Theoretical and experimental studies of relationship between pinwheel centers and ocular dominance columns in the visual cortex

Neuroscience Research 55 (2006) 370–382 www.elsevier.com/locate/neures

Theoretical and experimental studies of relationship between pinwheel centers and ocular dominance columns in the visual cortex Hayato Nakagama, Toshiki Tani, Shigeru Tanaka * Laboratory for Visual Neurocomputing, Brain Science Institute, RIKEN, Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan Received 29 October 2005; accepted 2 May 2006

Abstract In the visual cortex, pinwheel centers, which appear as point singularities in orientation maps, are likely to be found at the centers of ocular dominance columns in normal cats and monkeys. To elucidate the mechanism underlying the geometrical relationship, we performed computer simulation based on our correlation-based self-organization model. The simulation showed that pinwheel centers tended to be located at the ocular dominance centers at higher correlations of activities between the left- and right-eye specific pathways, whereas they tended to appear along the borders of ocular dominance columns at lower correlations. This tendency was mathematically analyzed with a formula describing the condition determining the geometrical relationship between pinwheel centers and ocular dominance columns. Moreover, to examine the effect of activity correlations in the eye-specific pathways on the column formation, we conducted intrinsic signal optical imaging using normally reared cats and dark-reared cats. The between-eye activity correlation in dark-reared cats is expected to be lower than that in normal cats due to the lack of common visual input in the two eyes. The statistical analysis of experimental data showed that while more pinwheel centers tended to be located in the center subregion of ocular dominance columns than in the border subregion in the normal cats, a weak tendency in the opposite direction was found in the dark-reared cats. Based on the consistent results from the model and experiment, it is suggested that the activity correlation between the left- and right-eye specific pathways has influence on the establishment of geometrical relationship in the cortical representation between orientation preference and ocular dominance. # 2006 Elsevier Ireland Ltd and the Japan Neuroscience Society. All rights reserved. Keywords: Pinwheel center; Ocular dominance columns; Orientation map; Model; Simulation; Optical imaging; Dark-reared cats

1. Introduction How much visual experience affects the formation of ocular dominance columns and orientation map has been a subject of debate. It has been demonstrated that ocular dominance columns (Wiesel and Hubel, 1965; Hubel et al., 1977; Shatz and Stryker, 1978; Olson and Freeman, 1980; Antonini and Stryker, 1998) and orientation preferences (Blakemore and Cooper, 1970; Hirsch and Spinelli, 1970; Rauschecker and Singer, 1981; Sengpiel et al., 1999; Tanaka et al., 2004, 2006) can be modified by monocular deprivation and orientation-restricted visual experience in an early life, respectively. These observations indicate that visual experience contributes to the formation of ocular dominance columns and orientation maps. On the other hand, structural

* Corresponding author. Tel.: +81 48 467 9667; fax: +81 48 467 9685. E-mail address: [email protected] (S. Tanaka).

relationships between the two maps have been found: (i) pinwheel centers, around which all orientations are circularly represented, tend to be located at the centers of ocular dominance columns (Bartfeld and Grinvald, 1992; Swindale, 1992; Crair et al., 1997; Hu¨bener et al., 1997; Matsuda et al., 2000) and (ii) iso-orientation contours tend to cross ocular dominance borders at right angles (Obermayer and Blasdel, 1993; Hu¨bener et al., 1997; Kim et al., 1999). These structural relationships suggest that the two functional maps develop with some interactions. Thus far, a lot of models have been proposed in an attempt to explain the coordinated development of different functional maps in the visual cortex (Erwin et al., 1995; Swindale, 1996). The self-organizing map (SOM) (Kohonen, 1984) or the elastic net model (Durbin and Mitchison, 1990) successfully provides frameworks, in which features in a higher dimensional space are mapped onto the two-dimensional (2D) cortical surface continuously. The resultant cortical maps appear biologically plausible in that most of the

0168-0102/$ – see front matter # 2006 Elsevier Ireland Ltd and the Japan Neuroscience Society. All rights reserved. doi:10.1016/j.neures.2006.05.004

H. Nakagama et al. / Neuroscience Research 55 (2006) 370–382

pinwheel centers are located along the middle of ocular dominance bands, and iso-orientation contours cross the ocular dominance borders at right angles (Obermayer et al., 1992). Another class of models called the correlation-based learning models has also been investigated in depth, assuming that cortical maps develop with Hebbian synaptic modification rather than mapping a priori features on the 2D cortical surface. Using a two-stage algorithm for the development of an orientation map and an ocular dominance map one after another in the framework of a correlation-based learning model, Erwin and Miller (1998) have shown a tendency for pinwheel centers to be located at the centers of ocular dominance columns. Mitchison and Swindale (1999) have proposed a Hebbian variant of SOM and shown that isoorientation contours intersect the borders of ocular dominance columns at right angles. In general, correlation-based learning models, however, do not reproduce the structural relationships in the representation between orientation preference and ocular dominance so clearly as SOM or the elastic net model. In the present study, first we examined the relationship between pinwheel centers and ocular dominance columns theoretically using our model that belongs to a class of correlation-based self-organization models. Computer simulations showed that pinwheel centers tended to appear at ocular dominance centers at higher activity correlations between the left- and right-eye specific pathways, and along the ocular dominance borders at lower activity correlations. From the analytical approach, we then postulated a mathematical formula that describes how rearing conditions affect the relationship between pinwheel centers and ocular dominance columns. To demonstrate the validity of the simulated results and the mathematical formula derived from the model, we performed optical imaging of intrinsic signals in normal and dark-reared cats. We found consistency between the theoretical results and experimental findings.

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Fig. 1. Schematic diagram of the present model. The upper square panel represents a postsynaptic layer corresponding to superficial layers in the visual cortex. Cells in a presynaptic layer corresponding to layer 4 are assumed to be orientation-selective and to respond specifically to either the left- or right-eye input. The presynaptic layer is fictitiously divided into the left- and right-eye specific layers, each of which is subdivided into eight layers according to preferred orientations (interval of orientations: p/8). The lower 16 square panels represent these fictitious layers.

2.2. Energy function The position of the receptive field center of a presynaptic neuron and the cortical position of its afferent input terminal are indicated by k and j, respectively. When a synaptic input from the neuron specified by (k, u, o) is terminated at the cortical position j, sj,k,u,o takes 1, and when the input is disconnected, sj,k,u,o takes 0. Each cortical position receives a single synaptic input due to the winner-takes-all mechanism based on the competition among afferent inputs at the same dendritic spine P (Tanaka, 1990). Each cortical position receives either input, satisfying k;u;o s j;k;u;o ¼ 1. The model is described by the energy function (Tanaka, 1990): H¼

XX X

V j; j0 G k;u;o;k0 ;u0 ;o0 s j;k;u;o s j0 ;k0 ;u0 ;o0 þ c

j; j0 k;u;o k0 ;u0 ;o0

X
X

2 s j;k;u;o

:

j

k;u;o

(1)

2. Materials and methods

Here, V j; j0 represents the cortical interaction between afferent inputs at positions j and j0 , which is given by the Gaussian function:

2.1. Model description V j; j0 ¼ The present model is built on the basis of the two-step map-formation algorithm for simplicity. That is, it is assumed that in the two-step algorithm, functional maps appear in the postsynaptic layer corresponding to superficial layers for the first time, rewiring afferent inputs from neurons in the presynaptic layer corresponding to layer 4, in which orientation selectivity and ocular dominance are established (Fig. 1). Although the orientation map may appear even in layer 4 of the visual cortex in cats, the algorithm assumes that neurons in the recipient layer of geniculocortical inputs possess orientation selectivity and ocular dominance irrespective of spatial segregation into map structure. Neurons in the presynaptic layer represented by 16
16 grids project axons to the postsynaptic layer represented by 256
256 grids. The periodic boundary condition was adopted to minimize finite-size effects. Neurons in one grid of the presynaptic layer are classified into two groups, one of which receives inputs from the left eye and the other receives inputs from the right eye. The ocularity is denoted by subscript o, where o = 1 for left-eye specific inputs and o = 1 for right-eye specific inputs. Each group of inputs is further subdivided into eight subgroups according to their preferred orientations denoted by the subscript u, which takes one of the eight values of 0, p/8, 2p/8, . . ., 7p/8.


 d 2j; j0 1 exp  ; 2l2v 2pl2v

(2)

where lv is the interaction length in the cortex (lv = 3.6), and d j; j0 represents the distance between cortical positions j and j0 . G k;u;o;k0 ;u0 ;o0 represents the correlation of firings between a pair of presynaptic neurons specified by (k, u, o) and (k0 , u0 , o0 ), and is given by

X 1 /2 g 1/0 2 0 1þb cos 2ðuu0 Þ 1þr 1r oo0 G k;u;o;k0 ;u0 ;o0 ¼ g k;u;o Rk;l
Rk0 ;l : / 1þb / 2 þ / 2 0 k ;u ;o l

(3) /2 g /0 2 of the correlation function G The coefficient g k;u;o k;u;o;k0 ;u0 ;o0 originates in k ;u0 ;o0 the assumption that presynaptic spike generation obeys the Poisson process, where gk,u,o indicates the average firing rate of presynaptic neurons specified by preferred orientation u, ocular dominance o, and receptive field center k. Since we consider only the normal or dark rearing condition, we can omit u and o dependence of the average firing rate. Since we are not interested in the retinotopic nonuniformity, either, we can omit k dependence. Without loss of generality, gk,u,o is set to be 1. b is a constant that regulates the activity correlation between a pair of presynaptic neurons with the same ocularity but 1

1

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different preferred orientations, u and u0 . r represents the strength of activity correlation between presynaptic neurons with different ocularities, which is given by r ¼ r0

1 þ a cos 2ðu  u0 Þ : 1þa

(4)

Here, r0 and a are positive constants. The activity correlation between the leftand right-eye specific pathways is assumed to depend on not only the absolute level of correlation between the two pathways but also the difference in preferred orientations of presynaptic neurons with different ocularities. r0 corresponds to the strength of the average activity correlation in the eye-specific pathways. We will call r0 simply the strength of between-eye correlation hereafter. Rk;l represents the normalized activity of a presynaptic cell whose receptive field center is located at position k in the visual field in response to the light-spot stimulus presented at position l in the visual field. This function is given by the Gaussian function with correlation length 6.4. The second term of the energy function given by Eq. (1) represents the constraint on the number of synaptic inputs from individual presynaptic neurons so that the total number of synaptic inputs from each presynaptic neuron tends to be as small as possible for positive constant c (c = 0.005).

2.3. Simulation At the beginning of the simulation, we randomize afferent inputs so that each cortical position receives an input from presynaptic neurons according to the probability:
 dJ2 ; j Pk; j ¼ exp  2k ; 2lArb

(5)

where d J k ; j represents the distance between cortical position j and position Jk that retinotopically corresponds to the position k in the visual field. lArb represents the extent of arborization of afferent inputs (lArb = 16.0). For each trial of rewiring in the simulation, a new candidate for an afferent input is selected according to the same probability (Eq. (5)), and then the present input is replaced with the new one on the basis of the probability determined by the energy difference between the states before and after the trial of replacement: Prðbefore ! afterÞ ¼

1 : 1 þ exp½bðH after  H before Þ

(6)

Here, Hbefore and Hafter represent the energy before and after the trial of replacement, respectively. The parameter b, which represents the inverse temperature in terms of statistical physics, was set at b = 1. Thereby the probability for afferent inputs to be updated takes 1 when Hafter is smaller than Hbefore, otherwise 0. The repetition of the update procedure likely decreases the energy of the system and realizes an equilibrium pattern of afferent inputs when the change in energy becomes negligibly small.

2.4. Experimental procedures The surgical operation and optical imaging were approved by the Institutional Animal Research Committee in RIKEN (No. H13-B040), and were performed in accordance with the ‘‘Guiding Principles for the Care and Use of Animals in the Field of Physiological Science’’ of the Japanese Physiological Society.

2.5. Manipulation of visual experience We used 12 kittens delivered from our own colony. Eight kittens were reared in standard animal cages (17-h light/7-h dark cycle). Four other kittens were reared in light-proof boxes placed in a dark room 1–3 days after birth.

2.6. Surgery for mounting an imaging chamber Initial anesthesia was induced by ketamine hydrochloride (5.0 mg/kg, i.m.) following the sedation with medetomidine hydrochloride (0.1 mg/kg, i.m.). The animals were fixed on a stereotaxic apparatus and were artificially ventilated

with a 60:40% mixture of N2O and O2 containing 0.5–1.0% isoflurane. The hart rate, end-tidal CO2 concentration, and rectal temperature were continuously monitored during surgery. A rectangular-shaped metal chamber was cemented on the skull using dental resin, and the skull and dura mater covering the recording area in the lateral gyrus were removed. Then, the chamber was filled with 2% agar desolved in a saline solution after the cortical surface was covered with a cellulose sheet for minimizing the adhesion of agar to the pia mater. Finally, the chamber was sealed with a polyvinylidene chloride thin film and a plastic plate.

2.7. Optical imaging Animals were anesthetized as described above, paralyzed with pancuronium bromide (0.1 mg/kg/h) and artificially ventilated. Contact lenses with appropriate curvatures were used to prevent the eyes from drying. The cortex was illuminated with a light of 700 nm wavelength. This wavelength of the light falls within a light-scattering region, such that the evoked intrinsic signal likely reflects subthreshold neuronal activity as well as spike activity (Cohen, 1973). The focal plane was adjusted 500 mm below the cortical surface using a tandem-lens macroscope arrangement (Bonhoeffer and Grinvald, 1996). Intrinsic optical signals were measured while the animals were exposed to visual stimuli displayed on a 20-in. CRT monitor screen placed 30 cm in front of the animal. Images were obtained using a CCD video camera and digitized and stored using CAPOS (Tsunoda et al., 2001) or IMAGER 2001 (Optical Imaging Inc.). The intrinsic optical signal in response to each stimulus was recorded for 5.0 s immediately after the stimulus onset, whereas a blank stimulus was presented for 5 or 15 s between the successive recordings of optical signals. Each stimulus was presented once in a pseudorandom sequence in a single trial of recordings.

2.8. Visual stimulation The animals were stimulated with full-screen square-wave gratings with a spatial frequency of 0.15 cycles/degree, which were drifted in two directions at six equally spaced orientations (interval, 308). The temporal frequency of the gratings was fixed at 2.0 Hz.

2.9. Data analysis After applying Gaussian low-pass filtering with a 150 mm standard deviation to eliminate high-frequency noise, we applied the generalized indicator function method (Yokoo et al., 2001), which extracts stimulus-evoked intrinsic signals effectively from noisy signals originating in blood vessels and spatially slowly varying fluctuation inherent in recorded signals (Ribot et al., 2006). Twelve single condition maps (6 orientations for each eye) were obtained by subtracting an image of responses to the blank stimulus from an image of stimulus-evoked responses after the application of the generalized indicator function method. Preferred orientations were determined by the vector sum method (Blasdel and Salama, 1986). Ocular dominance was calculated by subtracting the summed signals in response to all stimulus orientations presented to the right eye from the summed signals in response to all stimulus orientations presented to the left eye.

2.10. Analysis of orientation pinwheel centers The positions of orientation pinwheel centers are determined automatically as the points around which the circular integration of the orientation difference amounts to just p, where the circular integration is always taken in the clockwise direction. The clockwise and counterclockwise pinwheel centers are defined by the sign of the integral. The relative number of pinwheel centers in an iso-ocular dominance subregion is analyzed according to the method introduced by Hu¨bener et al. (1997). In particular, the center subregions of ocular dominance columns are defined as the domains whose pixels are classified into the top 10% of pixels exhibiting monocularity, either left- or right-eye dominance. The border subregions of ocular dominance columns are defined as the domains whose pixels exhibit binocularity within 10% of the pixels towards either left- or right-eye dominance. Thus, the combined center subregions

H. Nakagama et al. / Neuroscience Research 55 (2006) 370–382 compose 20% of the region of interest, and the border subregions occupy 20% of the area in the region of interest.

3. Results 3.1. Computer simulation Fig. 2 shows simulation results obtained after ten thousand trials for updating afferent inputs for different values of between-eye correlation (r0 = 0.2, 0.5 and 0.8). The periodicity of arrangements of preferred orientations does not seem to depend on the between-eye correlation (Fig. 2A, E and I). In contrast, the width of ocular dominance columns decreases with the increase in between-eye correlation (Fig. 2B, F and J), as previously reported (Goodhill, 1993; Nakagama and Tanaka,

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2004). At sufficiently low values of between-eye correlation (r0  0.2), the orientation maps tend to be independent in the two eyes, and preferred orientations do not necessarily match across the borders of ocular dominance columns. At r0 = 0.2, preferred orientations are barely continuous across the ocular dominance borders (Fig. 2A), where we can identify remnant discontinuities of preferred orientations at some border locations. At higher values of between-eye correlation, on the other hand, the preferred orientations in the two eyes match along the borders of ocular dominance columns (Fig. 2E and I). Since orientation line singularities running along the ocular dominance borders have not been found yet, the physiologically possible range of between-eye correlation may be in the range of 0.2  r0  1. In the present study, we limit ourselves to simulation and mathematical analysis within this range.

Fig. 2. Simulation results. The top, middle and bottom rows represent simulation results obtained at between-eye correlations r0 = 0.2, 0.5 and 0.8, respectively. The leftmost column (A, E and I) represents simulated orientation maps on the postsynaptic layer in the model cortex, in which the preferred orientations are indicated by different colors. The second column (B, F and J) represents patterns of ocular dominance columns, in which black and white pixels denote areas receiving synaptic inputs from the left- and right-eyes, respectively. The third column (C, G and K) represents orientation contour maps superimposing the borders of ocular dominance columns, in which the grey and black lines indicate iso-orientation contour lines and ocular dominance borders, respectively. The rightmost column (D, H and L) shows the histograms of the average number of pinwheel centers for different ocular dominance classes (see Section 2 for detail). Because the ocular dominance is represented by binary values in the present model (1 for left-eye inputs, 1 for right-eye inputs), we applied Gaussian low pass filtering with a standard deviation of 18 pixels to the binary ocular dominance columns to determine ocular dominance in neuronal responses, which changes smoothly over the model visual cortex.

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Spatial relationships between ocular dominance columns and orientation contour maps are visualized (Fig. 2C, G and K), in which pinwheel centers are observed as point singularities where all iso-orientation contour lines converge. The isoorientation contours are likely to cross the borders of ocular dominance columns at right angles. Fig. 2D, H and L shows the

histograms of the average numbers of pinwheel centers in subregions of different ocular dominance (see Section 2 for detail). At a higher between-eye correlation (Fig. 2L), the pinwheel centers tend to be at the centers of ocular dominance columns. On the other hand, pinwheel centers tend to be located along the borders of ocular dominance columns at a lower

Fig. 3. Relation of orientation gradient to ocular dominance in simulation results. The top, middle and bottom rows represent results simulated at between-eye correlations r0 = 0.2, 0.5 and 0.8, respectively. The left column shows patterns of orientation gradients and patterns of borders of Gaussian-filtered ocular dominance columns. The solid lines represent ocular dominance borders, and the greyscale indicates the magnitude of orientation gradient. The right column shows the average orientation gradients for different ocular dominance classes.

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between-eye correlation (Fig. 2D). At intermediate values of between-eye correlation, pinwheel centers are located either at ocular dominance centers or along ocular dominance borders (Fig. 2H). From the simulated results, the borders and centers of ocular dominance columns seem to be specific places from which pinwheel centers emerge (see Section 3.2). In Fig. 3, orientation gradients instead of pinwheel centers are shown superimposing ocular dominance columns (Fig. 3A, C and E), using the same simulated results as shown in Fig. 2. We can see the tendency that the average orientation gradient is high along the borders of ocular dominance columns at low betweeneye correlation (Fig. 3B), and it is high at the ocular dominance centers at low between-eye correlation (Fig. 3F). At an intermediate value of between-eye correlation, the average orientation gradient tends to be high at both ocular dominance centers and borders (Fig. 3D). The distributions of the average orientation gradient, shown in Fig. 3B, D and F, have similar features in the distributions of the relative number of pinwheel centers, shown in Fig. 2D, H and L. Thus, it is indicated that the behavior of orientation gradient is parallel to that of pinwheel centers.

375 2

~ ~x Þ ) along the ocular dominance change rapidly (large ðru 2 borders (small o~x ), and to be smoothly represented (small ~ ~x Þ2 ) around the ocular dominance centers (large o2 ). As seen ðru ~ x in the previous subsection, the spatial relationship between the average orientation gradient and ocular dominance resembles that between the number of pinwheel centers and ocular dominance. Consequently, the sign of the coefficient determines whether more pinwheel centers tend to appear along the ocular dominance borders or in the vicinity of the ocular dominance centers. The tendency that the optimal locations of pinwheel centers are either along the ocular dominance borders or near the ocular dominance centers (Fig. 2D, H, and L) originates in the specific form of interaction between the preferred orientation and ocular dominance given by Eq. (7), which maximizes or minimizes the overlap of orientation gradient with ocular dominance depending on the sign of the coefficient. To evaluate the above theoretical prediction, we examine how the coefficient of the interaction energy correlates with the

3.2. Mathematical analysis To analyze the relationship between pinwheel centers and ocular dominance, we performed a mathematical analysis for the energy function used in the computer simulation. In the continuum approximation, the first term of the energy function H given by Eq. (1) is expressed as the sum of the two energy functions: H0 + H1 (see Appendix A for derivation). H0 represents the energy function for the independent selforganization of ocular dominance and orientation maps, and H1 represents the energy function describing interaction between the ocular dominance and the preferred orientation. The preferred orientation u~x and ocular dominance o~x at cortical position ~ x change such that the value of energy function H decreases. Now, we focus particularly on H1 given by H1 ¼

l2v



bð1  r 0 Þ ar 0  1þb 1þa

Z

~ ~x Þ2 o2 ; d~ xðru ~ x

(7)

because it should regulate the geometrical relationshipR between ocular dominance and orientation gradient. In Eq. (7), d~ x is the ~ ~x is the 2D integration inside the region of interest, and ru gradient of the preferred orientation at cortical position ~ x. r0, a, b, and l2v are constants defined in Eqs. (2)–(4). The coefficient l2v ½ðbð1  r 0 Þ=ð1 þ bÞÞ  ðar0 =ð1 þ aÞÞ can be positive or negative depending on the values of parameters a, b, and r0. When the coefficient is negative, i.e., + a))/ R r0 > ((b(1 ~ ~x Þ2 o2 stabi(a + b + 2ab)), the increase of the value of d~ xðru lizes the system by decreasing the energy. After the self-organization is completed for this parameter setting, the orientation ~ ~x Þ2 ) near the centers of tends to change rapidly (large ðru 2 monocular domains (large o~x ). On the contrary, when the coefficient is positive, i.e., Rr0 < ((b(1 + a))/(a + b + 2ab)), the ~ ~x Þ2 o2 stabilizes the system. decrease of the value of d~ xðru At the final state of self-organization, the orientation tends to

Fig. 4. Comparison between theory and simulation. (A) The difference in the average number of pinwheel centers between the binocular class (Histogram(80–100)) and the monocular class (Histogram(0–20)) is plotted against b and r0. in the 3D representation, where nearby points are connected with red lines. The blue curve indicates the zero-crossing of the difference in the average number of pinwheel centers between the binocular and monocular classes. On the green (purple) curve, 10% more pinwheel centers are located in the monocular (binocular) class. (B) The red curve derived from the rigorous analysis indicates the curve on which the numbers of pinwheel centers are the same between the monocular and binocular classes. The blue circles and the vertical lines, respectively, show the average values and standard deviations of r0, where the same number of pinwheel centers is located in the monocular and binocular classes.

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difference in the number of pinwheel centers between the border regions and the center regions of ocular dominance columns. Fig. 4A shows b and r0 dependencies on such a difference in the number of pinwheel centers for a fixed value of a = 1. The blue line in Fig. 4A represents a contour line along which the number of pinwheel centers is exactly the same in the border regions and in the center regions. In the region left to the blue line in (b, r0) space, pinwheel centers tend to appear at the ocular dominance centers. Fig. 4B shows the comparison between the above results from simulations and the theoretical curve. For simulations conducted at a = 1.0, we plot points at which the number of pinwheel centers are balanced between the border regions and the center regions. The circles indicate the average positions of the points determined by 12 simulations with standard deviations indicated by the vertical lines. The red line represents the theoretical curve r 0 ¼ ð2b=ð1 þ 3bÞÞ for a = 1.0. The figure shows that the rigorous analysis describes well the spatial relationship between pinwheel centers and ocular dominance observed in the simulation results. 3.3. Orientation maps and ocular dominance columns in normal and dark-reared cats To examine how much visual experience affects the relationship of ocular dominance columns and orientation

map in the visual cortex, we performed optical imaging experiments in cats reared in the normal visual environment and cats reared in the dark. In normally developed cats, it has been reported that pinwheel centers tended to appear at the centers of ocular dominance columns (Crair et al., 1997; Hu¨bener et al., 1997; Matsuda et al., 2000). It is expected that common visual images presented to the two eyes probably contributed to keeping the activity correlation between the left- and right-eye specific pathways high. In contrast, while cats are raised under the visually deprived condition, the between-eye correlation may be reduced due to the lack of common visual images in the two eyes, although orientation maps and ocular dominance columns may develop only from spontaneous activities evoked at early stages of visual pathways. This suggests that betweeneye correlation in dark-reared cats should be lower than that in normal cats. Therefore, it is predicted that more pinwheel centers are located along the ocular dominance borders than at the ocular dominance centers. Despite dark rearing decreases the strength of cortical responses to visual stimuli (Mower and Christen, 1985), orientation columns (Blakemore and Price, 1987) and ocular dominance columns (Swindale, 1998) are observed in area 18 of dark-reared cats. Using grating stimuli with a spatial frequency of 0.15 cycle/degree, which is optimal to neurons in area 18, we reconstructed orientation maps in both hemispheres of dark-reared cats (Fig. 5A and E) and a normal

Fig. 5. Orientation maps and ocular dominance columns in the visual cortex of three cats examined. The leftmost column shows orientation maps observed in both hemispheres of two dark-reared cats (A and E) and a normally reared cat (I). The preferred orientation is color-coded according to the key shown on the right. The second column shows the positions of pinwheel centers superimposing patterns of ocular dominance columns for the dark-reared cats (B and F) and for the normally reared cat (J). The red and blue dots indicate clockwise and counterclockwise pinwheel centers, respectively. The black and white patches represent regions of the cortex more strongly responding to the left- and right-eye stimuli, respectively. In the third column, iso-orientation contour lines and ocular dominance borders are indicated by the grey and black lines, respectively, for the dark-reared cats (C and G) and for the normal cat (K). In the rightmost column, histograms of the relative number of pinwheel centers for different ocular dominance classes are illustrated for the dark-reared cats (D and H) and for the normal cat (L).

H. Nakagama et al. / Neuroscience Research 55 (2006) 370–382

cat (Fig. 5I). In these orientation maps, it can be seen that preferred orientations continuously changed along the cortical surface except at pinwheel centers (Bonhoeffer and Grinvald, 1991). The positions of pinwheel centers and ocular dominance columns are plotted in Fig. 5B and F for the dark-reared cats and in Fig. 5J for the normal cat. The red and blue circles denote clockwise and counterclockwise pinwheel centers, respectively. The black and white patches are regions of the cortex more strongly activated by stimuli presented to the left eye and the right eye, respectively. The ocular dominance borders and orientation contour maps are illustrated in Fig. 5C and G for the dark-reared cats and in Fig. 5K for the normal cat. The black and grey lines in the figures indicate the ocular dominance borders and iso-orientation contour lines, respectively. Pinwheel centers are at positions at which all orientation contour lines converge. To examine the relationships between the arrangements of preferred orientations and ocular dominance, relative number

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of pinwheel centers against ocular dominance is plotted in Fig. 5D and H for the dark-reared cats and in Fig. 5L for the normal cat. In the normal cat, more pinwheel centers tended to be located in the center subregion of ocular dominance columns than in the border subregion (Fig. 5L). In the dark-reared cats, the number of pinwheel centers was not reduced in the border subregion (Fig. 5D and H). We then carried out statistical analyses of the experimental data. In the histogram of the relative number of pinwheel centers against ocular dominance for normal cats (n = 8) (Fig. 6A), more pinwheel centers tended to appear significantly in the center subregion of ocular dominance columns than in the border subregion ( p < 0.0001; Chi-square test), which agrees with previous studies (Crair et al., 1997; Hu¨bener et al., 1997; Matsuda et al., 2000). Fig. 6B shows the histogram for dark-reared cats (n = 4), which exhibits a peak at the border region. A significant difference in the relative number of pinwheel centers between normal cats and dark-reared cats was found ( p < 0.01; Chi-square test).

Fig. 6. The relative number of pinwheel centers and the average orientation gradient for the ocular dominance class. A and B show histograms of the relative number of pinwheel centers for different ocular dominance classes in the normal cats (n = 8) and in the dark-reared cats (n = 4), respectively. C shows values of the normalized overlap index (NOI) calculated from the simulated maps for different values of between-eye correlation (0.2 < r0 < 0.8). D shows values of the NOI calculated from the data of the dark-reared cats (n = 4) and normal cats (n = 8). All the error bars indicate standard deviations.

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In Section 3.2, we have shown that the sign of the coefficient of the interaction energy given by Eq. (7) determines stable locations of pinwheel centers with respect to ocular dominance. When the coefficient is negative the increase of the R (positive), ~ ~x Þ2 o2 ) stabilizes (destavalue of the overlap integral ( d~ xðru bilizes) the system. Therefore, the value of the overlap integral should reflect the between-eye correlation through the sign of the coefficient. Likewise, it is expected that the overlap integral also differ between dark-reared cats and normal cats. To compare the values of the overlap integral among different simulations and different cats, which contain variability of the regions of interest and the amplitudes of orientation gradient and ocular dominance, here we introduce the normalized overlap index (NOI) defined by NOI ¼

P ~ ~x j2 ð1=NÞ ~x o~2x
jru : P 2 P ~ ~x j2 ð1=NÞ ~x o~x
ð1=NÞ ~x jru

(8)

where N represents the total number of pixels in the region of interest. We can give another interpretation of the value of the NOI: when the magnitude of monocularity o~2x and the magni~ ~x Þ2 are distributed with no tude of orientation gradient ðru spatial correlation, the value of the NOI becomes 1. When they have positive correlation, the value is larger than 1, otherwise it is smaller than 1 (see Appendix B). Fig. 6C shows values of the NOI calculated from the simulated maps for different values of between-eye correlation r0 at b = 1.0. For this value of b, r0 > 0.5 (or r0 < 0.5) corresponds to the range in which the coefficient of the interaction energy is negative (or positive) (Fig. 4B). The NOI for larger r0 indicating the negative coefficient is larger than the NOI for smaller r0 indicating the positive coefficient. Particularly, the fact that the NOI is smaller than 1 for smaller r0 implies that binocular domains and domains of high orientation gradient tend to overlap each other. Reversely, the NOI larger than 1 for larger r0 implies that monocular domains overlap domains of high orientation gradient. Fig. 6D shows values of the NOI estimated from the data of the dark-reared cats and the normal cats. The values of the NOI for normal cats are apparently larger than those for darkreared cats. The difference of the NOI between the dark-reared cats and normal cats was statistically significant ( p < 0.01, ttest). Furthermore, the NOI takes the value larger than 1 for the normal vats, whereas the NOI takes about 1.0 for dark-reared cats. This indicates that orientation gradient tended to be high in the monocular domains for the normal cats, whereas the magnitudes of orientation gradient and monocularity were spatially uncorrelated. Taken together, we can see that the between-eye correlation was higher in the normal cats than in the dark-reared cats, although the between-eye correlation in the dark-reared cats was not so small that the value of the NOI was smaller than 1.

modification of synaptic inputs based on the Hebbian rule (Hebb, 1949). In those models, correlation in activity between pre and postsynaptic neurons determines the layout of stimulus features such as ocular dominance and orientation preference. Using the two-stage development algorithm in the correlationbased learning model, in which the ocular dominance map starts to be formed after the orientation map is established, Erwin and Miller (1998) have demonstrated that pinwheel centers tended to be at the center of ocular dominance columns, although this tendency was very weak. However, isoorientation contour lines did not tend to intersect the borders of ocular dominance columns at right angles. A Hebbian variant of Kohonen’s algorithm (Mitchison and Swindale, 1999) has reproduced structural relationships between orientation and retinotopic maps in the visual cortex. Pinwheel centers tended to appear on the borders of ocular dominance columns, however. Osam and Ermentrout (2002) have attempted to solve the problem of the localization of pinwheel centers along the ocular dominance borders within a framework of a correlationbased learning model with a presumptive mechanism that the development of ocular dominance is retarded where orientation selectivity is weak. Swindale (1992) has already proposed a similar idea that the development of orientation selectivity is retarded where the ocular dominance grows rapidly. The objective of the present study is to attempt to explain the geometrical relationship between pinwheel centers (or locations of high orientation gradient) and ocular dominance from the viewpoint of the possible interaction between orientation gradient and ocular dominance under visual experience during development. For this purpose, we employed a correlationbased learning model on the assumption that presynaptic neurons have already acquired orientation selectivity and ocular dominance. The preysynaptic correlation function is often assumed to be given by the product of the correlation function of the orientation preference and the correlation function of the ocular dominance (Tanaka, 1996; Osam and Ermentrout, 2002). In the present model, the total correlation function given by Eq. (3) is not necessarily factorized into the correlation functions of ocular dominance and orientation preference, because the activity correlation between the leftand right-eye specific pathways r depends on the stimulus orientation, as shown in Eq. (4). In the case where the correlation function is factorized (a = 0), the coefficient of the interaction energy (Eq. (7)) always takes a positive value, and pinwheel centers are located along the borders of ocular dominance columns. In the case where the correlation function is not factorized (a > 0), the coefficient can be positive or negative depending on the other parameters (b and r0), and pinwheel centers can appear at ocular dominance centers. 4.2. Related experimental works

4. Discussion 4.1. Correlation-based self-organization model Correlation-based self-organization models (Miller et al., 1989; Tanaka, 1990) have reproduced cortical maps through the

In macaque monkeys (Blasdel and Salama, 1986; Bartfeld and Grinvald, 1992; Swindale, 1992) and cats (Crair et al., 1997; Hu¨bener et al., 1997; Matsuda et al., 2000) reared under the normal visual condition, there is a clear tendency that pinwheel centers lie at the centers of ocular dominance

H. Nakagama et al. / Neuroscience Research 55 (2006) 370–382

columns. Our computer simulation and mathematical analysis demonstrated that the decrease of between-eye correlation relocated pinwheel centers from the centers to the borders of ocular dominance columns (Fig. 2D and L). This suggests that the geometric tendency observed in normally raised animals can be disrupted in animals reared under some condition where the activity correlation between the left- and right-eye specific pathways is decreased. A typical example of such animals raised under low between-eye correlation may be strabismic cats. They did not show consistent topographical relationship of pinwheel centers to the peaks of ocular dominance (Lo¨wel et al., 1998; Engelmann et al., 2002), and the distribution of pinwheel centers with respect to ocular dominance columns was more heterogeneous in strabismic cats, either esotropic or exotropic, than in normally raised cats (Engelmann et al., 2002). Such a finding in the strabismic cats is not inconsistent with our theoretical prediction. Another example of animals raised under low between-eye correlation may be dark-reared cats, which are lack of common visual input to the two eyes. In our dark-reared cats, the tendency of pinwheel centers being at ocular dominance centers was evidently weakened and more pinwheel centers were located along the ocular dominance borders, although this tendency was not statistically significant ( p < 0.4; Chi-square test) (Fig. 6B). However, based on the comparisons with our theory, it is indicated that the betweeneye correlation during development affects the establishment of the geometric relationship between pinwheel centers and ocular dominance centers, and that the between-eye correlation is higher in normal rearing than in dark rearing or strabismic rearing How much does dark rearing reduce the between-eye correlation? When we set the value of between-eye correlation r0 to be smaller than 0.2 in the simulation, the discontinuity in the arrangements of preferred orientations appeared along the ocular dominance borders. r0 = 0.2 was near the marginal point at which continuous orientation maps barely emerge (Fig. 2A and C). On the other hand, monocular orientation maps of the dark-reared cats were almost identical in the two eyes, and no such orientation discontinuity was observed (data not shown). Therefore, even for dark-reared cats, the between-eye correlation should be higher than r0 = 0.2. Furthermore, the observation that the value of the NOI was about 1 indicates that the values of between-eye correlation in dark-reared cats was not so small even if they were significantly lower than those in normal cats. This suggests that some neural mechanism actively keeps the between-eye correlation nonzero. It is speculated that the cortico-geniculate feedback may work as common inputs to the left- and right-eye specific LGN cells (Murphy and Sillito, 1996), because the eye-specific laminas of the LGN (i.e., A and A1) share axonal projections from neurons in layer 6 of the visual cortex with some biased arborization of axons from monocular cells of the same-eye dominance. It is likely that the common inputs from the visual cortex in turn induce correlated activities of geniculo-cortical afferent inputs in the two eyes (Weliky and Katz, 1999), which keeps the betweeneye correlation at a certain positive level.

379

4.3. Pinwheel annihilation It was studied theoretically that nearby pinwheel centers with opposite singularities likely come close together and annihilate (Wolf and Geisel, 1998). It can be speculated that most of the pinwheel centers disappear, and stripe-like orientation and ocular dominance maps known as the ice-cube layout are realized after a long run of simulation. Therefore, one might argue that the presence of pinwheel centers in the orientation maps is a consequence of an incomplete minimization of the energy function. The wire length minimization approach can explain the preservation of pinwheel centers in orientation maps (Koulakov and Chklovskii, 2001). In most of the self-organization models proposed to date (Kohonen, 1984; Durbin and Mitchison, 1990; Erwin et al., 1995; Swindale, 1996), a final columnar pattern would be a stripelike layout if we perform simulation only on the basis of the models for either orientation preference or ocular dominance. Pinwheel centers in orientation maps, however, would be stable in a model of concurrent development of orientation maps and ocular dominance columns (Wolf and Geisel, 1998), because the preferred orientation does not change much where ocular dominance changes rapidly, and visa versa. As a result, pinwheel centers always lie at the centers of ocular dominance columns (Carreira-Perpinan and Goodhill, 2004). In those models, even in the case where two pinwheel centers with opposite singularities are placed straddling a border of ocular dominance columns at an intermediate stage of simulation, these two pinwheel centers cannot approach each other across the borders of ocular dominance columns to annihilate. In the present model, pinwheel centers appear either at the centers or along the borders of ocular dominance bands at high or low between-eye correlation. At intermediate values of between-eye correlation, which would be more plausible in the real cortex, one may expect that pinwheel centers appear at any positions between the centers and borders of ocular dominance columns. If it is the case, the annihilation of opposite pairs of pinwheel centers can occur, which leads to an ice-cube layout of the orientation preference and ocular dominance after the long-run of simulation. Nevertheless, orientation and ocular dominance maps obtained after a million of iterations starting from the maps in Fig. 2, the map structure did not exhibit an icecube layout and the pinwheel structure was preserved (data not show). This is attributed to the model’s property that center and border subregions of ocular dominance columns are favorite places for pinwheel centers and opposite pairs of pinwheel centers cannot approach to each other in the intermediate subregions of ocular dominance columns. Such a low probability of pinwheel pair annihilation in our model is consistent with the observation by intrinsic signal optical imaging that the basic structure of orientation maps first detectable at the age of 17 remained unchanged during the maturation period of the maps until the age of 30 when the maps become adultlike (Godecke et al., 1997). Our surmise that between-eye correlation is kept at a certain positive level even in the dark may be supported by the experimental finding that orientation maps of the two eyes were virtually identical for

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alternating monocularly deprived cats (Goedecke and Bonhoeffer, 1996). It is suggested that the intrinsically correlated activities work to match orientation maps of the two eyes even when the two eyes have never experienced common visual images. 5. Conclusion Our correlation-based self-organization model demonstrated that pinwheel centers tend to appear at the centers or along the borders of ocular dominance columns at high or low between-eye correlation. Our optical imaging study showed a statistically significant tendency for pinwheel centers to be located at the centers of ocular dominance columns in normal cats, but to be located along the borders of ocular dominance columns as well as at the ocular dominance centers in darkreared cats. Because the between-eye correlation is presumably high for normal cats and low for dark-reared cats, the experimental finding justifies the theoretical prediction. The consistency between the theoretical prediction and experimental results supports an idea that orientation maps and ocular dominance columns are formed with mutual interaction under the influence of visual experience. Acknowledgements We thank Je´roˆme Ribot for his technical advice in the analysis of optical imaging data, and Masanobu Miyashita for his advice in the analysis of simulation results. We also thank Yuji Akimoto and Katsuya Ozawa for their assistance in surgery and optical imaging and their daily maintenance of all the kittens examined. Appendix A. Continuum approximation of energy function The first term of the energy function (Eq. (1)), which provides a driving force for arranging the preferred orientation and ocular dominance, can be rewritten as the continuum approximation as follows: Z Z H ¼  d~ x d~ x0 Vð~ x ~ x0 ÞCðo~x ; u~x ; o~x0 ; u~x0 Þ; (A.1) R where d~ x is a surface integral within the region of interest, and Vð~ x ~ x0 Þ is a continuum approximation of cortical interaction function, and expressed as an inverse of the Fourier transform as follows: Z ˜ qÞÞ ¼ Vð~ ˜ qÞ
ei~qð~x~x0 Þ d~ Vð~ x ~ x0 Þ ¼ F 1 ðVð~ q; (A.2) ˜ qÞ is a Fourier transform of cortical interaction funcwhere Vð~ ˜ qÞ tion Vð~ xÞ, and up to the second order of the Taylor series of Vð~ can be written as follows:   ˜ qÞ  ˜ qÞ  @Vð~ 1 @2 Vð~   ~ ˜ qÞ  Vð ˜~ ~ Vð~ 0Þ þ  q þ (A.3) q2 : @~ q ~q¼~0 2 @~ q2 ~q¼~0

Because the cortical interaction function Vð~ x ~ x0 Þ is assumed to originate from finite-range symmetric excitatory connections (Eq. (2)), Eq. (A.2) can be reduced to the differential operator:   
1  ˜ ~  ~ 2 2 0 0 ˜ Vð~ x ~ x Þ ¼ ð2pÞ dð~ x ~ x Þ Vð0Þ þ D~q Vð0Þr~x ; (A.4) 2 ˜~ ˜ qÞ=@~ 0Þ ¼ @2 Vð~ q2 j~q¼~0 . To obtain the above equation, where D~q Vð ˜ qÞ=@~ we used the following equations: @Vð~ qj~q¼~0 ¼ 0, 0 0 ˜ qÞ=@~ q2 j~q¼~0 < 0, R ~ @2 Vð~ q2 ei~qð~x~x Þ ¼ ð@2 =@~ x2 Þ ei~qð~x~x Þ , and 0 2 0 dð~ x ~ x Þ ¼ 1=ð2pÞ ei~qð~x~x Þ d~ q. In energy function (Eq. (1)), Cðo~x ; u~x ; o~x0 ; u~x0 Þ is a continuum approximation of correlation function rewritten as follows: Cðo~x ; u~x ; o~x0 ; u~x0 Þ ¼

1 ½1 þ b cos 2ðu~x  u~x0 Þð1 þ o~x o~x0 Þ 2ð1 þ bÞ r0 ½1 þ b cos 2ðu~x  u~x0 Þ þ 2ð1 þ aÞð1 þ bÞ
½1 þ a cos 2ðu~x  u~x0 Þð1  o~x o~x0 Þ;

(A.5)

where u~x and o~x are variables representing preferred orientation and ocular dominance at cortical position~ x, respectively. Using the continuum cortical interaction and the continuum correlation function, the energy function H is rewritten as follows: Z ~ ˜ H ¼ Vð0Þ d~ xCðoð~ xÞ; uð~ xÞ; oð~ xÞ; uð~ xÞÞ Z Z 1 ˜~  jD~q Vð 0Þj d~ x d~ x0 dð~ x 2 ~ 2 Cðoð~ xÞ; uð~ xÞ; oð~ x0 Þ; uð~ x0 ÞÞ: ~ x0 Þr ~ x

(A.6)

After a short calculation, we obtain the following energy function: Z Z 1 þ r0 ˜ ~ 1  r0 ˜ ~ Vð0Þ d~ x  ð2pÞ2 Vð0Þ d~ xo~2x H ¼ ð2pÞ2 2 2   Z ð2pÞ2 1  r0 ~ 2 2bð1 þ r 0 Þ ~ ˜ x jD~q Vð0Þj d~ ðro~x Þ þ þ 1þb 2 2     2ar 0 ~ 2 2bð1  r 0 Þ 2ar 0 ~ 2 2  þ ðru~x Þ þ ðru~x Þ o ; 1þb 1þa 1þa (A.7) ~ ~x are vectors representing the gradient of ~ ~x and ru where ro ocular dominance and the gradient of preferred orientation at cortical position ~ x, respectively. To obtain the above equation, we used the following formula: Z Z Z ~ ~ 2 d~ oDo d~ x¼ oro
~ n d‘  ðroÞ x; ~ x

l

R

~ x

where d‘ is a line integral along the border of the region of interest, and ~ n is a unit vector normal to the border. The first term on the right-hand side of the above equation can be zero under the periodic boundary condition. Moreover, in the case of

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the Gaussian cortical interaction function
 1 j~ x ~ x0 j2 0 Vð~ x ~ xÞ¼ exp  ; 2l2v 2pl2v energy function H ¼ H 0 þ H 1 can be re-expressed as follows: Z Z 1 þ r0 1  r0 d~ x d~ xo~2x H0 ¼  2 2  Z 2 l2v 2 bð1 þ r 0 Þ ~ þ ð1  r 0 Þ d~ xðro~x Þ þ lv 1þb 4 Z ar 0 ~ ~x Þ2 ; d~ xðru þ (A 8) 1þa  Z ar 0 2 bð1  r 0 Þ ~ ~x Þ2 o2 :  H 1 ¼ lv d~ xðru (A.9) 1þb 1þa R The first term of the energy function H0, ð1 þ r 0 Þ=2 d~ x, which takes a constant value, does not have any effect on the map formation. The coefficient of the second term of the energy function is negative (1  r0)/2(0  r0  1),R thus increasing the amplitude of the ocular dominance d~ xo~2x stabilizes the energy function. Because the coefficients of the third term þl2v ð1  r 0 Þ=4 and the fourth term l2v ½ðbð1 þ r 0 ÞÞ=ð1 þ bÞ þ ðar 0 Þ=ð1 þ aÞ always take positive values, these terms render the ocular dominance and preferred orientation to be represented over the cortex as continuously as possible. In the energy function H1, the coefficient l2v ½ðbð1  r 0 ÞÞ=ð1 þ bÞ  ðar 0 Þ=ð1 þ aÞ can be positive or negative depending on the values of parameters a, b, and r0. For a negative coefficient, the increase of the value of R ~ ~x Þ2 o2 decreases the energy, and hence the orientation d~ xðru ~ x ~ ~x Þ2 ) near the centers of can change rapidly (large ðru monocular domains (large o~2x ). On the contrary, for a positive ~ ~x Þ2 ) coefficient, the orientation can change rapidly (large ðru along the ocular dominance borders (small o~2x ), and is ~ ~x Þ2 ) around the ocular smoothly represented (small ðru dominance centers (large o~2x ). Appendix B We consider the meaning of the normalized overlap index defined by Eq. (8) in Section 3. Generally, the spatially modulating variable is given as the sum of the average component and fluctuation. Therefore, the magnitude of monocularity and the magnitude of orientation gradient are reexpressed by o~x 2 ¼

1X 2 o~x þ dðo~x 2 Þ; N ~x

~ ~x j2 ¼ jru

1X ~ 2 ~ ~x j2 Þ: jru~x j þ dðjru N ~x

(A.10)

(A.11)

Using these expressions, the NOI is transformed to P ~ u~x j2 Þ ð1=NÞ ~x dðo~2x Þ
dðjr NOI ¼ 1 þ : (A.12) P 2 P ~ u~x j2 ð1=NÞ ~x o~x
ð1=NÞ ~x jr

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This formula indicates that when there is no correlation between the spatial modulation of the magnitude of mono~ u~x j2 Þ, the cularity dðo~2x Þ and that of orientation gradient dðjr NOI becomes 1. When the correlation is positive (negative), the NOI is larger (smaller) than 1.

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