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1Department of Mathematics and 2Department of Neuroscience, University of Pittsburgh, Pennsylvania; 3The Center for the Neural Basis of Cognition, Pittsburgh, Pennsylvania; and 4Laboratory of Biological Modeling, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland
Submitted 14 February 2005; accepted in final form 26 May 2005
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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, 1992). Rollenhagen and Olson (2005) recently observed that 5-Hz oscillations are enhanced by the presence of another stimulus in the visual field. In the presence of an eccentric flanking image, displaying a central preferred image tends to elicit a strong initially positive oscillatory response (see Fig. 1A). Conversely, in the presence of a central preferred image, an eccentric flanking image, although ineffective in isolation, tends to elicit a strong initially negative oscillatory response (see Fig. 1B). They demonstrated that similar phenomena arise in an extremely simple network in which fatiguing neurons responsive to the central image and the eccentric flanker inhibit each other. They did not, however, systematically explore the dependency of oscillatory activity on the properties of the network nor did they consider the possible relation of oscillatory activity to general principles that govern visual processing in IT. These issues are taken up in the present paper.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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We studied a model network of excitatory and inhibitory Hodgkin–Huxley-type conductance-based neurons. The network was an abstracted canonical cortical circuit (Douglas and Martin 2004; Raizada and Grossberg 2003) consisting of two excitatory and two inhibitory pools of 40 neurons each (Fig. 2A). Increases beyond this number had no appreciable effect on the results. Varying the numbers of inhibitory versus excitatory neurons also did not alter the qualitative results. Excitatory cells were coupled by 
-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA)–like excitatory synapses to other cells within their pool. There were excitatory projections from excitatory pool 1 (E1) to inhibitory pool 2 (I2) and from excitatory pool 2 (E2) to inhibitory pool 1 (I1). Neurons from the inhibitory pools formed 
-aminobutyric acid-A (GABA)–like inhibitory connections on neurons in the excitatory pools, with I1 neurons projecting to E1 neurons and I2 projecting to E2. External inputs (active in response to visual stimulation) terminated on neurons of the excitatory pools. Neurons in E1 and E2 show much higher responses for presentations of preferred visual stimuli compared to nonpreferred stimuli (in Fig. 2A, stimulus 1 "Object" is the preferred stimulus for E1 and nonpreferred for E2, and stimulus 2 "Flanker" is the preferred stimulus for E2 and nonpreferred for E1).
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; McCormick and Huguenard 1992; Wang et al. 2003). The second was synaptic depression with a time constant that was long compared to that of spike rate adaptation (Abbott et al. 1997; Grossberg, 1972). Conductance-based neuronal dynamics
For our simulations, the excitatory and inhibitory membrane potentials obeyed
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The membrane currents had the form
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= 3.
The calcium-dependent potassium adaptation current obeyed
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AHP = 100, and gAHP varied from 0 to 0.5.
The synaptic current to the jth excitatory neuron with voltage Ve[j] was
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k se[k] · 
e[k] and gi = k si[k] · i[k]; Jei and Jee are the synaptic strengths. The sum in ge ran over the corresponding excitatory pool, with k = 1 to 40 for E1 and k = 41 to 80 for E2. The sum in gi for inhibitory inputs ran within the ipsilateral inhibitory pool, with k = 1 to 40 for E1 and k = 41 to 80 for E2.
The synaptic gating variables se[k] and si[k], and the depression factors e[k] and i[k] obeyed
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e = 8, i = 10, ge = 1,000, gi = 800, Jei and Jee varied from 0 to 1, and fe and fi varied from 0 to 0.05.
Similarly, the synaptic current to the jth inhibitory neuron with voltage Vi[j] (inhibitory neurons receive inputs only from excitatory neurons) was
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k se[k] · 
e[k], Vei = 0, and Jie varied from 0 to 1. The sum in ge ran over the contralateral excitatory pool with k = 41 to 80 for I1 and k = 1 to 40 for I2. As described above se[k] was the gating variable of the synapses from the kth excitatory neuron with depression factor e[k]. External currents
The external current Iexte to the excitatory neurons consisted of a constant input I, which ranged from 0 to 2.5. The relative value of the external inputs to E1 and E2 neurons depended on the stimulus presented. If we presented stimulus 1 (Object), which was preferred by E1 and not preferred by E2, then the external current to E1 was higher than the external current to E2, and for a presentation of stimulus 2 (Flanker), which was preferred by E2 and not preferred by E1, the external current to E2 was higher than the external current to E1. For simultaneous presentation of both stimuli external currents to both populations were equal.
To reproduce experimental data observed by Rollenhagen and Olson (2005) we also added additional white noise to the external inputs, although all features of the model discussed in the paper such as different modes and transitions between them can be obtained in the absence of noise. For the external input to excitatory neurons we used Iexte = 2 + noise(0.5), where function noise(0.5) generated a random number uniformly distributed between –0.5 and 0.5. The external input Iexti to the inhibitory neurons was noise(0.25).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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The neuronal pools in our model were intended to represent two pools embedded in a network of many interconnected pools in various brain regions. Any given visual stimulus activated neurons in some pools and not in others. We considered specifically the case of two excitatory pools maximally activated by different visual stimuli and competing with each other by projections mediated by inhibitory interneurons. We presumed that different pools would inhibit each other with varying strengths of inhibition determined during development or by experience-dependent learning. By varying only the strength of the mutual inhibition, with all other parameters fixed to biophysically plausible values, we demonstrated that it was possible to elicit a broad range of distinct behaviors.
THREE FUNCTIONAL MODES DEPENDENT ON STRENGTH OF CROSS-INHIBITION. To simulate the simultaneous onset of two visual stimuli, one optimal for each excitatory pool, we applied currents I1 = 2.5 to E1 and I2 = 2.5 to E2 and applied white-noise inputs to the inhibitory pools. We then examined the impact on network behavior of varying the strength of the inhibitory parameter Jei, which represented the strength of inhibition exerted by inhibitory neurons on their excitatory targets. As inhibition increased, the behavior of the network passed through three modes. At low inhibitory strength, both pools of excitatory neurons were continuously active (Fig. 3A) because inhibition was too weak to allow one of the pools to suppress the other one. Although both pools of excitatory neurons were active, neither was as active as it would have been in the sole presence of its preferred stimulus, and thus the network operated in what we termed normalization mode. If the strength of the inhibition was increased, the network entered an oscillatory mode in which the two pools were alternately active (Fig. 3B). At high inhibitory strength, the network operated in a winner-take-all mode in which only one pool remained active and the other pool was suppressed (Fig. 3C). There was a transient time (different for different modes) that was necessary for the network to settle down into a mode. It was around 100 ms for the oscillation mode and 400 ms for the winner-take-all mode. Once the network reached a given mode it was stable in it for the remaining simulation time.
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Oscillatory mode. At higher strengths of inhibition, when simultaneous equal inputs to both pools were applied, the network responded with a form of oscillatory behavior in which pools of excitatory neurons preferring one stimulus or the other were alternately active. To demonstrate this effect, we maintained the input currents I1 = 2.5 and I2 = 2.5 while varying the strength of the inhibitory synapses Jei. We observed that the frequency of the oscillations depended on the strength of the inhibition, with stronger inhibition giving rise to oscillations at a lower frequency (Fig. 5, A–D).
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; Laing and Chow 2002).
We found that as the oscillations became slower, they also became more stochastic. This stochasticity occurred even in the absence of any noise inputs. The origin of this stochasticity was discussed in Laing and Chow (2002). Also, similar to what was shown in Laing and Chow (2002), the model was able to reproduce all the phenomena of binocular rivalry including: a lack of correlation in dominance times from epoch to epoch, a dominance time that obeyed a Gamma-like distribution, and Levelt's Second Proposition.
SIMULATION OF 5-HZ OSCILLATIONS IN IT. The above analyses of network behavior considered responses to the simultaneous onset of two visual stimuli. However, the model also reproduced firing patterns observed by Rollenhagen and Olson (2005) in response to staggered onset of two visual stimuli. While recording from a given neuron, they assessed how the response to onset of stimulus 2 was affected by the ongoing presence of stimulus 1, where stimulus 2 was the central image preferred by the neuron and stimulus 1 was an eccentric nonpreferred stimulus (Fig. 1A) or vice versa (Fig. 1B). As a basis for direct comparison to results obtained in the experiment, we assessed the responses of the spike-based network to successive onset of the two stimuli. When the recorded pool's preferred stimulus appeared against the backdrop of its nonpreferred stimulus, firing took the form of an initially positive oscillation (Fig. 6, top) just as in the physiological experiment (Fig. 1A). When the recorded population's nonpreferred stimulus appeared against the backdrop of its preferred stimulus, the response was an initially negative oscillation (Fig. 6, bottom), again just as in the physiological experiment (Fig. 1B).
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Rate-based model
To gain further insight into the behavior of the networks described above (two pools of excitatory conductance-based neurons reciprocally suppressing each other through inhibitory interneurons) we considered a classic rate-based model, consisting of two units inhibiting each other in the presence of fatigue (see Fig. 2B) (Carpenter and Grossberg 1983; Laing and Chow 2002). The units represented pools of excitatory neurons that inhibited each other through inhibitory neurons. For simplicity we did not include dynamics of the inhibitory units. As in the conductance-based model, we implemented two different fatigue mechanisms: spike-rate adaptation and synaptic depression. The equations describing the dynamics were
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, and parameters had values adap = 100, sd = 500, kadap = 0.5, ksd = 0.5, and I1 = I2 = 5.
The results of the simulations were not critically dependent on the choice of the gain function. We observed the same three modes with any reasonable gain function including step functions, sigmoidal functions, and piecewise linear functions. The choice of the gain function with the general form 
, where parameters can control the slope and threshold was out of convenience.
BIFURCATION DIAGRAM. The advantage of the reduced model was that the dependency of network behavior on parametric variables could be analyzed mathematically. A standard method is to construct a bifurcation diagram that presents the dependency of the system's behavior, including points of transition between qualitatively different behaviors, on a system parameter. As we observed in the conductance-based network, a critical parameter governing the transition between different kinds of dynamics was the strength of inhibition. We examined the behavior of the rate-based model with varying inhibition strength b and with varying input strengths to both units. The results, summarized in the bifurcation diagram of Fig. 7, were similar to those obtained with the conductance-based model. At low values of b, the network exhibited normalization behavior (curve between point 1 and point 2 in Fig. 7): in the presence of both stimuli, the excitatory units maintained a steady state of activation less than would have occurred if only the preferred stimulus were on. At intermediate values of inhibition, oscillatory responses occurred (open circles forming a closed curve joining points 2, 4, and 5): the activities of the units were alternating in time, when one unit was active the other was suppressed, and after a period of time they exchanged states. At high values, winner-take-all responses occurred (top curve between points 4 and 6 or bottom curve between points 5 and 8): one unit won the competition and the other unit could not recover from suppression. Points 2, 4, and 5 are transition points (bifurcation points) between different kinds of dynamics.
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The existence of an oscillatory region depended entirely on the adaptive mechanisms of spike-rate adaptation or synaptic depression. In a classic winner-take-all network consisting of units that inhibit each other in the absence of any fatigue mechanisms, the bifurcation diagram would take the form of a stable line, corresponding to normalization mode, at low values of inhibition (curve between points 1 and 3) and two branches of winner-take-all mode, at high values of inhibition (curves between points 3 and 6 and points 3 and 8), and there would be no oscillatory region.
ROBUSTNESS OF THE MODES. The results of simulations in the rate model were not critically dependent on the other parameters. For example, they persisted across a wide range of input strength as shown by the two-parameter diagram (Fig. 9) for the strength of the inhibition b and equal external currents to both units I1 = I2 = I. At any fixed value of external currents I1 = I2 = I, as the inhibition strength b increased, the network passed through normalization, oscillation, and winner-take-all modes. Changes of the other parameters deformed the regions but did not alter their topology.
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Thus we next probed whether similar phenomena would arise from the competition of multiple neuronal pools evoked by the presentation of multiple stimuli or by natural scenes. We ran simulations of a rate model of three units reciprocally inhibiting each other. We again found the same three distinct modes of behavior—normalization, oscillations, and winner-take-all depending on the strength of the inhibition. For weak inhibition, all three units were active simultaneously as in normalization mode. If inhibition was strong enough, one unit was active and suppressed the other two, with the choice of the dominating unit being random.
The oscillatory regime showed richer behavior than that in the two-unit network. With an increase in inhibition, the network switched from normalization mode to oscillation mode. Immediately after the transition, the oscillations took the form of alternations between one very active unit and two simultaneously less active units (each half as active as the very active unit). As the inhibition was increased, one unit became permanently suppressed while the other two activated alternately. For appropriately chosen parameters, we also observed a pattern in which the units fired in a sequential manner with the order determined by initial conditions. On the basis of these simulations, we conclude that multiple competitive pools operate in regimes qualitatively similar to those revealed by studying the two-unit model. The main difference arises from the existence of multiple oscillatory submodes. The complexity of these modes increased as we increased the number of competing pools.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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It is widely thought that visual stimuli actively compete for neuronal representation in inferotemporal cortex and other areas of the visual system (Desimone and Duncan 1995; Sheinberg and Logothetis 1990; and Rollenhagen and Olson, 2005). However, competition has been envisioned as having three different functional consequences. 1) In the biased competition framework (Desimone and Duncan 1995), competition takes the form of normalization: two simultaneously presented images elicit levels of activation intermediate between those elicited by the images in isolation (Carandini and Heeger 1994; Reynolds et al. 1999). 2) It has been suggested that competition could help to resolve ambiguous displays through a winner-take-all mechanism (Douglas and Martin 2004; Usher and McClelland 2001). In this scenario, when complicated images are presented, the representations of the various components compete with the outcome that the unit representing one component remains active and suppresses the others. 3) Oscillatory neural activity could arise from the combination of a classic winner-take-all mechanism with an adaptive process. This has been proposed as a mechanism for the alternating binocular rivalry that occurs when incompatible images are presented to the two eyes (Blake 1989; Blake and Logothetis 2002; Grossberg 1994; Laing and Chow 2002). Oscillatory activity has been observed over a wide range of frequencies for binocular rivalry and ambiguous stimuli (Carter and Pettigrew 2003; Logothetis et al. 1996) but generally on the order of a few hertz or less. However, oscillations with frequencies of approximately 5 Hz have also been observed in the visual responses of inferotemporal neurons (Nakamura et al. 1991; Sheinberg and Logothetis 1990; and Rollenhagen and Olson, 2005).
Here, we argue that the varied responses observed when disparate images are presented in various configurations could arise in a simple cortical circuit combining recurrent excitation and opponent inhibition with spike-frequency adaptation and synaptic depression (Carpenter and Grossberg 1983; Douglas and Martin 2004; Laing and Chow 2002; Raizada and Grossberg 2003; Wang et al. 2003). In such a circuit, neurons selective for a specific image mutually excite each other and inhibit pools of neurons selective for other images. We have shown through simulations and analysis that, as the effective strength of inhibition between pools is steadily increased, there is a transition from normalization to oscillatory behavior and, finally, to winner-take-all behavior. Which phenomenon is observed in a given area and under a given stimulus regime will depend critically on the intensity of inhibitory interactions as determined both by the strength of inhibitory connections and the degree to which the display activates inhibitory interneurons. These conclusions are in good agreement with the observation that two comparable kinds of rivalry can be observed in a network consisting of two layers of reciprocally inhibitory units, with the particular form dependent on the effective strength of inhibition (Wilson 2003). A bifurcation diagram of this model reveals three regimes: with weak inhibition, simultaneous activity of units at both levels; with moderate inhibition, oscillatory behavior; and, with strong inhibition, winner-take-all behavior.
Comparison between recurrent and feedforward models of normalization
It has been noted by Reynolds et al. (1999) that normalizing behavior (the tendency for a neuron to fire at an intermediate rate when presented with a preferred and a nonpreferred image) is intrinsic to a model based on feedforward shunting inhibition first proposed by Grossberg and colleagues (Grossberg 1973; Grossberg and Levine 1975) as a means for maximizing the dynamic range of a neuron. Our model differs in that normalization arises from mutual inhibition between two populations of neurons. It can account for phenomena that the model based on feedforward shunting inhibition cannot easily account for, notably oscillatory and winner-take-all behavior. However, with respect to normalization, there are many commonalities between the two models.
To exhibit normalization, we find that a network should obey a set of general conditions. Consider a given neuron (or a pool of neurons) that receives input from two stimuli 1 and 2 and is highly responsive to stimulus 1 but weakly responsive to stimulus 2. We separate the inputs into excitatory and inhibitory components. Thus for stimulus 1 alone, the neuron receives inputs E1 and I1 and fires at rate R1, for stimulus 2 alone the neuron receives inputs E2 and I2 and fires with rate R2, and for the stimuli presented together the inputs are E3 > E1, E2 and I3 > I2, I1 and the rate is R3. The inputs represent the total input arising from both feedforward and feedback sources (in the case of no excitatory feedback, E3 = E1 + E2). We define normalization as the case where R2 < R3 < R1. [We do not require perfect normalization or averaging, which would have R3 = (R1 + R2)/2].
Normalization will occur in any network provided the response of the neuron to excitatory and inhibitory inputs satisfies the following conditions (Moldakarimov and Chow, unpublished observations): 1) the firing rate increases with excitation and decreases with inhibition, 2) the firing rate is minimal for zero excitatory input (larger or the same for other input combinations), and 3) if E1 > E2 then I2 > I1. Although these conditions are sufficient to ensure normalization they are definitely not necessary. However, they can be satisfied by biophysically plausible networks.
The first condition is satisfied by almost all neuron models. The second condition is not as straightforward. In our simulations, when no excitation is present, the neurons are essentially shut off and fire only randomly because of noise. Inhibition does not decrease the firing rate any further. Here, the firing rate of the pool presented with the nonpreferred image is similar to that if no image is presented. The third condition can be satisfied by adjusting the synaptic weights of the inputs to the neuron. It implies that when a neuron receives strong excitation it is accompanied by weak inhibition and vice versa.
The three conditions are satisfied by the feedforward networks of Grossberg (1973), Grossberg and Levine (1975), and Reynolds et al. (1999), which have a firing rate of the form R 
E/(E + I + C), where C is a constant. As we can see, the firing rate increases with excitation and decreases with inhibition. Second, the firing rate without excitation is a unique minimum. The third condition in the feedforward network can be relaxed to E1I2 > E2I1.
In our recurrent model, the first two conditions are satisfied automatically by our conductance-based model and by the choice of gain function in our rate model. The third condition cannot be externally imposed as in a feedforward model because the net excitatory input and net inhibitory input are not independent. However, it can be satisfied for weak reciprocal inhibition, which agrees with our previous conclusions about normalization. The recurrent network could be considered to be a generalization of the feedforward network. The basic principles of normalization are the same.
Future directions
Although the model that we have described accounts in a broad way for oscillatory phenomena observed in IT by Rollenhagen and Olson (unpublished observations), it is worthwhile to point out that there are subtleties of oscillatory activity in IT for which it does not provide a ready explanation. One example is that the slight but consistent difference in frequency between oscillations elicited by presenting the object against the backdrop of the flanker and vice versa (Fig. 8 of Rollenhagen and Olson). Our model failed to capture this nicety because the connections of neuronal pools representing the flanker and object were perfectly symmetric. It will be of interest to ask in future studies whether, by introducing asymmetries in the strengths of inhibitory and excitatory synapses on neurons in the two pools, it is possible to achieve a match to the pattern observed in IT.
Another potential discrepancy between the behavior of the model and the behavior of neurons in IT concerns the results of turning on the two stimuli simultaneously. In our model, simultaneous onset elicits oscillatory activity. However, in several physiological studies, the simultaneous onset of two stimuli has not elicited obvious oscillatory activity, if the absence of any mention of oscillations is to be taken as evidence (Miller et al. 1993; Missal et al. 1999). In the model, there was generally a delay (around 100 ms for the oscillatory mode and 400 ms for the winner-take-all mode) before the network settled into its final state. It is possible that the failure to observe oscillatory activity in the physiological experiments was a result of not leaving the stimuli on long enough. It is also possible, however, that the discrepancy signals the presence in IT of features not captured in the model.
Electrophysiological experiments conducted under independent conditions in different laboratories have demonstrated that interstimulus competition can give rise to both normalizing responses (Heeger et al. 1990; Reynolds et al. 1999) and oscillatory responses (Rollenhagen and Olson, unpublished observations). We have argued that both phenomena could arise from a single neural mechanism, with the phenomenon observed in a given area under a given set of circumstances dependent on the intensity of inhibitory interactions. However, to demonstrate that this is the case will require further experiments involving systematic manipulation of the strength of inhibition between pools of neurons responsive to different stimuli. A direct approach might be the use of GABA agonists or antagonists. An indirect approach would be to control stimulus strength through systematic distortion or manipulation of contrast.
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GRANTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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FOOTNOTES |
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Address for reprint requests and other correspondence: C. C. Chow, Department of Mathematics, Thackeray 505, University of Pittsburgh, Pittsburgh, PA 15260 (E-mail: ccchow{at}pitt.edu)
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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