{"title": "An Architectural Mechanism for Direction-tuned Cortical Simple Cells: The Role of Mutual Inhibition", "book": "Advances in Neural Information Processing Systems", "page_first": 104, "page_last": 110, "abstract": null, "full_text": "An Architectural Mechanism for \n\nDirection-tuned Cortical Simple Cells: \n\nThe Role of Mutual Inhibition \n\nSilvio P. Sabatini \nsilvio@dibe.unige.it \n\nFabio Solari \n\nfabio@dibe .unige.it \n\nGiacomo M. Bisio \nbisio@dibe.unige.it \n\nDepartment of Biophysical and Electronic Engineering \n\nPSPC Research Group \nGenova, 1-16145, Italy \n\nAbstract \n\nA linear architectural model of cortical simple cells is presented. \nThe model evidences how mutual inhibition, occurring through \nsynaptic coupling functions asymmetrically distributed in space, \ncan be a possible basis for a wide variety of spatio-temporal simple \ncell response properties, including direction selectivity and velocity \ntuning. While spatial asymmetries are included explicitly in the \nstructure of the inhibitory interconnections, temporal asymmetries \noriginate from the specific mutual inhibition scheme considered. \nExtensive simulations supporting the model are reported. \n\n1 \n\nINTRODUCTION \n\nOne of the most distinctive features of striate cortex neurons is their combined \nselectivity for stimulus orientation and the direction of motion. The majority of \nsimple cells, indeed, responds better to sinusoidal gratings that are moving in one \ndirection than to the opposite one, exhibiting also a narrower velocity tuning with \nrespect to that of geniculate cells. Recent theoretical and neurophysiological stud(cid:173)\nies [1] [2] pointed out that the initial stage of direction selectivity can be related \nto the linear space-time receptive field structure of simple cells. A large class of \nsimple cells has a very specific space-time behavior in which the spatial phase of \nthe receptive field changes gradually as a function of time. This results in receptive \nfield profiles that are tilted in the space-time domain. To account for the origin \nof this particular spatio-temporal inseparability, numerous models have been pro(cid:173)\nposed postulating the existence of structural asymmetries of the geniculo-cortical \nprojections both in the temporal and in the spatial domains (for a review, see [3] \n\n\fAn Architectural Mechanism/or Direction-tuned Cortical Simple Cells \n\n105 \n\ninhibitory [~i:h01 \n: m \n\npool\n\n! \n\nLayer 1 \n\n' T oJ \neq \n\nLayer 2 \n\n{;l4.\\D------t \n\nI \n~ el \n\neo \n\nez \n\nge~iculate \n\nmput \n\n+ \n\nk1d \n~ k1a m \nKZd \nI \nI~I \n!ez \n\neO \n\n(a) \n\n(b) \n\nFigure 1: (a) A schematic neural circuitry for the mutual inhibition; (b) equivalent \nblock diagram representation. \n\n[4]) . Among them, feed-forward inhibition along the non-preferred direction, and \nthe combination of lagged and non-lagged geniculate inputs to the cortex have been \ncommonly suggested as the major mechanisms. \n\nIn this paper, within a linear field theory framework, we propose and analyse an \narchitectural model for dynamic receptive field formation, based on intracortical \ninteractions occurring through asymmetric mutual inhibition schemes. \n\n2 MODELING INTRACORTICAL PROCESSING \n\nThe computational characteristics of each neuron are not independent of the ones \nof other neurons laying in the same layer, rather, they are often the consequence of \na collective behavior of neighboring cells. To understand how intracortical circuits \nmay affect the response properties of simple cells one can study their structure \nand function at many levels of organization, from subcellular, driven primarily by \nbiophysical data, to systemic, driven by functional considerations. In this study, we \npresent a model at the intermediate abstraction level to combine both functional \nand neurophysiological descriptions into an analytic model of cortical simple cells. \n\n2.1 STRUCTURE OF THE MODEL \n\nFollowing a linear neural field approach [5] [6], we regard visual cortex as a con(cid:173)\ntinuous distribution of neurons and synapses. Accordingly, the geniculo-cortical \npathway is modeled by a multi-layer network interconnected through feed-forward \nand feedback connections, both inter- and intra-layers. Each location on the corti(cid:173)\ncal plane represents a homogeneous population of cells, and connections represent \naverage interactions among populations. Such connections can be modeled by spa(cid:173)\ntial coupling functions which represent the spread of the synaptic influence of a \n\n\f106 \n\nS. P. Sabatini, F. Solari and G. M. Bisio \n\npopulation on its neighbors, as mediated by local axonal and dendritic fields. From \nan architectural point of view, we assume the superposition of feed-forward (i.e., \ngeniculate) and intracortical contributions which arise from inhibitory pools whose \nactivity is also primed by a geniculate excitatory drive. A schematic diagram show(cid:173)\ning the \"building blocks\" of the model is depicted in Fig. 1. The dynamics of each \npopulation is modeled as first-order low-pass filters characterized by time constants \nT'S . For the sake of simplicity, we restrict our analysis to 1-0 case, assuming that \nsuch direction is orthogonal to the preferred direction of the receptive field [7]. This \n1-0 model would produce spatia-temporal results that are directly compared with \nthe spatio-temporal plots usually obtained when an optimal stimulus is moved along \nthe direction orthogonal to the preferred direction of the receptive field. \nGeniculate contributions eo(x, t) are modeled directly by a spatiotemporal convo(cid:173)\nlution of the visual input s(x, t) and a separable kernel ho(x, t) characteri~d in the \nspatial domain by a Gaussian shape with spatial extent 0'0 and, in the temporal \ndomain, by a first-order temporal low-pass filter with time constant TO. The out(cid:173)\nput el(x, t) of the inhibitory neuron population results from the mutual inhibitory \nscheme through spatially organized pre- and post-synaptic sites, modeled by the \nkernels kIa(x -~) and kId(X - ~), respectively: \n\ndel(X,t) \n\nTl \n\ndt =-el(x,t)+ \n\nkId(x-~)[eo(-~,t)-bm(-~,t)]d~ \n\nJ \n\nm(x,t) = J k~a(X-~)el(-~)~ \n\n(1) \n\n(2) \n\nwhere the function m(x, t) describes the spatia-temporal mutual inhibitory inter(cid:173)\nactions, and b is the inhibition strength. The layer 2 cortical excitation e2(x, t) is \nthe result of feed-forward contributions collected (k2d) from the inhibitory loop, at \naxonal synaptic sites, and the geniculate input (eo(x, t)) . To focus the attention on \nthe inhibitory loop, in the following we assume a one-to-one mapping from layer 1 \nto layer 2, i.e., k2d(x -~) = 6(x - ~), consequently: \n\nde2( x, t) \n\ndt = -e2(x, t) + eo(x, t) - bm(x, t) \n\nT2 \n\n(3) \n\nwhere TI and T2 are the time constants associated to layer 1 and layer 2, respectively. \n\n2.2 AVERAGE INTRACORTICAL CONNECTIVITY \n\nWhen assessing the role of intracortical circuits on the receptive field properties \nof cortical cells, one important issue concerns the spatial localization of inhibitory \nand excitatory influences. In a previous work [8] we evidenced how the steady(cid:173)\nstate solution of Eqs. (1 )-(3) can give rise to highly structured Gabor-like re(cid:173)\nceptive field profiles, when inhibition arises from laterally distributed clusters of \nIn this case, the effective intrinsic kernel kl(x), defined as kl(x _~) ~f \ncells. \nJ J k1a( -x', -e)k1d( x - x', ~ - e)dx'de, can be modeled as the sum of two Gaus(cid:173)\nsians symmetrically offset with respect to the target cell (see Fig. 2): \n\nk1(x) = z= -\n\n( WI \nv2~ ~ \n\n1 \n\nexp -(x - d l ) /20'd + -\n\n2 \n\n2 \n\n[ \n\nexp -(x + d2) /20'2] \n2 ) \n\n2 \n\n[ \n\n. \n\n(4) \n\nW2 \n0'2 \n\nThis work is aimed to investigate how spatial asymmetries in the intracortical cou(cid:173)\npling function lead to non-separable space-time interactions within the resulting \ndischarge field of the simple cells. To this end, we varied systematically the ge(cid:173)\nometrical parameters (0', W, d) of the inhibitory kernel to consider three different \n\n\fAn Architectural Mechanism/or Direction-tuned Cortical Simple Cells \n\n107 \n\nw \n\nFigure 2: The basic inhibitory kernel used k1(x -e). The cell in the center receives \ninhibitory contributions from laterally distributed clusters of cells. The asymmetric \nkernels used in the model derive from this basic kernel by systematic variations of \nits geometrical parameters (see Table 1). \n\ntypes of asymmetries: (1) different spatial spread of inhibition (i.e., 0'1 i= 0'2); (2) \ndifferent amount of inhibition (W1 i= W2); (3) different spatial offset (d1 i= d2). A \nmore rigorous treatment should take care also of the continuous distortion of the \ntopographic map [9] . In our analysis this would result in a continuous deformation \nof the inhibitory kernel, but for the small distances within which inhibition occurs, \nthe approximation of a uniform mapping produces only a negligible error. \n\nArchitectural parameters were determined from reliable measured values of recep(cid:173)\ntive fields of simple cells [10] [11]. Concerning the spatial domain, we fixed the size \n(0'0) of the initial receptive field (due to geniculate contributions) for an \"average\" \ncortical simple cell with a resultant discharge field of\"\" 20 ; and we adjusted, ac(cid:173)\ncordingly, the parameters of the inhibitory kernel in order to account for spatial \ninteractions only within the receptive field. \n\nConsidering the temporal domain, one should distinguish the time constant T1, \ncaused by network interactions, from the time constants TO and T2 caused by tem(cid:173)\nporal integration at a single cell membrane. In any case, throughout all simulations, \nwe fixed TO and T2 to 20ms, whereas we\"varied T1 in the range 2 - lOOms. \n\n3 RESULTS \n\nSince visual cortex is visuotopically organized, a direct correspondence exists be(cid:173)\ntween the spatial organization of intracortical connections and the resulting recep(cid:173)\ntive field topography. Therefore, the dependence of cortical surface activity e2(x, t) \non the visual input sex, t) can be formulated as e2(x, t) = h(x,t) * sex, t), where \nthe symbol * indicates a spatia-temporal convolution, and hex, t) is the equiva(cid:173)\nlent receptive field interpreted as the spatia-temporal distribution of the signs of \nall the effects of cortical interactions. In this context, hex, t) reflects the whole \nspatio-temporal couplings and not only the direct neuroanatomical connectivity. \n\nTo test the efficacy of the various inhibitory schemes, we used a drifting sine wave \ngrating s(x,t) = Ccos[211'{fxx \u00b1 Itt)] where C is the contrast, Ix and It are the \nspatial and temporal frequency, respectively. The direction selectivity index (DSI) \nand the optimal velocity (vopt) obtained from the various inhibitory kernels of Fig.2 \nare summarized in Table 1, for different values of T1 and b. The direction selectivity \nindex is defined as DS] = ~:~~::, where Rp is the maximum response amplitude \nfor preferred direction, and Rnp is the maximum amplitude for non-preferred di(cid:173)\nrection. The optimal velocity is defined as I;pt / lx, where Ix is chosen to match \nthe spatial structure of the receptive field, and I?t is the frequency which elicits \nthe maximum cell's response. As expected, increasing the parameter b enhances \nthe effects of inhibition, thus resulting in larger DSI and higher optimal velocities. \nHowever, for stability reason, b should \"remain below a theshold value strictly re-\n\n\fJ08 \n\nII \n\nS. P. Sabatini, F. Solari and G. M. Bisio \n\nT1 = 2 \nb II DSI I Vopt \n0.00 \n0.00 \n1.82 \n1.82 \n\n0.00 \n0.00 \n0.08 \n0.17 \n\n0.25 \n0.60 \n0.80 \n0.91 \n\nTl = 10 \nII DSI I vopt \n0.00 \n0.00 \n0.00 \n0.00 \n0.24 \n1.82 \n0.34 1.82 \n\nTable 1: \nTl = 20 \nII DSI I Vopt \n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\nII \n\nT1 = 100 \nII DSI I Vopt \n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.00 \n1.82 \n1.82 \n\n0.00 \n2.07 \n4.14 \n6.00 \n\n0.00 \n0.00 \n1.82 \n1.82 \n\n0.50 \n0.85 \n1.00 \n1.30 \n\n0.50 \n1.00 \n5.00 \n9.00 \n\n0.25 \n0.50 \n0.60 \n0.72 \n\n0.25 \n0.60 \n0.80 \n0.88 \n\n0.50 \n0.85 \n1.00 \n1.33 \n\n0.00 \n0.04 \n0.06 \n0.07 \n\n0.00 \n0.00 \n0.02 \n0.01 \n\n0.00 \n0.06 \n0.06 \n0.07 \n\n0.00 \n0.00 \n0.08 \n0.14 \n\n0.00 \n0.04 \n0.05 \n0.06 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.00 \n1.82 \n1.82 \n\n0.00 \n2.07 \n2.07 \n2.00 \n\n0.00 \n00.0 \n1.82 \n1.82 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.17 \n0.19 \n0.28 \n\n0.00 \n0.00 \n0.05 \n0.03 \n\n0.00 \n0.20 \n0.39 \n0.66 \n\n0.00 \n0.00 \n0.23 \n0.26 \n\n0.00 \n0.16 \n0.18 \n0.26 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.23 \n0.26 \n0.35 \n\n0.00 \n0.00 \n1.82 \n0.25 \n0.28 \n1.82 \n0.37 1.82 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.09 \n0.09 \n0.06 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.00 \n0.16 \n1.82 \n0.39 3.64 \n0.38 \n3.64 \n\n0.00 \n0.32 \n0.40 \n0.65 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n2.07 \n2.07 \n4.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n1.82 \n1.82 \n1.82 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\n0.00 \n0.00 \n0.00 \n0.00 \n\nASY-IA \n\nASY-lB \n\nASY-2A \n\nASY-2B \n\nASY-3A \nA ::::,. \nt$LA \n\n;. \n\nASY-3B \n\nlated to the inhibitory kernel considered. Moreover, we observe that, except for \nASY-2A, the strongest direction selectivity can be obtained when the intracortical \ntime constant Tl\u00b7 has values in the range of 10 - 20 ms, i.e., comparable to TO and T2. \nLarger values of Tl would result, indeed, in a recrudescence of the velocity low-pass \nbehavior. For each asymmetry, Figs. 3 show the direction tuning curves and the x-t \nplots, respectively, for the best cases considered (cf. bold-faced values in Table 1). \nWe have evidenced that appreciable DSI can be obtained when inhibition arises \nfrom cortical sites at different distance from the target cell (i.e. , ASY-2B, dl i= d2 ). \nIn such conditions we obtained a DSI as high as 0.66 and an optimal velocity up \nto '\" 6\u00b0/s, as could be inferred also from the spatia-temporal plot which present a \nmarked motion-type (i.e., oriented) non-separability (see Fig. 3ASY-2B). \n\n4 DISCUSSION AND CONCLUSIONS \n\nAs anticipated in the Introduction, direction selectivity mechanisms usually relies \nupon asymmetric alteration of the spatial and temporal response characteristics of \nthe geniculate input, which are presumably mediated by intracortical circuits. In \nthe architectural model presented in this study, spatial asymmetries were included \nexplicitly in the extension of the inhibitory interconnections, but no explicit asym(cid:173)\nmetric temporal mechanisms were introduced. It is worth evidencing how temporal \nasymmetries originate from the specific mutual inhibition scheme considered, which \noperates, regarding temporal domain, like a quadrature model [12] [13]. This can \n\n\fAn Architectural Mechanism/or Direction-tuned Cortical Simple Cells \n\n109 \n\n\" \n~ 1.0 \ne- 0.8 \n0 \n~ \n\" 0.6 \nco \n\u2022 !::! 0.4 \n\"6 \nE 0.2 \n0 \nZ 00 \n\n0 . 1 \n\n\" \n~ 1.0 \n0 \n~ 0.8 \n\" \n~ 0.6 \nco \n.!::' 0.4 \n'0 \nE 0.2 \n0 \nZ 0 .0 \n\n0.1 \n\n1 \n\n10 \n\n100 \n\nVelocity deg/s \n\n1 \n\n10 \n\n100 \n\nVelocity deg/s \n\n\"-' \n\n400 \n......... \nen e: \ns \n0 \n0 space (deg) 2 \n\n':;j \n\n400 \n......... \nen \nS \n\"-' \n\n'J: \n\ns \n0 \n0 space (deg) 2 \n\nASY-IB \n\n----\" \" -----' . \n'. \n\nco \n~ 1.0 \n& \n:l O.B \n~ 0.6 \nco \n~ 0.4 \n0 \nE 0.2 \n0 \nZ 0.0 \n\n0. 1 \n\n1 \n\n10 \n\n100 \n\nVelocity deg/s \n\nco \n~ 1.0 \n~ 0.8 -.. ~ ........ -. \n0 \n.., 0.6 \n~ \nco \n.!::! 0.4 \n'0 \nE 0.2 \n0 \nZ 0.0 \n\n0. 1 \n\n1 \n\nASY-3A \n\n10 \n\n100 \n\nVelocity deg/s \n\n400 \n\n'