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## Abstract

Covert attention affects prestimulus activity in the visual cortex. Although most studies investigating neural mechanisms of attention have focused on the effects of spatial attention, attention can also be directed to particular features. To investigate the spatiotemporal nature of feature attention, we measured subjects’ brain activity using magnetoencephalography (MEG) and functional magnetic resonance imaging (fMRI) while subjects attended to color or motion of a stimulus based on a visual cue, which was presented 1 s before the stimulus onset. We used the hierarchical Bayesian method that allows us to estimate cortical currents with MEG and fMRI data in the order of millimeters and milliseconds. When subjects attended to color, activity within the color-sensitive area (fusiform gyrus) was selectively enhanced within the prestimulus period. By contrast, when subjects attended to motion, activity within the motion-sensitive area (middle temporal gyrus) was selectively enhanced during this period. This effect was not seen in frontal, parietal, and lower visual areas. Additionally, this effect was transient rather than sustained, suggesting that it differs from temporal aspects of spatial attention. These results suggest that, although both spatial and feature attention modulate prestimulus activity within specific visual areas, neural mechanisms underlying these effects might be different.

## Introduction

Visual attention enables us to extract behaviorally significant information from a cluttered environment. Single-unit studies for nonhuman primates and neuroimaging studies over the past 15 years have provided substantial insight into the neural basis of information selection by visual attention. Spatial attention modulates the response gain of neurons in the visual cortex in a retinotopic fashion (Luck et al. 1997, 2000; Hillyard and Anllo-Vento 1998; Hopfinger et al. 2000), whereas feature attention modulates a response gain of neurons within feature-sensitive visual areas such as V4 in the ventral pathway and middle temporal (MT) visual area in the dorsal pathway (Motter 1994; Chawla, Ress and Friston 1999; Treue and Martinez Trujillo 1999; Maunsell and Treue 2006; Schoenfeld et al. 2007). Combining these insights from spatial and feature attention, it is possible to infer that common neural mechanisms are involved in both spatial and feature attention (Maunsell and Treue 2006). On the other hand, recent psychophysical (Kanai et al. 2006) and physiological studies (McAdams and Maunsell 2000; Hayden and Gallant 2005) have suggested that spatial attention and feature attention are based on distinct neural substrates.

Neural activity in the visual cortex is affected by covert attention even before a stimulus onset (Luck et al. 1997; Chawla, Ress and Friston 1999; Kastner et al. 1999; Hopfinger et al. 2000; Fries et al. 2001; Wylie et al. 2006; Silver et al. 2007). It has been claimed that this neural modulation may play a causal role in the attentional effects on sensory activity for a subsequently presented stimulus (Driver and Frith 2000). For spatial attention, this effect is observed in a retinotopic fashion and sustained while subjects keep attending to the corresponding visual field (Luck et al. 1997; Worden et al. 2000; Thut et al. 2006; Silver et al. 2007). For feature attention, functional magnetic resonance imaging (fMRI) studies have described a detailed picture of the anatomical regions related to this effect and identified feature-sensitive visual areas such as V4 and MT (Chawla, Ress and Friston 1999; Wylie et al. 2006). However, little is known about the temporal nature of neural activity associated with feature attention before the stimulus onset.

In the present study, we examined the effects of feature attention on neural activity, particularly within feature-sensitive visual areas before the stimulus onset, using magnetoencephalography (MEG) in combination with fMRI. Subjects attended to color (color condition) or motion (motion condition) of the stimulus and discriminated the attribute of the attended feature of the stimulus. The stimulus consisted of 1000 dots whose color and motion changed with the stimulus onset. To observe the effects of feature attention before and after the stimulus onset separately, a visual cue indicating the feature to be attended was presented to subjects 1 s before the stimulus onset. As it has been suggested that color and motion flow into different visual pathways and activate different areas (Zeki et al. 1991; Tootell et al. 1995; Toyama et al. 1999; Kuriki et al. 2000, 2005), this experimental paradigm allows us to investigate the effects of attention to color and attention to motion within color-sensitive visual areas and motion-sensitive visual areas separately.

It has proved difficult to estimate distributed sources of electrical activity within cortical areas from MEG data as the estimation of cortical activity based on MEG data is fundamentally ill posed due to the essential difficulty of the inverse problem. Recently, to overcome this difficulty, Sato et al. (2004) proposed a hierarchical Bayesian method for source-current estimation with MEG data. By constraining cortical regions based on fMRI information, this technique enables us to estimate the source currents on the cortical surface in the order of milliseconds and millimeters. We used this method to investigate the spatiotemporal nature of feature attention in the brain.

We conducted 3 experiments. In the first experiment, we measured subjects’ cortical activity using MEG while they attended to the color or motion of the stimulus (selective attention MEG experiment). In the second experiment, the subjects’ blood oxygen level–dependent (BOLD) signals were measured during the same task (selective attention fMRI experiment). In the last experiment, to determine color- and motion-sensitive areas for each subject, we conducted a separate control experiment (visual area specification fMRI experiment). Cortical currents were estimated by combining the results of the first and second experiments. Then, we compared the effects of attention to color and motion in frontal, parietal, and visual areas. We found no significant difference within frontal, parietal, and lower visual areas. On the other hand, during the 1-s prestimulus period, activity within the color-sensitive area and within the motion-sensitive area was selectively increased by attention to color and attention to motion, respectively. In addition, we found that this effect is transient rather than sustained. This result supports the idea that the neural mechanism underlying feature attention is different from that underlying spatial attention.

## Materials and Methods

### Subjects

Six healthy subjects (aged 23–27 years) participated in the experiments, which were approved by the ATR Human Subjects Review Committee. All subjects gave informed consent and had normal or corrected-to-normal visual acuity and normal color vision.

### Stimuli

The stimulus consisted of 1000 colored moving dots, which were arranged randomly in a circular area with a 5° radius from the center of the screen on a white background (Commission Internationate de l'Eclairage [CIE] coordinates x = 0.28, y = 0.33). The mean luminance of the background and stimulus was the same (70 cd/m2). The size of each dot was 0.06° square. Twenty percent of the dots moved toward a target direction, whereas the others moved in random directions at a speed of 10°/s. The target direction was from the center to the outside (expansion) or from the outside to the center (contraction). Each dot was vanished within 2 frames (33 ms) and rearranged randomly within the same circular area. Separately, 20% of the dots were colored in a target color, whereas the others were colored randomly in the 1 of 16 alternative colors. The target color was red (CIE coordinates x = 0.30, y = 0.32) or green (CIE coordinates x = 0.26, y = 0.34). Results from pretesting determined that the percentage of dots with the target color or motion equalizes the subjects’ task performance (percent correct) between the 2 conditions. The stimulus was generated using a VSG2/5 graphics board from Cambridge Research Systems (Rochester, UK).

### Selective Attention MEG Experiment

#### Procedure

The task procedure is depicted schematically in Figure 1. Subjects were instructed to discriminate an attribute of an attended feature of the stimulus and were asked to respond as soon as possible but accurately. There were 2 attention conditions: attention to color (color condition) and attention to motion (motion condition). Subjects were asked to maintain fixation on the center of the screen throughout the entire experiment, and 1000 white stationary dots were presented around the central fixation point. One second before the stimulus onset, subjects directed their attention to 1 of the 2 features of the stimulus, depending on a preceding visual cue presented on the center of the screen (“C” for color and “M” for motion). Then, the stimulus was presented for 500 ms. In the color condition, subjects reported whether the stimulus looked reddish or greenish, and in the motion condition, they reported whether the stimulus featured motion of expansion or contraction. A report was issued by the subjects pressing 1 of 2 buttons held in the left and right hand. The button assignment for target colors and motions varied across subjects. Subjects practiced the task extensively (more than 300 trials) before the main experiment.

Figure 1.

Schematic illustration of a trial sequence. Subjects covertly attended to either color or motion depending on a visual cue (for color: C, for motion: M) and discriminated an attribute of the attended feature of the following stimulus as quickly and accurately as possible.

Figure 1.

Schematic illustration of a trial sequence. Subjects covertly attended to either color or motion depending on a visual cue (for color: C, for motion: M) and discriminated an attribute of the attended feature of the following stimulus as quickly and accurately as possible.

Subjects underwent 2 color sessions in which all trials were under the color condition, and 2 motion sessions in which all trials were under the motion condition. One session contained 52 trials. The order of sessions varied across subjects. Four types of stimulus (2 target colors and 2 target motions) were presented in the same frequency and pseudorandom order. To help maximize retention of the subjects’ attention, the stimulus was presented after an interstimulus interval (ISI) of either 1000 ms (44 trials in 1 session) or 500 ms (8 trials in 1 session). The 2 ISI conditions were mixed in a pseudorandom order in each session. The trials containing an ISI of 500 ms were not included in the analysis of MEG data. The intertrial interval (ITI) was randomly varied between 2000 and 3000 ms. The stimulus was projected (ELP-710 projector; Epson, Tokyo, Japan) from outside a magnetically shielded room (Tokin Corp., Sendai, Japan) onto a semitranslucent screen inside the dark room.

#### MEG Acquisition

Magnetic responses were recorded using a 201-channel, whole-head biomagnetic imaging system (Shimadzu, Kyoto, Japan). The responses were sampled at 1000 Hz for 2048 ms, beginning 300 ms before the visual cue onset. Simultaneous recording with an electrooculogram (EOG) was used to continually monitor eye movements, and a chin rest was used to minimize head movement. All MEG data were band-pass filtered at 1–100 Hz (a linear-phase Finite Impulse Response filter using the Parks-McClellan algorithm, 63 degree). Because of an artifact in the band-pass filter, signals recorded during the first 50 ms and the last 248 ms in each trial were discarded. Trials were rejected if 1) the EOG exceeded 100 μV and 2) the max/median ratio of the MEG data exceeded 13. The max/median ratio was calculated using the maximum amplitude of each trial divided by the median amplitude of the trial.

### Selective Attention fMRI Experiment

#### Procedure

The task procedure was the same as that in the selective attention MEG experiment except for a fixed ITI (2000 ms) and a fixed ISI (1000 ms). Therefore, the length of 1 trial in this experiment was 5 s.

There were 88 scans in 1 session (time repetition [TR] = 3 s). The data from the first 4 and last 4 scans were not included in the analysis. The remaining 80 scans were alternately divided into 4 task blocks and 4 fixation blocks. Each block contained 10 scans. In each task block, subjects performed the task 6 times. Four types of stimulus (2 target colors and 2 target motions) were presented in the same frequency and in a pseudorandom order. Two of the 4 task blocks were for the color condition; the other 2 were for the motion condition. The subjects were informed the order of task blocks before the experiment, and the order was varied between subjects. In each fixation block, 1000 white stationary dots were presented around a central fixation point, and subjects were asked to maintain central fixation during the block. Each session was repeated 5 times for each subject.

#### fMRI Acquisition

A 1.5-tesla scanner (Shimadzu-Marconi Magnex Eclipse) was used to acquire both structural T1-weighted images (TR = 20 ms, time echo [TE] = 2.3 ms, flip angle = 40°, matrix = 256 × 256, field of view [FoV] = 256 mm, thickness = 1 mm, slice gap = 0 mm) and $T2*$-weighted echo planar images (TR = 3 s, TE = 49 ms, flip angle = 90°, matrix = 64 × 64, FoV = 192 mm, thickness = 4 mm, slice gap = 0 mm, 30 slices), showing BOLD contrasts. A custom-made bite bar was used to minimize head motion.

### Visual Area Specification fMRI Experiment

#### Procedure

The task procedure was the same as that in the selective attention fMRI experiment, except for the stimulus. Under the color condition, 50% of the dots changed their color to the target 1 with the stimulus onset but remained stationary. Under the motion condition, 50% of the dots started moving in the target direction with the stimulus onset but their color remained white.

There were 128 scans performed in 1 session (TR = 3 s). Data from the first 4 and last 4 scans were not included in analysis, and the remaining 120 scans were alternately divided into 6 task blocks and 6 fixation blocks. Each block contained 10 scans. Three of the 6 task blocks were for the color condition; the other 3 blocks were for the motion condition.

#### fMRI Acquisition

Acquisition was performed as it was in the selective attention fMRI experiment.

### Analysis

#### fMRI Analysis

The images were preprocessed and analyzed using an SPM 99 (http://www.fil.ion.ucl.ac.uk/spm/spm99.html). After correcting the slice timing, the images were realigned to the first image as a reference, spatially normalized to the Montreal Neurological Institute (MNI) space and spatially smoothed with a Gaussian kernel (9 mm, full-width at half-maximum). The block design analysis was conducted using boxcar regressors, convolved with a hemodynamic response function as the reference waveform for both color and motion conditions.

#### Cortical Current Estimation

A polygon model of the cortical surface was constructed based on individual MRI image slices for each subject using Brain Voyager software (Brain Innovation, Maastricht, Netherlands). The total number of vertex points for each subject was 26,783 ± 585 (mean ± standard error [SE]). The task-related cortical regions were determined by logical OR operation between the results of the color and motion conditions in the selective attention fMRI experiment. Within these regions, a single-current dipole was assumed perpendicular to the cortical surface at each vertex point. Here, the total number of dipoles for each subject was 3871 ± 291 (mean ± SE). The lead field matrix was calculated using a sphere model. Cortical currents were estimated with the recorded MEG data under 2 conditions using the hierarchical Bayesian method (Sato et al. 2004). Because the forward model was identical between color and motion conditions, the difference of the estimated currents between the 2 conditions reflects the difference of the MEG data between the 2 conditions.

Using the hierarchical Bayesian estimation, a linear spatial filter to estimate the cortical current for each dipole was calculated using the MEG data (see Supplementary Methods for more details). We assumed that the pattern of cortical activity changed during the cue period (from the cue onset to the stimulus onset: 1000 ms) and the stimulus period (from the stimulus onset to the stimulus offset: 500 ms). To prevent these states being mixed, we divided the time of 1 trial (from 250 ms before cue onset to 500 ms after stimulus onset: 1750 ms) into 34 time windows (100 ms in length: 50-ms overlap); the linear spatial filter was calculated separately for each time window. The cortical current was then estimated with the MEG data using an estimated liner spatial filter at each time window. In the overlap period, the cortical currents were averaged between 2 time windows.

#### Estimated Current Normalization

A cortical current estimation with the MEG data is necessarily inexact due to several factors, including noise in the measured data. Therefore, it is essential to quantitatively evaluate the uncertainties and potential errors of the estimated current by normalization (Dale et al. 2000). Thus, we normalized the current by dividing the estimated currents by the posterior variance of the cortical current at each dipole (Friston et al. 2002). We then conducted a statistical comparison of the normalized estimated currents between the 2 conditions.

## Results

### Behavioral Results

We compared the subjects’ task performance in the selective attention MEG experiment between 2 conditions. When the ISI was 1000 ms, reaction time (color condition, 757 ± 55 ms [mean ± SE]; motion condition, 833 ± 111 ms; Wilcoxon signed rank test, P = 0.29) and accuracy (color condition, 96 ± 2% [mean ± SE]; motion condition, 93 ± 2%; Wilcoxon signed rank test, P = 0.16) were not significantly different between 2 conditions. Similarly, when the ISI was 500 ms, reaction time (color condition, 783 ± 37 ms; motion condition, 834 ± 97 ms; Wilcoxon signed rank test, P = 0.56) and accuracy (color condition, 95 ± 3%; motion condition, 93 ± 4%; Wilcoxon signed rank test, P = 0.38) were not significantly different between 2 conditions. These results suggest that the task difficulty was similar in both conditions, and subjects kept their attention on the assigned feature in both ISI trials.

### Task-Related Areas Defined by fMRI

Figure 2A and B show the results of the selective attention fMRI experiment for the color condition (Fig. 2A) and the motion condition (Fig. 2B) for subject YS. Under both conditions, significantly activated areas were found in the bilateral dorsolateral prefrontal cortex (DLPFC; left and right), bilateral precentral gyrus (prCG; left and right), supplementary motor area (SMA), anterior part of the bilateral intraparietal sulcus (aIPS; left and right), posterior part of the bilateral intraparietal sulcus (pIPS; left and right), and occipital cortex. All subjects showed similar activity patterns as shown in Figure 2A and B. These areas were used as a spatial constraint for the cortical current estimation. Nine significant areas within the extravisual cortex were used to specify the regions of interest (ROIs) for frontal and parietal areas.

Figure 2.

Activated brain areas in the fMRI experiments for subject YS. (A) Results of the selective attention fMRI experiment in the color condition (P < 0.001, uncorrected; extent threshold of 30 voxels). (B) Results from the selective attention fMRI experiment in the motion condition (P < 0.001, uncorrected; extent threshold of 30 voxels). (C) Bilateral responses to the color stimulus (left) and motion stimulus (right) in the visual area specification experiment (P < 0.001, uncorrected; extent threshold of 30 voxels).

Figure 2.

Activated brain areas in the fMRI experiments for subject YS. (A) Results of the selective attention fMRI experiment in the color condition (P < 0.001, uncorrected; extent threshold of 30 voxels). (B) Results from the selective attention fMRI experiment in the motion condition (P < 0.001, uncorrected; extent threshold of 30 voxels). (C) Bilateral responses to the color stimulus (left) and motion stimulus (right) in the visual area specification experiment (P < 0.001, uncorrected; extent threshold of 30 voxels).

Figure 2C shows the results of the visual area specification fMRI experiment for subject YS. The responses to the color stimulus were seen in the bilateral fusiform gyrus (Fig. 2C, left), and the responses to the motion stimulus were seen in the bilateral MT gyrus (Fig. 2C, right). All subjects showed similar activity patterns as shown in Figure 2C. These areas were used to specify the ROIs for color- and motion-sensitive visual areas.

### MEG Channel-Level Analysis

#### Global Field Magnetic Power

Figure 3 shows the recorded MEG data for subjects YS and MS plotted as global magnetic field power (GFP) over time. GFP was calculated as:

where Si(t) is the trial average of the observed magnetic signal in channel i at time t, and N is the number of channels. For each subject, GFPs for both color and motion conditions remained virtually superimposed throughout the entire time period, and peaks of GFP to the stimulus ranged from 150 to 200 ms after the stimulus onset (gray area in Fig. 3), in agreement with the results of previous MEG studies using a color or motion stimulus (Toyama et al. 1999; Kuriki et al. 2000, 2005).

Figure 3.

GFP plotted as a function of time for subject YS and MS. The onset times of the visual cue (0 ms) and stimulus (1000 ms) are demarcated by vertical dotted lines. GFP is shown for attention to color (red line) and attention to motion (blue line). The gray shading shows the period of time used for the specification of ROIs.

Figure 3.

GFP plotted as a function of time for subject YS and MS. The onset times of the visual cue (0 ms) and stimulus (1000 ms) are demarcated by vertical dotted lines. GFP is shown for attention to color (red line) and attention to motion (blue line). The gray shading shows the period of time used for the specification of ROIs.

#### Channel-Level Analysis

We compared the MEG signal amplitudes between 2 conditions for all channel. To separately examine the effects of feature attention before and after the stimulus onset, we calculated the mean signal amplitudes for each period and condition and compared them between 2 conditions. In each period, several channels showed significant differences between 2 conditions within each subject (t-test, P < 0.05). For the cue period (from the cue onset to the stimulus onset), there were 26 ± 9 significant channels (mean ± SE), and for the stimulus period (from the stimulus onset to the stimulus offset), there were 16 ± 4 significant channels (mean ± SE) for each subject. However, between subjects, we did not find any common tendency in the number of the channels that showed significant differences or in the spatial configuration of the channels showing significant difference.

### Time Series of the Spatial Map of the Estimated Currents

Cortical currents in task-related areas defined by fMRI were estimated with MEG signals using the hierarchical Bayesian method (Sato et al. 2004). Figure 4 shows the spatial map of the estimated current amplitude for subject YS (see Supplementary Movie for the detailed time series). In the cue period (from the cue onset to the stimulus onset), activity was observed in the parietal and frontal cortex as well as in the visual cortex (Fig. 4, left). In the stimulus period (from the stimulus onset to the stimulus offset), strong responses to color and motion stimuli were observed in the visual cortex (Fig. 4, right). Spatiotemporal patterns of the brain activity were generally similar in the color and motion conditions, and this tendency was observed in all subjects.

Figure 4.

The spatial map of the mean current amplitude in the left inflated brain of subject YS during the cue period (left) and the stimulus period (right) in the color condition (top) and the motion condition (bottom).

Figure 4.

The spatial map of the mean current amplitude in the left inflated brain of subject YS during the cue period (left) and the stimulus period (right) in the color condition (top) and the motion condition (bottom).

### ROI Specification

#### Parietal and Frontal Areas

In the selective attention fMRI experiment, we found 9 significant areas in the parietal and frontal cortex (Fig. 2A,B). The estimated currents involved with the task were located in these areas, but they were predominantly focalized within the area. Thus, for further analysis, we determined the ROIs in frontal and parietal areas for each subject as follows: 1) specify 9 areas based on the results of the selective attention fMRI experiment, 2) find the dipole with the largest intensity in the spatial pattern of the estimated current amplitude temporally averaged from the cue onset to the stimulus offset within each area, and 3) specify a circular region with a 6-mm radius centered at the dipole found in step 2 and set it as an ROI. Nine ROIs (left DLPFC, right DLPFC, left prCG, right prCG, SMA, left aIPS, right aIPS, left pIPS, right pIPS) were determined for each subject. Each ROI contained 19 ± 2 dipoles (mean ± SE). These areas are known to belong to the front-parieto attention network (Corbetta and Shulman 2002; Shulman et al. 2002; Giesbrecht et al. 2003; Liu et al. 2003; Serences et al. 2005). As the time courses of the estimated currents were very similar among the dipoles within each ROI (correlation coefficient averaged more than 0.8), we averaged the estimated currents of dipoles in each trial and regarded this as the estimated current for the ROI.

#### Lower Visual Areas

To examine the effect of feature attention within the lower visual areas, Brodmann areas (BAs) 17 and 18, 4 ROIs (left BA17, right BA17, left BA18, right BA18) were determined for each subject. We used the results of selective attention fMRI experiment and the probabilistic cytoarchitectonic map in the SPM Anatomy toolbox (Eickhoff et al. 2005) to specify lower visual areas. For each subject, we determined the ROIs within lower visual areas as follows: 1) specify BA17 and BA18 in the left and right hemispheres by taking the common area between the results of the selective attention fMRI experiment and Brodmann areas 17 and 18 specified by the probabilistic cytoarchitectonic map, 2) find the dipole with the largest intensity in the spatial pattern of the estimated current amplitude temporally averaged from 150 to 200 ms after the stimulus onset within each area, and 3) specify a circular region with a 6-mm radius centered at the dipole found in step 2 and set it as an ROI.

#### Color- and Motion-Sensitive Areas

To examine the effect of feature attention within feature-sensitive areas, 4 ROIs (left and right color-sensitive area and left and right motion-sensitive area) were determined for each subject. In step 1, we used the results of the visual area specification fMRI experiment (Fig. 2C) to specify color- and motion-sensitive areas. The subsequent procedure (steps 2 and 3) was the same as that used in ROI specification for lower visual areas. Figure 5 shows the location of each ROI for subjects YS and MS. These locations were roughly consistent with the locations of color-sensitive areas (V4 complex) and motion-sensitive areas (MT+) as reported in previous fMRI studies (Zeki et al. 1991; Tootell et al. 1995; Chawla, Ress and Friston 1999; Bartels and Zeki 2000; Schoenfeld et al. 2003; Grill-Spector and Malach 2004).

Figure 5.

Four ROIs within feature-sensitive areas for subject YS and MS. Each ROI (indicated by green enclosure) was placed within the region determined by the visual area specification fMRI experiment (indicated by red broken lines). The Talairach coordinates of the ROI center for subject YS are (−34, −64, −16) for the left color-sensitive area; (45, −62, −14) for the right color-sensitive area; (−44, −67, 0) for the left motion-sensitive area; and (46, −66, 2) for the right motion-sensitive area. The Talairach coordinates of the ROI center for subject MS are (−32, −50, −20) for the left color-sensitive area; (36, −54, −18) for the right color-sensitive area; (−49, −61, −2) for the left motion-sensitive area; and (58, −60, −1) for the right motion-sensitive area.

Figure 5.

Four ROIs within feature-sensitive areas for subject YS and MS. Each ROI (indicated by green enclosure) was placed within the region determined by the visual area specification fMRI experiment (indicated by red broken lines). The Talairach coordinates of the ROI center for subject YS are (−34, −64, −16) for the left color-sensitive area; (45, −62, −14) for the right color-sensitive area; (−44, −67, 0) for the left motion-sensitive area; and (46, −66, 2) for the right motion-sensitive area. The Talairach coordinates of the ROI center for subject MS are (−32, −50, −20) for the left color-sensitive area; (36, −54, −18) for the right color-sensitive area; (−49, −61, −2) for the left motion-sensitive area; and (58, −60, −1) for the right motion-sensitive area.

### Comparison of Estimated Current Amplitudes within the ROI

Previous studies have reported that covert spatial attention continually modulates the neural activity within specific cortical areas before a stimulus onset (Luck et al. 1997; Worden et al. 2000; Yamagishi et al. 2005; Thut et al. 2006). For this reason, for the cue period, we calculated the time-averaged amplitude of the estimated current for each trial and for both conditions and then compared it between the 2 conditions. On the other hand, previous EEG studies (Hillyard and Anllo-Vento 1998; Luck et al. 2000) have reported that the attentional effects on the sensory stimulus were observed at peaks of stimulus-evoked potentials. Thus, for the stimulus period, we calculated and compared the peak amplitudes and latencies of the estimated current between conditions.

#### Parietal and Frontal Areas

We examined the effect of feature attention on 9 ROIs within frontal and parietal areas: left DLPFC, right DLPFC, left prCG, right prCG, SMA, left aIPS, right aIPS, left pIPS, and right pIPS. For the cue period, there was no significant difference between the 2 conditions in current amplitude for each ROI (Wilcoxon signed rank test, P > 0.094). For the stimulus period, no significant difference in current amplitudes was observed (P > 0.156). Similarly, latencies were not significantly different between conditions for each ROI (P > 0.156).

#### Lower Visual Areas

Regarding the frontal and parietal areas, we compared the time-averaged amplitudes of the estimated current between conditions for the cue period, whereas we compared the peak amplitudes and latencies of the estimated current between the 2 conditions for the stimulus period within 4 lower visual areas: left BA17, right BA17, left BA18, and right BA18. We found no significant difference between conditions for both the cue and stimulus periods in each ROI (Wilcoxon signed rank test, P > 0.156). There was also no significant difference between conditions in the latency in each ROI during the stimulus period (P > 0.219).

#### Color- and Motion-Sensitive Areas

We conducted the same comparisons within each feature-sensitive area. During the cue period, within the left color-sensitive area, the estimated current amplitude was significantly greater in the color condition than in the motion condition (Wilcoxon signed rank test, P = 0.031). Within the right color-sensitive area, although not statistically significant (P = 0.312), a similar tendency was observed in 5 out of 6 subjects. Within the right motion-sensitive area, the estimated current amplitude was significantly greater in the motion condition than in the color condition (P = 0.031). Within the left motion-sensitive area, although not statistically significant (P = 0.062), a similar tendency was observed in 5 out of 6 subjects. During the stimulus period, within both the left and right color-sensitive areas, the peak of the estimated current amplitude was significantly greater in the color condition than in the motion condition (P = 0.031), whereas within both the left and right motion-sensitive areas, the peak of the estimated current amplitude was significantly greater in the motion condition than in the color condition (P = 0.031). Figure 6 summarizes the amplitude (i.e., root-mean-square) comparison between the color condition (red bar) and the motion condition (blue bar) within each feature-sensitive area (see Supplementary Fig. 1 for the individual subject statistics). The average latencies of the peak amplitude from the stimulus onset were 234 ± 24 ms (mean ± SE, left color-sensitive area), 253 ± 29 ms (right color-sensitive area), 202 ± 33 ms (left motion-sensitive area), and 183 ± 28 ms (right motion-sensitive area). We did not find any significant differences in peak latency (P > 0.312).

Figure 6.

Comparison of the mean current amplitudes within each feature-sensitive area and each period (6 subjects). Each bar (+SE) shows the mean current amplitudes for color (red bars) and the motion (blue bars) conditions.

Figure 6.

Comparison of the mean current amplitudes within each feature-sensitive area and each period (6 subjects). Each bar (+SE) shows the mean current amplitudes for color (red bars) and the motion (blue bars) conditions.

Figure 7 shows the trial averages of the estimated current time courses for the color (red line) and motion (blue line) within each feature-sensitive area for subjects YS and MS (note that it is not root-mean-square). In each area, after the stimulus onset, the estimated currents peaked at approximately 200 ms, and the peak amplitudes differed between experimental conditions as shown in Figure 6. All subjects showed the same pattern of results. In contrast, during the stimulus period, we did not find any consistent peaks across subjects during the cue period. In addition, the pattern of experimental differences shown in Figure 6 was not always observed around the prominent peak but was observed around smaller peaks for some cases. For example, in the left color-sensitive area of subject YS, the peak amplitude around 200 ms after the stimulus onset was greater in the color condition than in the motion condition as shown Figure 6. In contrast, in the right motion-sensitive area of subject MS, the peak amplitude around 300 ms after the cue onset was rather smaller in the motion condition than in the color condition. Nevertheless, time-averaged amplitude was significantly greater in the motion condition (see the individual statistics of subject MS in Supplementary Fig. 1). In the next section, Temporal Characteristics of the Effect of Feature Attention within Color- and Motion-Sensitive Areas during the Cue Period, we will focus on temporal characteristics of the amplitude differences between conditions during the cue period.

Figure 7.

Trial average of the estimated current time courses within each feature-sensitive area for subject YS and MS. The onset times of the cue (0 ms) and the stimulus (1000 ms) are demarcated by vertical broken lines. Current is shown for attention to color (red line) and attention to motion (blue line). Red and blue shadings indicate the SE for the color and motion conditions, respectively.

Figure 7.

Trial average of the estimated current time courses within each feature-sensitive area for subject YS and MS. The onset times of the cue (0 ms) and the stimulus (1000 ms) are demarcated by vertical broken lines. Current is shown for attention to color (red line) and attention to motion (blue line). Red and blue shadings indicate the SE for the color and motion conditions, respectively.

### Temporal Characteristics of the Effect of Feature Attention within Color- and Motion-Sensitive Areas during the Cue Period

Figure 8 shows the difference of current amplitudes calculated by subtraction of the amplitude in the motion condition from that in the color condition for each ROI and averaged across subjects. Consistent with our previous statistics, the differences tended to be positive for color-sensitive areas and negative for motion-sensitive areas. We divided the estimated current amplitudes during the cue period into bins of 500 ms and compared time-averaged amplitudes between conditions in each bin. No significant difference was observed in each bin or each ROI between subjects (Wilcoxon signed rank test, P > 0.062). Similarly, when the length of the time bin was 250 ms, no significant difference was observed (P > 0.094).

Figure 8.

Mean differences of current amplitudes in the left hemisphere (left) and right hemisphere (right) across subjects, as calculated by subtraction of the amplitude of the motion condition from that in the color condition. Red and blue lines show the time course for color- and motion-sensitive areas, respectively.

Figure 8.

Mean differences of current amplitudes in the left hemisphere (left) and right hemisphere (right) across subjects, as calculated by subtraction of the amplitude of the motion condition from that in the color condition. Red and blue lines show the time course for color- and motion-sensitive areas, respectively.

Figure 9A shows the amplitude difference between 2 conditions for each ROI and each subject. Each colored broken line corresponds to 1 subject, and each square marker corresponds to the amplitude difference averaged over 100 ms, calculated by subtraction of the amplitude in the motion condition from that in the color condition. Because magnitude of the difference varied across subjects, for the purpose of data visualization, we normalized them by dividing the amplitude difference by the absolute value of the maximum difference for each subject. As shown in Figure 9A, the time-points when differences were observed during the cue period varied among subjects. In addition, the amplitude difference was rather transient, not sustained, during the cue period in each subject.

Figure 9.

(A) Normalized difference of the estimated current amplitudes plotted for each subject within each feature-sensitive area. Each colored broken line corresponds to 1 subject, and each square marker corresponds to the amplitude difference between 2 conditions averaged over 100 ms. (B) Histogram of a continuity index for all feature-sensitive areas and all subjects. See text for a definition of the continuity index.

Figure 9.

(A) Normalized difference of the estimated current amplitudes plotted for each subject within each feature-sensitive area. Each colored broken line corresponds to 1 subject, and each square marker corresponds to the amplitude difference between 2 conditions averaged over 100 ms. (B) Histogram of a continuity index for all feature-sensitive areas and all subjects. See text for a definition of the continuity index.

To show this characteristic quantitatively, we examined individual subject data and quantified how long the conditional difference lasted during the cue period. Figure 9B shows a histogram representing the continuity index for all feature-sensitive areas and all subjects. The continuity index was calculated for each feature-sensitive area and for each subject as follows: 1) calculate significance levels of the amplitude difference between the 2 conditions in each 100-ms bin, 2) count how many bins successively showed significant differences (t-test, P < 0.05, corrected for multiple comparisons with 10 of 100-ms bins) between conditions (note that only positive differences were accounted for color-sensitive areas and only negative differences were accounted for motion-sensitive areas), and 3) the maximum number of such successive bins was regarded as a continuity index for the area and for a subject. As shown in Figure 9B, the distribution of the continuity index is skewed, and the index less than 3 contains 75% of the total cases, indicating that significant amplitude differences typically lasted 100–300 ms during the cue period.

## Discussion

In the present study, we investigated the effects of feature attention on neural activity before stimulus onset, using MEG in combination with fMRI. First, we compared the signal amplitudes of the recorded MEG data between the 2 conditions at the channel level. We found that, in several channels, the signal amplitudes were significantly different between the 2 conditions. However, we did not find a common tendency in most subjects. Next, cortical currents were estimated in the order of millimeters and milliseconds in each cortical area using the hierarchical Bayesian method. We found that spatiotemporal patterns were generally similar between the 2 conditions (Fig. 4). Then, we examined the effects of feature attention for each ROI and found that no significant difference between conditions was observed in frontal, parietal, and lower visual areas. By contrast, the current amplitude within the color-sensitive area was larger when subjects directed their attention to color than when they directed their attention to motion. On the other hand, the current amplitude within the motion-sensitive area was greater when subjects directed their attention to motion than when they directed their attention to color. Furthermore, within each feature-sensitive area, and for the averaged data across subjects, we did not find any significant differences between the 2 conditions when we divided current amplitudes during the cue period into several time bins (500 or 250 ms) and compared them in each bin. This, at first, seems puzzling and in contrast with the significant differences (Fig. 6) obtained from the averaged data across subjects during the cue period. Plots of the differences between the 2 conditions for each subject (Fig. 9A) explain the seemingly contradictory statistical results for time-averaged data (significant) versus short time bin analyses (nonsignificant). That is, for each individual, the feature-sensitive area activity was generally greater when a subject attended to that feature, but the time for the difference differed from subject to subject. In addition, the difference between the 2 conditions was transient, not sustained, during the cue period (Fig. 9B). Thus, when the average is taken for both time and subjects, we obtained significant results, but the difference within each time bin did not reach significance because of individual fluctuations in the difference time.

The effect prior to the stimulus onset was observed while white stationary dots were presented at the center of a display without any changes in the visual presentation. Therefore, this effect may reflect top-down attention control processes to sensory processing areas. Regarding research on nonhuman primates, no study has measured single-unit activity from different feature-sensitive areas simultaneously and examined the effects of feature attention within each area. To date, only 1 study has reported activity changes in the visual cortex induced by motion attention without a stimulus in the context of working memory (Ferrera et al. 1994). Recently, some fMRI studies (Chawla, Ress and Friston 1999; Wylie et al. 2006) have reported that attention to a specific feature selectively modulates the activity in the corresponding feature-sensitive areas, even before stimulus onset. However, the BOLD signal does not have sufficient resolution to investigate the temporal nature of the attentional effect that may change in the order of a few hundred milliseconds (Muller et al. 1998; van Velzen and Eimer 2003; Slagter et al. 2005; Yamagishi et al. 2005). Although EEG and MEG have high temporal resolution, it has been difficult to examine the effect of feature attention within particular cortical areas by using these techniques because of the ill-posed inverse problem. In this study, we overcame these difficulties using the MEG source estimation in combination with the fMRI data (Sato et al. 2004). However, there might be some worries about how much fMRI predefined area information affects current estimation results, even though it is soft constrained. To answer this question, we conducted the current estimation without the fMRI data and compared the results with and without fMRI data (see Supplementary Results for details). The current estimation results without fMRI data showed very similar pattern of activities to the results with fMRI data. The resultant spatial patterns of activities were similar (compare Supplementary Fig. 2A,B), and the amplitude comparisons showed the same pattern of results with and without fMRI data (compare Supplementary Fig. 3A,B). This supports the reliability of the hierarchical Bayesian method that we used in this study.

Figure 6 shows that attention to color and motion selectively enhanced the current amplitude within color- and motion-sensitive visual areas within a 1-s prestimulus period. Previous neuroimaging studies of spatial attention have suggested that the effect of spatial attention starts several hundred milliseconds after subjects attended to 1 visual field within the corresponding visual areas (Muller et al. 1998; Worden et al. 2000; Slagter et al. 2005; Yamagishi et al. 2005; Thut et al. 2006). Our results suggest that, like spatial attention, the effect of feature attention starts within several hundred milliseconds after subjects direct their attention to a particular feature within the corresponding feature-sensitive visual areas.

However, the temporal nature of the effect of feature attention might be different from that of spatial attention. When we compared current amplitudes throughout the cue period, time-averaged amplitudes were significantly different between conditions. By contrast, there was no significant difference when we divided the estimated current amplitudes into shorter time bins and compared them. Moreover, the time-points in which conditional differences were observed during the cue period varied between subjects (Fig. 9A). The amplitude differences between conditions lasted mostly for a short period time (100–300 ms, Fig. 9B). To clarify that this result was not led by a particular estimation window size, we calculated the continuity index of the estimated current for 3 different time windows: time window 1 (200 ms in length, 100-ms overlap), time window 2 (100 ms in length, 50-ms overlap), and time window 3 (50 ms in length, 25-ms overlap) (see Supplementary Results for details). Resultant histograms of the continuity index show similar pattern of continuities and peaked at 2 (corresponding to 200 ms) for all time windows (A, B, and C in Supplementary Fig. 4). This suggests that, therefore, temporal characteristics of the estimated current we obtained were not particularly led by the time window we used.

It is unclear why an observed amplitude modification occurred at a different time for each subject even though we asked subjects to maintain their attention to color or motion during the entire cue period, with randomly interleaved catch trials (an ISI was 500 ms). Our results suggest that the effect of feature attention before the stimulus onset is transient rather than sustained, at least, for the frequency components that are higher than 1 Hz because of a 1- to 100-Hz band-pass filter we used. By contrast, previous EEG and MEG studies have suggested that the effect of spatial attention starts from several hundred milliseconds after subjects attend to 1 visual field and persists until the stimulus onset (Worden et al. 2000; Yamagishi et al. 2005; Thut et al. 2006). Previous psychophysical (Kanai et al. 2006) and physiological (McAdams and Maunsell 2000; Hayden and Gallant 2005) studies indicated that spatial and feature attentions are based on distinct neural substrates. Taken together, although both spatial and feature attention modulate neural activity within specific visual areas before the stimulus onset, mechanisms underlying these attentional effects may differ.

There was no significant amplitude difference between the conditions within the lower visual areas as well as the parietal and frontal areas. Does this mean that the difference of the effect of feature attention is represented within the corresponding feature-sensitive areas only? According to previous studies, this is not necessarily the correct view. Watanabe et al. (1998) reported that the effect of attention to motion within V1 is task dependent. They found that attention to expanding motion enhanced the activity in MT/MST, whereas attention to translating motion enhanced the activity not only in MT/MST but also in V1. We used expanding or contracting motion in our study. Additionally, the effect of attention can be represented by a spatial pattern of cortical activity. Kamitani and Tong (2006) showed that the pattern of activity in lower visual areas can predict whether subjects attend to clockwise or counterclockwise motion. Although we cannot conclude that the difference in feature attention is represented only at the corresponding feature-sensitive visual areas, at least, we can assert that feature-sensitive visual areas are central to the effect of feature attention.

One might argue that the difference of current amplitudes between the conditions during the cue period is not derived from a difference in top-down processes, but from the differences in bottom-up visual information, as the visual cue preceding the stimulus presentation was different in each condition (color condition: C and motion condition: M). However, the difference in cue shapes is not sufficient to explain the differences in current amplitude for the cue period as it does not seem plausible that color-sensitive areas and motion-sensitive areas selectively respond to C and M, respectively. Rather, top-down processes, which shift subjects’ attention to a particular feature, are important factors that help to explain the differences between the conditions during the cue period.

Consistent with previous fMRI (Chawla et al. 1999) and EEG/MEG (Schoenfeld et al. 2007) studies, the amplitude of the stimulus-evoked response within each ROI was significantly different between the 2 conditions. Becuase the visual stimulus was identical under both conditions, this difference should also reflect the effects of feature attention. As there was no significant difference in peak latency within feature-sensitive visual areas, the effects of feature attention to a sensory input may be represented by an increase or a decrease in the neural firing rate or the degree of synchronization between neurons, rather than the latency of neural firing or neural synchronization. These effects of feature attention may help to facilitate our visual processing by improving information transmission to subsequent processing in the visual cortex (Fries et al. 2001; Salinas and Sejnowski 2001) or decoding accuracy of the stimulus in higher cognitive brain functions (Nakahara and Amari 2002).

How do attentional effects before and after the stimulus onset relate to each other? One hypothesis is that the effect of feature attention before the stimulus onset acts as a preparatory activity to enhance the stimulus-evoked signal within feature-sensitive areas (Driver and Frith 2000). Some computational (Chawla, Lumer and Friston 1999, 2000; Borgers et al. 2005) and physiological studies (Cossart et al. 2003; McCormick et al. 2003; Shu et al. 2003) have proposed that background activity before sensory inputs contribute to the modulation of stimulus-locked transients. Previous fMRI studies of both spatial and feature attention have reported changes in the BOLD signal within each feature-sensitive or retinotopic area without a visual stimulus (Chawla et al. 1999; Kastner et al. 1999; Hopfinger et al. 2000; Wylie et al. 2006). These effects may reflect a change in background neural activity. However, the difference between spatial and feature attention has remained largely unclear as the BOLD signal reflects time-integrated neural activity within the local neural population. According to our data, it is possible that feature and spatial attention before the stimulus onset contribute to the modulation of the stimulus-evoked signal via different neural mechanisms.

## Supplementary Material

Supplementary Figs 14, Methods, Results, and Movie can be found at: http://www.cercor.oxfordjournals.org/.

## Funding

This research was supported in part by The National Institute of Information and Communications Technology (NICT-KARC).

We thank T. Tomita, S. Kajihara, and Y. Furukawa for their support with the MEG recordings and K. Amano and K. Toyama for their helpful comments for our study. Conflict of Interest: None declared.

## References

Bartels
A
Zeki
S
The architecture of the colour centre in the human visual brain: new results and a review
Eur J Neurosci
,
2000
, vol.
12
(pg.
172
-
193
)
Borgers
C
Epstein
S
Kopell
NJ
Background gamma rhythmicity and attention in cortical local circuits: a computational study
,
2005
, vol.
102
(pg.
7002
-
7007
)
Chawla
D
Lumer
ED
Friston
KJ
The relationship between synchronization among neuronal populations and their mean activity levels
Neural Comput.
,
1999
, vol.
11
(pg.
1389
-
1411
)
Chawla
D
Lumer
ED
Friston
KJ
Relating macroscopic measures of brain activity to fast, dynamic neuronal interactions
Neural Comput.
,
2000
, vol.
12
(pg.
2805
-
2821
)
Chawla
D
Rees
G
Friston
KJ
The physiological basis of attentional modulation in extrastriate visual areas
Nat Neurosci
,
1999
, vol.
2
(pg.
671
-
676
)
Corbetta
M
Shulman
GL
Control of goal-directed and stimulus-driven attention in the brain
Nat Rev Neurosci
,
2002
, vol.
3
(pg.
201
-
215
)
Cossart
R
Aronov
D
Yuste
R
Attractor dynamics of network UP states in the neocortex
Nature
,
2003
, vol.
423
(pg.
283
-
288
)
Dale
AM
Liu
AK
Fischl
BR
Buckner
RL
Belliveau
JW
Lewine
JD
Halgren
E
Dynamic statistical parametric mapping: combining fMRI and MEG for high-resolution imaging of cortical activity
Neuron
,
2000
, vol.
26
(pg.
55
-
67
)
Driver
J
Frith
C
Shifting baselines in attention research
Nat Rev Neurosci
,
2000
, vol.
1
(pg.
147
-
148
)
Eickhoff
SB
Stephan
KE
Mohlberg
H
Grefkes
C
Fink
GR
Amunts
K
Zilles
K
A new SPM toolbox for combining probabilistic cytoarchitectonic maps and functional imaging data
Neuroimage
,
2005
, vol.
25
(pg.
1325
-
1335
)
Ferrera
VP
Rudolph
KK
Maunsell
JH
Responses of neurons in the parietal and temporal visual pathways during a motion task
J Neurosci
,
1994
, vol.
14
(pg.
6171
-
6186
)
Fries
P
Reynolds
JH
Rorie
AE
Desimone
R
Modulation of oscillatory neuronal synchronization by selective visual attention
Science
,
2001
, vol.
291
(pg.
1560
-
1563
)
Friston
KJ
Penny
W
Phillips
C
Kiebel
S
Hinton
G
Ashburner
J
Classical and Bayesian inference in neuroimaging: theory
Neuroimage
,
2002
, vol.
16
(pg.
465
-
483
)
Giesbrecht
B
Woldorff
MG
Song
AW
Mangun
GR
Neural mechanisms of top-down control during spatial and feature attention
Neuroimage
,
2003
, vol.
19
(pg.
496
-
512
)
Grill-Spector
K
Malach
R
The human visual cortex
Annu Rev Neurosci
,
2004
, vol.
27
(pg.
649
-
677
)
Hayden
BY
Gallant
JL
Time course of attention reveals different mechanisms for spatial and feature-based attention in area V4
Neuron
,
2005
, vol.
47
(pg.
637
-
643
)
Hillyard
SA
Anllo-Vento
L
Event-related brain potentials in the study of visual selective attention
,
1998
, vol.
95
(pg.
781
-
787
)
Hopfinger
JB
Buonocore
MH
Mangun
GR
The neural mechanisms of top-down attentional control
Nat Neurosci
,
2000
, vol.
3
(pg.
284
-
291
)
Kamitani
Y
Tong
F
Decoding seen and attended motion directions from activity in the human visual cortex
Curr Biol.
,
2006
, vol.
16
(pg.
1096
-
1102
)
Kanai
R
Tsuchiya
N
Verstraten
FA
The scope and limits of top-down attention in unconscious visual processing
Curr Biol.
,
2006
, vol.
16
(pg.
2332
-
2336
)
Kastner
S
Pinsk
MA
De Weerd
P
Desimone
R
Ungerleider
LG
Increased activity in human visual cortex during directed attention in the absence of visual stimulation
Neuron
,
1999
, vol.
22
(pg.
751
-
761
)
Kuriki
I
K
Takeda
T
Nenonen
J
Ilmoniemi
RJ
Katila
T
MEG response of human V4 to chromatic visual stimulus
Proceedings of the 12th International Conference on Biomagnetism

Espoo (Finland)
Helsinki University of Technology
(pg.
177
-
180
)
Kuriki
I
K
Takeda
T
MEG recording from the human ventro-occipital cortex in response to isoluminant color stimulation
Vis Neurosci
,
2005
, vol.
22
(pg.
283
-
293
)
Liu
T
Slotnick
SD
Serences
JT
Yantis
S
Cortical mechanisms of feature-based attentional control
Cereb Cortex
,
2003
, vol.
13
(pg.
1334
-
1343
)
Luck
SJ
Chelazzi
L
Hillyard
SA
Desimone
R
Neural mechanisms of spatial selective attention in areas V1, V2, and V4 of macaque visual cortex
J Neurophysiol
,
1997
, vol.
77
(pg.
24
-
42
)
Luck
SJ
Woodman
GF
Vogel
EK
Event-related potential studies of attention
Trends Cogn Sci.
,
2000
, vol.
4
(pg.
432
-
440
)
Maunsell
JH
Treue
S
Feature-based attention in visual cortex
Trends Neurosci
,
2006
, vol.
29
(pg.
317
-
322
)
CJ
Maunsell
JH
Attention to both space and feature modulates neuronal responses in macaque area V4
J Neurophysiol
,
2000
, vol.
83
(pg.
1751
-
1755
)
McCormick
DA
Shu
Y
Hasenstaub
A
Sanchez-Vives
M
M
Bal
T
Persistent cortical activity: mechanisms of generation and effects on neuronal excitability
Cereb Cortex
,
2003
, vol.
13
(pg.
1219
-
1231
)
Motter
BC
Neural correlates of attentive selection for color or luminance in extrastriate area V4
J Neurosci
,
1994
, vol.
14
(pg.
2178
-
2189
)
Muller
MM
Teder-Salejarvi
W
Hillyard
SA
The time course of cortical facilitation during cued shifts of spatial attention
Nat Neurosci
,
1998
, vol.
1
(pg.
631
-
634
)
Nakahara
H
Amari
SI
Attention modulation of neural tuning through peak and base rate in correlated firing
Neural Netw
,
2002
, vol.
15
(pg.
41
-
55
)
Salinas
E
Sejnowski
TJ
Correlated neuronal activity and the flow of neural information
Nat Rev Neurosci
,
2001
, vol.
2
(pg.
539
-
550
)
Sato
MA
Yoshioka
T
Kajihara
S
Toyama
K
Goda
N
Doya
K
Kawato
M
Hierarchical Bayesian estimation for MEG inverse problem
Neuroimage
,
2004
, vol.
23
(pg.
806
-
826
)
Schoenfeld
M
Hopf
JM
Martinez
A
Mai
H
Sattler
C
Gasde
A
Heinze
HJ
Hillyard
S
Spatio-temporal analysis of feature-based attention
Cereb Cortex
,
2007
, vol.
17
(pg.
2468
-
2477
)
Schoenfeld
MA
Tempelmann
C
Martinez
A
Hopf
JM
Sattler
C
Heinze
HJ
Hillyard
SA
Dynamics of feature binding during object-selective attention
,
2003
, vol.
100
(pg.
11806
-
11811
)
Serences
JT
Liu
T
Yantis
S
Itti
L
Rees
G
Tsotsos
J
Parietal mechanisms of attentional control: locations, features, and objects
Neurobiology of attention
,
2005
San Diego (CA)
(pg.
35
-
41
)
Shu
Y
Hasenstaub
A
McCormick
DA
Turning on and off recurrent balanced cortical activity
Nature
,
2003
, vol.
423
(pg.
288
-
293
)
Shulman
GL
Tansy
AP
M
Petersen
SE
McAvoy
MP
Corbetta
M
Reactivation of networks involved in preparatory states
Cereb Cortex
,
2002
, vol.
12
(pg.
590
-
600
)
Silver
MA
Ress
D
Heeger
DJ
Neural correlates of sustained spatial attention in human early visual cortex
J Neurophysiol
,
2007
, vol.
97
(pg.
229
-
237
)
Slagter
HA
Kok
A
Mol
N
Kenemans
JL
Spatio-temporal dynamics of top-down control: directing attention to location and/or color as revealed by ERPs and source modeling
Brain Res Cogn Brain Res.
,
2005
, vol.
22
(pg.
333
-
348
)
Thut
G
Nietzel
A
Brandt
SA
Pascual-Leone
A
Alpha-band electroencephalographic activity over occipital cortex indexes visuospatial attention bias and predicts visual target detection
J Neurosci
,
2006
, vol.
26
(pg.
9494
-
9502
)
Tootell
RB
Reppas
JB
Kwong
KK
Malach
R
Born
RT
TJ
Rosen
BR
Belliveau
JW
Functional analysis of human MT and related visual cortical areas using magnetic resonance imaging
J Neurosci
,
1995
, vol.
15
(pg.
3215
-
3230
)
Toyama
K
Yoshikawa
K
Yoshida
Y
Kondo
Y
Tomita
S
Takanashi
Y
Ejima
Y
Yoshizawa
S
A new method for magnetoencephalography: a three-dimensional magnetometer-spatial filter system
Neuroscience
,
1999
, vol.
91
(pg.
405
-
415
)
Treue
S
Martinez Trujillo
JC
Feature-based attention influences motion processing gain in macaque visual cortex
Nature
,
1999
, vol.
399
(pg.
575
-
579
)
van Velzen
J
Eimer
M
Early posterior ERP components do not reflect the control of attentional shifts toward expected peripheral events
Psychophysiology
,
2003
, vol.
40
(pg.
827
-
831
)
Watanabe
T
Harner
AM
Miyauchi
S
Sasaki
Y
Nielsen
M
Palomo
D
Mukai
I
Task-dependent influences of attention on the activation of human primary visual cortex
,
1998
, vol.
95
(pg.
11489
-
11492
)
Worden
MS
Foxe
JJ
Wang
N
Simpson
GV
Anticipatory biasing of visuospatial attention indexed by retinotopically specific alpha-band electroencephalography increases over occipital cortex
J Neurosci
,
2000
, vol.
20
(pg.
1
-
6
)
Wylie
GR
Javitt
DC
Foxe
JJ
Jumping the gun: is effective preparation contingent upon anticipatory activation in task-relevant neural circuitry?
Cereb Cortex
,
2006
, vol.
16
(pg.
394
-
404
)
Yamagishi
N
Goda
N
Callan
DE
Anderson
SJ
Kawato
M
Attentional shifts towards an expected visual target alter the level of alpha-band oscillatory activity in the human calcarine cortex
Brain Res Cogn Brain Res.
,
2005
, vol.
25
(pg.
799
-
809
)
Zeki
S
Watson
JD
Lueck
CJ
Friston
KJ
Kennard
C
Frackowiak
RS
A direct demonstration of functional specialization in human visual cortex
J Neurosci
,
1991
, vol.
11
(pg.
641
-
649
)


text:
# Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become identical right?

• Could this just be a rounding issue? I ran your code and it looks like the largest error was on the order of 10^-5 percent. – Atul Ingle Oct 10 '17 at 16:28
• Not really because usually round up errors are on the order of $10^{-16}$ and exhibit strong irregularities. – Ronan Tarik Drevon Oct 10 '17 at 16:37
• I'm not sure I agree that rounding errors are always on the order of 10^-16. – Atul Ingle Oct 10 '17 at 16:39
• Maybe but the curve is very smooth as well which is not typical of round of errors – Ronan Tarik Drevon Oct 10 '17 at 16:54
• can you please precisely state the error that you consider? And what was your expectation ? Since there are a number of error sources it's important to fix the one you want to discuss. – Fat32 Oct 10 '17 at 17:10

First consider the theoretical DTFT of $x[n]$ $$X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j \omega n} = \sum_{n=0}^{\infty} a^n e^{-j \omega n} = \frac{1}{1 - a e^{-j \omega} }$$

Then consider the N-point DFT of the truncated signal: $$X[k] = \sum_{n=0}^{N-1} x[n] e^{-j \frac{2\pi}{N} n k} = \sum_{n=0}^{N-1} a^n e^{-j \frac{2\pi}{N} n k} = \frac{1 - a^N}{1 - a e^{-j \omega} }$$

A simple inspection of those DTFT abd DFT yields the following conclusion: $$\boxed{ X[k] = (1-a^N) X(e^{j\omega_k}) }$$

Where the DTFT is evaluated at the DFT frequency of $w_k = \frac{2\pi}{N}k$.

And the error between $X[k]$ and $X(e^{j \omega_k})$ is: $$E[k] = X(e^{j \omega_k}) - X[k] = a^N X(e^{j \omega_k})$$

As can be seen, the error due to the finite length DFT truncation is multiplied by the constant $a^N$ and the frequency dependent variable $X(e^{j \omega_k})$ for each DFT frequency of $\omega_k$. It's obvious from this that the error plot will have a shape of the actual DTFT $X(w)$ scaled by $a^N$.

In order to make this error smaller than the floating point round-off errors (such as $10^{-16}$ for 64 bit IEEE floating point numbers) you have to make sure that for all frequencies $\omega_k$ for $0 \leq k \leq N-1$ you should satisfy $E[k] < 10^{-16}$ that is:$$E[k]=a^N X(e^{j \omega_k}) < 10^{-16}$$

Note that this error is frequency dependent due to $X(e^{j \omega_k})$ term: the higher the DTFT at that frequency, the higher will be the error and vice versa. So for a randomly fluctuating DTFT $X(e^{j \omega_k})$ the error will also randomly fluctuate.

In the analysis of the Gaussian signal $x[n] = e^{-\alpha n^2} = a^{n^2}$ where $0 < a = e^{-\alpha} < 1$ , an analytic expression for the error will be difficult to simplify as you have to evaluate the following sum:

$$X_g(e^{j\omega}) = \sum_{n=-\infty}^{\infty} a^{n^2} e^{-j \omega n} = \beta a^{\gamma w^2}$$ where $\beta$ and $\gamma$ should be determined.

And the N-point DFT of the truncated (and shifted by $N/2$) Gaussian signal: $$X_g[k] = \sum_{n=0}^{N-1} a^{(n-N/2)^2} e^{-j \frac{2\pi}{N} n k} = ?$$

Unfortunately since the series sum may not have a closed form expression (or it will be quite complicated) therefore a simple formulation of the error does not exist as were the case for the simpler signal $x[n] = a^n u[n]$.

EDIT: The following exposition shows the same results as attained above however from a different perspective which may be less intuitive to follow. I suggest the above perspective for a clear understanding and the below perspective a complementary case.

Consider the following infinite length signal $x[n] = a^n u[n]$, with $|a|<1$, where the discrete-time Fourier tranform (DTFT) $X(e^{j\omega})$ of $x[n]$ is:

$$X(e^{j\omega}) = \text{DTFT} \{x[n] \} = \sum_{n=0}^{\infty} a^n e^{-j \omega n} = \frac{1}{1-a e^{-j\omega} }$$

In order to practically compute the DTFT $X(e^{j\omega})$ using a digital computer, the relative transform, DFT (discrete Fourier transform) is used. For finite length signals it's well known that DFT $X[k]$ will be samples of DTFT $X(e^{j \omega})$. But for infinite length signals (such as $x[n]=a^n u[n]$) the computed N-point DFT (which should practically use a finite length truncated version of the infinite length signal) will be an approximation to the true theoretical DTFT. The question is ; what's the relation between the true DTFT spectrum $X(e^{j\omega})$ and the computed DFT samples $X[k]$ ?

Now it's obvious that for infinite length signals the values of DFT $X[k]$ will not be the samples of the function $X(e^{j\omega})$. For example given $x[n]=a^nu[n]$ and $X(e^{j \omega}) = \frac{1}{1-a e^{-j\omega}}$ then we see that $$X[k] \neq X(e^{j\frac{2 \pi}{N}k}) = \frac{1}{1-a e^{-j\frac{2 \pi}{N}k}}$$

First lets consider the following, assume you have artificially obtained $N$ uniform samples from the function $X(e^{j\omega})=\frac{1}{1-a e^{-j\frac{2 \pi}{N}k}}$ as
$$X_f[k] = \frac{1}{1-a e^{-j\frac{2 \pi}{N}k}}$$ where $X_f[k]$ stands for the DFT samples of a finite length signal $\tilde{y}[n]$ whose N-point DFT would be $X_f[k]$. Therefore we can assert that the inverse DFT of the $X_f[k]$ would yield the N-point discrete-time sequence $\tilde{y}[n]$ as:

$$\tilde{y}[n] = IDFT\{ X_f[k] \} = \frac{1}{N} \sum_{k=0}^{N-1} \frac{1}{1-a e^{-j\frac{2 \pi}{N}k}} e^{ j \frac{2\pi}{N} k n}$$

It can be shown from the periodicity assumption of the IDFT that the result is also equivalent to: $$\tilde{y}[n] = IDFT\{ X_f[k] \} = \sum_{r=-\infty}^{\infty} x[n-rN]$$

Where the signal $x[n]$ is nothing but the theoretical infinite length inverse of the DTFT $X(e^{j\omega}) = \frac{1}{1-ae^{-j\omega}}$ which is as we know $$x[n] = IDTFT \{\frac{1}{1-ae^{-j\omega}} \} = \frac{1}{2\pi} \int_{-\pi}^{\pi} \frac{1}{1-a e^{-j\omega}} e^{j\omega n} d\omega = a^n u[n]$$

Therefore we have the following:

$$\tilde{y}[n] = \sum_{r=-\infty}^{\infty} x[n-rN] = \sum_{r=-\infty}^{\infty} a^{n-rN}u[n-rN]$$

It will be interesting to consider the value of $\tilde{y}[n]$ for the range $0 \leq n \leq N-1$ for which we have:

$$\tilde{y}[n] = a^n \sum_{r=-\infty}^{\infty} a^{-rN}u[n-rN] = a^n \sum_{r=-\infty}^{0} a^{-rN} = a^n \sum_{r=0}^{\infty} (a^N)^r = a^n \frac{1}{1-a^N}$$

Which shows the relation between finite length signal $\tilde{y }[n]$ and the infinite length signal $x[n] = a^n u[n]$ as: $$\tilde{y}[n] = \frac{x[n]}{1-a^N}$$ where $x[n] = a^n u[n]$.

One can see that the modification involved on the computed value of finite length approximation to the infinite length signal $x[n]$ is a constant weight of $K = \frac{1}{1-a^N}$ for the signal $x[n]= a^n u[n]$

Now using the linearity properties of DFT one can deduce the following: Given the pair of N-point sequence $\tilde{y}[n]$ and its N-point DFT $X_f[k]$ which is the uniform samples of the true theoretical DTFT $X(e^{j\omega})$ associated with infinite length signal $x[n]=a^n u[n]$ we have the following

$$\tilde{y}[n] \leftrightarrow X_f[k]$$ $$\tilde{y}[n] = K x[n] \leftrightarrow X_f[k] \Rightarrow x[n] \leftrightarrow \frac{1}{K} X_f[k] = (1-a^N) X_f[k]$$

This last statment is concluded as follows: given an N-point finite length truncation $x_N[n]$ of the infinite length signal $x[n] = a^n u[n]$ for the range $0 \leq n< N$, then the N-point DFT of the N-point sequence $x_N[n]$ will be $(1-a^N)X[k]$ where $X[k]$ are the samples of the true DTFT spectrum $X(e^{j\omega}) = \frac{1}{1 -a e^{-j \omega}}$ associated with the infinite length signal $x[n]=a^n u[n]$. So the error associated with the DFT computation is the fixed weight $1-a^N$ per sample of the DFT.

You should perform a similar analysis for the Gaussian signal $x[n] = \beta e^{-\alpha n^2}$ to find the error associated per computed DFT sample.

• That's looks rigorous indeed but could you just say that the computed DFT being the finite sum of geometrical series is then $\Sigma_1^N (a\Omega)^n=\frac{1- (a\Omega)^N}{1-a\Omega}$ where $\Omega=e^{-j\omega}$? So you confirm that the difference DTFT-DFT should rapidly get below round-off errors. – Ronan Tarik Drevon Oct 10 '17 at 23:57
• Th edifference $X(e^{jw}) - X[k] = X(e^{jw}) - (1-a^N)X(e^{jw}) = a^N X(e^{jw})$ So the difference (error) depends on 1-) the factor $a^N$, 2-) the value of $X(e^{jw})$ at the frequency of investigation for the signal of $x[n] = a^n u[n]$. So for a roundoff error you look for $a^N X(e^{jw}) = 10^{-16}$ at a given frequency. – Fat32 Oct 11 '17 at 8:18
• you can upvote the answer too if you found it useful. – Fat32 Oct 11 '17 at 8:21
• That was useful indeed, although I realised that line "df = 1/((N-1)dt) " in the code should be replaced by "df = 1/(N*dt)" so the error decrease uniformly with increasing sampling length N. – Ronan Tarik Drevon Oct 11 '17 at 10:38
• I'll put a few extra lines and discuss the error pattern in the answer asap. – Fat32 Oct 11 '17 at 12:04

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## Let's settle this once and for all /co/

No.105151600
Will it flop YES or NO


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# Tag Info

1

But what about the 180° gap in the middle? This implies a negative real part of the impedance, i.e., a negative resistance. Express the impedance $Z$ in polar form $$Z = R + jX = |Z|e^{j\phi}$$ where $$|Z| = \sqrt{R^2 + X^2}$$ and $$\tan \phi = \frac{X}{R}$$ For $R > 0$ $$-90^{\circ} \lt \phi \lt 90^{\circ}$$ but for $R < 0$ $$90^{\circ} ... 0 If the phase angle is greater then 90 degrees the load would effectively become the source and vice-versa. 1 In 1999, the president of the IEEE Power Engineering Society, Robert Dent, noted that: "The degree or intensity of the corona discharge and the resulting audible noise are affected by the condition of the air--that is, by humidity, air density, wind and water in the form of rain, drizzle and fog. Water increases the conductivity of the air and so ... 2 I think EnergyNumbers's Answer is an excellent one, but could leave some people a bit mystified by what exactly a "direct ray" is and what exactly its relevance is. The essential idea here is that a Fresnel lens is an imaging machine: it puts a curvature on a low aberration wavefront so that that wavefront converges. Its working depends on there being ... 1 I think the range of answers you have got from friend, teacher and web reflect that there is not a straightforward response. Without high voltage it would not be possible to drive the dangerous current through the body, but high voltage itself is not lethal - it depends how much current can be delivered at high voltage. Another question is how high does the ... 0 They are not generally used with the photovoltaic cells as they concentrate the sun light and heat to a temperature so high that many of the cells get, for lack of a better word, fried. 1 Some corrosion always takes place (pure gold is not used for transmission lines, AFAIK:-) ), so the conductivity decreases with time, although for some materials this effect can be very small ... 2 The line itself does not change much over years. What changes and therefore needs maintenance on power transmission lines is insulators, connectors and spacers. Insulators get dirty or simply break, connectors work loose due to thermal expansion and contraction, mechanical stresses and oxidation, and spacers can be damaged by wear due to these same ... 0 "Stange" harmonics cannot appear. Two 50Hz sinus waves always produce another 50Hz sinus wave :$$cos(\omega t)+cos(\omega t+\phi)=2cos(\omega t+\frac{\phi}{2})cos(\frac{\phi}{2}) $$About synchronization, you should remember that the voltage in the network is fixed, and thus only the current changes. I don't see any problem with combining two different ... 1 Cores of the transformers a made of ferromagnetic material, which can change the shape due to magnetostriction phenomena. When transformer works with typical grid frequency of 50Hz..60Hz, it can be heard. As it was answered above, generated noise depends on mechanical construction. New power supplies use frequency converters, which drive the transformers ... 6 Transformers generate oscillating magnetic fields at the mains frequency and the fields produce an oscillating force on: anything nearby that's ferromagnetic (like the core) anything nearby that is carrying a current (like the windings) The sound you hear is because various bits of the transformers are moving in response to the oscillating fields and ... 0 Current is basically the flow of charges.Well what i think so whatever charge it is it will flow and will cause the production of current and if there is a production of current electricity is automatically formed.For example the current positron generator.Here is a link where there is a PDF which is all about Dirac current generator ... 4 You are being imprecise about electricity. It's probably better to just think of electricity as current. You have a current whenever you have a charge moving. To your question, yes, positrons are just as good as electrons for carrying a charge. There is no difference between "positron electricity" and "electron electricity". Another way to see this is ... 1 The exact equations for I-V characteristics of transistors are derived using quantum-mechanics. Several approximations can be used, one of which is based on the shottky barrier analysis This reference here derives the I-V linear and quadratic approximation (in saturation) for FET transistors. Another reference here UPDATE: As @QMechanic pointed, ... 0 Situation 1., the ideal wire with an ideal voltage source, is an idealization: it is not a situation that can occur in nature. One should not be surprised that physics, which aims to describe nature, cannot describe a situation that cannot occur in nature. I will note that you specified a battery, not an ideal voltage source. In this case one can develop ... 0 The drift velocity does not depend on the length or the cross sectional area of the wire, when dealing with a macroscopic (ordinary, everyday life) wire. However, if the wire is, say, too short, e.g. comparable to the average distance a charge carrier travels before undergoing a collision, then it might begin to depend on the wire length, but for all ... 0 For simplicity, I'm going to assume a 0.7V diode drop in the ON state. In your diagram, you can call the low node ground (0V), in that case, you have one of two situations: The diode is on (closed switch) In this state, current can flow through the resistor and therefore, through the diode. In this case, 0.7V is dropped across the diode and the remaining ... 4 You are familiar with the concept of static electricity. When you have a DC line with many thousands of volts on it, there will be a polarizing effect, and that will attract things to it - especially dielectrics. Now if your piece of paper is thus attracted to the line, and you try to grab it, it is quite possible that you get peripheral nerve stimulation ... 0 The first expression refers to definition of polarization in terms of density of physical electric dipoles. The second expression refers to (approximate) physical law, which says the polarization is proportional to total electric strength. 0 In the first equation \Delta L itself is typically proportional to E. That establishes the connection between the two formulas. However, \Delta L might not be proportional to E. It might be permanent. The first equation allows for this case, but the second does not. If \Delta L is permanent, the material has a permanent electric dipole ... 0 Electrons indeed move extremely slow.However it is not the really the electrons' motion for which the light goes on in a bulb as you switch it.Just as you find water coming out of a pipe as you open the tap (the water flows;the water at the mouth of the pipe comes out as you open the tap.Similarly the electrons push other electrons and so on and this ... 2 The first mistake is directly taking dq=\lambda dy. If you look closely, as you go up from the origin to higher y, you find that charge density per unit y increases. Take instead \lambda as charge per unit circumference of the wire. i.e \lambda={dq \over dl} = \lambda={dq \over rd\theta} Now,$$dE= {k\lambda dl \over r^2} \cos{\theta} ...

0

Individual electrons may have a range of speeds in a circuit (thermal motion, scattering, absorption, photon etc..) However the current (or drift velocity) gives the average speed of the whole electron cloud (not a single electron). Note again single electrons may have a range of speeds (from slow to very fast, near $c$). It is the electron cloud that ...

1

The speed of an individual electron cannot be c: an electron is a massive particle and can therefore never achieve the speed of light Furthermore, an individual electron moves very slowly in a current (mm/h scale). The question i think you're trying to ask is how fast do electric signals travel through a wire. (e.g. if I had two lightyears of copper wire ...

1

You've begun with this: $P_l = I^2R$ $P_l = IV$ This is correct, but the $V$ here is not the line voltage, but instead the voltage drop across the resistor under consideration. Increasing the line voltage does not increase the voltage drop. Your diagram with a single resistance and a power station implies that the current in the line depends on that ...

0

The diode has a central barrier at the PN junction which allows charge to flow across it only when a sufficiently positive voltage is applied to the P part with respect to the N part so that the barrier potential is overcome.In this case the diode is said to be forward biassed and conducts current.At all times when the P part has a voltage less than the ...

0

From the information you have put in your question I think that in the reverse bias case the 'diode is open' means that it behaves like an 'open circuit' or 'open switch' and no current passes through it and thus no current goes through the battery. The load resistor has current passing through it that goes to circuit/components not shown on the right hand ...

1

The pendulum motion can be converted to energy any number of ways. For example, if the pendulum bob is a magnet simply placing a coil of wire near it as it swings will induce a current which can be siphoned off and used.

0

Look, for example the page at http://farside.ph.utexas.edu/teaching/302l/lectures/node57.html . It seems that the EMF E is a characteristic of the cell. So it should be constant, whatever hapens that is non-destructive. Regarding the second question, both formulae are correct, and both refer to the same voltage V between the points A and B in the figure of ...

0

I'll try to keep it short and sweet; Stranded wire is capable of delivering high amperage without overheating because the strands devide the load..I.E. battery cables on your car. stranded wire is superior to solid but to expensive for long runs, so solid wire is used for long runs like for your house (easy to snake or bend) solid but flexible electric ...

1

But recall that power dissipated $P= VI$ is also , from Ohm's law, expressible as $P = I^2 R$ So the dependency of power dissipated is linear in voltage, but quadratic in current, given the same resistance. Also remember that the voltage supplied from the power station, and the voltage drop across the transmission line - which is what is important in ...

0

What you're likely seeing are the leafs returning to their natural, uncharged state. They are not attracting each other. Their close proximity is due in part to gravity, as you say, but also the particular shape of the supporting apparatus.

2

In addition to the good answers above here is something to think about. If the resistance is zero then for a current to flow there does not need to be a battery - the emf can be zero. No work is done as the current flows. This may sound like a strange case, but it is how many strong magnets work in nmr (nuclear magnetic resonance) spectrometers. The have a ...

6

Your original text admitted three interpretations, and I'm leaving the answers here: 1: What happens with a toy model when there's a circuit with an ideal battery and no resistance? All the charge moves around the circuit at one moment in time (infinite current). The energy must leave the system as Electromagnetic radiation - accelerating charges radiate, ...

2

Yes, there is a fundamental reason why electricity is so universal. It is because matter is made of electric charges bound together (protons and electrons). When you think of non-electric technologies such as the wheel, realize that the wheel relies on the rigidity of matter which depends on the bonds between atoms which are electric in nature. So even ...

1

You should be able to get the same result by using the electrical work formula - but note that you need to integrate since the force changes with position. That's really all the potential is - it is the integral of force for unit charge. That's why force has the $1/r^2$ relationship while potential has $1/r$ (with appropriate signs and constants...). ...

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Do not open the cone. Think of it in the profile view : You have an isosceles triangle. Now move along the axis of the cone, say a distance $x$ and take an element $\mathrm{d}x$. Somewhat like this : This small element is similar to the rectangle you described. With length as $2\pi r(x)$ and width $\mathrm{d}x$. You also know the velocity with which the ...

0

Others have provided mathematical explanations, so I'll try an intuitive explanation. You've probably seen the electric field represented like one of these: (image source) At the top of the mountain there is a positive charge, and at the bottom of the valley is a negative charge. A positive test charge in this field experiences a force analogous to a ...

3

As you said, current is like water flow similarly voltage is like water level and voltage difference like difference of water level. We know that water flow from higher level to lower level like current flow from higher voltage to lower voltage. Voltage difference means there is a difference of charge i.e. difference in number of electrons. Now we consider a ...

5

Suppose your pipes form a loop i.e. water can flow through the pipes and get back to where it started. As the water flows round the loop there will be some places where pressure rises (e.g. a pump = battery) and some places where pressure falls (e.g. a restriction = resistor). However if the water goes all the way round the loop back to its starting point ...

2

The Kirchoff Voltage law states that the sum of emfs in a circuit is equal to the total potential drop in the circuit. So for a simple example, where you a 6V cell, for example, and 2 resistors in series. The 6V cell can be seen as a place where the water is given potential energy - if we imagine a ramp, it would be the water being pumped up to the top of ...

1

AC can be made from DC, and vice versa. DC is a steady voltage, with AC the voltage fluctuates from negative to positive and back, many times per second. Most electricity generators generate AC. The reason for preferring AC is that, historically, it is easier change voltage with AC: all you need is a transformer. The generators generate a high voltage, ...

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Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Optimal Relay Network for Aerial Remote Inspections

Version 1 : Received: 30 November 2021 / Approved: 2 December 2021 / Online: 2 December 2021 (12:49:03 CET)

How to cite: Pinto, L.R.; Almeida, L. Optimal Relay Network for Aerial Remote Inspections. Preprints 2021, 2021120037 (doi: 10.20944/preprints202112.0037.v1). Pinto, L.R.; Almeida, L. Optimal Relay Network for Aerial Remote Inspections. Preprints 2021, 2021120037 (doi: 10.20944/preprints202112.0037.v1).

## Abstract

Unmanned Aerial Vehicles (UAVs) in particular multirotors are becoming the {\it de facto} tool for aerial sensing and remote inspection. In large industrial facilities, a UAV can transmit an online video stream to inspect difficult to access structures, such chimneys, deposits and towers. However, the communication range is limited, constraining the UAV operation range. This limitation can be overcome with relaying UAVs placed between the source UAV and the control station, creating a line of communication links. In this work we assume the use of a digital data packet network technology, namely WiFi, and tackle the problem of defining the exact placement for the relaying UAVs that creates an end-to-end channel with maximal delivery of data packets. We consider asymmetric communication links and we show an increase as large as $15$\% in end-to-end packet delivery ratio when compared to an equidistant placement. We also discuss the deployment of such a network and propose a fully distributed method that converges to the global optimal relay positions taking, on average, 1.4 the time taken by a centralized method.

## Keywords

Multi-hop network; , packet delivery ratio; relay network; TDMA; throughput; UAV; wireless networks

## Subject

ENGINEERING, Electrical & Electronic Engineering

We encourage comments and feedback from a broad range of readers. See criteria for comments and our diversity statement.

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6:46 AM
The tag also has tag-excerpt and tag-wiki.
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Is the set $(e_n)_{n>0}$ a (vector space) basis for the sequence Hilbert space $l^2$? It is a Hilbert space basis anyway. I would say no, because the sequence $\left(\frac{1}{n}\right)_{n>0}$ is in $l^2$ but it can't be written as a finite linear combination of $e_i$'s. Is that right?

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Let $V$ be a vector space with infinite dimensions. A Hamel basis for $V$ is an ordered set of linearly independent vectors $\{ v_i \ | \ i \in I\}$ such that any $v \in V$ can be expressed as a finite linear combination of the $v_i$'s; so $\{ v_i \ | \ i \in I\}$ spans $V$ algebraically: this is...

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I seem to have tripped on the common Hamel/Schauder confusion. If $X$ is any vector space (not necessarily finite dimension) and $B$ is a linearly independent subset that spans $X$, then $B$ is a Hamel basis for $X$. If there exists a sequence $(e_n)$ such that for every $x \in X$ there exists ...

4

Alright, so I'm reading a book on Hilbert spaces and functional analysis, and here it defines a "Hamel basis" to be a "maximal linearly independent set". I take this to mean that $S$ is a Hamel basis if $S$ is linearly independent, and there is no linearly independent set $T$ for which $|T|>|S|$....

There were also these two new tags: and .


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PDF

Given a probability density function find the mean, median and mode of the distribution.

Scale Invariance

By exploring the concept of scale invariance, find the probability that a random piece of real data begins with a 1.

Into the Exponential Distribution

Get into the exponential distribution through an exploration of its pdf.

Circle pdf

Why do this problem?

This problem gives an opportunity to explore the properties of pdfs using the mathematics of sectors of circles.

Possible approach

There are two main parts to this problem.

The first is to understand why certain shapes are firstly valid pdfs and secondly how they satisfy the technical requirement of the question. This would benefit from a discussion approach.

The second part, calculating the maximum value, will lead students into the mathematics of sectors and segments of circles.

Key questions

What properties must a pdf have?
How would the requirements of the questions relate to a graph?
In order to obtain the maximum possible value for the case of the circle of radius $1$, what do we know about the arc?

Possible support

Point students in the direction of the formula for the segment of a circle and that the area must be 1.


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# Math Help - 1-D Inequality Help

1. ## 1-D Inequality Help

Solve: 6x^4 - 10x^2 - 12 > 10x^3 + 38x
where x is a member of the real numbers.

I rearranged it as an equality equaling zero, as:
6x^4 - 10x^3 - 10x^2 - 38x - 12 = 0

I then factored out a 2, giving:
(2)(3x^4 - 5x^3 - 5x^2 -19x - 6) = 0

Using factor theorum and synthetic division, I determined that:
(2)(3)(x + 1/3)(x - 3)(x^2 + x + 2) = 0

However, the quadratic equation above does not yield any real solutions (unless I'm doing it wrong). Where do I go from here?

2. "Solve: $6x^4 - 10x^2 - 12 > 10x^3 + 36x$" but you wrote: $6x^4 - 10x^3 - 10x^2 - 38x - 12$. It's 36, not 38.

Are you sure you wrote the inequality correct?
x=2: 96 - 40 - 12 > 80+72 is false.

3. It's 38, not 36. I wrote it incorrectly the first time. The bold line at the top is what was given to me on the assignment sheet.

4. Originally Posted by Kangles
Solve: 6x^4 - 10x^2 - 12 > 10x^3 + 36x
where x is a member of the real numbers.

I rearranged it as an equality equaling zero, as:
6x^4 - 10x^3 - 10x^2 - 38x - 12 = 0

I then factored out a 2, giving:
(2)(3x^4 - 5x^3 - 5x^2 -19x - 6) = 0

Using factor theorum and synthetic division, I determined that:
(2)(3)(x + 1/3)(x - 3)(x^2 + x + 2) = 0

However, the quadratic equation above does not yield any real solutions (unless I'm doing it wrong). Where do I go from here?
Hi Kangles,

In your first line, you have 36x, but when you "rearranged" it, it changed to 38x.

Assuming you meant 38x, and your factorization is correct, you have two real roots showing.

If (x + 1/3) is a factor (and it is!), then x = -1/3 is a root.

If (x - 3) is a factor (and it is, too!), then x = 3 is a root.

Ignore the two complex roots. Your beginning statement said x is a member of the real numbers.

Plot these two real values on a number line and check the intervals that will make your original inequality true.

I found the solution to be $\{x|x < -\frac{1}{3} \r\: x > 3\}" alt="\{x|x < -\frac{1}{3} \r\: x > 3\}" />