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Doi:10.1016/j.neucom.2006.10.095

Neurocomputing 70 (2007) 1977–1982 A model of the illusory contour formation based on dendritic computation Drazˇen Domijan, Mia Sˇetic´, Domagoj Sˇvegar Department of Psychology, Faculty of Philosophy, University of Rijeka, I. Klobucˇaric´a 1, 51000 Rijeka, Croatia Available online 9 November 2006 We proposed a new model of illusory contour formation based on the properties of dendritic computation. The basic elements of the network are a single-excitatory cell with two dendritic branches and an inhibitory cell. Both dendritic branches behave as an independentlinear unit with a threshold. They sum all excitatory input from the nearby collinear cells, and the inhibition from one collateral of thecorresponding inhibitory cell. Furthermore, the output of dendritic branches multiplicatively interacts before it is sent to the soma. Themultiplication allows the excitatory cell to be active only if both of its branches receive enough excitation to reach the threshold.
Computer simulations showed that the presented model of the illusory contour formation is able to perform perceptual grouping ofnonadjacent collinear elements. It shows a linear response relationship with the input magnitude because dendritic inhibition counteractsrecurrent excitation. The model can explain why illusory contours are stronger with irregular placement of inducing elements rather thanregular placement and why top-down influences may prevent the illusory contour formation.
r 2006 Elsevier B.V. All rights reserved.
Keywords: Contour integration; Dendritic computation; Illusory contours; Recurrent excitation; Top-down processing graded strength or clarity of illusory contours with therelative size of inducing elements .
Illusory (or subjective) contours are vivid percepts of How is sensitivity to the illusory contour formed in the lines between the inducing elements that do not have neural tissue? Grossberg and Mingolla proposed a support in physical stimulation. They give rise to even model of boundary contour system (BCS), which is able to more complex illusory figures such as Kanizsa square perform perceptual grouping. BCS is a two-dimensional () or Ehrenstein figure ) . Neurophysio- network that simulates properties of neurons observed in logical investigations showed that many neurons in a the primary visual cortex including simple, complex, and monkey V2 cortex are sensitive to illusory contours. That hyper complex neurons. In order to explain illusory is, neurons respond as if the real contour is presented in contour formation, they introduce a bipole cell that forms their receptive field. These neurons respond to real the illusory contour. The bipole cell is modelled as a contours as well and they show similar orientation tuning recurrent excitatory network that computes the logical curves for real and for illusory contours. Therefore, they AND function between two collinear but spatially sepa- could not distinguish between real and illusory contours.
rated parts (or poles) of their receptive field. Bipole cells Later it was found that the strength of their response receive input from a network of complex cells that compete depends linearly on the size of the illusory line or the with each other in order to achieve a sharp contour number of inducing elements up to a saturation point .
detection. However, the model was not able to simulate the Psychophysical investigations with humans also revealed finding that the strength of illusory contour varies withinput amplitude or number of inducing elements. The model also does not account for the fact that strength of Corresponding author. Tel.: +385 51 315 233; fax: +385 51 315 228.
the illusory contour depends on the placement of inducing E-mail addresses: elements. Irregular lines produce stronger impression of the 0925-2312/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
D. Domijan et al. / Neurocomputing 70 (2007) 1977–1982 Fig. 2. A model of the illusory contour formation based on the propertiesof dendritic computation.
inhibition from one collateral of the corresponding inhibi-tory cell. The branches also receive direct input from a Fig. 1. Illusory figures: (a) Kanizsa square, (b) Ehrenstein figure; illusory complex cell at corresponding locations. Furthermore, the contours formed by (c) regular inducers or (d) irregular inducers.
output of dendritic branches multiplicatively interacts beforeit is sent to the soma. The multiplication allows excitatorycell to be active only if both of its branches receive enough illusory contour (than regular inducing lines excitation to reach the threshold. The excitatory cell must . Furthermore, illusory contour formation is receive suprathreshold activation from both sides because not an obligatory process and it could be prevented by illusory contours are possible only if at least two inducing cognitive or top-down influences . Moreover, attention elements are present. Therefore, basic property of the bipole may prevent collinear interaction between neighbouring cell is achieved using the dendritic multiplication.
line segments .
We also introduced two different spatial scales in order to achieve smooth contour representation. The large-scale 2. Model description bipole cells detect the presence of the illusory contour.
Their output is sent to the small-scale cells that smooth the In this paper, we propose a new model of bipole cells and contour by local recurrent interactions. Formally, the show how is illusory contour formed based on the network is described as properties of dendritic computation We suggest that the bipole cells use dendritic inhibition in order to achieve analogue sensitivity to the input magnitude . First, 1ip þ I i w1ii  hðyLiÞ bipole cells sum inputs along its dendritic branches.
Outputs from dendritic branches multiplicatively interact (logical AND computation) before they reach cell's body.
qÞw1iq þ I i w1ii  hðyLiÞ Therefore, bipole cell will become active only if bothdendritic branches receive enough stimulation. When it is active, bipole cell sends inhibition to its dendritic branches 1ii gðxLi Þ þ TDi .
in order to reduce the effect of recurrent excitation. In this way, the proposed model is able to group collinear i denotes activity of the large-scale bipole cell at the spatial location i and yL elements into the illusory contour with analogue sensitivity i is activity of the correspond- ing inhibitory cell. The parameter A describes passive decay to input magnitude.
that drives the cell toward the zero when no input is The basic elements of the network are an excitatory cell presented. Function f() describes output of the computa- with two dendritic branches and an inhibitory cell tion from a single dendritic branch. In the simplest case it is The excitatory cell (empty circle in ) receives recurrent input from other excitatory cells, whose receptivefields are collinear with it and, therefore, forming a f ½a ¼ max½a  Tr; 0n, recurrent excitatory network (horizontal arrows at the where Tr is a threshold, and n could be smaller, larger or top of the ). Also it sends signal to other excitatory equal to 1. We also studied model behaviour with a more cells and to the inhibitory cell (filled circle in complex and biologically more realistic sigmoid function Excitatory and inhibitory cells are modelled as continuous- time linear units with a threshold. The inhibitory cell has two collaterals which contacts dendritic branches of the excitatory cell. Both dendritic branches behave as an Functions g() and h() describe an output from the independent linear unit with a threshold. They summed all excitatory and the inhibitory cells, respectively. They are excitatory input from nearby collinear cells, and the defined as linear above threshold as shown in Eq. (3) with D. Domijan et al. / Neurocomputing 70 (2007) 1977–1982 n ¼ 1. Feedforward input is denoted with Ii and the and their corresponding inhibitory cells ySi are given by recurrent excitatory input is denoted with xp and xq, where indices p and q describes the left and the right pole of the 2ii gðxSi Þ þ TDi .
bipole cell. Indices are given by p ¼ {j: 1, y, joi} andq ¼ {j: j4i, The small-scale bipole cells receive input from the large- y, N} and N is a network dimension. Term, scale bipole cells. Their lateral connections are restricted to ij, denotes strength of excitatory feedback and, wii, denote strength of feedforward excitatory connections. Recurrent the nearest neighbours. Therefore, p ¼ i1 and q ¼ i+1.
connections are described with Gaussian fall-off from thecentre of the receptive field In order to test properties of the proposed network, we performed computer simulations by solving a set ofnonlinear differential equations (). For simplicity, where r is either p or q, D is amplitude and s is a spatial we ignored the second spatial dimension. Also, we ignored spread of the Gauss kernel. Wii denotes strength of computations in the retina and the primary visual cortex.
excitatory to the inhibitory connection. The multiplication Instead, we focused on the properties of recurrent between two dendritic branches is denoted with  sign. In excitatory network and its ability to connect nonadjacent Eq. (2), term TDi describes top-down influences on input signals into a unique perceptual group. The model inhibitory cells which could prevent collinear contour parameters were set to: N ¼ 30; A1 ¼ .1; A2 ¼ .001; D ¼ 150; s ¼ 10; W1ii ¼ W2ii ¼ 1; w1ii ¼ .8; w2ii ¼ .2; The small-scale bipole cells xSi are given by 2ip ¼ w2iq ¼ 1 for all i. Output functions f(), g() and h() are described as threshold linear (Eq. (3)) with Tr ¼ 0 and n ¼ 1. First, we showed that the model cells indeed behave 2ip þ gðxLi Þw2ii  hðySi Þ as the bipole cells because they do not respond when one of its poles do not receive stimulation (dotted line).
gðxSqÞw2iq þ gðxLiÞw2ii  hðySiÞ , Furthermore, computer simulations revealed that proposed network is sensitive to the input magnitude as shown by Fig. 3. Computer simulations illustrating the network behaviour.
D. Domijan et al. / Neurocomputing 70 (2007) 1977–1982 different responses to different input magnitude of the right bipole cells and consequently lead to weaker illusory inducer , solid and dashed lines). shows a contour when stimulus is a regular set of inducing linear relationship between the number of inducing elements and excitatory cell response. In this simulation, According to Grossberg and Mingolla , illusory inducers are added from the middle two toward ends. In contour formation is an obligatory process that always this way, we simulate the neurophysiological experiment produces strong contour. However, Albert showed described in showing increase in cell's response with examples where illusory contours are weak or absent increase in the number of inducing elements. These despite the existence of collinear inducers (e.g., when simulations illustrate that the input magnitude could be crosses are used as inducers in We suggest that the defined either as a strength of individual contour elements top-down influences (attending to or recognizing certain or as a number of contour elements within the same perceptual groupings) may reduce the strength of the dendritic branch. In both cases, cell response increases with illusory contour. This may happen because inducers could input magnitude.
be recognized as independent figures (e.g., crosses) or they We also simulated the difference in strength or clarity of could draw attention to itself due to their symmetry or the illusory contour for regular or irregular placement of parallelism. The top-down influence on illusory contour the inducing elements described by Gillam ( formation and on collinear facilitation is explained in the With irregular placement of the inducing elements some of presented framework by the operation of the inhibitory the bipole cells will receive stronger input because the cells. We may assume that inhibitory cell receives excitation competition between the complex cells is distance depen- from higher visual centres when attention is directed in the dent. Consider a case in which certain inducing element is part of the visual scene where its receptive field is located.
isolated or placed far apart from other elements. The This is described by the term TD in Eq. (2). The inhibitory complex cell positioned over it will receive less inhibition cell will distribute this top-down signal to the dendritic compared to other complex cells. This situation is branches of the corresponding excitatory cell and effec- illustrated in the input vector of (dashed line). On tively raise their threshold for activation. If top-down the other hand, with regular placement of the inducing signal is sufficiently strong it will completely prevent elements, all complex cells will receive the same amount of formation of the illusory contour and cells show no inhibition (solid line). Due to the fact that the response although both parts of its receptive fields are distance between regularly spaced elements is smaller than stimulated. This may occur when a stronger perceptual the distance between the most isolated elements in irregular grouping overrides the illusory figure The same arrangement, the total amount of inhibition will be greater mechanism also helps to explain why attention prevents with regular placement of elements. A strong inhibition collinear contour facilitation as observed by Ito and between complex cells will provide weaker input to the Fig. 4. Parametric simulations with different dendritic output functions f(). (a) Power function with exponent n ¼ .5 (solid line), 1 (dashed line), or 2(dash-dotted line). (b) Sigmoid output function with C ¼ .5 (solid line), 1 (dashed line), or 2 (dash-dotted line).
D. Domijan et al. / Neurocomputing 70 (2007) 1977–1982 We checked the robustness of the model in parametric contour. We also used local interaction between excitation simulations shown in First, we studied how the and inhibition on isolated dendritic branches in order to network behaves with a different setting of exponent n in achieve the MAX function computation. In it is Eq. (3), which describes the output function f() of the shown that feedforward dendritic inhibition allows com- dendritic branches. showed cell responses of a putation of MAX function on cell's input. Here, a large-scale bipole cell, xLi, as a function of the input recurrent extension of the network with dendritic inhibition strength for n ¼ .5 (solid line), 1 (dashed line), and 2 (dash- is used. A disadvantage of the proposed model is that a dotted line). In these simulations, the input vector desirable behaviour is achieved when the output of a single represents real contour which has the same value at all dendritic branch is linear above threshold. On the other spatial locations i. All other parameters are set to the same hand, theoretical analysis of the detailed model of the values as in the simulations shown in As can be seen, pyramidal neuron suggests sigmoid output function as a the model shows the linear response to the input magnitude better description for dendrites .
when the dendritic output function is linear or slower In conclusion, the presented model of the illusory then linear. For the faster then linear output function, contour formation is able to perform perceptual grouping response becomes linear at higher input magnitudes. In of nonadjacent collinear segments. It shows a linear we showed that the analogue sensitivity is not response relationship with input magnitude because the preserved for the sigmoid output function (Eq. (4)) with dendritic inhibition regulates the signal flow and prevents B ¼ 2 and C ¼ .5 (solid line), 1 (dashed line), or 2 (dash- unbounded growth of activity in the recurrent excitatory dotted line).
network Analogue sensitivity is achieved under a wide Furthermore, we studied the model behaviour when the variety of parameter settings and input–output functions as strength of synaptic weights between excitatory cells (wij) shown in parametric simulations. The model can explain or excitatory and inhibitory cells (Wii) are systematically why illusory contours are stronger with irregular placement varied (not shown). For simplicity, we set all wij to be of of inducing elements rather than regular placement and equal strength. The basic finding is that the model achieves why the attention or other top-down signals may prevent analogue sensitivity as long as the synaptic weights between the illusory contour formation. These findings could not be the excitatory cells are kept weaker than the weights modelled with a recent extension of the BCS proposed by between the excitatory and the inhibitory cells. When Wii Grossberg and Raizada and with the model of V1–V2 is set to 0, an unbounded growth of excitatory activity is interaction proposed by Neumann and Sepp .
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D. Domijan et al. / Neurocomputing 70 (2007) 1977–1982 [14] S. Grossberg, R.D.S. Raizada, Contrast-sensitive perceptual group- Domagoj Sˇvegar is a Ph.D. student at the ing and object-based attention in the laminar circuits of primary University of Ljubljana, Slovenia. He received visual cortex, Vis. Res. 40 (2000) 1413–1432.
B.A. in Psychology from the University of [15] H. Neumann, W. Sepp, Recurrent V1–V2 interaction in early visual Rijeka, Croatia. His research interests include boundary processing, Biol. Cybern. 81 (1999) 425–444.
behavioural testing and computational modellingof visual working memory.
Drazˇen Domijan is an Assistant Professor ofPsychology at the University of Rijeka, Croatia.
He received B.A. in Psychology from theUniversity of Rijeka, Croatia, and Ph.D. inPsychology from the University of Zagreb,Croatia. His research interests include computa-tional modelling of visual perception, attentionand cognition. In particular, he studied howmodels of neural networks with dendrites mayhelp to explain various cognitive phenomena and how artificial dendrites may increase computational power of neuralnetworks. He is a member of International Neural Network Society andAssociation for Psychological Science.
Mia Sˇetic´ is a Ph.D. student at the University ofZagreb, Croatia. She received B.A. in Psychologyfrom the University of Rijeka, Croatia. Herresearch interests include behavioural investiga-tions of interaction between perception andabstract conceptual processing. Also she isinterested in computational modelling of percep-tion, attention, and numerical cognition. She is amember of Association for Psychological Science.

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