Department of Cognitive and Neural Systems and Center for Adaptive Systems, Boston University, Boston, MA 02215, USA
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Abstract |
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Introduction |
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A related question concerns why visual cortex, indeed all neocortex, is organized into layers. What functional properties are achieved by laminar computing? The model clarifies how such laminar computing abets both infant development and adult perception by enabling cortex to select and complete correct groupings of visual signals, while actively suppressing incorrect groupings, without losing sensitivity to the relative contrasts and spatial positions of these signals.
The model proposes developmental rules whereby cortical circuits grow whose excitatory and inhibitory signals are balanced. Several model studies have shown how balanced excitation and inhibition can produce the highly variable interspike intervals that are found in cortical data (Shadlen and Newsome, 1998; van Vreeswijk and Sompolinsky, 1998
). The present study suggests that such variability may reflect mechanisms that are needed to ensure stable development and learning by cortical circuits.
Cells in cortical area V1 are arranged into columns whose local circuits link together cortical layers. Cells in each column have similar orientational tuning and sensitivity to eye of origin, or ocular dominance. The columns are arranged into two- dimensional maps of orientation and ocular dominance (Hubel and Wiesel, 1962, 1963
, 1968
). Cortical simple cells are sensitive to the contrast polarity of oriented image contrasts, whereas complex cells pool signals from like-oriented opposite contrast polarities. The classical, oriented receptive fields of these cells are derived from local interactions between cells in nearby cortical columns. A number of models have studied how simple cells and complex cells develop their orientationally tuned receptive fields within maps of orientation and ocular dominance (von der Malsburg, 1973
; Grossberg, 1976a
; Willshaw and von der Malsburg, 1976
; Swindale, 1980
, 1982
, 1992
; Linsker, 1986a
, 1986b
; Rojer and Schwartz, 1989
, 1990
; Durbin and Mitchison, 1990
; Obermayer et al., 1990
, 1992
; Miller, 1992
, 1994
; Grossberg and Olson, 1994
; Sirosh and Miikkulainen, 1994
; Olson and Grossberg, 1998
) [these models have been reviewed by Olson and Grossberg (Olson and Grossberg, 1998
)].
None of the models has investigated the functional utility of organizing visual cortex into layers. Nor have they modeled development of the longer-range horizontal and interlaminar interactions that link cells in different cortical columns, or how such development may be stabilized by self-balancing excitatory and inhibitory signals. These interactions are often cited as the basis of non-classical receptive fields that are sensitive to the context in which individual features are found (von der Heydt et al., 1984; Peterhans and von der Heydt, 1989
; Born and Tootell, 1991
; Knierim and van Essen, 1992
; Sillito et al., 1995
). The present article assumes that receptive fields of individual simple and complex cells have already substantially developed in their respective layers and columns, and models how longer-range horizontal and interlaminar connections develop between columns. This type of study extends the functional understanding of cortical organization, because the perceptual units that visual cortex processes are spatially distributed patterns of luminance or color that are processed in parallel by multiple, interacting columns. The model was briefly reported previously (Grossberg and Williamson, 1997
; Williamson and Grossberg, 1998
).
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Methods |
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Perceptual grouping is the process whereby the brain organizes image contrasts into emergent boundary structures that segregate objects and their backgrounds in response to texture, shading and depth cues in scenes and images (Julesz, 1971; Ramachandran and Nelson, 1976
; Beck et al., 1983
; Polat and Sagi, 1994
). Perceptual grouping is a basic step in solving the binding problem, whereby spatially distributed features are bound into representations of objects and events in the world. Illusory contours are a particularly vivid form of perceptual grouping, since they illustrate how perceptual groupings can form over image locations that contain no contrastive scenic elements.
The model suggests that many aspects of cortical design have evolved to carry out perceptual grouping. In particular, the model proposes how the laminar circuits of visual cortex enable it to develop connections capable of actively selecting and completing the perceptual grouping which best represents a visual scene, and suppressing the weaker groupings which represent the scene less well. The winning grouping that is chosen in this way can also represent the relative contrasts and spatial positions of objects in the scene.
Such a linkage between brain and behavior typically requires a demonstration of how interactions among many model cells give rise to emergent properties that match behavioral data. Several types of emergent properties are simulated by the model. The model assumes that the classical receptive fields of simple and complex cells have already developed. This hypothesis is consistent with data showing that the oriented pattern of lateral geniculate nucleus (LGN)-to-V1 connections develops prior to eye opening and structured visual input (Chapman et al., 1991; Antonini and Stryker, 1993a
; Chapman and Stryker, 1993
). The model focuses upon how the longer-range non-classical connections between cortical columns develop both prior to eye opening and after structured visual inputs occur. We propose rules whereby such cortical development is controlled. Several such rules work together to control stable growth of model connections by ensuring that balanced excitatory and inhibitory connections develop. The emergent properties of this developmental process are the adult anatomical and neurophysiological circuits into which the model develops. After model development stabilizes, visual inputs activate cells within the developed anatomy, thereby leading to a second type of emergent properties, namely the cell activity patterns that match data about adult visual perception.
Classical Receptive Fields
The model assumes that three types of circuits with (primarily) classical receptive field properties develop, at least in part, before the circuits that subserve non-classical receptive fields. We call the circuits that have already developed pre-developed circuits. The circuits that develop through model dynamics are called self-organized circuits. The model analyzes one important combination of intracortical and intercortical pathways. It does not attempt to model all cortical connections, or the variations that exist across species. It also models the pre-developed circuits in the simplest possible way, since they are not the focus of the study, and the computational demands of the simulations are great even with these simplifications. Preliminary studies indicate, however, that the computational principles modeled herein can be elaborated and adapted to handle these variations.
Model analyses will be restricted to cortical area V1, and more particularly to the interblob organization of V1 that we propose interacts with area V2 to carry out perceptual grouping of boundary contours. Converging evidence suggests that area V2 replicates the structure of area V1, but at a larger spatial scale (van Essen and Maunsell, 1983; von der Heydt et al., 1984
; Felleman and van Essen, 1991
; Grosof et al., 1993
; Kisvarday et al., 1995
). We therefore assume that similar developmental processes may be operative in both V1 and V2. The model's predeveloped and self-organized properties are described below, first intuitively and then mathematically. Figure 1
and Table 1
schematize the model's connections. The proposed role of the blob stream in forming surface representations, and its predicted interactions with the boundaries formed in the interblob stream, are discussed elsewhere (Gove et al., 1995
; Grossberg, 1994
, 1997
; Grossberg and McLoughlin, 1997; Grossberg and Pessoa, 1998
).
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In both the brain and the model, the retina activates the LGN, which, in turn, inputs to cortical area V1. LGN inputs directly excite layer 4C in both the cat and macaque, as well as layer 4A in the macaque (Hubel and Wiesel, 1962; Chapman et al., 1991
; Reid and Alonso, 1995
). In the model, a single, generic, layer 4 is used for simplicity; see the pathways with open triangles in Figure 1a
. These inputs play a key role in establishing the orientational tuning of V1 simple cells.
Simple cells in the brain respond to a given orientation and contrast polarity, i.e. they respond best to visual inputs that have a prescribed orientation and whose luminance preference, across this oriented axis, goes either from dark-to-light, or from dark-to-light, but not both. Simple cells in the model are pre-developed and are represented by circular symbols with half white and half black hemidisks in Figure 1a. Model simple cell properties arise as follows from model LGN inputs and intracortical interactions: LGN ON cells (cells that are turned on by input onset; see symbols with white disks and black annuli in Fig. 1a
) and LGN OFF cells (cells that are turned off by input onset; see symbols with black disks and white annuli in Fig. 1a
) both input to layer 4. They are organized into spatially offset arrays, with the ON cell inputs spatially displaced with respect to the OFF cell inputs, as in Figure 1a
. Due to this input array, layer 4 simple cells can respond to an oriented input whose luminant area excites the ON cells, and whose dark area excites the OFF cells.
Selectivity of simple cell responses to oriented contrasts is improved by including mutually inhibitory interactions between cells that are sensitive to the same orientation but opposite contrast polarities (Palmer and Davis, 1981; Pollen and Ronner, 1981
; Ferster, 1988
; Liu et al., 1992
; Gove et al., 1995
); see the pathways with black triangles in Figure 1a
. Then, when model cells that code opposite contrast polarities are equally activated by a uniform pattern of activation in the LGN, they shut each other off by mutual inhibition. On the other hand, when there is an oriented transition from ON to OFF activations in the LGN, the simple cells that best match its position, orientation and polarity will be most activated. Olson and Grossberg (Olson and Grossberg, 1998
) have modeled how mutually inhibitory simple cells develop which are sensitive to the same orientation and opposite contrast polarities, at the same time that a cortical map develops whose orientation and ocular dominance columns exhibit the fractures, singularities and linear zones reported by others (Blasdel, 1992a; 1992b; Obermayer and Blasdel, 1993
).
Balanced LGN Inputs to Layer 4 via Layer 6
In both brain and model, LGN inputs also directly excite layer 6 (Ferster and Lindström, 1985), which then indirectly influences layer 4 via an on-center off-surround network of cells (Grieve and Sillito, 1991a
,b
, 1995
; Ahmed et al., 1994
, 1997
). In both brain and model, cells in the on-center receive excitatory inputs from layer 6, whereas those in the spatially broader off-surround, which spans more than a single hypercolumn (Grieve and Sillito, 1995
), receive inhibitory inputs from layer 6 via inhibitory interneurons in layer 4. In Figure 1b
, open triangles designate excitatory connections and black triangles designate inhibitory connections. Such a combination of direct and indirect input pathways to layer 4 is found in many neocortical areas (van Essen and Maunsell, 1983
; Felleman and van Essen, 1991
). The model suggests that it helps to preserve stable development and learning in all these areas, while also allowing them to be activated by bottom-up inputs. In particular, the model predicts that the excitation and inhibition within the on-center of the 6-to-4 pathway are approximately balanced. The model also predicts that, if the on-center inputs from layer 6 get too strong relative to the off-surround inputs from layer 6 to 4, then development does not self-stabilize. Instead, the non-classical receptive fields of the model proliferate uncontrollably. On the other hand, if the inhibition gets too strong, then it can inhibit the inputs arriving at layer 4 too much, thereby preventing the cortex from becoming activated at all.
Maintaining a balance between the excitation and inhibition within the on-center from layer 6 to 4 has important implications for cortical design. Direct activation of layer 6 is predicted to modulate, prime or subliminally activate, cells in layer 4, but not to fire them vigorously. This prediction is consistent with the finding that layer 4 EPSPs elicited by layer 6 stimulation are much weaker than those caused by stimulation of LGN axons or of neighboring layer 4 sites (Stratford et al., 1996), and also with the fact that binocular layer 6 neurons synapse onto monocular layer 4 cells of both eye types without reducing these cells' monocularity (Callaway, 1998, p. 56). Other compatible data have been reported (Hupé et al., 1997
; Wittmer et al., 1997
). We suggest that the on-center excitation is inhibited down into being modulatory by the overlapping and broader off-surround. Thus, although the center excitation is weak, the suppressive effect of the off-surround inhibition can be strong. The need to maintain the on-center excitatoryinhibitory balance also predicts why direct inputs to layer 4 are needed, in addition to the indirect on-center inputs via layer 6, in many cortical areas. The model predicts that, by themselves, the indirect 6-to-4 inputs cannot activate layer 4 cells without destabilizing cortical development and learning. Hence the direct bottom-up inputs to layer 4 are predicted to be necessary to initiate cortical firing.
Given that strong direct inputs from LGN to layer 4 do exist, the combined effect of both the direct and indirect pathways from LGN to layer 4 is to form an on-center off-surround network whose net on-center excitatory input can fully activate layer 4 cells. When cells in such a network obey the membrane equations of neurophysiology, they can maintain their sensitivity to input intensities that may vary over a large dynamic range (Grossberg, 1973, 1980b
; Heeger, 1993
; Douglas et al., 1995
). This is because the membrane equations contain shunting, or automatic gain control terms, that respond to properly balanced on-center and off-surround inputs by normalizing the activities of target cells without destroying their sensitivity to the relative sizes of the inputs. In the present instance, such a model network maintains the sensitivity of cells in layer 4 to inputs from the prior processing level, whether it be cells in V1 responding to LGN inputs, cells in V2 responding to inputs from V1, or any other combination of inputs. The layer 6-to-4 network is also used to preattentively select and attentively modulate the perceptual groupings that form in layer 2/3 (Grossberg, 1999
; Grossberg and Raizada, 2000
).
In summary, the model predicts that the mechanism whereby the balance between excitation and inhibition is maintained in the layer 6-to-4 circuit is of the greatest importance for achieving stable cortical development and later visual perception. This issue has hardly been explored experimentally. This prediction implies that a key cortical design problem is the following: as more and more cells in the off-surround become activated by increasingly dense patterns of inputs, what prevents the total inhibition that is converging on a layer 4 cell from growing linearly? If there was just enough inhibition to balance the excitation when just a few inputs were active, then why would not the inhibition become much too strong when many inputs were active, thereby shutting down the network? On the other hand, if the inhibition is well balanced when many inputs are active, then why does not runaway excitation occur when just a few inputs are active?
Development of Self-normalizing Inhibitory Interneurons in Layer 4
The model solves this problem by assuming that the inhibitory inter- neurons in layer 4 inhibit one another, as well as target cells in layer 4 [see Ahmed et al. (Ahmed et al., 1994, 1997
) for consistent data]. In particular, the model suggests how layer 4 inhibitory interneurons connect to layer 4 spiny stellate excitatory cells as well as to other nearby layer 4 inhibitory interneurons during development. These connections eventually span all the orientation columns within a hypercolumn, as well as all the orientation columns of neighboring hypercolumns. This recurrent inhibition converts the network of inhibitory interneurons into a recurrent feedback network. Because the cells of this network obey membrane equations, the inhibitory interneurons within such a population of recurrent interactions tend to normalize their total activity across the entire interneuron population (Grossberg, 1973
, 1980b
). The total inhibition that converges on a target cell thus tends to be conserved as the total number of inputs varies, thereby preventing the problems stated above. If this property is experimentally confirmed, then it will be an interesting example of how less order on one level of biological organization generates more order on a higher level. In particular, the crucial self-normalization property can be achieved simply by allowing the inhibitory interneurons to randomly inhibit all cells within their range, rather than restricting their inhibition to excitatory target cells. As a result of this less ordered growth of inhibitory connections, the stability of the total network is facilitated.
Maintaining the balance between excitation and inhibition within the layer 6-to-4 on center does not imply that inhibition is weak. In fact, layer 4 cells that receive only off-surround inputs can be strongly inhibited. The model suggests below how the on-center off-surround network from layer 6-to-4 can use this property to selectively amplify the strongest perceptual groupings in layer 2/3 while using the off-surround to actively suppress LGN inputs to layer 4 that correspond to weaker groupings in layer 2/3. The weaker groupings hereby collapse. This is proposed to happen as follows.
Columnar Organization via Folded Feedback
Active model layer 4 cells are assumed to generate inputs to pyramidal cells in layer 2/3 via pre-developed pathways. These layer 2/3 cells initiate the formation of perceptual groupings via horizontal connections that self-organize during model development. How these horizontal connections develop in the model is described below. Before describing this, we first note what happens when layer 2/3 cells are activated. Throughout the developmental process, all cells that are activated in layer 2/3, whether by bottom-up or horizontal inputs, send excitatory feedback signals to layer 6 via layer 5 (Gilbert and Wiesel, 1979; Ferster and Lindström, 1985
), as in Figure 1b
. (The model does not attempt to discuss any other functional role for layer 5, notably its role in generating signals to motor control centers.) Layer 6, in turn, once again activates the on-center off-surround network from layer 6-to-4. This process is called folded feedback (Grossberg, 1999
), because feedback signals from layer 2/3 get transmitted in a feedforward fashion back to layer 4. The feedback is hereby folded back into the feedforward flow of bottom-up information within the laminar cortical circuits.
Folded feedback is predicted to be a mechanism that binds the cells throughout layers 2/3, 4, 5 and 6 into functional columns (Mountcastle, 1957; Hubel and Wiesel, 1962
, 1977
). The on-center off-surround network from layer 6-to-4 responds to its layer 2/3 inputs by helping to control which combinations of cells remain simultaneously active during development, and thus which cells will wire together, because cells that fire together wire together.
In particular, early during the development of model horizontal connections in layer 2/3, the activation of layer 2/3 cells can cause horizontal activations that are relatively unselective for colinear position and orientation (Galuske and Singer, 1996; Ruthazer and Stryker, 1996
). Without further selection among the possible activations, cortical interactions could remain both spatially and orientationally dispersed. This is corrected in the model via the intracortical folded feedback loop. In particular, suppose that a combination of bottom-up inputs and horizontal connections activates one subset of layer 2/3 cells a little more than a nearby subset of cells. Then, other things being equal, the favored layer 2/3 cells more vigorously activate their layer 2/3-to-5-to-6 pathway, and then their on-center off-surround layer 6-to-4 circuit. As a result, the cells whose activities form the strongest layer 2/3 grouping will suppress the activities of other cells via the layer 6-to-4 off-surround. The winning cells then get connected together via development, leading to a progressive increase in the projection range and orientational selectivity of these cells, as simulated in the Results section.
This refinement process exploits the fact that orientationally tuned simple cells in the model and the brain can bias development to favor long-range horizontal connections that are colinear with the preferred orientations of spatially aligned simple cells (Fitzpatrick, 1996; Schmidt et al., 1997a
). It is shown below how such oriented and colinear horizontal connections develop from an initial state in which no horizontal connections exist at all. It is also shown that, after development self-stabilizes, the same properties play a key role in generating perceptual groupings which exhibit properties of adult neurophysiological and psychophysical data.
Horizontal Connections and Perceptual Grouping
How these developing horizontal connections are prevented from generating run-away excitation and uncontrollable growth is one of the key properties of the model. A clue may be derived from properties of adult horizontal connections. In areas V1 and V2 of the adult, layer 2/3 pyramidal cells excite each other using monosynaptic long-range horizontal connections. They also inhibit each other using short-range disynaptic inhibitory connections that are activated by the excitatory horizontal connections (Hirsch and Gilbert, 1991; McGuire et al., 1991
) (see Fig. 1c
). The excitatory connections, which span several hyper- columns (Gilbert and Wiesel, 1979
, 1989
), are hereby balanced by inhibitory connections, which span a single hypercolumn (Lund and Yoshioka, 1991
). Thus, the excitatory connections within layer 2/3 have a broader spatial extent than the inhibitory off-surround connections from layer 6-to-4, which in turn have a broader spatial extent than the inhibitory interneurons in layer 2/3. These relative relationships are also simulated in the model. A range of numerical values could be chosen which obey these qualitative constraints without disrupting the model's key properties. In fact, in simulations of how the model performs perceptual grouping and attention, the excitatory connections within model layer 2/3 in cortical area V2 are chosen to be longer than those in cortical area V1, as is also true in anatomical data (Grossberg and Raizada, 2000
).
We show below how both types of connections can develop to generate perceptual groupings inwardly between two or more image contrasts that are aligned colinearly across space (von der Heydt et al., 1984; Redies et al., 1986
; Peterhans and von der Heydt, 1989
; Grosof et al., 1993
), but not outwardly from a single image contrast (Hirsch and Gilbert, 1991
; Knierim and van Essen, 1992
; Cannon and Fullenkamp, 1993
; Somers et al., 1995
; Stemmler et al., 1995
). This is called the bipole property (Grossberg and Mingolla, 1985
). Illusory contours provide an excellent example of the bipole property: if a single image contrast could generate outward groupings, then our percepts would become crowded with webs of illusory contours spreading out from every feature in a scene. On the other hand, percepts of illusory contours between two or more colinear inducers are commonplace (Kanizsa, 1979
, 1985
).
We now describe how a balance between layer 2/3 excitation and inhibition develops that helps to stabilize cortical development and leads to the bipole property in the adult. We call layer 2/3 pyramidal cells that receive bottom-up input from layer 4 supported cells, and those that do not unsupported cells. In the model, if an unsupported cell, or cell population, receives a sufficient amount of horizontal excitation, then it will be driven above its firing threshold. The cell population will then output horizontal excitation to itself as well as to other pyramidal cell populations. Unsupported cells can generate suprathreshold excitation if they receive enough horizontal excitation from supported cells. Turning off input support from layer 4 causes all supported cells, and then all layer 2/3 activities, to decay to zero. Therefore, boundaries can group across a gap provided the gap is small enough and the grouping signals from the supported cells on each end of the gap are sufficiently strong to drive the interior, unsupported cells above threshold.
The horizontal excitation from a single supported cell population cannot cause runaway excitation and outward grouping among un- supported cells because it also activates balanced disynaptic inhibition from smooth stellate cells. In this situation, the disynaptic inhibition is proportional to the horizontal excitation because both pyramidal and smooth stellate cells receive the same horizontal input signal. Given that horizontal excitation from a single supported cell population is inhibited by disynaptic inhibition, how do groupings ever span a region of unsupported cells? One factor is that inhibition from smooth stellate cells to pyramidal cells can lag behind the direct excitation between pyramidal cells due to the time it takes the smooth stellate cells to integrate their inputs. Therefore, synchronized inputs to layer 2/3 facilitate grouping because they allow the horizontal signals to summate at the target pyramidal cells before inhibition from local smooth stellate cells takes effect. This property is consistent with the finding of Usher and Donnelly that visual groupings are facilitated when inducers are presented synchronously (Usher and Donnelly, 1998).
This argument about synchrony is not sufficient, however, to explain how inward grouping succeeds whereas outward grouping does not. The model notes that when two or more pyramidal cell populations are activated at positions that are located at opposite sides of an unsupported pyramidal cell, then excitation from these cells more easily summates at the unsupported cell, which can therefore exceed its firing threshold. In addition, this excitation activates the corresponding disynaptic inhibitory interneurons. As in the case of the layer 4 off-surround, the model disynaptic inhibitory interneurons are predicted to inhibit each other as well as the pyramidal cells. This model hypothesis is consistent with anatomical data showing that inhibitory layer 2/3 interneurons synapse on both pyramidal cells and other interneurons (McGuire et al., 1991; Kisvarday et al., 1993
). Hence the total activation within such a population of inhibitory interneurons is predicted to be at least partially normalized. As a result, total activation may grow less quickly than summating activation of the pyramidal cells. The model hereby predicts that recurrent inhibition may influence the excitatoryinhibitory balance in both layer 2/3 and layer 4. In summary, due to a combination of spatial summation factors in the sources of excitation, and delays and amplitude properties of inhibition, net activation of the target pyramidal cells is possible, and grouping can occur inwardly but not outwardly, thereby realizing the bipole property (Grossberg and Mingolla, 1985
), which has been used to explain and predict many perceptual grouping data (Born and Tootell, 1991
; Shipley and Kellman, 1992
; Watanabe and Cavanagh, 1992
; Field et al., 1993
; Grossberg, 1994
, 1997
; Polat and Sagi, 1994
; Gove et al., 1995
; Dresp and Grossberg, 1997
; Grossberg and Pessoa, 1998
).
There is more neurophysiological evidence for the bipole property in cortical area V2 (von der Heydt et al., 1984; von der Heydt and Peterhans, 1989
) than in V1. In V1, just a few unsupported cells have, to date, been found that show full activation of unsupported cells by pairs of supporting cells. More V1 cells show a modulatory influence from neighboring pyramidal cells (Redies et al., 1986
; von der Heydt and Peterhans, 1989
; Grosof et al., 1993
; Kapadia et al., 1995
). These are challenging experiments to do in V1 because of the shorter horizontal connections there, and the existence of feedback from V2, which has longer horizontal connections. Unsupported V2 cells could be fully activated by stimuli that fall outside the V1 receptive fiel/ds, and could modulate V1 cells by top-down feedback. For simplicity, the present model assumes that the bipole property holds in both V1 and V2. Altering the model to allow only modulatory bipole influences in V1 can be accomplished by changing the model parameters that control whether convergent horizontal connections can fire the cell in the absence of bottom-up input; see the Appendix, equations (26
)-(28
).
Developmental Growth Rules
These properties of adult grouping arise in the model by specializing two well-known developmental rules. The first rule is that axons are attracted to cell targets when the source and target cells are both active (Gunderson and Barrett, 1979, 1980; Letourneau, 1978; Purves and Lichtman, 1980
; Lichtman and Purves, 1981
). The second rule is that axons compete intracellularly for growth resources (Purves and Lichtman, 1980
; Lichtman and Purves, 1981
). In the present instance, the first rule enables horizontal connections to form if activations in a source pyramidal cell and a target pyramidal cell are sufficiently correlated in particular, if the target cell satisfies the bipole property and removed if they are not (Callaway and Katz, 1990
, 1991
; Lowel and Singer, 1992
). This rule is realized by an activity-dependent morphogenetic gradient whose strength decreases with distance from the target cell that emits it. The gradient influences horizontal growth only in active source cells. As contact between two cells is achieved, a synaptic learning law strengthens the synaptic contact by continuing to sense the correlation between presynaptic and postsynaptic activity.
The second rule prevents uncontrolled proliferation of horizontal connections by withdrawing connections from target cells that are receiving more poorly correlated signals than other target cells. The two rules work together to withdraw connections from cells that may be activated by weakly correlated image features or statistically insignificant noise. These model mechanisms for axonal growth and synaptic tuning dynamically stabilize cortical development as the developing cortical structure matches the statistics of its environmental inputs. If this match is disrupted later in life, then a new bout of development and/or learning can be triggered by the same mechanisms. Because of this property, the model can be used to clarify data about shared molecular substrates of neonatal development and adult learning (Bailey et al., 1992; Kandel and O'Dell, 1992
; Mayford et al., 1992
), plasticity of adult cortical representations after lesions (Merzenich et al., 1988
; Chino et al., 1992
; Gilbert and Wiesel, 1992
; Darian-Smith and Gilbert, 1994
; Kapadia et al., 1994
; Das and Gilbert, 1995
; Schmidt et al., 1996), dynamical reorganization of long-range connections in the visual cortex (Gilbert and Wiesel, 1992
; Zohary et al., 1994
), and perceptual learning in the adult (Karni and Sagi, 1991
; Poggio et al., 1992
). In fact the model equations for activity- dependent control of synaptic strength have already been used to explain properties of adult learning (Grossberg, 1980a
; Carpenter and Grossberg, 1991
).
Top-down Feedback from V1 to LGN
Layer 6 of model area V1 sends top-down feedback to the LGN via an on-center off-surround network, as also occurs in vivo (Murphy and Sillito, 1987; Weber et al., 1989
; Murphy and Sillito, 1996
) (see Fig. 1d
). The feedback on-center reinforces the activities of those LGN cells which have succeeded in activating V1 cells, notably V1 cells whose activations represent the strongest perceptual groupings. The feedback off-surround suppresses the activities of other LGN cells. As in the brain, this model feedback circuit increases the useful visual information that is transmitted from LGN to cortex by enhancing contextually significant differences between LGN responses (McClurkin et al., 1994
), and also influences the length tuning of LGN cells (Murphy and Sillito, 1987
). The LGN to V1 circuit is also known to be modulatory (Sillito et al., 1994
). Earlier modeling work predicted that this feedback pathway plays a role in stabilizing the development of bottom-up connections from LGN to V1, as well as the reciprocal top-down connections from V1 to LGN (Grossberg, 1976b
, 1980b
). Grunewald and Grossberg have modeled how the normal development of bottom-up disparity tuning can occur at V1 complex cells when such top-down feedback is operative, and have shown how this development may break down when it is not (Grunewald and Grossberg, 1998
). Further experimental study of this question is needed. For purposes of the present modeling analysis, it is assumed that these top-down connections are pre-developed and are available to facilitate activation of the correct combinations of simple and complex cells.
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Results |
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The next three sections summarize how the model simulates data about the development of long-range horizontal connections in area V1. After development self-stabilizes, the resultant network can, without further change, simulate adult neurophysiological and psychophysical data. As in the brain, the model undergoes two stages of development (Fig. 2). One occurs prior to eye opening, when endogenous random geniculate and cortical activity determine the initial specificity of horizontal connections (Ruthazer and Stryker, 1996
). The other occurs after eye opening, when patterned visual inputs can strengthen and refine these connections (Galuske and Singer, 1996
).
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Increase of Projection Range
The Galuske and Singer investigation of long-range projections in cat area 17 (the analog of monkey area V1) at different stages of postnatal development yielded a similar conclusion (Galuske and Singer, 1996). These authors also reported quantitative data about the projection range of pyramidal cells (Fig. 3
, top). Soon after eye opening, the projection range doubled over a period of 12 days (from P15 to P26). Presumably, the increase in projection range is due to the greater correlations in activity over large spatial distances that occurs in natural, structured images. Figure 3
(bottom) shows the simulated projection range in the model. Before eye opening, the short-range spatial correlations of the unstructured inputs are reflected in the relatively short-range extent of horizontal projections. Soon after eye opening, the long-range spatial correlations in the structured visual inputs cause the projection range to double, just as in the data of Galuske and Singer (Galuske and Singer, 1996
). These results exploit the developmental rules described above by causing a larger projection range to grow when the statistics of visual imagery provided more long-range correlations.
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A similar pattern of exuberant growth followed by slow refinement of projections has also been found in the ferret. Because the ferret is born 3 weeks earlier in development than the cat, it has more stable orientation-selective cortical cell responses than the cat during the period in question (Durack and Katz, 1996; Ruthazer and Stryker, 1996
). Ruthazer and Stryker reported quantitative data about the growing orientational selectivity of horizontal clustering over time, using a statistic called the cluster index (CI) (Ruthazer and Stryker, 1996
). The CI measures the log of the average nearest-neighbor distance between horizontal projections within a measurement window, divided by the average distance between a randomly selected point in the window and the nearest horizontal projection. Therefore, a uniform distribution of horizontal projections would lead to a CI of log(1) = 0. As clustering becomes more refined, CI increases. Figure 4
(top) shows the CI obtained by Ruthazer and Stryker from 21 days postnatal up to adult age (Ruthazer and Stryker, 1996
). Before eye opening, which is about 31 days postnatal, there is a positive CI, indicating a clustering bias, presumably favoring iso-orientation connections. After eye opening, the CI rapidly increases to reflect the strong, adult bias in favor of iso-orientation connections.
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After development, horizontal projections preferentially connect columns with similar orientation preferences that are aligned colinearly with their orientation preference (Fitzpatrick, 1996; Schmidt et al., 1997a
). Figure 5
(left) shows a polar plot by Fitzpatrick of the projection field from a site in layer 2/3 of tree shrew striate cortex (Fitzpatrick, 1996
). The distance of each point from the center of the projection field represents the number of labeled terminals at that angle (in 10° increments). The orientation of the projection field is aligned with the orientation preference of its source neuron. Figure 5
(right) shows the analogous projection field from a horizontally tuned column in layer 2/3 of the model after development has equilibrated. The size of each circle represents the strength of the connection to each iso-orientation column. The anisotropy of the model's projection field is qualitatively consistent with Fitzpatrick's data. These results derive from the fact that visual cues are, with high probability, locally linear across space, so that the largest correlations would be generated by cells whose orientations match those of the input and are colinearly aligned across space. The developmental rules enable the network to sense these correlations and to selectively amplify the growth of those connections which best match them.
|
Projection Field versus Receptive Field
This section shows that the model network that develops has neurophysiological properties that have been recorded from adult animals. One such property shows, remarkably, that the extent of a cell's total anatomical projection field is much greater than that of its classically recorded receptive field (Fitzpatrick, 1996). Fitzpatrick found that the projection fields in tree shrew extend for >2 mm from the injection site, a distance that corresponds to 15° eccentricity, whereas the dimensions of classically defined receptive fields at that eccentricity are <5°. The dwarfing of classical receptive fields by projection fields was also shown in neurophysiological data recorded from cats by Das and Gilbert (Das and Gilbert, 1995
). These authors compared cortical point spread (PS) distributions, measured with optical recording, which reflect both spiking and subthreshold activity, with spiking distributions measured with extracellular electrodes. A small oriented visual stimulus produced a PS distribution 20 times larger than the spiking distribution. Moreover, the close match of the PS distribution with columns whose orientation preference agrees with the orientation of the visual stimulus suggests that the distribution arises from iso- oriented long-range horizontal projections.
A similar property holds in the model after development equilibrates: Figure 5 (right) shows the size of a layer 2/3 cell classical receptive field (dashed-line circle) with respect to its projection field in the model. This discrepancy between projection field and receptive field can be traced to the model's bipole property: the classical receptive field reflects mainly bottom-up properties of the cortical network in the model, whereas the subthreshold activations reflect the fact that the bipole requirement for firing the cells via long-range horizontal connections was not satisfied.
Cortical Point Spread Functions
The measurement of cortical point spread functions (PSFs) provides additional evidence about the strength of horizontal connections. In this regard, optically recorded signals are believed to arise from subthreshold dendritic activity in the superficial layers (Grinvald et al., 1994). These dendrites may belong to cells in both the superficial and deep layers. Grinvald et al. measured an asymmetric PSF in macaque monkeys, with twice as much spread along the axis parallel to the V1/V2 border as along the perpendicular axis (Grinvald et al., 1994
). The axis parallel to the V1/V2 border is perpendicular to the direction of OD columns in this cortical region. Therefore, the explanation given by Grinvald et al. for this asymmetry is that a spread in activity among the equivalent number of same-eye OD columns would traverse twice as much cortical surface in the axis perpendicular to the direction of OD columns. Accordingly, they modeled the PSF with an asymmetric two-dimensional exponential distribution, having a space constant of 3.0 mm in the axis perpendicular to OD columns, and 1.5 mm in the axis parallel to OD columns.
In comparing the PSF obtained by our model with this experimentally derived distribution, it is appropriate to use the 1.5 mm space constant because our model is monocular. The only remaining step is to map the metric of cortical surface distance into the metric of model hypercolumns. We found the best-fitting match by assuming that a hypercolumn in our model would have a diameter of 450 µm parallel to the direction of OD columns, and therefore 900 µm perpendicular to the direction of OD columns. Given this assumption, Figure 6 compares the experimentally derived PSF with the PSF generated by our model following stimulation in the center hypercolumn. These PSFs closely resemble each other out to four hypercolumns away from the central one.
|
Psychophysical Data and Simulations
After model cortical development stabilizes, the cortical network that is formed in this way, without further change, simulates key psychophysical data about adult perceptual grouping. Facilitation of cortical responses by oriented, colinearly arrayed inducers has been found by a number of researchers (von der Heydt et al., 1984; von der Heydt and Peterhans, 1989
; Field et al., 1993
; Grosof et al., 1993
; Polat and Sagi, 1993
, 1994
; Kapadia et al., 1995
). Many of these facilitory effects may be explained by colinear groupings mediated by layer 2/3 connections in V1 and V2, or by groupings that form perpendicular to line ends. That is why the current self-organized model was restricted to horizontal and vertical orientations, and was used to study how grouping strength changed when the spatial separation of inducers was varied.
Illusory Contour Formation
Figure 7 illustrates the model's grouping behavior. Figure 7a
shows an input image consisting of a 5 x 5 pixel square and a 5 x 3 pixel rectangle, separated by a 5-pixel gap. Figure 7b,c
shows the network's equilibrated suprathreshold layer 4 and layer 2/3 activities, respectively, in response to this image. In these line-segment displays, the orientation of each boundary segment denotes the orientation preference of the cell at that location, and the length of the segment denotes the cell's activity.
|
Contour Sensitivity to Spatial Context
The model's context-sensitivity also includes the property of spatial impenetrability (Grossberg and Mingolla, 1987), in which boundary groupings in one orientation inhibit weaker potential groupings in other orientations at the same position. Figure 8
illustrates this property. Figure 8
(top left) shows two aligned vertical bar inputs, and Figure 8
(top right) shows how the network's equilibrated suprathreshold layer 2/3 pyramidal cell activities vertically group the two bars together. Figure 8
(bottom left) shows a modified input with same two bars augmented by several squares aligned horizontally in the gap between them. These squares do not individually favor horizontal over vertical grouping. In fact, the vertical sides of two of the squares are colinear with the vertical sides of the bars. Other things being equal, they would facilitate vertical grouping. On the other hand, the set of all the squares, taken together, generates a strong horizontal grouping. The emergent horizontal orientation of this grouping inhibits the vertical grouping, as shown in the equilibrated suprathreshold layer 2/3 pyramidal cell activities (bottom right). The network's spatial impenetrability is due to the cross-orientational inhibition that develops in layer 2/3 and layer 4. This simulation shows that the self- organized balance between the layer 2/3 horizontal excitation and disynaptic inhibition that achieves the bipole property is also well-balanced against the interlaminar 6-to-4 connections that help to select which groupings will survive.
|
It should be noted that the simulations describing perceptual grouping consider interactions within the interblob stream of visual cortex, which has been predicted to support such grouping dynamics (Grossberg and Mingolla, 1985). In fact, there is neurophysiological evidence that locally aligned contours can produce illusory contour-like responses in area V2 (Peterhans and von der Heydt, 1989
). Grouping strength does not, however, necessarily covary with visibility, as measured, say, by a brightness or color difference. In fact, it has been predicted that the perceptual groupings which are formed in the interblob stream are amodal, i.e. they themselves do not represent a visible brightness or color signal. Visible brightness or color signals have been predicted to occur within the blob cortical stream as part of the process whereby three-dimensional surface representations are formed (Grossberg, 1994
).
Contour Sensitivity to Support Ratio
Figure 9 shows how the illusory contours formed by the model, either colinear to edges or perpendicular to line ends, vary in strength as the inducing features are parametrically varied. These simulations illustrate that the developed layer 2/3 connections do not saturate; instead, they enable the network to exhibit the type of spatial context-sensitivity found in human psychophysical data. Figure 9
(top) plots data of Shipley and Kellman (Shipley and Kellman, 1992
) which show the effect of increasing the length of the inducers while decreasing the gap between them, keeping the total length of inducers-plus-gap constant: illusory contour clarity increases roughly linearly. In other words, contour clarity increases with support ratio. Figure 9
(top) shows that the clarity of the model's illusory contours also increases linearly as the support ratio is increased. The mapping from network activities to clarity ratings is described below. This result is due to the fact that, as the gap between two inducers is made smaller, the grouping signal becomes stronger, due to the monotonically increasing magnitude of the layer 2/3 grouping kernel towards its center (see Figs 5 and 6
).
|
Figure 9 (bottom) summarizes psychophysical data obtained by Lesher and Mingolla showing that, if support ratio is increased in a different way, then an inverted-U in illusory contour clarity strength is obtained (Lesher and Mingolla, 1993
). In this study, parallel bars with aligned ends were used to form four Pacman figures with which to induce an illusory Kanizsa square percept. The square formed perpendicular to the bars through their aligned ends. Contour clarity of the illusory square was measured as the numbers of bars, and hence the support ratio, varied. The inducing Pacmen had a circular radius of 128 pixels, and the gap between Pacman pairs in which the Kanizsa square percept formed was 128 pixels. The support ratio was computed as the number of bar inducers (1, 2, 4, 8, 16 per Pacman) times bar width, divided by the length of the side of the square (384 pixels). As the width of the bar inducers is increased, the number of possible inducers becomes limited, which is why there are only results for up to 16 inducers in the 9-pixel-wide case, and up to 8 inducers in the 17-pixel-wide case. Figure 9
(bottom) shows that the model simulates the inverted-U in contour strength as a function of bar density. This inverted-U result is due to an interaction between the long-range excitatory horizontal connections in layer 2/3 and the medium-range inhibitory connections from layer 6-to-4. The Shipley and Kellman data (Shipley and Kellman, 1992
), and our simulation thereof, show that decreasing the distance between inducers, up to a certain point, increases grouping strength as a result of layer 2/3 horizontal cooperation. As the inducers get even closer together, however, layer 6-to-4 inhibition increasingly inhibits the net excitation caused at layer 4 by each LGN input. Thus, although more inputs activate the cooperating layer 2/3 pyramidal cells, the net effect of each input on layer 2/3 gets smaller as the inducers get denser. This simulation shows that the self-organized connections preserve a good balance between layers 6, 4 and 2/3. As in the psychophysical data in Figure 9
, the model's illusory contour strength is affected more strongly by variations in support ratio than in bar density.
Due to the implementational limitations of the model described above, the network simulated these data using bars that are relatively wide with respect to the length of the gap (2-pixel-wide bars, 4-pixel -ong gap). Figure 9 (bottom) shows results obtained by the model with inter-bar gap size decreasing from 3 to 0, with the total length spanned by the inducers and gaps held roughly constant. The model's inverted-U curve is shifted to the right of the data curves, reflecting the fact that the model used inducers that were wider relative to the gap size. Note that, in the data as well, the curves shift to the right as the width of the inducers increases. Using short bars instead of line ends to simulate this result, due to limitations of cell density in the model, does not and should not alter the qualitative results because the model predicts that they are due to an interaction between how the bottom-up layer 6-to-4 off-surround attenuates the activation of layer 4 cells in response to bottom-up inputs before layer 4 cells can activate a horizontal grouping in layer 2/3, in response to any input pattern, as its inducers get closer together.
Mapping Network Activity into Illusory Contour Clarity
The model's layer 2/3 activities were mapped into the psycho- physical illusory contour clarity metric (ICC) via the following equation:
![]() | (1) |
![]() | (2) |
Detection Threshold Context Sensitivity
Figure 10 (top) shows bandpass-limited inducers (Gabor patches) similar to those used by Polat and Sagi in their psychophysical experiments (Polat and Sagi, 1993
). If the spatial separation of the flanking patches is sufficiently small that the patches overlap, then the threshold for detecting the target Garbor patch is greater than the baseline threshold for the target patch alone. This threshold increase is due to mutual inhibition between representations of the nearby stimuli and/or a reduction in the signal to-noise ratio for the central stimulus. If the separation is increased such that the patches do not overlap, then the detection threshold decreases below the baseline threshold. As the separation is further increased, the detection threshold gradually returns to the baseline level. Figure 10
shows an optimal separation that yielded a reduced threshold (middle) in Polat and Sagi's experiments, and a larger separation for which this reduction was diminished (right).
|
Contrast-sensitive Temporal Dynamics of Perceptual Grouping
Using the input stimuli of Figure 7a, the contrast-sensitive temporal dynamics underlying grouping were analyzed by examining the activities over time of horizontally oriented layer 4 and layer 2/3 cells whose receptive fields are placed: along an object contour (case 1), in the middle of a gap between two colinear contours (case 2), and the same distance from a single colinear contour (case 3).
Figure 11 (top) plots a layer 4 cell's activation as a function of time in case 1 for three different input levels Iij in equations (1
) and (2
). The cell has an initial burst of activity which is largely attenuated by subsequent inhibition from layer 4 interneurons, which are activated from layer 6. As the input increases, the layer 4 activity peaks earlier, and equilibrates at a higher level. Figure 11
(bottom) plots the activation of the corresponding layer 2/3 cell. The layer 2/3 activity also peaks earlier, and equilibrates at a higher level.
|
|
Destabilization of Perceptual Grouping by Unbalanced Excitation and Inhibition
The bipole rule requires that layer 2/3 cells remain subthreshold, as in Figure 12 (bottom), unless they are surrounded by colinear inputs on both sides. This requirement is enforced by inhibition, both within layer 4 (mediated by excitatory interlaminar input from layer 6), and within layer 2/3 (mediated by excitatory horizontal input at the disynaptic inhibitory interneurons). The model predicts that the bipole rule fails if either source of inhibition is lost. This prediction suggests that the selectivity of colinear facilitation is not just a property of layer 2/3, but rather a property of how intercellular interactions are balanced across several cortical layers.
Figure 13 (top) shows how a model layer 2/3 cell in case 3 is affected as the layer 6-to-4 inhibition is reduced. If this happens, then layer 4 cells can be activated solely by input from layer 6 because layer 6-to-4 excitation becomes stronger than 6-to-4 inhibition. If the W+ inhibitory kernel (see equation 21
) is reduced by 50%, then the bipole rule is still maintained. However, if it is reduced by 60%, then the layer 2/3 cell becomes suprathreshold and the bipole property is lost. This means that layer 4 cells without bottom-up input can become activated by excitatory feedback via layers 2/3-to-5-to-6-to-4. Destroying the balance of excitation and inhibition between layers 6-to-4 enable this feedback to act like a spurious bottom-up input, thereby leading to a slow spread of activity away from all line ends and corners. This simulation dramatizes the importance of balancing the layer 6-to-4 on-center and off-surround so that its main effects on layer 4 are either to modulate the excitability of layer 4 cells in the on-center or to strongly inhibit the activation of layer 4 cells in the surround. The simulation hereby provides a strong test of the model's ability to self-organize connections that maintain this balance.
|
Destabilization of Development by Unbalanced Excitatory and Inhibitory Learning
Figure 13 shows that reducing the inhibitory kernels in layer 4 or layer 2/3 to about half their present values can lead to a loss of the bipole property. Once this happens, cortical development and adult perceptual learning can spiral out of control, as more excitatory learning (via equations 29
and 33
) leads to greater average cortical activity, which in turn leads to more excitatory learning. Therefore, inhibitory learning that balances excitatory learning is needed to stabilize cortical development and learning, and, in so doing, cause the grouping properties with which we are familiar in the adult. The key parameters that guarantee network stability are C3 in equation (38
) and
in equation (35
).
These parameters need to be set so that the W+ and T+ kernels that regulate layer 4 and layer 2/3 inhibition, respectively, reach sufficiently large values. The results depicted in Figure 13 show why network development and learning is robust with regard to changes in these parameters, since the bipole property that maintains grouping, and thus learning, selectivity is maintained for a wide range of values in both inhibitory kernels.
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Discussion |
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---|
Model development leads to a network that is capable of simulating adult psychophysical data about context-sensitive perceptual grouping, notably data that depend upon non- classical receptive field properties. Further simulations of adult psychophysical data can be found elsewhere (Grossberg et al., 1997; Grossberg and Raizada, 2000
; Raizada and Grossberg, 2000
). One of the model's key lessons is that the same mechanisms which stabilize development also control properties of perceptual grouping and learning in the adult. In particular, connections which grew to stably reflect robust statistical properties of the visual world define the properties of adult perceptual grouping as we know them. We claim that these grouping properties help to dynamically maintain the match between world statistics and the brain's ability to process them.
As noted above, in both the brain and the model, layer 2/3 boundary signals feed back via connections to layer 6 via layer 5 (Gilbert and Wiesel, 1979; Ferster and Lindström, 1985
). Layer 6, in turn, activates the on-center off-surround network from layer 6-to-4. This feedback has been called folded feedback (Grossberg, 1999
) because the feedback signals from layer 2/3 to layer 6 get transmitted in a feedforward fashion back to layer 4 and thereupon to layer 2/3. The feedback is hereby folded back into the feedforward flow of bottom-up information within the laminar cortical circuits. Folded feedback links cells in layers 2/3, 6, 5 and 4 into functional columns (Mountcastle, 1957
; Hubel and Wiesel, 1962
, 1977
). In so doing, it enables the strongest grouping signals in layer 2/3 to use the on-center off-surround network from layer 6-to-4 to reinforce the strongest groupings and to inhibit weaker groupings, during both early development and adult grouping and learning.
The 6-to-4 folded feedback pathway is thus predicted to do several things: (i) maintain contrast sensitivity to bottom-up inputs from LGN; (ii) help to select the strongest groupings that initially get formed in layer 2/3; (iii) receive top-down attentional modulation from V2 and other cortical areas; and (iv) deliver top-down attentional signals to LGN. The spatial scale of the inhibition from layer 6-to-4, being smaller than the spatial extent of the excitatory horizontal connections within layer 2/3, and larger than the disynaptic inhibition within layer 2/3 that maintains the bipole grouping property, is well-suited to these tasks. In particular, the model uses this pathway to simulate how attention can propagate along a curve (Grossberg and Raizada, 2000), as found in neurophysiological recordings from macaque area V1 (Roelfsema et al., 1998
), thereby illustrating how attention can selectively enhance an entire object. This layer 6-to-4 on-center off-surround attentional circuit in the model also clarifies other important properties of attention, such as its on-center off-surround characteristics (Bullier et al., 1996
) and the property that the V1 layer whose activation is most reduced by cutting off V2 feedback is layer 6 (Sandell and Schiller, 1982
). An interface in layer 6 for top-down attention also clarifies how attention can, in principle, propagate across multiple brain regions via layer 6-to-6 top-down connections, modulating each target cortical area without fully activating, or driving, its grouping cells.
The model provides a mechanistic account of how adult perceptual learning and the plasticity of cortical representations after lesions may arise from developmental mechanisms when the dynamic equilibrium between input statistics and cortical circuitry is upset (Grossberg, 1980; Merzenich et al., 1988; Carpenter and Grossberg, 1991
; Karni and Sagi, 1991
; Bailey et al., 1992
; Gilbert and Wiesel, 1992
; Kandel and O'Dell, 1992
; Mayford et al., 1992
; Poggio et al., 1992
; Zohary et al., 1994
). As noted above, the model predicts that the stability of model development and adult learning requires an approximate balance between excitation and inhibition within layer 2/3 and between layers 6 and 4.
In particular, the model predicts that the balanced layer 6-to-4 on-center off-surround circuit modulates layer 4 cells, but cannot, by itself, fully activate them. This prediction is consistent with neurophysiological data from ferret visual cortex showing that the layer 6-to-4 circuit is functionally weak (Wittmer et al., 1997). The model also predicts that the layer 6-to-4 modulatory circuit is used by top-down signals from higher cortical areas to attentionally prime layer 4 cells in area V1, without fully activating them (Grossberg, 1999
). Thus the same modulatory property that is needed to ensure stable development is predicted to control the ability of higher-order processes to attentionally prime lower areas, without fully activating them. The rules of stable development are thus predicted to define what we mean by adult attention, as well as adult grouping and learning. This hypothesis is consistent with neurophysiological data of Hupé et al. who have shown that feedback connections from area V2 modulate but do not create center-surround inter- actions in V1 neurons (Hupé et al., 1997
). Such intercortical feedback connections from V2 to V1 can modulate the circuits of V1 with higher-order boundary completion and figure-ground perception properties of area V2 (Grossberg, 1994
, 1997
; Lamme, 1995
; Zipser et al., 1996
), and/or other cortical areas (Hupé et al., 1998
; Watanabe et al., 1998
). Taken together, these properties open the way towards a unified mechanistic model of infant cortical development and adult neurophysiology, perceptual grouping, attention and learning.
The model hereby provides a simple functional explanation of why there are direct bottom-up inputs to layer 4, as well as indirect bottom-up inputs to layer 4 via layer 6, in many cortical areas (van Essen and Maunsell, 1983; Felleman and van Essen, 1991
). The proposed explanation is that direct inputs to layer 4 are needed to supraliminally activate layer 4 cells because the indirect layer 6-to-4 inputs cannot do so: they must be merely modulatory in order to stabilize cortical development and learning.
Comparison with Other Models
It is informative to compare the properties of our developed model and its precursors (Grossberg and Mingolla, 1985, 1987
; Gove et al., 1995
; Grossberg et al., 1997
) with alternative models of visual cortex and perceptual grouping. The present model's grouping properties have several advantages over those proposed by other computational models of visual cortex. Models which do not address the formation of illusory contours (Stemmler et al., 1995
; Li, 1998; Somers et al., 1998
; Yen and Finkel, 1998
) not only fail to account for neurophysiological data (von der Heydt et al., 1984
; Sheth et al., 1996
) but also are unable to exploit the computational advantages that follow from closing incomplete boundaries: use of closure to guide surface reconstruction, boundary completion over the blind-spot and retinal veins, and more complete information for the recognition of partially occluded objects (Grossberg, 1994
). Layer 2/3 bipole cells in the present model (Figure 1c
) respond to both real and illusory contour stimuli of similar orientations, consistent with neurophysiological data (Sheth et al., 1996
), and are connected by horizontal axons which are coaxial with the receptive fields' preferred orientation (Bosking et al., 1997; Schmidt et al., 1997a
), not orthogonal, as has also been proposed (Peterhans and von der Heydt, 1991). Because groupings are explicitly represented by connected regions of above-threshold layer 2/3 firing, the model shows how a high-contrast item can group with its neighbors while still having its net neural response suppressed by their presence, as found by Polat and co-workers (Polat et al., 1998
) [see Grossberg and Raizada for simulations of such data (Grossberg and Raizada, 2000
).] Models in which grouping is represented only by lateral facilitation (Somers et al., 1998
; Stemmler et al., 1995
) cannot account for this, and force the paradoxical conclusion that high-contrast items would never group with each other, which is demonstrably not the case (Elder and Zucker, 1998
). The present model's representation of grouping as distinct from visible stimulus contrast [as reviewed by Grossberg (Grossberg, 1994
)] also receives support from recent psychophysical work (Hess et al., 1998
).
The Li model, in particular, uses bipole-like grouping cells in a single-layer recurrent network (Li, 1998). The surround inhibition originating from layer 6 in our model produces important functionality which is lacking in Li's model. Due to surround inhibition in both layer 4 and LGN, instigated by input from layer 6, our model performs spatial contrast enhancement on the boundary representation. The surround inhibition enhances boundaries at line ends and corners, setting the stage for the formation of illusory contours perpendicular to discontinuities. Li (Li, 1998) also considers synchronous oscillations during grouping of a type that has earlier been simulated in precursors of the present model (Grossberg and Somers, 1991), leading to quantitative simulations of human psychophysical data that may be linked to such oscillations (Grossberg and Grunewald, 1997
).
Many predictions follow from our model and its extension to V2 (Grossberg, 1999; Grossberg and Raizada, 2000
). For example: test if top-down V1 to LGN feedback helps to stabilize the development of disparity tuning in V1 during the visual critical period; test if a long-range horizontal grouping in layer 2/3 of V2 can inhibit vertically oriented simple cells at the midpoint of this grouping in layer 4 of V1; and test if layer 4 simple cells cannot be supraliminally activated if only the LGN-to-6-to-4 input pathway is active.
In their No Strong Loops Hypothesis, Crick and Koch (Crick and Koch, 1998) suggested that a strong excitatory [feedback] loop would throw the cortex into uncontrolled oscillations, as in epilepsy' (p. 248). They used this argument to suggest why modulatory brain circuits exist. However, there are many mathematical theorems which prove that neural networks with strong excitatory feedback can readily converge to stable patterns of activation that do not oscillate (Grossberg, 1969
, 1973
, 1978b
, 1978c
, 1980a
; Ellias and Grossberg, 1975
; Grossberg and Levine, 1975
; Cohen and Grossberg, 1983
; Hopfield, 1984
). We propose that the reasons for modulatory circuits are more subtle than the CrickKoch hypothesis: such circuits help to stabilize development in the infant and learning in the adult (Ito et al., 1998
).
We propose that variants of these laminar circuits may be used in other perceptual and cognitive systems to achieve self- stabilizing learning and development. For example, long-range horizontal connections are known to occur in the auditory and language areas of human temporal cortex (Schmidt et al., 1997b). Specializations of these connections may be used to group information in several neocortical areas. The present results may thus be viewed as a first step towards showing how laminar neocortex develops and learns connections and weights with which to optimally carry out many information-processing tasks.
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Notes |
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Address correspondence to Professor Stephen Grossberg, Department of Cognitive and Neural Systems, Boston University, 677 Beacon Street, Boston, MA 02215, USA. Email: steve{at}cns.bu.edu.
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Appendix: Model Equations |
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Retina
At each retinal position (i,j), an ON cell activity uij+ is defined by an on-center off-surround network that possesses narrow on-center and Gaussian off-surround kernels (see Fig. 14a). An OFF cell activity uij is also defined by an off-center on-surround network with narrow off-center and Gaussian on-surround kernels. The retinal cell activities caused by the constant visual inputs I have the equilibrium values:
![]() | (3) |
![]() | (4) |
![]() | (5) |
|
The LGN ON cell activity vij+ and OFF cell activity vij, at each position (i,j) obey membrane equations (Hodgkin, 1964) that interact via on-center off-surround networks. As noted above, such a network realizes a contrast gain-control process which retains cell sensitivity to image contrasts while compensating for variable illumination and normalizing network activity (Grossberg, 1982
, 1983
):
![]() | (6) |
![]() | (7) |
![]() | (8) |
![]() | (9) |
Cortical Simple Cells
Simple cell responses derive from arrays of ON cell and OFF cell outputs from the LGN (see Fig. 1a). These ON cell and OFF cell outputs are filtered by a pair of Gaussian receptive fields, with each Gaussian offset to the right (R) or left (L) of the simple cell oriented axis. For simplicity, only vertical and horizontal orientations were simulated. The Gaussians were defined by equation (5
). The individual Gaussian inputs from LGN to layer 4 equal:
![]() | (10) |
![]() | (11) |
![]() | (12) |
![]() | (13) |
![]() | (14) |
![]() | (15) |
In order to make the simulations manageable, some simplifications of known biological interactions were made. Even so, each simulation of model development (described below) took 11 days to run on a Silicon Graphics workstation. The simplifications that were made should not influence our results on cortical development or adult grouping. In particular, in vivo layer 4 simple cells that are sensitive to opposite contrast polarities pool their outputs at layer 2/3 complex cells (Alonso and Martinez, 1998). In order to cut the number of simulated cells in half, and with it the run time, we assumed that these simple cell outputs were pooled in layer 4:
![]() | (16) |
![]() | (17) |
![]() |
![]() | (18) |
![]() | (19) |
![]() | (20) |
![]() |
![]() | (21) |
Layer 6 Cells
A layer 6 cell at position (i,j) and orientation k is assumed to receive oriented input, Cijk, from LGN and feedback, zijk, from layer 2/3 complex pyramidal cells (see Fig. 1b). It is assumed that the Cijk inputs are registered at layer 6 cells due to a prior stage of development during which LGN inputs learned an oriented connection to layer 6 cells due to correlations induced by layer 2/3-to-6 and/or layer 4-to-6 feedback from cells of both contrast polarities. Ringach et al. have reported that the responses of layer 6 cells in V1 do, in fact, exhibit a polarity-independent response component as well as a polarity-dependent component (Ringach et al., 1999
). In all, layer 6 cell activation, xijk, obeys:
![]() | (22) |
![]() | (23) |
![]() | (24) |
Layer 4 Inhibitory Interneurons
Layer 4 inhibitory interneuron activities, mijk, receive on-center input from layer 6 activities xijk and off-surround inhibition mpqr from other layer 4 inhibitory interneurons:
![]() | (25) |
Layer 2/3 Complex Cells and Long-range Horizontal Connections
In the adult model, layer 2/3 pyramidal cells receive excitatory input [yijk]+ from layer 4 at the same position, excitatory input pqr Hpqrijk F(zpqr,
) via the long-range horizontal kernels Hpqrijk from other layer 2/3 pyramidal cell signals F(zpqr,
) at different positions, and inhibitory input
rTrksijr+ from the layer 2/3 disynaptic interneuronal activities sijr at the same position (i,j) and all orientations r (see Fig. 1c
). Layer 2/3 pyramidal cells thus obey the equation:
![]() | (26) |
The horizontal connection strength, Hpqrijk, is the product of the axonal strength, Upqrijk, from cell (p,q,r) to cell (i,j,k) and the synaptic weight, Vpqrijk, that abuts cell (i,j,k) after the axon from (p,q,r) contacts it. Thus:
![]() | (27) |
Layer 2/3 Disynaptic Inhibitory Interneurons
In the adult network, layer 2/3 inhibitory interneurons are excited by horizontal connections pqr Hpqrijk F(zpqr,
) from layer 2/3 pyramidal cells, and are inhibited by layer 2/3 disynaptic interneuronal activities sijr that represent all orientations r at the same position (i,j). This recurrent inhibitory network tends to normalize the total output signal from each inhibitory interneuronal population:
![]() | (28) |
Parameters for Equations (3)(28
)
Cell activation parameters in all simulations, except those that were varied to demonstrate model robustness, are: 1 = 1.0,
= 10.0,
= 6.0,
= 0.1, C1 = 1.5, C2 = 0.075,
2 = 0.5,
C = 0.25,
= 0.5,
= 2.0,
= 2.0,
= 1.25.
Development of Layer 2/3 Excitatory Horizontal Axons
At the beginning of model development, layer 2/3 pyramidal cells have no horizontal axons. An intracellular process calibrates an amount of potential axonal growth, Utotal. The variables Upqrijk represent the strength of axonal connections from a layer 2/3 source cell at position (p,q) with orientational tuning r to a target cell at position (i,j) with orientational tuning k. At each position, there are four two-dimensional U kernels corresponding to the four orientational combinations of r and k (vertical-to-vertical, vertical-to-horizontal, etc.). These variables are initialized at zero, and updated via the equation
![]() | (29) |
When the source cell is above threshold, its axonal growth to a target cell is driven by activity-dependent morphogenetic gradients zijkA(Epqrijk) in (29). Quantity Epqrijk represents the distance between the axonal growth cone of source cell (p,q) and target cell (i,j). It is defined by the difference between the length Upqrijk of the growth cone and the distance Dpqij between the source cell and the target cell:
![]() | (30) |
The morphogenetic gradient A(Epqrijk) that influences growth to cell (i,j,k) increases as the growth cone Upqrijk approaches (i,j,k), i.e. as Epqrijk decreases. This property is captured by the equation:
![]() | (31) |
Development of the connections Upqrijk is restricted to a local, circular window, such that:
![]() | (32) |
Parameter Hrange determines the spatial extent in which the growth of horizontal connections is possible. The actual extent of growth may be less than Hrange. In our simulations, the horizontal connections grew to a length of nine iso-orientation columns (see Fig. 5). Figure 14d,h
shows how the vertical-to-vertical U kernel has developed just before eye opening and after visual development self-equilibrates, respectively.
Equations (29)(31
) determine how, over time, activity correlations among layer 2/3 cells produce a spatial distribution of axonal connections. The four key parameters that influence this process are as follows. (i) Parameter Utotal in (29) determines the total amount of axonal growth out of each layer 2/3 cell. (ii) Parameter
in (29) determines the level of intracellular competition for axonal resources between different axons. Reducing
causes the distributions of axonal connections to become more isotropic. (iii) Parameter
in equation (30
) maps units of axon growth into units of spatial distance, in order to calculate the distance of an axon from its target. Increasing
increases the length to which axons are capable of growing. (iv) Parameter ß in (31) affects the shape of the chemical gradient that attracts an axon to its target. Raising ß flattens this gradient and thus reduces the effect of distance on competition in equation (29
). Raising ß thus causes the axonal distribution to become more isotropic with respect to distance.
Learning of Layer 2/3 Excitatory Horizontal Synaptic Weights
Once a horizontal axon reaches its target cell, then the strength of its synaptic connection can be modified by activity-dependent correlations. We use an instar learning law, which has become the standard law for learning self-organizing maps (Grossberg, 1976a, 1980b
; Kohonen, 1989
). During instar learning, the activity in the postsynaptic target cell turns on learning, and the adaptive weight learns the expected value of the presynaptic source cell's signals during intervals when the target cell is active. Disynaptic inhibition can prevent a postsynaptic cell from firing and thus, by instar learning, prevent the learning of irrelevant horizontal connections (Hess and Donahue, 1994). The synaptic weights Vpqrijk carried by the excitatory horizontal axons equal zero at the beginning of training, and are updated using the instar learning equation:
![]() | (33) |
![]() | (34) |
Development of Layer 2/3 Disynaptic Inhibitory Connections
Layer 2/3 disynaptic inhibition (Fig. 1c) is mediated by two weights. Weight Trijk- calibrates the mutual inhibition between layer 2/3 inhibitory interneurons in equation (28
), and weight Trijk+ calibrates the inhibition of layer 2/3 pyramidal cells by layer 2/3 disynaptic inhibitory interneurons in equation (26
). These weights have only two spatial indices because they are short-range interactions whose spatial extent is limited to a single hypercolumn, indexed by (i,j). The indices r and k denote the orientations of the source cell and target cell, respectively. The weights start with zero values and develop using an outstar learning law (Grossberg, 1968
, 1980b
).
Outstar learning of inhibitory connections is used to maintain the balance between inhibition and excitation. Outstar learning accomplishes this by causing the inhibitory synaptic weights to track the expected activation of the excitatory cells. If instar learning had been used, then the inhibitory weights would have tracked the expected value of the inhibitory cells. If an excitatory cell got more and more active, this would not necessarily cause a balanced increase in inhibition. Instar learning of excitatory connections was used in (33) to offset imbalances in cell activation patterns while maintaining the selectivity of cell connections. Had outstar learning been used for that purpose, then problems could have ensued. For example, suppose that two cells, A and B, are learning excitatory connections to each other. Let cell A be active 90% of the time, and cell B be active 10% of the time. For simplicity, assume the activities are independent and always equal to one. If outstar learning were used, then the synaptic weight to cell A would approach 0.9, and the synaptic weight to cell B would approach 0.1. Therefore, the discrepancy between the cells would increase. Outstar learning at the horizontal layer 2/3 connections could hereby cause some cells to get stronger and stronger, and could end up using all the axonal resources to support connections to them, at the expense of other cells. However, with instar learning, the weight to cell A would approach 0.1, and the weight to cell B would approach 0.9, thereby reducing the discrepancy between their mutual activations, without a loss of selectivity. It is for these reasons that instar and outstar learning were used to control excitatory and inhibitory connections, respectively, in the model.
The outstar learning laws that are used to connect the inhibitory kernels T+ and T in equations (35) and (36
) are:
![]() | (35) |
![]() | (36) |
Development of Layer 4 Inhibitory Connections
The layer 4 surround inhibition (Fig. 1b) is mediated by Wpqrijk- weights that carry mutual inhibition between layer 4 inhibitory interneurons in equation (25
), and Wpqrijk+ weights that carry inhibition of layer 4 excitatory cells by layer 4 inhibitory interneurons in equation (20
). These inhibitory weights start from zero values and develop using an outstar learning rule, in which learning is activated when the source cell turns on. During these sampling intervals, weight strength approaches the expected value of the target cell's activity at a rate that covaries with the source cell's activity. Such a learning law incorporates both Hebbian and anti-Hebbian properties (Singer, 1983), since weight strength can either increase or decrease to track and thereby balance its postsynaptic target activity. Inhibitory learning rules of this type have also been used to model dynamic receptive field changes produced by scotomas (Kalarickal and Marshall, 1999
). The learning laws for the W and W+ weights are:
![]() | (37) |
![]() | (38) |
![]() | (39) |
![]() | (40) |
Parameters for the Developmental Equations (36) and (37
)
Axon growth and synaptic weight update parameters are: W = 1.0, C3 = 6.0, Wrange = 7,
A = 0.25, Utotal = 44.0,
= 0.01,
= 6.0, ß = 8.0, Hrange = 11,
V = 0.5,
= 0.01,
= 1.5.
Training Procedure
Model development occurred during two successive stages, as also occurs in vivo. The initial coarse specification of horizontal connections prior to eye opening is followed by a strengthening and increase in the selectivity of these connections after structured vision begins. The unstructured vision phase was modeled using uniformly distributed random noise inputs which are Gaussianly filtered to induce local correlations (Fig. 2, left). This Gaussian filtering uses the same standard deviation (
= 0.5) that was used to define each lobe of the simple cell receptive field in equations (10
) and (11
). Following eye opening, inputs contain spatial structure that is determined by objects in the world. We modeled these structured visual inputs with randomly sized and positioned rectangles (Fig, 2
, right), in keeping with the idea that essentially all visual objects have linear contours on a sufficiently small spatial scale. Rectangles were appropriate in the present simulation study because the model only represented horizontal and vertical orientations. Later work will use more orientations and will train the model using real-world images. In the present study, each input image contained seven rectangles, each with a contrast that was randomly distributed between 0 and 2. The length and width of each rectangle was determined by an iterative random process in which each dimension started at zero pixels, grew (independently) by one pixel at each iteration, and stopped growing with probability 0.1 at each iteration. The images were processed with wrap-around in both the x and y dimensions in order to avoid spurious boundary effects.
The training procedure consisted of presenting each randomly generated image (see Fig. 2) and integrating cell activation equations (3
)(28
), using the fourth-order RungeKutta method, until equilibrium, while keeping all the weights fixed. Equilibrium was considered achieved when the average absolute activity change of layer 2/3 pyramidal cells, defined in equation (26
), fell below a threshold of 0.002. This typically occurred after ~20 iterations through equations (3
)(28
). Then, a single integration step of the developmental equations (36
) and (37
) was run using the fourth-order RungeKutta method. This scheme captured the main idea that development occurs slowly relative to the time-scale of cell activation. Using this procedure, the network was trained with 20 000 unstructured images (Fig. 2
, left) followed by 30 000 structured images (Fig. 2
, right). After presentation of ~10 000 unstructured images, the learning equations stabilized, and little change took place until the structured images were presented. A burst of new learning then took place due to the different statistics of the structured images, and did not stabilize until presentation of ~15 000 structured images.
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