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Computational Neurophysics Laboratory, Physics and Astronomy Department, University of North Carolina, Chapel Hill, North Carolina
Submitted 9 September 2007; accepted in final form 14 February 2008
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: THE MODEL EQUATIONSGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: THE MODEL EQUATIONSGRANTSACKNOWLEDGMENTSREFERENCES |
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). Selective attention is the process by which stimuli are selected based on current behavioral goals. For instance, when you are looking for a friend at the airport, and you know that she will be wearing a red shirt, red objects will be processed preferentially (Saenz et al. 2002; Serences and Boynton 2007). Experiments conducted to determine the single neuron correlate of selective attention have revealed attention-induced changes in the firing rate (Moran and Desimone 1985; Reynolds and Chelazzi 2004; Maunsell and Treue 2006) and in the gamma-frequency-range (35–50 Hz) coherence (Bichot et al. 2005; Fries et al. 2001; Taylor et al. 2005). Pathologies in the selection and processing of visual information underlie diseases such as attention-deficit disorder and schizophrenia. Patients suffering from these conditions show differences in gamma and beta (15–35 Hz) frequency oscillatory activity (Uhlhaas and Singer 2006; Uhlhaas et al. 2006) as well as postmortem abnormalities in interneuron densities (Benes and Berretta 2001). These results suggest a causal link between cognitive processing in the healthy human and the oscillations generated by interneurons.
Cortical area V4 is one of the best places to start the investigation of this link. The receptive field (RF) of a V4 neuron is large enough so that it fits multiple stimuli, only one of which may be behaviorally relevant at a given time. Stimulus selection by selective attention should therefore cause changes at the level of V4, both in terms of firing rate as well as coherence. The response of V4 neurons is the outcome of a competition between bottom-up salience, interaction with other neurons in V4 responding to the same stimuli, and the effect of top-down projections mediating selective attention (Ogawa and Komatsu 2004; Reynolds and Desimone 2003). Experiments have provided an extensive, yet incomplete, quantitative characterization of response modulation for a variety of stimulus configurations and attention conditions. Phenomenological models have been constructed to account quantitatively for these results (Boynton 2005; Reynolds et al. 1999). However, a mechanistic understanding of how these changes are achieved at the circuit level is still lacking. With the advent of experimental techniques to record from multiple neurons simultaneously during a behavioral task, this lack of understanding is rapidly becoming a serious impediment to further progress.
We study the mechanism for attentional modulation in a biophysical network of spiking neurons based on the hypothesis that attention effects are mediated by interneurons (Tiesinga and Sejnowski 2004; Tiesinga et al. 2004a,b). Our goal is to determine what types of interneurons are involved in attention and predict their behavior under various tasks and stimulus conditions.
Our model simulations suggest that at least two types of interneurons are involved in attention, referred to as feedforward interneurons (FFI) and top-down interneurons (TDI). They are differentially modulated by attention as follows. The firing rate of the FFI increases with spatial attention (SA) and decreases with feature-based attention (FBA), whereas the TDI increase their firing rate with FBA and shift the network synchrony from the beta to the gamma frequency range. The phase lag between interneurons and excitatory cells also increases with FBA.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: THE MODEL EQUATIONSGRANTSACKNOWLEDGMENTSREFERENCES |
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To keep the model simple, only the necessary synaptic connections were simulated. These were as follows (see Fig. 1 ).
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Second, there were cross-columnar excitatory projections to the FFI: E1 to FFI2 and E2 to FFI1; and there was local feedforward (FF) inhibition: FFI1 to E1 and FFI2 to E2. The label E1 stands for the E cells in the first column, the rest of the labels, E2, FFI1 and FFI2, are defined accordingly. These two projections together, E1 to FFI2 to E2 and E2 to FFI1 to E1, mediated stimulus competition. The excitatory projection activated a mixture of AMPA and N-methyl-D-aspartate (NMDA) receptors, whereas the inhibition was mediated by GABAA receptors (Shepherd 1998).
Last, the TDI projected fast GABAergic inhibition to the FFI and the E cells in their column. The combination of the direct pathway, TDI1 to E1 (TDI2 to E2) and the indirect pathway, TDI1 to FFI1 to E1 (TDI2 to FFI2 to E2) mediated FBA on the preferred feature of column 1 (column 2).
Sensory information entered the columns via a FF projection implemented as a driving current to the E and FFI neurons (the TDI did not receive direct stimulus-related inputs). The expression for this current is (using the same convention for subscripts as in the preceding paragraph)
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). We made the current linear in contrast so that we could sample the entire dynamical range of the network responses at a uniform density. Presynaptic activity varying nonlinearly with contrast can be incorporated by plotting a rescaled x coordinate, c1new = f(c1), in our graphs. Spatial attention at the location of stimulus 1 is implemented by replacing, for both columns, c1 by 
Ec1 and FFIc1 in Eq. 1 for the E and FFI neurons, respectively. This implements the contrast gain model (Reynolds and Desimone 2003; Reynolds et al. 2000) for spatial attention in V2, but for a linear current-contrast relationship, this is also consistent with the response gain model (Williford and Maunsell 2006).
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). In the default parameter setting, the AMPA component was zero, but simulations with a nonzero AMPA component are discussed in RESULTS. Because presynaptic firing rates exceeding 20 Hz saturated the current through the NMDA channels, an increase in E rate did not yield a sufficient increase in the drive to the FFI necessary for stimulus competition. This problem was solved by having a pool of 250 excitatory neurons in each column (E1b and E2b) that were made less excitable by injecting an additional hyperpolarizing current so that they were only active when there were two stimuli present at sufficient contrast. The number of synaptic connections made between the different groups are listed in Table 2.
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; Douglas and Martin 2004): local recurrent excitation (E1 to E1 and E2 to E2), cross-columnar excitation (E1 to E2 and E2 to E1), and excitatory feedback onto the FFI (E1 to FFI1 and E2 to FFI2). The potential impact of these omissions is discussed in the DISCUSSION.
For the excitatory neurons, we used the model in (Golomb 1998; Golomb and Amitai 1997), whereas for the TDI and FFI, we used the model given in (Tiesinga and Jose 2000; Wang and Buzsaki 1996). Full details of the model setup and its implementation are given in the APPENDIX.
The simulation output was analyzed using custom-written programs in Matlab (The Mathworks, Natick, MA). Spike times were detected when the membrane potential crossed 0 and –20 mV from below for the inhibitory and excitatory neurons, respectively. For each group, the firing rate was calculated as the spike count divided by the measurement period, and it was averaged across all neurons in the group. A typical simulation was 1,000 ms long, of which the first 200 ms was discarded as a transient. To determine coherence (see following text), longer runs with a duration of 10 s were sometimes used. The spike time histogram was the number of spikes in a time bin from all neurons in the group, normalized by the number of neurons and the bin width in seconds (typical value: 0.001 s). This yielded the mean firing rate of a single neuron as a function of time.
Linear relationships between attention-induced changes in firing rate and other quantities were quantified using the Matlab routine polyfit. The strength of linear correlation and the significance thereof were determined using the Matlab routine corrcoef.
The local field potential (LFP) was modeled as the spike time histogram of the E cells. The multi-taper power spectrum of the E-cell, FFI, and TDI histogram was calculated using the Matlab routine mtm (with time-bandwidth product NW = 3.5). We extracted the power in the beta and gamma band by averaging the power spectrum in a 10-Hz interval around the strongest beta peak (this peak usually was between 15 and 30 Hz) and around either 40 or 45 Hz for the gamma band. To calculate the coherence, the spike train of each neuron was binned at a resolution of 1 ms. A pair of binned spike trains was passed to the Matlab routine coherencypb which was part of the CHRONUX toolbox (parameters: NW = 3, the FFT padding parameter was 1, and we averaged across 5 tapers). The coherency between two groups, or within a group, was averaged across 10 pairs randomly picked from all possible distinct pairs. In addition, we determined the coherence between the spike time histograms of the respective cell groups. Significance was reached when the coherence exceeded the shift-predicted coherence by 3 SE. The shift predictor was obtained by calculating the coherence between histograms obtained from different simulation runs. The SE was estimated using ten independent simulation runs (which were started from different initial conditions and using different seeds for the white noise currents, see APPENDIX).
The strength of stimulus competition was measured using the parameter 
= [R(cp,cnp) – R(0,cnp)]/[R(cp,0) – R(0,cnp). Here R(cp,cnp) is the firing rate of the excitatory neurons in column 1 in response to stimulus 1 at P contrast cp and stimulus 2 at NP contrast cnp. Stimulus competition occurred when 0 
1, with lower values indicating stronger stimulus competition because then the NP stimulus had a stronger suppressive effect. Problems arise with this expression when the denominator is zero or when the response to the NP stimulus exceeds that to the P stimulus. These data points were not included in the analyses reported here.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONAPPENDIX: THE MODEL EQUATIONSGRANTSACKNOWLEDGMENTSREFERENCES |
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Our goal is to construct a model that reproduces experimental results on the attentional modulation of the response of V4 neurons with one or two stimuli in their RFs; predicts the corresponding response of interneurons and the modulation of their coherence; and provides quantitative insight into what model features are responsible for the strength and frequency of emergent oscillations. To help the reader, the experimental data are summarized in two tables (Tables 3 and 4). They can also be summarized in words using phenomenological theories that have been proposed to account for these data (see also DISCUSSION). For a single stimulus, spatial attention approximately results in a change in either the contrast gain or the response gain, whereas feature-based attention results in a response gain (also referred to as feature gain). A quantitative definition is given in Tables 3 and 4. For multiple stimuli, the biased competition framework predicts that the response to two stimuli is less than the response to the strongest single stimulus and that attention drives this response toward the response to the attended stimulus by itself. Rather than discuss the underlying experiments in RESULTS, we use the predictions of the phenomenological frameworks to constrain the model. To facilitate our description of the model results, we abbreviate several frequently appearing phrases: spatial attention (SA), feature-based attention (FBA), preferred (P), nonpreferred (NP), and feedforward (FF).
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The model is driven by two distinct stimuli. The P stimulus strongly activates the E cells in the first column. It is denoted by a rectangle in the lower right corner of the box that represents the neuron's RF (Fig. 2 Ac). The NP stimulus is less effective in driving the E cells in the first column, but it does strongly activate the ones in the second column. It is denoted by a circle in the upper-left corner of the RF square (Fig. 2Ab). When these two stimuli are either presented at full contrast or not at all, there are four different stimulus conditions as shown in Fig. 2A, subpanels a–d. In the model, stimuli can be presented at various values for the contrast, denoted by P contrast and NP contrast, respectively. The NP-P contrast plane is spanned by a 9 x 9 grid (which corresponds to a contrast resolution of 0.125), yielding 81 contrast pairs. The four stimulus conditions (Fig. 2A, a–d) correspond to the four corners of the grid.
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The baseline conditions, in which there is no SA (Fig. 2Ba) or FBA (Fig. 2Ca), are considered the same in the model. This may not be the same in experiment because it would depend on whether the behavioral task is the same. For instance, for SA you would compare attend into the RF to the attend-away condition. But for FBA you may instead use a task where you attend to a particular orientation and compare the responses of visual neurons to their baseline response when attending a different modality. Thus in total, we consider 20 different combinations of four stimulus and five attention conditions. To make the figures easier to interpret, these conditions are represented by icons, which are placed at only two possible positions: either just below a bar in a bar graph or next to the corresponding curve in a contrast-response-function graph.
Overview of the model mechanisms
To guide the reader through RESULTS, we briefly summarize the model mechanisms using Fig. 3. The model network (Fig. 1) is described in METHODS and contains E cells and two types of interneurons, the FFI and TDI.
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FBA to a feature increases the response of a neuron when it is the P feature and decreases the response, otherwise (see summary in Table 4). In the model, we hypothesize that this is mediated by a top-down activation of all columns that prefer the attended feature (Fig. 3B). This increases the E-cell rate via disinhibition (TDI1 to FFI1 to E1, projections 4 and 2), and the network will display an oscillation in the gamma frequency range. The suppressive effects of FBA to a NP feature originate in the second column and are mediated by the cross-columnar projection (not shown).
SA to a location within the RF is mediated by a change in contrast gain of the stimulus at the attended location, which changes the FF input to the E cells and the FFI (Fig. 3C).
The quantitative results of these hypothesized mechanisms are illustrated using only one parameter setting of the model. Simulations conducted to test the robustness of the model and oscillation frequencies against parameter changes will also be summarized in RESULTS.
Stimulus competition
As the results will be presented from the point of view of column 1, unless stated otherwise, the column number (for instance, 1 in E1) will be suppressed for notational simplicity. Furthermore, unless stated otherwise, results are for the E and FFI subpopulation, without including the Eb and FFIb subpopulation (see METHODS).
Stimulus competition was mediated by cross-columnar activation of the FFI
The model was designed to display stimulus competition in the E-cell rate when both stimuli were presented at full contrast (Fig. 4). Here we determine how the interneurons behave under these conditions. When the P stimulus or NP stimulus was presented by itself, it activated the E cells, although the former did so more than the latter. However, when both were presented simultaneously, the E-cell rate was less than that in response to the P stimulus by itself (Fig. 4C) even though the FF inputs corresponding to the P and NP stimulus were added (Fig. 4D). By design, the firing rate of the TDI did not vary with stimulus condition (Fig. 4A); hence, their activity was not responsible for stimulus competition. The FFI were not activated by the NP stimulus presented by itself (compared with their rate without a stimulus, as indicated by the solid line in Fig. 4B), but they were activated by the P stimulus (Fig. 4B). When both the FF projection and the cross-columnar excitatory projection were fully activated, the FFI fired at their highest rate so that the resulting E-cell rate was suppressed, thus yielding stimulus competition. In the bar graph, Fig. 4B, the combined activity of the FFI and FFIb in a given column is shown. The FFI received cross-columnar excitatory inputs but were not spontaneously active. The FFIb were spontaneously active, irrespective of stimulus condition. Hence the FFI and FFIb population together had a nonzero firing rate even when there was no stimulus present. In the other figures (Fig. 5, 6, and 11), we show only the FFI rate because it varies as a function of contrast.
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E-cell firing rate varied nonmonotonously with stimulus contrast
The cross-columnar excitatory projection, which mediates stimulus competition, is only effective when the P stimulus is presented at sufficient contrast, which has consequences for the behavior of the E-cell firing rate as a function of contrast. First, consider the response to a single stimulus. The E-cell firing rate increased with NP contrast (Fig. 5A), without any sign of saturation because the FFI were not activated (the FFI activity is plotted in Fig. 5C). By contrast, for the P stimulus by itself, the E-cell firing rate increased rapidly with contrast (Fig. 5B) but saturated for larger contrast because the FF projection activated the FFI (Fig. 5D). Second, consider the response to a pair of stimuli. When the P stimulus was present at full contrast and the NP contrast was varied, the E-cell firing rate decreased with NP contrast (Fig. 5A), showing that the NP stimulus was suppressive in that case. When the NP stimulus was at full contrast and the P contrast was varied instead, there was a switch from facilitatory to suppressive effects. For low P contrast, adding the NP stimulus increased the E-cell firing rate (Fig. 5B, 
), whereas, when the P contrast was sufficiently high, adding it decreased the rate (Fig. 5B, 
).
The model predicts that stimulus interactions switch from facilitatory for low P contrasts to suppressive for high contrast values. The biased competition framework predicts only suppressive effects.
During stimulus competition the model response was dominated by FFI activity
The FFI mediate suppressive interactions, which would imply a correlation between FFI activity and the degree of stimulus competition. The strength of suppressive interaction is measured by (see METHODS). For stimulus competition, the response to the pair is higher than the response to the NP stimulus and lower than the response to the P stimulus. This range of firing rates is mapped to values between 0 and 1, respectively. Thus 0 corresponds to the strongest competition and 1 to the weakest competition in which case the response is equal to the maximum of the individual responses. For given values of the contrast, it is possible that the response to the NP stimulus is larger than that to the P stimulus, yielding a negative value for . However, in that case, the stimulus interaction in the model was facilitatory, and it was not included in our analysis. It is possible that is not defined because the terms in the denominator are equal, but this did not occur in our simulations except for the no-stimulus condition.
The strongest suppression of E-cell rate was obtained when both stimuli were present at high contrast (indicated by the arrow in Fig. 6A), which corresponded to the highest level of FFI activity (arrow in Fig. 6B). We therefore analyzed how the strength of suppression increased with FFI rate for all contrast pairs satisfying 0 < < 1. Indeed, decreased linearly with FFI rate (Fig. 6C). The strength of competitive interactions could also be represented by a weighted contrast with more weight given to the P contrast than to the NP contrast. The FFI rate was strongly correlated with the weighted contrast (Fig. 6D, NP weight was 32%, P weight was 68%, and the correlation coefficient was 0.97). The E-cell rate decreased with a weighted contrast, but a linear relationship explained less of the variance than for the FFI (correlation coefficient: –0.72). For lower values of the weighted contrast, there were facilitatory interactions, such as those illustrated in Fig. 5B, but these contrast pairs were not included in Fig. 6D.
The model predicts a relationship between the strength of stimulus competition () and the FFI rate, which was approximately linear for the parameter settings used here.
Feature-based attention
EXCITATORY FIRING RATE EFFECTS OF FBA WERE MEDIATED BY DISINHIBITION. Our hypothesis is that FBA to the P feature of a column activates the TDI in that column (Fig. 3B). This means that FBA to the NP feature does not directly affect the TDI in the column, rather the effect on the E cells is indirect and involves the cross-columnar projection from the column for which the attended feature is preferred. The model (Fig. 7) reproduced the predictions of the biased competition framework as follows (Table 4). FBA to the P feature led to an increased TDI rate (Fig. 7A), which reduced the FFI rate (Fig. 7B), which in turn led to an increase in E-cell rate (Fig. 7C). The activity of the FF projection was not modified by FBA (Fig. 7D). FBA to the NP feature did not change the TDI rate in column 1 (Fig. 7A), rather the FFI rate was increased (Fig. 7B) by way of the cross-columnar projection, which resulted in a decreased E-cell rate.
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SUPPRESSIVE EFFECTS OF FBA WERE ONLY PRESENT AT SUFFICIENTLY HIGH STIMULUS CONTRAST. The excitatory and suppressive effects of FBA both involved the FFI, which limits the contrast pairs for which these effects can be obtained. One prediction, common to both the contrast gain and the response gain model (Table 4), is that FBA to the P-feature does not decrease the E-cell firing rate. In the model this implies that in the single stimulus condition, for all values of the contrast, the disinhibition should be stronger than the direct inhibition (Fig. 3B). This required two subtypes of FFI in the model: those that were spontaneously active without any stimulus (FFIb) and those that required the presence of the P stimulus to be active (FFI) with both types receiving inputs from the TDI. It is important to emphasize that the FFIb were not necessary to obtain the effects of biased competition for when the P stimulus was presented at full contrast. The CRF for a P stimulus presented by itself is shown in Fig. 8 A for the baseline (—) and FBA to P condition (- - -). For low contrast, there is an approximately constant increase in firing rate due to a constant disinhibition mediated by the FFI1b population. The strength depends primarily on the spontaneous firing rate of the FFIb and also on the strength of both the TDI to FFIb and FFIb to E projection. This dependence makes it possible to modulate the low-contrast response change via the spontaneous firing rate of the FFIb. For high contrasts, when the FFI were active and saturation was present in the baseline condition, the increase in firing rate with FBA was larger. Because the increase in firing rate with FBA was higher for higher firing rates than it was for lower firing rates, we label this effect as a response gain (see DISCUSSION). In the presence of a weak (low contrast) NP stimulus, the same happens except that both the saturation as well as the large effects of FBA started at slightly lower values of the contrast (Fig. 8B).
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). This is still consistent with response gain because the attentional modulation was strongest when the firing rate was highest. There was no effect at low contrast because the FFI were not active, and the FFIb were not driven by the cross-columnar projection. Overall, increases in E-cell rate with FBA to the P feature were obtained for all contrast values, but significant decreases with FBA to NP were only obtained for high enough contrasts (Fig. 8D). Significant means that the difference between means exceeds two SDs when calculated based on 10 independent simulation runs.
The model predicts that FBA is more similar to a response gain with an independent low and high contrast component. Furthermore, it predicts that suppressive interactions only occur for higher values of the contrast.
Spatial attention
SA WAS MEDIATED BY A CHANGE IN CONTRAST GAIN OF THE FF INPUT. Our hypothesis is that spatial attention at a resolution less than the size of the typical RF is mediated by an increase in the FF input corresponding to the stimulus at the locus of attention (Fig. 3C). The increase in FF input with SA was higher for the P stimulus than for the NP stimulus. The model reproduced the effects predicted by the biased competition framework (Table 3, Fig. 9 C). The issue addressed here is how the FFI rate changes with SA.
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The model predicts that the firing rate of the FFI increases with SA, irrespective of whether SA is directed to the location of the P or NP stimulus.
IN THE SINGLE STIMULUS CONDITION, SA RESULTED APPROXIMATELY IN A CHANGE OF CONTRAST GAIN.
In the model, there was no saturation in the FF input and saturation arose because of the normalization provided by the FFI neurons (solid lines in Fig. 5, B and D). SA was implemented as a change in contrast gain for the stimulus at the attended location, for the FF input to both the E cells and the FFI. For the single stimulus condition, the contrast gain model (Table 3) suggests that the firing rate does not need to increase with SA at high contrast. Furthermore (Table 3), neither the response gain nor the contrast gain model predicts that the firing rate actually decreases with SA. In the model, the suppressive effects due to the FFI dominated during saturation. This implies that the change in contrast gain with attention should be less for the FFI than that for the E cells. The model therefore predicts that for low P contrast, and for all values of NP contrast, when the FFI are not active, the effects of attention are exactly that of a change in contrast gain (Fig. 10, A and C). For higher contrast, when the FFI are active, the effective contrast for the FFI is lower than that for the E cells, and the change is not exactly a contrast gain. Nevertheless, the largest change in rate occurred below the contrast value with the highest firing rate (Fig. 10A, ).
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SUPPRESSIVE EFFECTS OF SA WERE ONLY PRESENT FOR A LIMITED NUMBER OF CONTRAST PAIRS.
In the model, suppressive effects on the E-cell rate were mediated through the cross-columnar projection, which involved the FFI, thus limiting the contrast pairs for which these effects can be obtained. The excitatory effects reflect a balance between the FF activation of the E cells and the FFI, thereby also limiting the contrast pairs for which excitatory effects are present. To further investigate these issues, we determined how SA affected the pair response. When the NP stimulus was presented at full contrast and the P contrast was varied, the rate change with attention appeared to be nonmonotonic (Fig. 10B, ···). Specifically, the largest change was not obtained for the largest firing rate, implying that it was different from a simple response gain. Overall, SA to the P location increased firing rate for all contrast pairs (Fig. 10D, 
) except when there was no P stimulus present (that is, a NP stimulus by itself). In contrast, SA to the NP location decreased the firing rate only in a narrow band of contrast pairs (Fig. 10D, 
). For a single NP stimulus and as well as for some pair conditions, the firing rate even increased (results not shown), which is inconsistent with the predictions of the biased competition framework.
The model predicts that suppressive effects of SA are only obtained for a limited range of contrast values.
ATTENTION-INDUCED CHANGES IN FIRING RATE WERE SOMETIMES CORRELATED WITH THE STRENGTH OF STIMULUS COMPETITION.
Attention-induced changes in E-cell and FFI rate were correlated because of the FFI to E-cell projection. The coordinated changes in E-cell and FFI rate were studied for the case when there was stimulus competition, that is, 0 < < 1 (Fig. 11, A and B). For FBA, the magnitude of the changes in the FFI and E-cell rate was correlated: an increased change in FFI rate was associated with an increased change in E-cell rate (Fig. 11A). The sign of the change in firing rate was opposite: FBA to the NP feature led to increases in the FFI rate and decreases in the E-cell rate. Likewise, for FBA to the P feature, the FFI rate decreased and E-cell rate increased. In both cases, the linear correlations were almost perfect, with correlation coefficients of –0.99 and –1.00, respectively. For SA, the FFI firing rate always increased with SA (Fig. 11B). The relationship between FFI and E-cell rate changes was noisier than compared with FBA (the correlation coefficients were –0.46 and –0.77 for SA to the NP and to the P stimulus, respectively).
A correlation between attention-induced firing rate changes and the degree of stimulus competition is also expected because both involve FFI activity. For FBA, the correlation coefficients between either the E-cell or the FFI rate changes and were small and not significant (Fig. 11C). For SA, there was a strong and significant correlation: the E-cell rate change was highest for the strongest stimulus competition, whereas the opposite was true for FFI rate changes (Fig. 11D).
The biased competition framework (Table 3) makes predictions for how the pair response is related to the responses to component stimuli and how this response changes with attention. We investigated how difficult it was within the context of the model to satisfy all the constraints implied by the framework. Contrast pairs satisfy the predictions of biased competition when FBA to the P feature increases the E-cell rate and when FBA to the NP feature decreases it. For SA, it means that SA to the P location increases the rate and SA to the NP location decreases the rate. In addition, the pair response should be in between the response to the NP and P stimulus when they are presented singly. In terms of the number of contrast pairs for which the predictions were satisfied, FBA was easiest to achieve (Fig. 11E, ), followed by stimulus competition (Fig. 11E, 
), with SA being the most difficult to achieve (Fig. 11E, ). For the parameter settings used, there were only 5 of 81 pairs for which all conditions were satisfied.
The effect of attention on stimulus competition can be studied by determining how it affects the distribution of values. Starting from all contrast pairs with 0 < < 1 in the baseline condition, SA to the location of the P stimulus made the distribution sharper and shifted its mean to a higher value (results not shown). As a result, the few values closest to one actually decreased. For FBA on the P-feature, the distribution broadened and values larger than one were obtained (results not shown). For these contrast pairs, there was stimulus competition in the baseline condition, but with FBA, this had disappeared. Overall, SA to the location of the P stimulus or FBA to the P feature reduced the degree of stimulus competition.
The model predicts correlations between attention-induced rate changes and the degree of stimulus competition, which are different for FBA compared with SA. This makes it possible to distinguish the effects of FBA from those of SA based on these correlations.
GAMMA AND BETA OSCILLATIONS EMERGED IN THE NETWORK MODEL.
The model network displayed two types of coherent oscillations: in the beta and gamma frequency range. The peak beta frequency ranged from 
15 to 30 Hz depending on the P and NP contrast values. The peak gamma frequency was either 40 or 45 Hz. We briefly describe how these two oscillations emerge. In the subsection that follows, we show how FBA changes the balance between the beta and gamma oscillations, which is followed by a description of how the various model parameters affect oscillation strength and frequency.
The TDI synchronize by way of mutual inhibition, commonly referred to as ING (Borgers and Kopell 2003, 2005; Whittington et al. 2000). Conditions for synchronization by mutual inhibition have been extensively studied theoretically and computationally (see DISCUSSION). For a network with identical neurons and without noise currents, synchrony can be obtained for any oscillation frequency. However, when there is heterogeneity so that the firing rate of uncoupled neurons varies (Tiesinga and Jose 2000) and when there is noise so that the spike train of an uncoupled neuron is not exactly periodic (Tiesinga 2002; Tiesinga and Jose 2000), the network will only robustly synchronize at specific oscillation frequencies related to the inhibitory time constant. This is reviewed in one of our previous publications (Tiesinga and Jose 2000). The key parameter is the driving current to the TDI neurons: only when the current is such that the firing rate of individual neurons matches a harmonic or subharmonic of these oscillation frequencies will synchrony be obtained. The parameter setting of the network was such that for the baseline state the current was too low. Hence only a weak and noisy gamma oscillation was obtained. The top-down activation of the TDI increased the driving current to a value where the network synchronized.
The beta oscillation emerged via different mechanism, which is related to the PING mechanism (PING stands for pyramidal-interneuron-gamma) (see Borgers and Kopell 2003). It starts when a synchronous volley from the E2 cells recruits a synchronous volley from the FFI1. This temporarily shuts down the E1 cells, but on recovering, they produce a synchronous volley that in turn recruits a synchronous FFI2 volley. This shuts down the E2 cells for a while, but on recovery of the E2 cells, the sequence starts over again. The oscillation is thus a consequence of a competition in which each stimulus tries to shut down the column for which it is not preferred.
FBA SHIFTED THE NETWORK OSCILLATION FREQUENCY FROM BETA TO GAMMA. The beta oscillation was strongest in the pair condition, where two stimuli were presented at high and equal contrast. The beta oscillations led to spike alignments in the E-cell rastergrams (Fig. 12 Aa, box) and were visible as peaks at the beta frequency in the E-cell spectral density (Ab, oblique arrow) and also in the coherence between E cells in the same column (results not shown). The spectrum of the FFI cells also had a small peak at the beta frequency (Fig. 12Ab). In addition, there were small peaks in the gamma frequency range, at 45 Hz, in the TDI and E spectra (Fig. 12Ab, vertical arrow) but not in the FFI spectrum (Ab).
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25 ms apart) but had alternating long and short intervals between spike alignments (Fig. 12Ba, box). The model thus showed competition between two oscillations. The relative strength of beta and gamma oscillations also depended critically on the nature of the excitatory synapses (see following text). As there was no direct connection between TDI cells of different columns, the gamma synchrony could only reach the other column via the cross-columnar excitatory projection, which was only effective when the FFI were active. In that case, beta oscillations were also present.
TDI SYNCHRONIZED THE E CELLS VIA THE DIRECT AND INDIRECT PATHWAY. During FBA to the P feature, the E cells received inhibitory inputs, synchronous in the gamma frequency range, from two sources, the FFI and TDI (Fig. 3B). Which of these two projections is most important to the synchronization of the E cells? To assess the contribution of each of these projections to the FBA-induced synchronization, we first introduced a constant delay in the synaptic connection. This did not change the results appreciably. Then a jitter (SD 10 ms) was introduced in the delays for the TDI to E connection but not in the FFI to E connection. This abolished the gamma coherence in response to the P stimulus presented by itself at full contrast but only weakly reduced the coherence in the pair condition when both stimuli were at full contrast. Note that for the single stimulus condition, the FFI rate was low, so that it only contributed a small fraction of the total inhibitory conductance that the E cells received. Only when the FFI to E connection was also jittered by the same amount was the coherence completely abolished in the pair condition. Thus both projections, when active, contribute to the observed synchronization of the principal cells.
OSCILLATION FREQUENCY AND POWER OF THE GAMMA OSCILLATIONS DEPENDED ON THE TIME SCALE OF INHIBITORY SYNAPSES.
The mechanism for beta and gamma is different but both depend critically on the inhibitory time scale. As mentioned before, gamma is generated by activation of the TDI by way of the ING mechanism, which works through mutual inhibition. The other cell groups in the column lock to this rhythm. Slice experiments in hippocampus show that pharmacological manipulations that increase the time constant of inhibition, reduce the frequency of gamma oscillations, whereas those that decrease the time constant, increase the oscillation frequency (Whittington et al. 1995). We explored whether the same was true for the model network when activated by FBA to the P feature. We covaried the time constant 
GABA and maximum conductance gii so that their product—an estimate of synaptic efficacy—remained constant. For time constants larger than the default, 8 ms, the oscillation frequency decreased as expected. For lower values of GABA, the oscillation frequency jumped from 40 to 60 Hz. The peak frequencies in that regime varied between 60 and 65 Hz with GABA. The low-synchrony baseline state (no FBA) was stable against small changes in synaptic time scale. However, for large GABA>10 ms, a synchronized oscillation at beta frequencies (30 Hz) appeared.
Taken together, the results show that the network can support FBA-mediated transitions to oscillations in the gamma frequency range for a range of parameter values.
OSCILLATION FREQUENCY AND POWER OF THE BETA OSCILLATIONS DEPENDED ON THE TIME SCALE OF INHIBITORY SYNAPSES.
In the model, beta oscillations were generated in a loop with four synaptic steps: E1 to FFI2 to E2 to FFI1 to E1. When excitation was mediated by NMDA only, effectively there were only two time scales: the time constant of inhibition and the delay between the emission of the inhibitory volley by the FFI and the arrival of the excitatory volley at the FFI. The latter involves both the level of depolarization of the FFI and axonal delays (with a default value of 0.05 ms). In the default parameter setting, the synaptic time scale for inhibition generated by the FFI was the same as for the TDI. Nevertheless, the beta oscillation had a longer period (lower frequency) because of the additional delays associated with the synaptic steps. We explored how the oscillation frequency obtained in response to both stimuli at full contrast was affected by manipulation of synaptic time scales and axonal delays. First, when GABA was varied, the frequency of the beta oscillation did not change but the power did. The peak height at the oscillation frequency in the spectral density was maximal for GABA values of 10 ms. Second, the level of depolarization of the FFI and E (varied via AFFI and AE, respectively, see METHODS) can affect the frequency by altering the delay between inhibitory spike emission and the arrival of excitatory inputs or by lengthening the interval after the inhibitory volley during which the E cells do not spike. The power as a function of AFFI was peaked with the strongest oscillation occurring at 30 Hz for AFFI = 0.7 µA/cm2 (the default value). The oscillation frequency did not vary much for AE values between 2 and 4 µA/cm2, but the height of the beta peak did, reaching a maximum for the default value of the current, AE = 3 µA/cm2. Third, when the axonal delays were varied, the oscillation frequency did not change significantly.
Thus the frequency of the beta oscillation in the pair condition was related to the time scale of inhibitory synapses and was robust against parameter variation.
STRENGTH AND FREQUENCY OF THE OSCILLATIONS DEPENDED ON STIMULUS CONTRAST. We investigated the behavior of the oscillation for all contrast pairs, by determining the fraction of power in a 10-Hz interval around gamma and beta frequencies. Because the frequency of the beta oscillation did vary with contrast, we picked the power in an interval around the highest peak present in the spectrum <35 Hz. For gamma, we calculated the fraction of the power in the 40- to 50-Hz frequency range for the baseline condition and between 35 and 45 Hz in the FBA condition. With FBA to the P feature, gamma power increased and beta power decreased for all contrast pairs (results not shown). An interesting structure becomes visible when the power is plotted as a function of firing rate (Fig. 12, C and D). The structure arises because the same E-cell firing rate is obtained in response to multiple different stimulus conditions (Fig. 6A). For instance, the response to a P stimulus at medium contrast can be the same as the response to the P and NP stimulus together at high contrast. For FBA to the P feature, the former stimulus condition has more gamma power than the latter. In the graph, this shows up as a general increase of gamma power with firing rate, but the curve then curls over to give two different gamma powers for the same firing rate. The high power branch is for the case with weak NP stimuli and the low power branch is for when the NP stimulus is stronger (box in Fig. 12D). The opposite happens for the beta power plotted as a function of the firing rate, there the high power branch of the curve corresponds to the "strong NP" pairs (Fig. 12C, box indicated by horizontal arrow).
In the baseline, or attend-away, condition there are two interesting features. First, gamma power increases with firing rate. Because, for the single stimulus condition and for low to medium contrasts in the pair condition, firing rate increases with contrast, this means that gamma power also increases with stimulus contrast. Second, beta power has a peak as a function of firing rate (Fig. 12C, box indicated by oblique arrow). The position of this peak corresponds to the firing rate obtained when two stimuli are present at high contrast, resulting in stimulus competition, with a firing rate less than that obtained in response to the P-stimulus at full contrast. There is a corresponding dip the gamma power as indicated by the oblique arrow in Fig. 12D.
We investigated the behavior of the coherence in the gamma frequency range between spike trains of individual E cells, averaged across 10 pairs. For a single P stimulus (results not shown), the coherence for FBA to the P feature exceeded the attend-away coherence when the contrast was >0.5. This difference increased with higher contrast values. This shows that behavior predicted by the model should be visible in the coherence between spike trains of two simultaneously recorded E cells when they are both in the same column. The coherence was increased even more when the NP stimulus was presented in addition to the P stimulus (with a P contrast equal to 1 and FBA to the P feature). This behavior is different from that of the gamma power shown in Fig. 12D, which decreased when the NP stimulus was added.
The model predicts that FBA to the P feature introduces a gamma oscillation that competes with the beta oscillation generated by two high contrast stimuli. Both in the baseline condition and the FBA to P condition, gamma power generally increased with contrast, but it decreased with stimulus competition. The latter was indicated by an increase in the beta power.
FBA INCREASED THE PHASE LAG BETWEEN INTERNEURONS AND E CELLS.
The coherent activity of E cells is often considered the main contribution to the LFP recorded in vivo. Recent experiments in the hippocampus show that during network oscillations different types of interneurons lock to the LFP at different phases (Klausberger et al. 2003; Tukker et al. 2007). In the model, this raises the issue of how the temporal response of interneurons is related to that of the E cells. In a fully synchronized network, each cell group will produce synchronized volleys. When the network oscillates at one frequency, it is possible to determine the average delay between the volleys of inhibitory and excitatory neurons as illustrated in our previous work (Buia and Tiesinga 2006). However, for the networks studied here, it is not as simple because there is frequency content in both the beta and gamma frequency range (Fig. 12Ba, box). The coherence between two time series yields
or 
inside the RF box indicates of which stimulus the contrast was varied. In B, 
