{"title": "Visual Cortex Circuitry and Orientation Tuning", "book": "Advances in Neural Information Processing Systems", "page_first": 887, "page_last": 893, "abstract": null, "full_text": "Visual Cortex Circuitry and Orientation \n\nTuning \n\nTrevor M undel \n\nDepartment of Neurology \n\nUniversity of Chicago \n\nChicago, IL 60637 \n\nmundel@math.uchicago.edu \n\nAlexander Dimitrov \n\nDepartment of Mathematics \n\nUniversity of Chicago \n\nChicago, IL 60637 \n\na-dimitrov@ucllicago.edu \n\nJack D. Cowan \n\nDepartments of Mathematics and Neurology \n\nUniversity of Chicago \n\nChicago, IL 60637 \n\ncowan@math.uchicago.edu \n\nAbstract \n\nA simple mathematical model for the large-scale circuitry of pri(cid:173)\nmary visual cortex is introduced. It is shown that a basic cor(cid:173)\ntical architecture of recurrent local excitation and lateral inhi(cid:173)\nbition can account quantitatively for such properties as orien(cid:173)\ntation tuning. The model can also account for such local ef(cid:173)\nfects as cross-orientation suppression. It is also shown that non(cid:173)\nlocal state-dependent coupling between similar orientation patches, \nwhen added to the model, can satisfactorily reproduce such ef(cid:173)\nfects as non-local iso--orientation suppression, and non-local cross(cid:173)\norientation enhancement. Following this an account is given of per(cid:173)\nceptual phenomena involving object segmentation, such as \"pop(cid:173)\nout\", and the direct and indirect tilt illusions. \n\n1 \n\nINTRODUCTION \n\nThe edge detection mechanism in the primate visual cortex (VI) involves at least \ntwo fairly well characterized circuits. There is a local circuit operating at sub(cid:173)\nhypercolumn dimensions comprising strong orientation specific recurrent excitation \nand weakly orientation specific inhibition. The other circuit operates between hy(cid:173)\nper columns , connecting cells with similar orientation preferences separated by sev(cid:173)\neral millimetres of cortical tissue. The horizontal connections which mediate this \n\n\f888 \n\nT. Mundel. A. Dimitrov and J. D. Cowan \n\ncircuit have been extensively studied. These connections are ideally structured to \nprovide local cortical processes with information about the global nature of stim(cid:173)\nuli. Thus they have been invoked to explain a wide variety of context dependent \nvisual processing. A good example of this is the tilt illusion (TI), where surround \nstimulation causes a misperception of the angle of tilt of a grating. \nThe interaction between such local and long-range circuits has also been inves(cid:173)\ntigated. Typically these experiments involve the separate stimulation of a cells \nreceptive field (the classical receptive field or \"center\") and the immediate region \noutside the receptive field (the non-classical receptive field or \"surround\"). In the \nfirst part of this work we present a simple model of cortical center-surround inter(cid:173)\naction. Despite the simplicity of the model we are able to quantitatively reproduce \nmany experimental findings. We then apply the model to the TI. We are able to \nreproduce the principle features of both the direct and indirect TI with the model. \n\n2 PRINCIPLES OF CORTICAL OPERATION \n\nRecent work with voltage-sensitive dyes (Blasdel, 1992) augments the early work of \nRubel & Wiesel (1962) which indicated that clusters of cortical neurons correspond(cid:173)\ning to cortical columns have similar orientation preferences. An examination of local \nfield potentials (Victor et al., 1994) which represent potentials averaged over corti(cid:173)\ncal volumes containing many hundreds of cells show orientation preferences. These \nconsiderations suggest that the appropriate units for an analysis of orientation se(cid:173)\nlectivity are the localized clusters of neurons preferring the same orientation. This \nchoice of a population model immediately simplifies both analysis and computation \nwith the model. For brevity we will refer to elements or edge detectors, however \nthese are to be understood as referring to localized populations of neurons with a \ncommon orientation preference. We view the cortex as a lattice of hypercolumns, \nin which each hypercolumn comprises a continuum of iso-orientation patches dis(cid:173)\ntinguished by their preferred orientation \u00a2. All space coordinates refer to distances \nbetween hypercolumn centers. The popUlation model we adopt throughout this \nwork is a simplified form of the Wilson-Cowan equations. \n\n2.1 LOCAL MODEL \nOur local model is a ring (\u00a2 = -900 to + 90\u00b0) of coupled iso-orientation patches and \ninhibitors with the following characteristics \n\n\u2022 Weakly tuned orientation biased inputs to VI. These may arise either from \nslight orientation biases of lateral geniculate nucleus (LGN) neurons or from \nconverging thalamocortical afferents \n\n\u2022 Sharply tuned (space constant \u00b17.5\u00b0) recurrent excitation between iso(cid:173)\n\norientation populations \n\n\u2022 Broadly tuned inhibition to all iso-orientation populations with a cut-off \n\nof inhibition interactions at between 45\u00b0 and 60\u00b0 separation \n\nThe principle constraint is that of a critical balance between excitatory and in(cid:173)\nhibitory currents. Recent theoretical studies (Tsodyks & Sejnowski 1995; Vreeswijk \n& Sompolinsky 1996) have focused on this condition as an explanation for certain \nfeatures of the dynamics of natural neuronal assemblies. These features include the \nirregular temporal firing patterns of cortical neurons, the sensitivity of neuronal \nassemblies in vivo to small fluctuations in total synaptic input and the distribu(cid:173)\ntion of firing rates in cortical networks which is markedly skewed towards low mean \n\n\fVISual Cortex Circuitry and Orientation Tuning \n\n889 \n\nrates. Vreeswijk & Sompolinsky demonstrate that such a balance emerges naturally \nin certain large networks of excitatory and inhibitory populations. We implement \nthis critical balance by explicitly tuning the strength of connection weights between \nexcitatory and inhibitory populations so that the system state is subcritical to a \nbifurcation point with respect to the relative strength of excitation/inhibition. \n\n2.2 HORIZONTAL CONNECTIONS \n\nWe distinguish three potential patterns of horizontal connectivity \n\n\u2022 connections between edge detectors along an axis parallel to the detectors \n\npreferred orientation (visuotopic connection) \n\n\u2022 connections along an axis orthogonal to the detectors preferred orientation, \n\nwith or without visuotopic connections \n\n\u2022 radially symmetric connection to all detectors of the same orientation in \n\nsurrounding hypercolumns \n\nRecent experimental work in the tree shrew (Fitzpatrick et al., 1996) and prelim(cid:173)\ninary work in the macaque (Blasdel, personal communication) indicate that vi(cid:173)\nsuotopic connection is the predominant pattern of long-range connectivity. This \nconnectivity pattern allows for the following reduction in dimension of the problem \nfor certain experimental conditions. \nConsider the following experiment. A particular hypercolumn designated the \"cen(cid:173)\nter\" is stimulated with a grating at orientation resulting in a response from the \nhdge detector. The region outside the receptive area of this hypercolumn (in the \n\"surround\") is also stimulated with a grating at some uniform orientation ' result(cid:173)\ning in responses from <,i>'-edge detectors at each hypercolumn in the surround. In \norder to study the interactions between center and surround, then to first order ap(cid:173)\nproximation only the center hypercolumn and interaction with the surround along \nthe visuotopic axis (defined by the center) and the ' visuotopic axis (once again \ndefined by the center) need be considered. In fact, except when = ' the effect \nof the center on the surround will be negligible in view of the modulatory nature \nof the horizontal connections detailed above. Thus we can reduce the problem (a \npriori three dimensional - one angle and two space dimensions) to two dimensions \n(one angle and one space dimension) with respect to a fixed center. This reduc(cid:173)\ntion is the key to providing a simple analysis of complex neurophysiological and \npsychophysical data. \n\n3 RESULTS \n\n3.1 CENTER-SURROUND INTERACTIONS \n\nAlthough, we have modeled the state-dependence of the horizontal connections, \nmany of the center-surround experiments we wish to model have not taken this \ndependence explicitly into account. In general the surround has been found to \nbe suppressive on the center, which accords with the fact that the center is usually \nstimulated with high contrast stimuli. A typical example of the surround suppressive \neffect is shown in figure 1. \n\nThe basic finding is that stimulation in the surround of a visual cortical cell's re(cid:173)\nceptive field generally results in a suppression of the cell's tuning response that \nis maximal for surround stimulation at the orientation of the cell's peak tuning \n\n\f890 \n\nT. Mundel, A. Dimitrov and 1. D. Cowan \n\n60 \u00ae \n30 ~ 40 \n\n+30 +60 +90 \n\nA \n\n-60 \n\n-30 \n\n0 \n\nORJENTATIONOFBAR (dell \n\n10 \n\nB \n\n-60 \n\n-30 \n\n+30 +60 +90 \nORJENfATION OF GRATINO (dell \n\n0 \n\nFigure 1: Non-local effects on orientation tuning - experimental data. Response to \nconstant center stimulation at 15\u00b0 and surround stimulation at angles [-90\u00b0 , 900J \n(open circles), Local tuning curve (filled circles). Redrawn from Blakemore and \nTobin (1972) \n\nresponse and falls off with stimulation at other orientations in a characteristic man(cid:173)\nner. Further examples of surround suppression can be found in the paper of Sillito \net al. (1995). Figure 2 depicts simulations in which long-range connections to lo(cid:173)\ncal inhibitory populations are strong compared to connections to local excitatory \npopulations. \n\nThese experiments and simulations appear to conflict with the consistent experimen(cid:173)\ntal finding that stimulating a hypercolumn with an orthogonal stimulus suppresses \nthe response to the original stimulus. \n\nThe relevant results can be summarised as follows: cross-orientation suppression \n(with orthogonal gratings) originates within the receptive field of most cells exam(cid:173)\nined and is a consistent finding in both complex and simple cells. The degree of \nsuppression depends linearly on the size of the orthogonal grating up to a critical \ndimension which is smaller than the classical receptive field dimension. It is possible \nto suppress a response to the baseline firing rate by either increasing the size or the \ncontrast of the orthogonal grating. \n\nThe model outlined earlier can account for all these observations, and similar mea(cid:173)\nsurements recently described by Sillitto et. al. (1995), in a strikingly simple fash(cid:173)\nion in the setting of single mode bifurcations. Orthogonal inputs are of the form \na/2[1 + cos2s(cP - cPo)J + b/2[1 + cos2s(cP - cPo + 90\u00b0)]' where a and b are ampli(cid:173)\ntudes with a > band cP E [-900,900J. By simple trigonometry this simplifies to \n(a + b) /2 + (a - b) /2 cos 2s( cP - cPo) Thus the input of amplitude b reduces the ampli(cid:173)\ntude of the orthogonal input and hence gives rise to a smaller response. This is then \nthe mechanism by which local double orthogonal stimulation leads to suppression. \n\nThe center-surround case is different in that the orthogonal input originates from \nthe horizontal connections and (in the suppressive setting) is input primarily to \nthe orthogonal inhibitory population. It can be shown rigorously that for small \namplitude stimuli this is equivalent to an orthogonal input to the excitatory popu(cid:173)\nlation with opposite sign. Thus we have a total input (a + b)/2[1 + cos(2s(cP - cPo)J \nwhere b arises from the horizontal input and hence increases the amplitude of the \nfundamental component of the input. \n\n\fVISual Cortex Circuitry and Orientation Tuning \n\n891 \n\nB \n\n, \n-90 -\u00ab> -30 \u00b0 30 60 90 \n\n-90 -\u00ab> -30 \u00b0 30 60 90 , \n\n.~1.5 \n~ \n\n~ 1 : ::10.5 \n-90 -\u00ab> -30 \u00b0 30 60 90 \n\n~ \n\nFigure 2: Non-local effects on orientation tuning. (