to an overall suppression of rather than a central feature. spike rate at the cell body (compare lowest curve in Fig. 6a with In other conditions tested, we found that distributing excitation-only tuning curve in Fig. 3c). We found this supinhibitory inputs over the full length of the dendrites yielded pression was due to an increased dependence of somatic spike results intermediate to those for proximal and distal conditions. generation on close temporal summation of dendritic spikes in We also obtained qualitatively similar results in simulations using two or more branches. slow, hyperpolarizing (GABAB) inhibitory synapses, although This effect accounted for the peculiar dip at the peak of the cell’s orientation-tuning curve seen at high contrast: optimally the differences between proximal and distal conditions were less oriented stimuli provided lopsided excitation to the cell, driving pronounced in this case. same contrast R
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mal inhibition or with no inhibition (data not shown) were more additive in nature. However, binocular-disparity kernels were similar with or without proximal or distal inhibition, consistent with the very weak dependence of this representation on the form of the underlying subunit nonlinearity (Fig. 5). The lack of subunit specificity of proximally distributed inhibition suggests that the ‘on-path’ condition for maximally effective shunting d e f 1.05 3 x/10 d x e x f e inhibition, derived in the context of passive dendritic integration19, may have limited applicability when dendrites are physiologically active. In our model, perisomatic inhibition primarily controlled output firing rates of the cell as a whole, but had practically no influence on dendritic spike initiation. We also found that, in the present model, proximal shunting inhibition in the absence Fig. 5. Binocular disparity kernels, showing the deviation of the cell’s response from linearity. Kernels were of an expansive boosting mechacalculated by subtracting the mean opposite-contrast response from the mean same-contrast response7. Axes in (a, c–f) were shifted and scaled to match data in (b). (a) Kernel generated from our simulation nism (when NMDA and sodium results. The outward curvature of the nonzero-disparity flanks is due to strong side lobes in the subunit ker- channels were blocked) could not nel. (b) Kernel from a disparity-tuned complex cell in cat (reprinted from ref. 7). (c) Kernel based on alge- produce orientation or disparitybraic subunit model: for each dendritic branch, the firing rates of all 16 synaptic terminals were summed, and tuned complex-cell-like responses. this sum was squared. The complex-cell response was then calculated by summing the four subunit When pure AMPA-type excitatory responses. (d–f) Binocular kernels using alternative expansive nonlinearities applied to each subunit output. conductances were powerful enough By comparison, a cell with linear subunits would result in a uniform value of zero. to evoke reasonable firing rates in response to an optimally oriented grating, the large synaptically induced depolarizations within the dendritic branches resulted in a significant loss of driving force—an DISCUSSION effect that opposed and, in fact, overwhelmed the veto-like nonWe demonstrate that a small number of excitatory, monocular, linearity provided by shunting inhibition. The balance in this comcenter–surround inputs to a few weakly excitable dendritic petition between the shunting veto and classical synaptic saturation branches could produce the phase-invariant orientation and discould probably be altered by spreading the excitatory and inhibitoparity tuning characteristic of many binocular complex cells in ry synapses on a more elaborate dendritic arbor, but this change cats and monkeys. In this model, each of the four dendrites of seems unlikely to generate a nonlinear effect as powerful as that the complex cell plays the role conventionally assigned to a binocassociated with inhibitory control of dendritic spiking. ular simple cell8. In particular, each of the subunits binds togethSurprisingly, binocular responses showed both boosting and er, under a separate ‘threshold’, swaths of the left and right visual suppressive nonlinearities when our model cell was driven exclufields at a fixed horizontal offset, thus ensuring that the cell sively by excitatory synapses (Fig. 4). A boosting interaction was responds best to a consistent horizontal disparity over the entire seen when two light bars were presented to the two eyes at the width of its binocular receptive field. The within-branch nonsame location. This binocular facilitation occurred because, for linearity is threshold-like in that it involves a rapid expansion in this stimulus configuration, synaptic input was spatially conbranch output as synaptic input intensity is increased (Fig. 2). centrated within one or two dendritic branches, effectively driThe expansive nonlinear regime in the excitation-only version of ving these branches up through their expansive nonlinear our model results from the combination of NMDA receptors, input–output regimes (Figs. 2 and 3). Suppressive binocular whose voltage-dependence amplifies synaptic currents primariinteractions, which peaked when bars of opposite polarity were ly in the subthreshold range, and voltage-dependent sodium and presented at zero disparity (Fig. 4), result from this expansive potassium channels, which generate dendritic spikes at a rate that dendritic nonlinearity operating in reverse: each synaptic input to increases supralinearly with locally injected synaptic current. the cell fired at a spontaneous background rate of 15 Hz, so each When distal synaptic inhibition was included, orientationbranch received significant input even without a visual stimulus. tuning curves were significantly sharpened, indicating that the When a stimulus bar was presented, the firing rate of some of the combination of excitatory ‘pushing’ and inhibition ‘pulling’ led to synaptic afferents was driven towards zero because of the oppoa more powerful branch nonlinearity than was seen with excitanency inherent in the underlying center–surround receptive fields. tion alone. With both excitation and distal inhibition, orientaSuch removal of excitation drove a branch-subunit down its tion-tuning curves were boosted in a quasi-multiplicative fashion expansive operating curve, thereby reducing the efficacy of other as contrast was increased, in agreement with empirical data31,32, excitatory inputs onto the same branch. A non-zero background whereas the corresponding increases in responses under proxi-
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zero disparity. There are several other types of disparity-tuned cells2; all could be reproduced with minor contrast 40 40 modifications of the present model. (arbitrary units) 30 30 A nonzero-disparity preference 25 20 (near or far) would result if the 20 20 15 receptive-field centers of the affer10 10 12 ents from one eye were shifted hori0 0 zontally. A ‘tuned-inhibitory’ cell –90° –45° 0° 45° 90° –90° –45° 0° 45° 90° would result if the afferents from Grating orientation Grating orientation one eye only were swapped, so that 45 Hz every ON-center input was replaced b cc hh ii b by an OFF-center input with the same receptive field location and vice versa. This manipulation produces same-contrast responses identical to the opposite-contrast responses 0 Hz shown in Fig. 4. In addition, some +25 Hz dd e jj kk neurons in macaque V1 have spae tially offset light and dark subre0 Hz gions, consistent with a simple-cell receptive field, but diagonal binocular interaction maps characteristic –25 Hz of complex cells15. We were able to replicate responses of this kind by ff ll modifying the model so that afferent contacts from left and right eyes were mapped preferentially to proximal versus distal dendritic sites. Our model critically depends on the assignment of synaptic contacts Fig. 6. Responses of two cells with different configurations of inhibitory synapses. (a) In the proximal con- to distinct dendritic subunits. This dition, 16 inhibitory synapses were placed within the proximal 20 µm of each branch as described in leads to the prediction that in visuMethods. Orientation tuning is shown at four contrasts (12, 15, 20 and 25, in arbitrary units, where a conal cortex, afferents with unoriented trast of 12 was equivalent to that used in excitation-only case of Fig. 3). Each data point is an average over 18 runs. Half width at half maximum varied from 30° at low contrast to 90° at high contrast. (b, c) Binocular monocular receptive fields, wherevresponses in same-contrast (mean of light-light and dark-dark) and different-contrast (mean of light-dark er they impinge on complex-cell and dark-light) conditions. Monocular responses were reduced relative to excitation-only case. dendrites, should, as an outcome of (d, e) Binocular interaction maps corresponding to (b, c), showing deviations from linear prediction based a Hebb-type developmental mechaon monocular firing rates. (f) Corresponding binocular disparity kernel. (g) In distal condition, 16 inhibitory nism, be appropriately grouped into synapses were distributed at random along the distal 180 µm of each branch. Half width at half maximum distinct postsynaptic compartments varied from 20°(low contrast) to 45°(high contrast). Binocular results (h–l) are defined as in (a–f). consistent with the orientation and disparity preferences of the cell. In the present model, each such group contained 16 synapses distributed throughout a dendritic branch 200 µm in length, which would firing rate thus enabled suppressive nonlinear interactions in the account for only a small fraction of the excitatory contacts norabsence of inhibitory synaptic input. mally present on a branch of this length33. Similar results were The simplified pyramidal-cell morphology developed for these experiments allowed us to examine in a controlled fashion both generated with a wide variety of synapse arrangements, demonthe nonlinear input–output behavior of a single dendritic branch strating that the effect is robust to changes in the number and and nonlinear interaction between branches. We found that a spacing of synapses25,27,30. The spatial grouping of afferents might model cell with two active dendrites behaved in key respects like be visualized physiologically in vivo using calcium- and/or volta quadratic function with two input variables (Fig. 2). This conage-sensitive dyes while driving a complex cell with optimal vernection was echoed in the striking resemblance between binocularsus non-optimal visual stimuli. disparity kernels generated by the biophysically detailed Our model also depends on boosting nonlinear interactions compartmental model, which involved numerical integration of between nearby synapses, leading to the prediction that, in complex hundreds of nonlinear differential equations, and those produced cells receiving a significant component of unoriented monocular by a simple algebraic formula akin to standard binocular ‘energy’ input from LGN (in cat) or layer 4C (in monkey), the nonlinear models. This successful reduction of the input–output behavior tuning properties of the cell, including both orientation and disof a neuron with complex intradendritic physiology to a simple parity tuning, should be significantly degraded by an intracellualgebraic formula could provide a method for computationally lar blockade of excitatory voltage-dependent currents (including efficient study of large networks of ‘dendritically realistic’ neurons. Na+, Ca2+ and NMDA-type synaptic currents). Conversely, a uniThe pattern of afferent connectivity in Fig. 1 was designed to form reduction of cellular responses under such a blockade withreplicate the responses of a ‘tuned excitatory’ cell selective for out a loss of tuning would constitute evidence against the present Proximal inhibition
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in the receptive field properties that result from intracortical feedback36. The impor0 tance of cortical feedback should not be Somatic underestimated: one modeling study voltage demonstrates that even the classical com(mV) plex-cell receptive field may fundamen–70 tally depend on intracortical feedback, so 0 50 100 200 250 that cells classified as simple based on their afferent drive would actually respond like complex cells by virtue of their network interactions37. 0 Our model included two simplifying soma Dendritic assumptions regarding the temporal primary branch voltage other branches dynamics of neuronal and synaptic (mV) responses. First, each center–surround –70 afferent providing input to the cell was 50 60 70 80 90 100 modeled as a stationary Poisson process time (ms) with mean firing rate determined by the response of a half-wave-rectified linear filter applied to the input image. This approach ignored the transient compobb nent of the responses of LGN cells and most cortical cells when a stimulus is flashed on or off. Second, the modeled 0 synapses lacked any form of activitydependent dynamics, whereas excitatory Voltage –20 synapses in neocortex typically show (mV) –40 some depression during trains of 200 0 input38,39. Our model must therefore be 1 150 viewed as representing the response of a cortical neuron during a short period 2 100 distance from over which the cell’s input rates are stable Time (ms) soma (µm) 3 50 and the synaptic transfer functions can be adequately approximated by a mean 4 0 value. As typical visual cortex recording Fig. 7. Mechanisms of intradendritic spiking with distal inhibition. (a) Somatic voltage trace in protocols involve counting only a few response to an optimal grating with simultaneous voltage recordings from the four dendritic branches spikes per stimulus7,15, this assumption of and the cell body, shown on an expanded time scale. Bold trace, primary subunit (that which received stationarity seems unlikely to have impormaximal excitation); dashed trace, somatic voltage; thin traces, other dendritic branches. Somatic tant consequences for our results. action potentials occurred most often following two or more closely spaced dendritic spikes, usually The set of somatic and dendritic conevoked first within the primary subunit, followed closely by a secondary, ‘piggyback’ spike in another ductances used in our model does not branch. Not all dendritic spikes evoked somatic spikes, giving rise instead to small EPSP-like depolarizations at the cell body. (b) Space–time plot of membrane potential from the soma to a distal dendritic include all channels known to exist in the tip showing a dendritic spike followed closely by a somatic spike. Time increases to the right; dendritic dendrites of cortical pyramidal cells, omitbranch length is mapped into depth (soma at front, distal tip at back). Dendritic spike is largest at the ting, for example, persistent Na+ channels, distal tip and strongly compressed at the soma–dendritic interface. Branch voltage moves up and down several types of voltage-dependent Ca2+ essentially as a unit (within 1 ms from leading to trailing edge). Attenuation of back-propagated somatic channels, calcium-dependent K+ channels spike was explained by Na+-channel inactivation by preceding dendritic spike in the branch. and so on28. However, simulation studies show that expansive nonlinear boosting of synaptic inputs can arise from several different kinds of voltage-dependent currents acting in various combinations17,25,27,30. Pilot simulations for model. All species of voltage-dependent conductances must be blocked in such an experiment: although NMDA and Na+ conthis study indicated that tuning persists over a large range of conductance values, for example, spanning two orders of magnitude in ductances in this model both contributed to the expansive nonNa+-channel density, with physiological estimates falling near the linearity on which the computation was based, we found in simulations that NMDA, Na+ or Ca2+ conductances alone (or in middle of this range40. Insensitivity to the specific choice, density any combination) could supply a suitable expansive nonlinearity. and implementation of ionic channels present in such models is Intracortical feedback, both excitatory and inhibitory, is imporconsistent with our observation that the nonlinear component of tant in shaping the responses of cortical neurons34,35. Our model, the disparity-tuned responses depended little on the particular functional form of the expansive nonlinearity present in the dendriven exclusively by feedforward connections, is therefore unable dritic branches (Fig. 5). to account for all the response properties of complex cells in visuOur results lend further support to the possibility that active al cortex. For example, our model does not account for the effects dendrites provide compartmentalized interactions among synapof stimuli outside the classical receptive field, presumably carried in tic inputs, where the the cell as a whole computes a sum over the part by long-range horizontal connections, or for dynamic changes 61
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outputs of these nonlinear subunits. Note that the basic structure of this nonlinear computation—sometimes expressed as a sum of product terms—occurs frequently in analyses of both classical and extraclassical receptive-field properties in visual cortex41. For example, beyond our illustration of phase-invariant orientation and disparity tuning, similar computations are performed by cells whose responses are facilitated by stimuli presented in the extraclassical receptive field42, by cells that respond to illusory contours43 and by cells that show multiplicative boosting under the influence of focal visual attention44. This commonality could indicate that active dendritic processing contributes widely to the information-processing functions of cortical neurons.
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METHODS Responses of a single cortical neuron were modeled using the NEURON simulation environment45. The cell morphology is described in Fig. 1; dendritic spines were not modeled, but previous simulations27,46 suggest that the inclusion of spines would not significantly affect the results. Biophysical parameters are summarized in Table 1. Stimulus images were presented to an LGN model consisting of 4 arrays (64 × 64) of spatial filters modeling ON- and OFF-center cells for left and right eyes and covering a 5° × 5° region of visual space. The spatial filter kernel used was a difference-of-Gaussians, adapted from the spatiotemporal LGN model of Wörgötter and Koch 47 , with Gaussian kernels G(x, y) = (K/2πσ2) exp[–(x2 + y2)/2σ2] and parameter values σcenter = 10.6′, σsurround = 31.8′ and Kcenter/Ksurround = 17/16. The resulting center diameter was approximately 0.8°. The firing rate of each LGN unit was calculated by applying the local spatial filter to the input image to obtain the stimulusdriven component of the firing rate, adding a spontaneous firing rate of 15 Hz, and setting all negative values to 0 Hz. The resulting value specified the mean rate of a Poisson train of presynaptic action potentials. A regular 4 × 4 grid covering 0.7° × 0.7° was centered on each of the 4 LGN arrays (see Fig. 3). Each of the 16 grid points specified the receptive field center of an LGN cell forming a single excitatory synapse onto one of the four branches of the cortical neuron. The LGN afferents were grouped into 4 subunits of 16, where each group projected to one of the 4 basal branches (Fig. 1). Odd-symmetric subunits (Fig. 1, upper left and right insets) included four ON-center and four OFF-center cells from each eye, and even-symmetric subunits (Fig. 1, lower left and right insets) included eight ON- or eight OFF-center cells from each eye. In each case, the 16 afferents projecting to a given branch were drawn from LGN cells of the same type and in the identical spatial layout in the left and right eyes. In cases involving inhibition, a GABAA-type inhibitory synapse was created to mirror the activity of each of the 64 excitatory synapses. Each synapse was presumed to arise from an inhibitory interneuron with tuning similar to that of the corresponding excitatory input. The inhibitory afferents thus had center-surround receptive fields; orientation-tuned inhibition has yielded similar results in pilot studies. Each of the 64 inhibitory synapses was assigned to a branch at random and subject to 2 constraints: each branch received 16 inhibitory synapses, and corresponding excitatory and inhibitory connections never fell on the same branch. Each inhibitory synapse was driven by an independent Poisson spike train, with mean rate one half that of its corresponding excitatory afferent. Three parameters determining the magnitude of inhibition—the number of inhibitory synapses per branch, their peak conductances and their firing rates—were only weakly connature neuroscience • volume 3 no 1 • january 2000
strained by available data. The values of these parameters were thus set within reasonable ranges, then adjusted until the inhibition produced strong, but not total, suppression of cell firing rates. Stimulus contrasts were then adjusted to drive the cell through a reasonable range of output rates. The cell began each simulation uniformly polarized at a resting potential near –70 mV. The firing rate typically stabilized after a brief initial period of depolarization from rest. Each stimulus was presented for 1.05 s, with all response rates calculated over a 1-s interval following the initial 50 ms simulation transient.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation. We thank Dan Ruderman for contributions to early stages of this work, and Margaret Livingstone and Gary Holt for comments.
RECEIVED 29 JUNE; ACCEPTED 5 NOVEMBER 1999 1. Barlow, H. B., Blakemore, C. & Pettigrew, J. D. The neural mechanism of binocular depth perception. J. Physiol. (Lond.) 193, 327–342 (1967). 2. Poggio, G. F. & Fischer, B. Binocular interaction and depth sensitivity in striate and prestriate cortex of behaving rhesus monkey. J. Neurophysiol. 40, 1392–1405 (1977). 3. Movshon, J. A., Thompson, I. D. & Tolhurst, D. J. Spatial summation in the receptive fields of simple cells in the cat’s striate cortex. J. Physiol. (Lond.) 283, 53–77 (1978). 4. Hubel, D. H. & Wiesel, T. N. Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. J. Physiol. (Lond.) 160, 106–154 (1962). 5. Movshon, J. A., Thompson, I. D. & Tolhurst, D. J. Receptive field organization of complex cells in the cat’s striate cortex. J. Physiol. (Lond.) 283, 79–99 (1978). 6. Ohzawa, I., DeAngelis, G. C. & Freeman, R. D. Stereoscopic depth discrimination in the visual cortex: neurons ideally suited as disparity detectors. Science 249, 1037–1041 (1990). 7. Ohzawa, I., DeAngelis, G. C. & Freeman, R. D. Encoding of binocular disparity by complex cells in the cat’s visual cortex. J. Neurophysiol. 77, 2879–2909 (1997). 8. Hubel, D. H. Exploration of the primary visual cortex, 1955–1978. Nature 299, 515–524 (1982). 9. Fleet, D. J., Wagner, H. & Heeger, D. J. Neural encoding of binocular disparity: energy models, position shifts and phase shifts. Vision Res. 36, 1839–1857 (1996). 10. Zhu, Y.-D. & Qian, N. Binocular receptive field models, disparity tuning, and characteristic disparity. Neural Comput. 8, 1611–1641 (1996). 11. LeVay, S. & Gilbert, C. Laminar patterns of geniculocortical projection in the cat. Brain Res. 113, 1–19 (1976). 12. Alonso, J.-M. & Martinez, L. M. Functional connectivity between simple cells and complex cells in cat striate cortex. Nat. Neurosci. 1, 395–403 (1998). 13. Hubel, D. H. & Wiesel, T. N. Receptive fields and functional architecture of monkey striate cortex. J. Physiol. (Lond.) 195, 215–243 (1968). 14. Poggio, G. F., Doty, R. W. Jr & Talbot, W. H. Foveal striate cortex of behaving monkey: single-neuron responses to square-wave gratings during fixation of gaze. J. Neurophysiol. 40, 1369–1391 (1977). 15. Livingstone, M. S. & Tsao, D. Y. Receptive fields of disparity-selective neurons in macaque striate cortex. Nat. Neurosci. 2, 825–832 (1999). 16. Koch, C. & Poggio, T. in Synaptic Function (eds. Edelman, G. M., Gall, W. E. & Cowan, W. M.) 637–697 (Wiley, New York, 1987). 17. Mel, B. W., Ruderman, D. L. & Archie, K. A. Translation-invariant orientation tuning in visual ‘complex’ cells could derive from intradendritic computations. J. Neurosci. 18, 4325–4334 (1998). 18. Rall, W. & Rinzel, J. Branch input resistance and steady attenuation for input to one branch of a dendritic neuron model. Biophys. J. 13, 648–688 (1973). 19. Koch, C., Poggio, T. & Torre, V. Retinal ganglion cells: a functional interpretation of dendritic morphology. Philos. Trans. R. Soc. Lond. B Biol. Sci. 298, 227–264 (1982). 20. Woolf, T., Shepherd, G. & Greer, C. Local information processing in dendritic trees: Subsets of spines in granule cells of the mammalian olfactory bulb. J. Neurosci. 11, 1837–1854 (1991). 21. Zador, A. M., Agmon-Snir, H. & Segev, I. The morphoelectrotonic transform: a graphical approach to dendritic function. J. Neurosci. 15, 1669–1682 (1995). 22. Koch, C., Poggio, T. & Torre, V. Computations in the vertebrate retina: Gain enhancement, differentiation and motion discrimination. Trends Neurosci. 9, 204–211 (1986). 23. Rall, W. & Segev, I. in Synaptic Function (eds. Edelman, G. M., Gall, W. E. & Cowan, W. M.) 605–636 (Wiley, New York, 1987).
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24. Mel, B. W. in Advances in Neural Information Processing Systems Vol. 4 (eds. Moody, J. E., Hanson, S. J. & Lippman, R. P.) 35–42 (Kaufmann, San Mateo, California, 1992). 25. Mel, B. W. NMDA-based pattern discrimination in a modeled cortical neuron. Neural Comput. 4, 502–517 (1992). 26. Borg-Graham, L. J. & Grzywacz, N. M. in Single Neuron Computation (eds. McKenna, T., Davis, J. & Zornetzer, S. 347–375 (Academic, San Diego, 1992). 27. Mel, B. W. Synaptic integration in an excitable dendritic tree. J. Neurophysiol. 70, 1086–1101 (1993). 28. Johnston, D., Magee, J. C., Colbert, C. M. & Christie, B. R. Active properties of neuronal dendrites. Annu. Rev. Neurosci. 19, 165–186 (1996). 29. Stuart, G., Spruston, N., Sakmann, B. & Häusser, M. Action potential initiation and backpropagation in neurons of the mammalian CNS. Trends Neurosci. 20, 125–131 (1997). 30. Mel, B. W., Ruderman, D. L. & Archie, K. A. in Advances in Neural Information Processing Systems Vol. 10 (eds. Jordan, M. I., Kearns, M. J. & Solla, S. A.) 208–214 (MIT Press, Cambridge, Massachusetts, 1998). 31. Sclar, G. & Freeman, R. D. Orientation selectivity in the cat’s striate cortex is invariant with stimulus contrast. Exp. Brain Res. 46, 457–461 (1982). 32. Skottun, B. C., Ohzawa, I., Sclar, G. & Freeman, R. D. The effects of contrast on visual orientation and spatial frequency discrimination: a comparison of single cells and behavior. J. Neurophysiol. 57, 773–786 (1987). 33. Beaulieu, C. & Colonnier, M. A laminar analysis of the number of roundasymmetrical and flat-symmetrical synapses on spines, dendritic trunks, and cell bodies in area 17 of the cat. J. Comp. Neurol. 231, 180–189 (1985). 34. Vidyasagar, T. R., Pei, X. & Volgushev, M. Multiple mechanisms underlying the orientation selectivity of visual cortical neurones. Trends Neurosci. 19, 272–277 (1996). 35. Sompolinksy, H. & Shapley, R. New perspectives on the mechanisms for orientation selectivity. Curr. Opin. Neurobiol. 7, 514–522 (1997). 36. Gilbert, C. D., Das, A., Ito, M., Kapadia, M. & Westheimer, G. Spatial integration and cortical dynamics. Proc. Natl. Acad. Sci. USA 93, 615–622 (1996). 37. Chance, F. S., Nelson, S. B. & Abbott, L. F. Complex cells as cortically amplified simple cells. Nat. Neurosci. 2, 277–282 (1999).
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A model for intradendritic computation of binocular disparity Kevin A. Archie1 and Bartlett W. Mel2 1
Neuroscience Program, University of Southern California, Los Angeles, California 90089-2520, USA
2
Department of Biomedical Engineering, University of Southern California, Los Angeles, California 90089-1451, USA
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Correspondence should be addressed to K.A.A. ([email protected])
Many complex cells in mammalian primary visual cortex are finely tuned to binocular disparity. In the prevailing model, several binocular simple cells drive each disparity-tuned complex cell. However, some cat complex cells receive direct LGN input, and binocular simple cells are rare in macaque. In our biophysically detailed compartmental model, active dendrites of a single neuron perform the multiple simple-cell-like subunit computations that underlie both orientation and disparity tuning. The responses of our detailed model could be predicted by a simple algebraic formula closely related to an ‘energy’ model. Adding inhibitory synapses led to sharper, more contrast-invariant tuning curves. Thus active dendrites could contribute to binocular-disparity tuning in complex cells.
Neurons in mammalian primary visual cortex are grouped into two broad categories: simple cells, sensitive to stimulus orientation and position (or phase), and complex cells, relatively insensitive to the spatial phase of an oriented stimulus. Many complex cells receive input from both eyes and are tuned to binocular disparity, the horizontal offset of features between the two eyes that conveys stimulus depth1,2. Whereas simple cells are often approximated as linear3, complex cells are fundamentally nonlinear: their responses to combinations of stimuli cannot be predicted by summing the responses to individual stimuli. The form of the response nonlinearity suggests a hierarchy in which outputs of multiple simple-cell-like subunits are pooled by complex cells4,5. In a disparity-tuned complex cell, those subunits seem to be binocular and oriented6,7. An early proposal held that simple cells receive input from unoriented cells in the lateral geniculate nucleus (LGN) in cat or cortical layer 4C in monkey, and complex cells pool over the oriented simple-cell outputs4,8. In similar models for disparity tuning, a set of binocular simple cells performs the requisite subunit computations6,7,9,10. However, evidence rules out such a strict hierarchy. In cat, some ‘first order’ complex cells receive direct LGN input, circumventing the simple-cell stage11,12. In macaque, the binocular simple cells required by conventional models are rare or nonexistent. Although many disparity-tuned neurons are found in macaque V1, a recent study found no binocular simple cells: a few had spatially offset light and dark subregions (accounting for the small proportion of binocular cells classified as simple in earlier studies13,14), but all binocular cells showed position-invariant disparity tuning15. If the simple-cell stage is bypassed, where might the required subunit computations be carried out? It has been suggested that simple-cell-like subunit computations might be performed directly within the dendrites of complex cells8,16,17. In theoretical studies, spatially extended dendrites electrotonically compartmentalize synaptic potentials18,19–21, potentially allowing different dendrites of a single neuron to perform quasi-independent nonlinear subunit 54
computations19,22–27. Also, dendrites of neocortical pyramidal cells contain voltage-dependent channels that could profoundly influence their integrative behavior, including NMDA channels and voltage-dependent Na+ and Ca2+ conductances that can amplify synaptic inputs and generate both fast and slow dendritic spikes28,29. Based on this evidence, two modeling studies show that a single neuron with active dendrites can reproduce the defining spatial nonlinearities of complex cells, including position-invariant orientation tuning17 and binocular-disparity tuning30. The morphologically realistic pyramidal cell and the random subunit allocation used in these studies, however, make it difficult to quantify contributions of active dendritic currents, or to compare the behavior of the biophysical model to existing mathematical models. Here we developed a simplified model, patterned after an existing ‘energy’ model for disparity tuning6,7, in which four binocular simple-cell-like subunits were mapped onto four basal branches of a dendritic tree. In addition, although it was previously shown that excitatory inputs onto active dendrites can produce nonlinear boosting interactions, we asked whether observed suppressive interactions between stimuli5–7,15 could likewise result from intradendritic computations, by assuming more realistic (nonzero) background firing rates of inputs, or by including synaptic inhibition16,19.
RESULTS We used compartmental simulations to study the response of a simplified pyramidal neuron with active dendrites (Fig. 1). Low input resistance at the soma (87 MΩ) meant that voltage signals generated within the thin basal dendrites were strongly attenuated when recorded at the cell body, electrically isolating the branches from one another such that a synaptic input changed voltage within its associated branch much more than in the other three. Nonlinear interactions between synaptic inputs were thus limited primarily to those among synapses on the same branch and mediated by dendritic Na+ and K+ currents and voltage-dependent NMDA-type synaptic currents. nature neuroscience • volume 3 no 1 • january 2000
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100 µm
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Fig. 1. Connectivity between LGN center–surround cells and the model complex cell. The cortical neuron had a simplified pyramidal-cell morphology, with a spherical soma 20 µm in diameter (1 compartment), 4 basal dendrites 1.5 µm in diameter and 200 µm long (16 compartments each), and one apical dendrite 5 µm in diameter and 500 µm long (8 compartments). Visual images were presented to the LGN, which consisted of four arrays of spatial filters (ON- and OFF-center cells in left and right eyes). Each of the 4 basal dendrites received input from 16 excitatory synapses (small black circles on dendrites), each provided by a different LGN cell. The collection of LGN cells providing input to each branch corresponded to one of the simple-cell-like subunits in a binocular energy model7. The type and spatial layout of LGN cells driving each branch were identical for each eye (insets).
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To determine the input–output relation of each branch and the degree of crosstalk between branches, we measured the mean firing rate of the cell as varying numbers of synapses were stimulated at 100 Hz on 2 of the 4 basal branches (Fig. 2). Under the conditions tested, the single-branch input–output relation was expansively nonlinear with a threshold of approximately nine active synapses (Fig. 2a). The cell showed significant functional compartmentalization, as indicated by the drop in the cell’s firing rate when a given number of active synapses on a single branch were dispersed over two branches. The branches were not perfectly independent (compare Fig. 2a with response predicted by the sum of two single-branch responses in Fig. 2b; difference in Fig. 2c). Significant facilitative crosstalk peaked when both branches were driven by 10–12 synapses. The two-branch response plot was compared to that of a hypothetical cell with perfectly isolated squaring subunits—the building blocks of an energy model (Fig. 2d). Whereas the compartmental model responds differently than a quadratic subunit model overall, many iso-synapse-count contours (constant levels of synaptic activity) are similar in the two cases. When the number of activated synapses is held constant across stimulus conditions, as in most of our studies, the biophysical model closely mimics a quadratic subunit model. In contrast, an electrically compact neuron does not support independent subunits and would therefore respond at a constant level along every iso-synapse-count contour (Fig. 2e). To test the orientation tuning of the model cell, we presented sinusoidal gratings at various orientations and phases to both eyes at zero horizontal disparity. A grating at the cell’s optimal (vertical) orientation directed most synaptic input to one branch, giving a high output firing rate (37 Hz, Fig. 3a). A grating at the null (horizontal) orientation led to a more diffuse arrangement of the same synaptic activation values, and a far weaker overall response (15 Hz, Fig. 3b). Based on this spatial concentration, the cell showed clear orientation tuning (Fig. 3c, diamonds). The modest variation in the cell’s response with grating phase (Fig. 3d) was due to subsampling of the LGN array, yielding 25% peak-to-peak modulation in 55
total input to the cell. This modulation was distinct from that seen in responses of typical simple cells in that the peaks and valleys of the modulation occurred at identical locations for light and dark stimuli, confirming precise registration rather than spatial segregation of ON and OFF subfields. To verify the importance of the subunit organization, we randomly scrambled the 64 afferents over the basal dendrites. Orientation tuning was abolished by spatial scrambling (Fig. 3c, squares) or by blocking NMDA and dendritic Na+ channels (Fig. 3c, crosses), replicating findings using a morphologically realistic cell17 and showing that a small number of excitatory afferents projecting to a few dendritic branches could produce clear complex-cell-like responses. Table 1. Biophysical simulation parameters. Parameter Rm Eleak Ra Cm Somatic gNa, gDR Dendritic gNa, gDR gAMPA τAMPA (on, off) gNMDA τNMDA (on, off) Esyn (excitory) gGABA τGABA (on, off) EGABA gGABA τAMPA (on, off) EGABA A
A
A
B
B
B
Value 10 kΩ•cm2 –70 mV 250 Ω•cm2 1.0 µF/cm2 0.02, 0.012 S/cm2 0.006, 0.0012 S/cm2 0.18–0.56 nS 0.1 ms, 2 ms 0.36–1.13 nS 0.1 ms, 80 ms 0 mV 0.33 nS 0.1 ms, 50 ms –65 mV 0.099 nS 62.5 ms, 213 ms –95 mV
Hodgkin-Huxley-style voltage-dependent Na+ and K+ channels were distributed throughout the cell, with higher concentrations in the soma than in the dendrites. The channel implementation27 was adapted from ref. 48. This set of conductances led to a somatic input resistance of 87 MΩ for small perturbations around a steady-state voltage of –70 mV, comparable to the measured input resistance of visual cortical neurons in vivo49. Synapses were modeled according to a described kinetic scheme50, with activation and inactivation time constants as shown. Excitatory synapses had both voltageindependent (AMPA/kainate) and voltage-dependent (NMDA) components; peak conductances were scaled inversely by the local input resistance to reduce the dependence of local EPSP size on synapse location. The local peak of an isolated EPSP varied from 0.94 mV near the soma to 2.15 mV at the distal tips, with the corresponding somatic peak varying from 0.81 mV (proximal synapse) to 0.23 mV (distal synapse).
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two bars were presented at corresponding locations in the two eyes (zero horizontal disparity). 80 80 The peak response region was elongated along the zero-disparity diagonal, indicating that the 60 60 Hz Hz cell’s disparity preference was invariant to the 40 40 absolute positions of the bars within the recep20 20 tive field. When the bars were in either of the 16 16 0 0 12 12 ‘opposite-contrast’ conditions (light and dark, 16 16 12 12 8 8 or dark and light), the response pattern was 8 8 4 n1 4 4 4 n2 n1 complementary, with weak responses along the 0 0 n2 0 0 zero-disparity diagonal, and strong responses in the nonzero-disparity flanks. c Our cell’s measured response differed from a linear prediction based on the sum of the 80 responses to the same stimuli presented monocularly (Fig. 4b). Responses to same-contrast con60 ditions showed a modest nonlinear Hz 16 12 40 enhancement along the zero-disparity diagonal n1 8 and somewhat stronger suppression at nonzero 20 4 disparities, whereas responses to opposite-con0 0 trast cases showed a complementary pattern of 0 4 8 12 16 n2 strong suppression at zero disparity and weak boosting at nonzero disparities. This pattern of boosting and suppression is consistent with e dd e recordings from disparity-tuned complex cells 80 80 in macaque V1 (ref. 15). 60 60 A ‘binocular kernel’ was obtained for the Hz Hz model cell by subtracting the mean opposite40 40 contrast response from the mean same-contrast 20 20 response (Fig. 5a). The diagonally elongated 16 16 0 0 12 12 16 16 structure with alternating excitatory and 12 12 8 8 8 8 inhibitory lobes closely resembled kernels gen4 4 4 4 n1 n1 erated from binocular complex cells in cat visun n 0 0 2 2 0 0 al cortex (Fig. 5b), which are often well fit by an Fig. 2. Interaction between synaptic inputs on two branches. Each frame shows a cell’s mean energy model consisting of four simple-cell-like firing rate (averaged over 64 simulations, 1 s each) as the number of fully active (100 Hz) subunits (negative and positive parts of evensynapses was increased on the 2 branches. Within each branch, synapse locations were ran7 dom. The other 2 branches each contained 16 synapses firing at a background rate of 15 Hz. and odd-symmetric Gabor filters) . We attempted to predict the binocular kernels of our bioCell parameters were as in Table 1. (a) Response of the model cell. Solid and dashed lines show two iso-synapse-count contours (see text; solid, 12 total synapses, dashed, 16 total physically modeled cell using a simple algebraic synapses). Dips in these contours reflect compartmentalization of cell. (b) Response of an formula inspired by the structure of the energy idealized cell with single-branch nonlinearity as in (a), but with perfect independence of the model. To do this, we calculated the linear stage two branches. Each branch was assumed to produce an independent nonlinear response f(n), of each of the four subunits by adding the mean where n was the number of active synapses on that branch. f(n) was taken as the average of firing rates of the 16 afferent inputs to each the values on the branch 1 and branch 2 axes in (a), which differed slightly because of the ran- branch. Each of the subtotals was then squared, domization of spike train inputs used in these simulations. The idealized cell output was the and the four resulting values were added to presum of the individual branch responses, f(n1) + f(n2). (c) Difference between (a) and (b). Raised portions represent cell responses above linear prediction. (d) Model cell whose out- dict the cell’s overall response. The resulting put is proportional to n12 + n22. The result is scaled for comparison with other frames. Iso- binocular disparity kernel (Fig. 5c) resembled synapse-count contours are similar to those shown in (a). (e) Response of a cell with branch corresponding plots generated by the compartnonlinearity f(n) as in (a), but with no subunit isolation, giving a response f(n1 + n2). All iso- mental simulations (Fig. 5a) as well as experisynapse-count contours are flat, indicating that the cell is insensitive to the arrangement of mental data (Fig. 5b). synapses across the two branches. To assess the importance of the squaring nonlinearity in determining the basic form of the binocular kernel, we applied three other expansive nonlinear functions to the branchsubunit outputs in the algebraic model: x1.05 (Fig. 5d), x3(Fig. 5e) We tested the binocular-disparity tuning of the cell using bars at its optimal orientation. Following experimental protocols used and ex/10(Fig. 5f). Binocular kernels produced by all cases were in cat7 and monkey15, we ran four stimulus conditions consistsimilar, and on this basis would be difficult to distinguish experimentally. However, the magnitude of the nonlinear response ing of light and dark bars presented simultaneously to the left components that make up the kernel depended strongly on the and right eyes in each of the four possible combinations (Fig. 4). subunit exponent: the output of the x1.05 model, for instance, For each condition, we presented the two bars at 64 horizontal locations in each of the two eyes to generate a two-dimensional contained a nonlinear component that reached only 3% of the binocular-response plot (Fig. 4a). In the ‘same-contrast’ condipeak cell response, whereas the contribution rose to 38% in the tions (both light or both dark), the cell responded best when the quadratic model. For comparison, the magnitude of the nonlinnature neuroscience • volume 3 no 1 • january 2000
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postsynaptic compartment. As in a previous proposal16, the functional role of the synaptic inhibition in this model was to define a subunit both in terms of a stimulus that should be present (carried by excita25 Hz tion), and a stimulus that should be absent (carried by inhibition). 50 Hz 0 0 We examined two different spatial dis75 Hz mV mV tributions of inhibitory synapses to test a 100 Hz –60 –60 biophysical principle formulated assuming 0 250 250 ms ms 0 passive dendritic integration19: that shunting inhibition is most suppressive—serving as a ‘veto’ mechanism—when physically interposed between a distal site of synaptic excitation and the axosomatic site of spike initiation. In the ‘proximal’ condition, the d 50 c 50 16 inhibitory contacts impinging on each branch were distributed over the first 20 40 40 active, ordered µm of the branch where it emerged from the cell body. In the ‘distal’ condition, 30 30 passive, ordered inhibitory contacts were distributed uni20 20 formly over the distal 180 µm of the branch, avoiding the 20 proximal µm. active, scrambled 10 10 Counter to expectations, proximal inhibition weakened orientation tuning relative 0 0 –90° –45° 0° 45° 90° 0° 120° 240° 360° to that seen in the excitation-only condition (compare Fig. 6a with Fig. 3c). Blunting of Grating orientation Grating phase orientation tuning was particularly evident Fig. 3. Response of the model cell to sinusoidal gratings. (a) Presynaptic firing rates are indicated at higher contrasts, and was due to both an by the size of the synapse marker; synapses with firing rate of 0 Hz are not shown. An input at the elevation of responses at null orientations preferred (vertical) orientation leads to grouping of the most active synapses on a single branch. This grouped input leads to robust firing at the cell body (inset voltage trace). The visual stimulus is and an anomalous suppression of responsshown above the voltage trace, with the grid of LGN cell centers indicated by black dots. One LGN es to optimal stimuli. Binocular-response receptive field is shown for scale. (b) An input at the null orientation leads to dispersal of the most plots (Fig. 6b and c) were qualitatively simactive synapses throughout the basal dendrites; the cell responds only weakly to this diffuse pattern ilar to those in the excitation-only condiof input. (c) Firing rate as a function of grating orientation shows clear orientation tuning (dia- tion, though monocular responses were monds) that is abolished when afferents were randomly reassigned to the four branches (squares) very weak. The binocular interaction plots or when the NMDA and dendritic Na+ channels were blocked (crosses). Higher synaptic conduc- differed from the excitation-only case in that tances were used in the latter case to boost firing rates. Each data point is an average over 1-s runs they consisted almost entirely of boosting, at each of 18 different phases. (d) Firing rate for an optimally oriented grating as a function of gratbut not suppresive nonlinearities (Fig. 6d ing phase (diamonds). Modulation is due to spatial variation in total LGN input to cell (see text). and e). Although suppressive binocular Each data point represents a single 1-s run. Two lower curves are for conditions as in part (c). interactions were almost absent in this cell, the binocular disparity kernel appeared normal in form, containing both positive and negative lobes (Fig. 6f). In contrast to proximal inhibition, distal inhibition produced ear component of the biophysically modeled cell response was both sharper orientation tuning than was seen with excitation in the range of 25–60% of the peak response of the cell (10–15 alone, and approximately multiplicative scaling of the tuning Hz boost over a 40-Hz peak above background in the same-concurve with increased contrast (Fig. 6g). The cell lacked monocutrast conditions; 20-Hz suppression relative to peak of 35 Hz in lar responses almost entirely, but showed strong, disparity-tuned different-contrast conditions), values comparable to those reportbinocular responses (Fig. 6h and i). Binocular interactions were ed for a disparity-tuned complex cell in monkey V1 (ref. 15). again exclusively boosting (Fig. 6j and k), and binocular kernels We added synaptic inhibition to the model to investigate seemed normal for a disparity-tuned complex cell, with prowhether it could also contribute to the dendritic subunit comnounced positive and negative lobes (Fig. 6l). putations underlying disparity tuning. Through an inhibitory To identify the source of differences between cases with and interneuron, each excitatory LGN afferent was assumed to drive without inhibition, we recorded simultaneously from each of the a single GABAA-type synapse placed on one of the four dendritfour dendritic branches and the soma while presenting an optiic compartments. This ‘shunting’ inhibitory synapse was mapped mally oriented grating (Fig. 7a). Somatic action potentials were onto a pyramidal cell dendrite at random and balanced so that most often triggered when two or more dendritic branches fired every branch received an identical number of inhibitory contacts in rapid succession, one spike riding ‘piggyback’ on the other. and direct excitatory and indirect inhibitory inputs driven by any For an optimally oriented stimulus, the initial spike was most given LGN afferent never fell on the same branch. This wiring often generated within the primary subunit, followed quickly by constraint reflected the assumption that a Hebb-type rule gova secondary spike in another branch. Isolated dendritic spikes erning synapse survival would discourage stabilization of correthat did not trigger a somatic spike led to EPSP-like somatic depolated excitatory and inhibitory synapses within the same
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one subunit far more than the other three (Fig. 3a). In this configuration, spikes continued to be generated frequently within the primary subunit, but weak activation of the other branches reduced the incidence of secondary spikes in a these branches and increased spike 45 Hz 2° failures at the cell body. In contrast to proximal inhibi0° tion, an identical set of inhibitory synapses distributed distally led to –2° a substantial reduction in the rate 0 Hz of spike initiation within the branch itself, particularly when +25 Hz 2° excitation to the branch was relab b tively weak. This accounted for the 0° strong suppression of responses to 0 Hz stimuli at non-preferred orientations, which tended to produce a –2° –25 Hz mixture of middling excitation –2° 0° 2° –2° 0° 2° –2° 0° 2° –2° 0° 2° and middling inhibition within Left stimulus position each of the four branches. The Fig. 4. Response of the cell to optimally oriented pairs of bars. The four stimulus conditions are shown at near elimination of dendritic top. Each gray box represents a full array of LGN center locations; stimulus bars are shown to scale. (a) Firing spikes in this condition meant that rate of the cell as a function of bar position in the left and right eyes. The elongated band of strong responses somatic action potentials were along the zero-disparity diagonal in the two same-contrast conditions indicates position-invariant disparity tuning. In opposite-contrast cases, strongest responses are in symmetrical off-diagonal flanks. Responses rarely triggered. Optimally orientwere smoothed by convolution with a Gaussian filter (σ = 0.08°) before contours were generated. ed stimuli continued to drive the Conventional disparity-tuning curve is shown above each frame, derived by integrating the binocular- cell strongly in this case, however, response plot along iso-disparity diagonals7. Monocular responses were computed in separate runs with bars as most of the excitation was conpresented to a single eye, but were nearly identical to responses shown here along the edges of the plot (in centrated within the primary substimulus configurations with a bar at either extreme end of the receptive field in one eye). (b) Nonlinear unit, whereas inhibition was component of the firing rate, computed by subtracting the linear prediction based on the monocular largely relegated to distal portions responses. Positive contours are solid, negative contours are dashed. Both suppressive and boosting interac- of the other three branches. tions are seen in complementary patterns for same-contrast and opposite-contrast conditions. Similar mechanisms accounted for the difference in the pattern of binocular interactions seen in cases with and without inhibition—in particular, the disappearance of suppressive nonlinear interaclarizations of several millivolts. The precise size of these events tions when inhibition was included in the model. As for stimdepended on many factors, including the assumed concentrauli at non-optimal orientations, monocular stimuli were poorly tions of Na+ and K+ channels in the dendritic branches. In gensuited to evoke dendritic spikes because they produced only eral, however, the impedance mismatch between the thin half the excitation of binocular stimuli within their respective dendritic branches and the much larger cell body led to a prosubunits; even at high contrast, monocular responses with disnounced attenuation of dendritic spike height as measured at the tal inhibition peaked at 3 Hz. Weak monocular responses led, in soma. The magnitude of this attenuation was illustrated by plotturn, to a floor effect: binocular responses had little room to ting the voltage along a dendritic branch during a dendritic spike drop below the monocular prediction. For such a purely binocfollowed by a somatic spike (Fig. 7b). ular cell, any nonzero response indicated a boosting interacObserved interactions between dendritic and somatic spikes tion between the two eyes. As a result, the binocular-interaction largely accounted for the differential effects of proximal versus plots were nearly identical to the corresponding binoculardistal inhibition. Whereas we expected proximal shuntingresponse plots under both inhibitory conditions. Because many inhibitory synapses to produce powerful branch-specific supalternative assumptions could produce more powerful monocpression, we found that they did little to impede production of ular responses in this model, the balance of boosting versus dendritic spikes in the ‘unclamped’ distal portions of their suppressive interactions in the cell’s binocular receptive field respective branches. Rather, the girdle of perisomatic inhibishould be viewed as a weakly specified parameter of the model, tion in the proximal case led mainly to an overall suppression of rather than a central feature. spike rate at the cell body (compare lowest curve in Fig. 6a with In other conditions tested, we found that distributing excitation-only tuning curve in Fig. 3c). We found this supinhibitory inputs over the full length of the dendrites yielded pression was due to an increased dependence of somatic spike results intermediate to those for proximal and distal conditions. generation on close temporal summation of dendritic spikes in We also obtained qualitatively similar results in simulations using two or more branches. slow, hyperpolarizing (GABAB) inhibitory synapses, although This effect accounted for the peculiar dip at the peak of the cell’s orientation-tuning curve seen at high contrast: optimally the differences between proximal and distal conditions were less oriented stimuli provided lopsided excitation to the cell, driving pronounced in this case. same contrast R
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mal inhibition or with no inhibition (data not shown) were more additive in nature. However, binocular-disparity kernels were similar with or without proximal or distal inhibition, consistent with the very weak dependence of this representation on the form of the underlying subunit nonlinearity (Fig. 5). The lack of subunit specificity of proximally distributed inhibition suggests that the ‘on-path’ condition for maximally effective shunting d e f 1.05 3 x/10 d x e x f e inhibition, derived in the context of passive dendritic integration19, may have limited applicability when dendrites are physiologically active. In our model, perisomatic inhibition primarily controlled output firing rates of the cell as a whole, but had practically no influence on dendritic spike initiation. We also found that, in the present model, proximal shunting inhibition in the absence Fig. 5. Binocular disparity kernels, showing the deviation of the cell’s response from linearity. Kernels were of an expansive boosting mechacalculated by subtracting the mean opposite-contrast response from the mean same-contrast response7. Axes in (a, c–f) were shifted and scaled to match data in (b). (a) Kernel generated from our simulation nism (when NMDA and sodium results. The outward curvature of the nonzero-disparity flanks is due to strong side lobes in the subunit ker- channels were blocked) could not nel. (b) Kernel from a disparity-tuned complex cell in cat (reprinted from ref. 7). (c) Kernel based on alge- produce orientation or disparitybraic subunit model: for each dendritic branch, the firing rates of all 16 synaptic terminals were summed, and tuned complex-cell-like responses. this sum was squared. The complex-cell response was then calculated by summing the four subunit When pure AMPA-type excitatory responses. (d–f) Binocular kernels using alternative expansive nonlinearities applied to each subunit output. conductances were powerful enough By comparison, a cell with linear subunits would result in a uniform value of zero. to evoke reasonable firing rates in response to an optimally oriented grating, the large synaptically induced depolarizations within the dendritic branches resulted in a significant loss of driving force—an DISCUSSION effect that opposed and, in fact, overwhelmed the veto-like nonWe demonstrate that a small number of excitatory, monocular, linearity provided by shunting inhibition. The balance in this comcenter–surround inputs to a few weakly excitable dendritic petition between the shunting veto and classical synaptic saturation branches could produce the phase-invariant orientation and discould probably be altered by spreading the excitatory and inhibitoparity tuning characteristic of many binocular complex cells in ry synapses on a more elaborate dendritic arbor, but this change cats and monkeys. In this model, each of the four dendrites of seems unlikely to generate a nonlinear effect as powerful as that the complex cell plays the role conventionally assigned to a binocassociated with inhibitory control of dendritic spiking. ular simple cell8. In particular, each of the subunits binds togethSurprisingly, binocular responses showed both boosting and er, under a separate ‘threshold’, swaths of the left and right visual suppressive nonlinearities when our model cell was driven exclufields at a fixed horizontal offset, thus ensuring that the cell sively by excitatory synapses (Fig. 4). A boosting interaction was responds best to a consistent horizontal disparity over the entire seen when two light bars were presented to the two eyes at the width of its binocular receptive field. The within-branch nonsame location. This binocular facilitation occurred because, for linearity is threshold-like in that it involves a rapid expansion in this stimulus configuration, synaptic input was spatially conbranch output as synaptic input intensity is increased (Fig. 2). centrated within one or two dendritic branches, effectively driThe expansive nonlinear regime in the excitation-only version of ving these branches up through their expansive nonlinear our model results from the combination of NMDA receptors, input–output regimes (Figs. 2 and 3). Suppressive binocular whose voltage-dependence amplifies synaptic currents primariinteractions, which peaked when bars of opposite polarity were ly in the subthreshold range, and voltage-dependent sodium and presented at zero disparity (Fig. 4), result from this expansive potassium channels, which generate dendritic spikes at a rate that dendritic nonlinearity operating in reverse: each synaptic input to increases supralinearly with locally injected synaptic current. the cell fired at a spontaneous background rate of 15 Hz, so each When distal synaptic inhibition was included, orientationbranch received significant input even without a visual stimulus. tuning curves were significantly sharpened, indicating that the When a stimulus bar was presented, the firing rate of some of the combination of excitatory ‘pushing’ and inhibition ‘pulling’ led to synaptic afferents was driven towards zero because of the oppoa more powerful branch nonlinearity than was seen with excitanency inherent in the underlying center–surround receptive fields. tion alone. With both excitation and distal inhibition, orientaSuch removal of excitation drove a branch-subunit down its tion-tuning curves were boosted in a quasi-multiplicative fashion expansive operating curve, thereby reducing the efficacy of other as contrast was increased, in agreement with empirical data31,32, excitatory inputs onto the same branch. A non-zero background whereas the corresponding increases in responses under proxi-
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zero disparity. There are several other types of disparity-tuned cells2; all could be reproduced with minor contrast 40 40 modifications of the present model. (arbitrary units) 30 30 A nonzero-disparity preference 25 20 (near or far) would result if the 20 20 15 receptive-field centers of the affer10 10 12 ents from one eye were shifted hori0 0 zontally. A ‘tuned-inhibitory’ cell –90° –45° 0° 45° 90° –90° –45° 0° 45° 90° would result if the afferents from Grating orientation Grating orientation one eye only were swapped, so that 45 Hz every ON-center input was replaced b cc hh ii b by an OFF-center input with the same receptive field location and vice versa. This manipulation produces same-contrast responses identical to the opposite-contrast responses 0 Hz shown in Fig. 4. In addition, some +25 Hz dd e jj kk neurons in macaque V1 have spae tially offset light and dark subre0 Hz gions, consistent with a simple-cell receptive field, but diagonal binocular interaction maps characteristic –25 Hz of complex cells15. We were able to replicate responses of this kind by ff ll modifying the model so that afferent contacts from left and right eyes were mapped preferentially to proximal versus distal dendritic sites. Our model critically depends on the assignment of synaptic contacts Fig. 6. Responses of two cells with different configurations of inhibitory synapses. (a) In the proximal con- to distinct dendritic subunits. This dition, 16 inhibitory synapses were placed within the proximal 20 µm of each branch as described in leads to the prediction that in visuMethods. Orientation tuning is shown at four contrasts (12, 15, 20 and 25, in arbitrary units, where a conal cortex, afferents with unoriented trast of 12 was equivalent to that used in excitation-only case of Fig. 3). Each data point is an average over 18 runs. Half width at half maximum varied from 30° at low contrast to 90° at high contrast. (b, c) Binocular monocular receptive fields, wherevresponses in same-contrast (mean of light-light and dark-dark) and different-contrast (mean of light-dark er they impinge on complex-cell and dark-light) conditions. Monocular responses were reduced relative to excitation-only case. dendrites, should, as an outcome of (d, e) Binocular interaction maps corresponding to (b, c), showing deviations from linear prediction based a Hebb-type developmental mechaon monocular firing rates. (f) Corresponding binocular disparity kernel. (g) In distal condition, 16 inhibitory nism, be appropriately grouped into synapses were distributed at random along the distal 180 µm of each branch. Half width at half maximum distinct postsynaptic compartments varied from 20°(low contrast) to 45°(high contrast). Binocular results (h–l) are defined as in (a–f). consistent with the orientation and disparity preferences of the cell. In the present model, each such group contained 16 synapses distributed throughout a dendritic branch 200 µm in length, which would firing rate thus enabled suppressive nonlinear interactions in the account for only a small fraction of the excitatory contacts norabsence of inhibitory synaptic input. mally present on a branch of this length33. Similar results were The simplified pyramidal-cell morphology developed for these experiments allowed us to examine in a controlled fashion both generated with a wide variety of synapse arrangements, demonthe nonlinear input–output behavior of a single dendritic branch strating that the effect is robust to changes in the number and and nonlinear interaction between branches. We found that a spacing of synapses25,27,30. The spatial grouping of afferents might model cell with two active dendrites behaved in key respects like be visualized physiologically in vivo using calcium- and/or volta quadratic function with two input variables (Fig. 2). This conage-sensitive dyes while driving a complex cell with optimal vernection was echoed in the striking resemblance between binocularsus non-optimal visual stimuli. disparity kernels generated by the biophysically detailed Our model also depends on boosting nonlinear interactions compartmental model, which involved numerical integration of between nearby synapses, leading to the prediction that, in complex hundreds of nonlinear differential equations, and those produced cells receiving a significant component of unoriented monocular by a simple algebraic formula akin to standard binocular ‘energy’ input from LGN (in cat) or layer 4C (in monkey), the nonlinear models. This successful reduction of the input–output behavior tuning properties of the cell, including both orientation and disof a neuron with complex intradendritic physiology to a simple parity tuning, should be significantly degraded by an intracellualgebraic formula could provide a method for computationally lar blockade of excitatory voltage-dependent currents (including efficient study of large networks of ‘dendritically realistic’ neurons. Na+, Ca2+ and NMDA-type synaptic currents). Conversely, a uniThe pattern of afferent connectivity in Fig. 1 was designed to form reduction of cellular responses under such a blockade withreplicate the responses of a ‘tuned excitatory’ cell selective for out a loss of tuning would constitute evidence against the present Proximal inhibition
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in the receptive field properties that result from intracortical feedback36. The impor0 tance of cortical feedback should not be Somatic underestimated: one modeling study voltage demonstrates that even the classical com(mV) plex-cell receptive field may fundamen–70 tally depend on intracortical feedback, so 0 50 100 200 250 that cells classified as simple based on their afferent drive would actually respond like complex cells by virtue of their network interactions37. 0 Our model included two simplifying soma Dendritic assumptions regarding the temporal primary branch voltage other branches dynamics of neuronal and synaptic (mV) responses. First, each center–surround –70 afferent providing input to the cell was 50 60 70 80 90 100 modeled as a stationary Poisson process time (ms) with mean firing rate determined by the response of a half-wave-rectified linear filter applied to the input image. This approach ignored the transient compobb nent of the responses of LGN cells and most cortical cells when a stimulus is flashed on or off. Second, the modeled 0 synapses lacked any form of activitydependent dynamics, whereas excitatory Voltage –20 synapses in neocortex typically show (mV) –40 some depression during trains of 200 0 input38,39. Our model must therefore be 1 150 viewed as representing the response of a cortical neuron during a short period 2 100 distance from over which the cell’s input rates are stable Time (ms) soma (µm) 3 50 and the synaptic transfer functions can be adequately approximated by a mean 4 0 value. As typical visual cortex recording Fig. 7. Mechanisms of intradendritic spiking with distal inhibition. (a) Somatic voltage trace in protocols involve counting only a few response to an optimal grating with simultaneous voltage recordings from the four dendritic branches spikes per stimulus7,15, this assumption of and the cell body, shown on an expanded time scale. Bold trace, primary subunit (that which received stationarity seems unlikely to have impormaximal excitation); dashed trace, somatic voltage; thin traces, other dendritic branches. Somatic tant consequences for our results. action potentials occurred most often following two or more closely spaced dendritic spikes, usually The set of somatic and dendritic conevoked first within the primary subunit, followed closely by a secondary, ‘piggyback’ spike in another ductances used in our model does not branch. Not all dendritic spikes evoked somatic spikes, giving rise instead to small EPSP-like depolarizations at the cell body. (b) Space–time plot of membrane potential from the soma to a distal dendritic include all channels known to exist in the tip showing a dendritic spike followed closely by a somatic spike. Time increases to the right; dendritic dendrites of cortical pyramidal cells, omitbranch length is mapped into depth (soma at front, distal tip at back). Dendritic spike is largest at the ting, for example, persistent Na+ channels, distal tip and strongly compressed at the soma–dendritic interface. Branch voltage moves up and down several types of voltage-dependent Ca2+ essentially as a unit (within 1 ms from leading to trailing edge). Attenuation of back-propagated somatic channels, calcium-dependent K+ channels spike was explained by Na+-channel inactivation by preceding dendritic spike in the branch. and so on28. However, simulation studies show that expansive nonlinear boosting of synaptic inputs can arise from several different kinds of voltage-dependent currents acting in various combinations17,25,27,30. Pilot simulations for model. All species of voltage-dependent conductances must be blocked in such an experiment: although NMDA and Na+ conthis study indicated that tuning persists over a large range of conductance values, for example, spanning two orders of magnitude in ductances in this model both contributed to the expansive nonNa+-channel density, with physiological estimates falling near the linearity on which the computation was based, we found in simulations that NMDA, Na+ or Ca2+ conductances alone (or in middle of this range40. Insensitivity to the specific choice, density any combination) could supply a suitable expansive nonlinearity. and implementation of ionic channels present in such models is Intracortical feedback, both excitatory and inhibitory, is imporconsistent with our observation that the nonlinear component of tant in shaping the responses of cortical neurons34,35. Our model, the disparity-tuned responses depended little on the particular functional form of the expansive nonlinearity present in the dendriven exclusively by feedforward connections, is therefore unable dritic branches (Fig. 5). to account for all the response properties of complex cells in visuOur results lend further support to the possibility that active al cortex. For example, our model does not account for the effects dendrites provide compartmentalized interactions among synapof stimuli outside the classical receptive field, presumably carried in tic inputs, where the the cell as a whole computes a sum over the part by long-range horizontal connections, or for dynamic changes 61
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outputs of these nonlinear subunits. Note that the basic structure of this nonlinear computation—sometimes expressed as a sum of product terms—occurs frequently in analyses of both classical and extraclassical receptive-field properties in visual cortex41. For example, beyond our illustration of phase-invariant orientation and disparity tuning, similar computations are performed by cells whose responses are facilitated by stimuli presented in the extraclassical receptive field42, by cells that respond to illusory contours43 and by cells that show multiplicative boosting under the influence of focal visual attention44. This commonality could indicate that active dendritic processing contributes widely to the information-processing functions of cortical neurons.
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METHODS Responses of a single cortical neuron were modeled using the NEURON simulation environment45. The cell morphology is described in Fig. 1; dendritic spines were not modeled, but previous simulations27,46 suggest that the inclusion of spines would not significantly affect the results. Biophysical parameters are summarized in Table 1. Stimulus images were presented to an LGN model consisting of 4 arrays (64 × 64) of spatial filters modeling ON- and OFF-center cells for left and right eyes and covering a 5° × 5° region of visual space. The spatial filter kernel used was a difference-of-Gaussians, adapted from the spatiotemporal LGN model of Wörgötter and Koch 47 , with Gaussian kernels G(x, y) = (K/2πσ2) exp[–(x2 + y2)/2σ2] and parameter values σcenter = 10.6′, σsurround = 31.8′ and Kcenter/Ksurround = 17/16. The resulting center diameter was approximately 0.8°. The firing rate of each LGN unit was calculated by applying the local spatial filter to the input image to obtain the stimulusdriven component of the firing rate, adding a spontaneous firing rate of 15 Hz, and setting all negative values to 0 Hz. The resulting value specified the mean rate of a Poisson train of presynaptic action potentials. A regular 4 × 4 grid covering 0.7° × 0.7° was centered on each of the 4 LGN arrays (see Fig. 3). Each of the 16 grid points specified the receptive field center of an LGN cell forming a single excitatory synapse onto one of the four branches of the cortical neuron. The LGN afferents were grouped into 4 subunits of 16, where each group projected to one of the 4 basal branches (Fig. 1). Odd-symmetric subunits (Fig. 1, upper left and right insets) included four ON-center and four OFF-center cells from each eye, and even-symmetric subunits (Fig. 1, lower left and right insets) included eight ON- or eight OFF-center cells from each eye. In each case, the 16 afferents projecting to a given branch were drawn from LGN cells of the same type and in the identical spatial layout in the left and right eyes. In cases involving inhibition, a GABAA-type inhibitory synapse was created to mirror the activity of each of the 64 excitatory synapses. Each synapse was presumed to arise from an inhibitory interneuron with tuning similar to that of the corresponding excitatory input. The inhibitory afferents thus had center-surround receptive fields; orientation-tuned inhibition has yielded similar results in pilot studies. Each of the 64 inhibitory synapses was assigned to a branch at random and subject to 2 constraints: each branch received 16 inhibitory synapses, and corresponding excitatory and inhibitory connections never fell on the same branch. Each inhibitory synapse was driven by an independent Poisson spike train, with mean rate one half that of its corresponding excitatory afferent. Three parameters determining the magnitude of inhibition—the number of inhibitory synapses per branch, their peak conductances and their firing rates—were only weakly connature neuroscience • volume 3 no 1 • january 2000
strained by available data. The values of these parameters were thus set within reasonable ranges, then adjusted until the inhibition produced strong, but not total, suppression of cell firing rates. Stimulus contrasts were then adjusted to drive the cell through a reasonable range of output rates. The cell began each simulation uniformly polarized at a resting potential near –70 mV. The firing rate typically stabilized after a brief initial period of depolarization from rest. Each stimulus was presented for 1.05 s, with all response rates calculated over a 1-s interval following the initial 50 ms simulation transient.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation. We thank Dan Ruderman for contributions to early stages of this work, and Margaret Livingstone and Gary Holt for comments.
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