# Significant changes in EEG neural oscillations during different phases of
three-dimensional multiple object tracking task (3D-MOT) imply different roles
for attention and working memory.

Yannick Roy, Jocelyn Faubert

###### Abstract

Our ability to track multiple objects in a dynamic environment enables us to
perform everyday tasks such as driving, playing team sports, and walking in a
crowded mall. Despite more than three decades of literature on multiple object
tracking (MOT) tasks, the underlying and intertwined neural mechanisms remain
poorly understood. Here we looked at the electroencephalography (EEG) neural
correlates and their changes across the three phases of a 3D-MOT task, namely
identification, tracking and recall. We recorded the EEG activity of 24
participants while they were performing a 3D-MOT task with either 1, 2 or 3
targets where some trials were lateralized and some were not. We observed what
seems to be a handoff between focused attention and working memory processes
when going from tracking to recall. Our findings revealed a strong inhibition
in delta and theta frequencies from the frontal region during tracking,
followed by a strong (re)activation of these same frequencies during recall.
Our results also showed contralateral delay activity (CDA) for the lateralized
trials, in both the identification and recall phases but not during tracking.
Keywords: Multiple-Object Tracking, MOT, CDA, EEG, Working Memory, Attention

Authors Information

Address: Faubert Lab, Université de Montréal, H3T 1P1, Québec, Canada
Corresponding author<EMAIL_ADDRESS>
Data and Code available: https://github.com/royyannick/3DMOT_EEG

##  1 Significance Statement

Multiple Object Tracking (MOT) is heavily studied both in human cognition and
computer vision. While MOT tasks have been used extensively in research and
even commercially as a cognitive training tool, our understanding of the roles
and relationship between attention and working memory remains vague. There is
no clear consensus yet as how separated or intertwined these two mechanisms
really are. Here, by looking at the brain activity both in the time domain and
frequency domain, we show a clear cognitive transition between the different
phases of a 3D-MOT task, showing some sort of a hand-off between attention and
working memory.

##  2 Introduction

Our ability to track multiple moving objects simultaneously in a dynamic
environment enables us to perform everyday tasks such as driving, playing team
sports, and walking in a crowded mall. In such tasks, it is required to manage
internal representations of relevant information in order to predict future
spatial positions of surrounding objects and optimize decision making
accordingly. A professional athlete being able to make a perfect pass to a
teammate in a high speed sport while avoiding players from the other team, is
a great example of the brain’s remarkable ability to track multiple objects
both in space and time. In order to study this ability in a laboratory
setting, researchers often use a variant of the multiple-object tracking (MOT)
task developed by Pylyshyn & Storm in 1988 (Pylyshyn and Storm, (1988)). In
typical MOT tasks we find two categories of visual objects: targets (objects
of interest) and distractors (objects to ignore) both sharing identical visual
properties. In order to modulate task difficulty, parameters usually include
the number of targets (Pylyshyn, (1989)), speed (Chen et al., (2013);
Iordanescu et al., (2009)), and distance between objects (Alvarez and
Franconeri, (2007); Shim et al., (2008)).

Multiple-object tracking is an active area of research in humans but also in
computer vision as we are observing an increasing demand for technology for
automated tracking of vehicles and people in various contexts (Milan et al.,
(2016); Dendorfer et al., (2020); Park et al., (2021)). Humans’ visual system
has inspired current neural network architectures driving most of the
artificial intelligence (AI) field as we know it today (Rajalingham et al.,
(2018)) and recent neural network architectures are trying to take advantage
of higher brain mechanisms such as attention (Luong et al., (2015)).
Therefore, better understanding the underlying mechanisms of the brain’s
ability to track multiple objects in time and space could also benefit AI-
related fields.

Despite the increasing literature on multiple-object tracking and evidence of
effective training showing transfer on real tasks in real environments
(Faubert and Sidebottom, (2012)), our understanding of the underlying neural
mechanisms remains fuzzy. Several studies have shown that our capacity to keep
individual items in working memory is limited to 3 or 4 (Cowan, (2001); Vogel
and Machizawa, (2004)) and over the last few decades, several cognitive models
have been brought forward trying to explain how individual items are being
encoded and deciphering the intertwined roles of attention and working memory
in such tasks. Older proposals suggested that early individuation of objects
(up to 3-4) does not require attention mechanisms (Trick and Pylyshyn,
(1993)), however, recent research indicates otherwise suggesting that
simultaneous indexing of items relies on attention mechanisms (Cavanagh,
(2011)) before involving subsequent mechanisms such as visual WM to encode the
individuated objects in greater details. Another proposal include multifocal
spatial attention, where attention can be split and work in parallel for
multiple targets (Cavanagh and Alvarez, (2005)). On the other hand, Oksama and
Hyöna suggested that rather than having a fixed-capacity parallel mechanism,
the tracking performance would be better explained by a serial model where the
maintenance of moving objects requires continuous serial refreshing of
identity-location bindings (Oksama and Hyönä, (2016)). Moreover, a previous
behavioural study also showed that one cognitive strategy is to process the
targets as one illusory object by mentally creating connections between the
targets to make, for example, a geometrical shape (Yantis, (1992)).

The confounding aspects of attention versus working memory and location-based
versus object-based tracking still remains fuzzy and recent EEG studies were
performed trying to provide more insights by disentangling them. For example,
Drew and colleagues tried to delineate the neural signatures of tracking
spatial position and working memory during attentive tracking. They found that
there was a unique contralateral negativity related to the process of
monitoring target position during tracking which was absent when objects
briefly stopped moving. These results suggest that the process of tracking
target locations elicits an electrophysiological response that is distinct and
dissociable from neural responses of the number of targets being attended
(Drew et al., (2011)). Also, Merkel and colleagues looked at the spectral
properties of the electrophysiological signal, mainly in the gamma range,
during tracking to find a difference between location-based and object-based
maintenance of visual information (Merkel et al., (2022)). Their results
suggest that object-based tracking is supported by enhanced encoding during
the initial presentation of the targets and location-based tracking is
characterized by the sustained maintenance of the individual targets during
the entire tracking period, in that same processing neural network. In a
previous study Merkel and colleagues also showed that neural networks involved
in both tracking processes (object-based and location-based) are at least
partly overlapping (Merkel et al., (2015)).

The experiment described in this manuscript is part of a larger study where we
seek to develop a passive brain-computer interface composed of an EEG closed-
loop system for a cognitively demanding task. We hypothesize that by looking
at different neural correlates related to attention and working memory as well
as other EEG signals in real-time while the user is performing such a task, we
can identify different cognitive states and adapt the task in real-time to
enhance the overall experience and outcome of the task. As part of the larger
study, all subjects participated to three different tasks: MOT task, N-Back
task, and flight simulator task. In this publication we share only our
findings related to cognitive processes during the MOT task, and we don’t
cover the other tasks nor the BCI system which will all be discussed in depth
in another publication.

Most MOT studies are conducted using a 2D-MOT task on a computer screen.
However, we live in a 3D environment and different cognitive processes might
be at play in a 3D environment. NeuroTrackerTM is a commercially available
3D-MOT task currently used by a multitude of users in many countries around
the world as a perceptual-cognitive training and assessment tool. It has been
used and studied in various fields such as sport (Faubert and Sidebottom,
(2012); Faubert, (2013); Mangine et al., (2014); Romeas et al., (2016)),
ESports (Benoit et al., (2020)), education (Tullo et al., (2018)), aviation
(Hoke et al., (2017)) and military (Vartanian et al., (2016)). 3D-MOT training
has been demonstrated to enhance attention, working memory and visual
information processing speed (Parsons et al., (2016)). Given the wide adoption
of the NeuroTrackerTM and existing literature showing effective transfer, we
opted for a modified version of the NeuroTrackerTM for our study.

According to existing literature and the nature of the task, different neural
correlates linked to working memory, attention, workload and visual processes
should be at play during 3D-MOT. In the time domain, event-related potentials
(ERPs) should be observed at the beginning of each phase of the task given the
sudden change in visual information displayed. For lateralized trials, where
the targets are displayed and moving only in one hemifield, we expect to see
contralateral delay activity (CDA). The CDA is a sustained negativity over the
hemisphere contralateral to the positions of the items to be remembered. As
mentioned before, the CDA has been shown to be linked with the number of items
held in WM (Unsworth et al., (2015); Luria et al., (2016); Roy and Faubert,
(2022)) and previous studies have shown a correlation between CDA amplitude
and the number of targets during a 2D-MOT task (Drew and Vogel, (2008); Drew
et al., (2011)). We hypothesise that during the indexing phase the participant
would load the targets in working memory, hold the targets in working memory
during tracking and then access them during recall to provide the answer. In
the frequency domain, we are expecting different changes in frequency bands
linked to working memory (e.g. Theta), attention (e.g. Alpha), workload (e.g.
Beta), and visual processes (e.g. Gamma) during the different phases of the
task.

##  3 Materials and Methods

###  3.1 Participants

Twenty-four participants (thirteen females) aged between 21 and 41 years of
age (M=29.3, SD=4.9) took part in this study. The participants were healthy
university students of various ethnicity from different universities in
Montreal. All participant self-reported normal or corrected-to-normal visual
acuity and passed the Randot Stereotest for stereo vision. The study was
reviewed and approved by the Université de Montréal ethics committee for
health research (Comité d’éthique de la recherche en santé; CERES #2018-334).
Recommended ethics procedures and guidelines were followed, and informed
consent was obtained from all participants. The three hour long session
included the 3D-MOT task discussed here but also included the recording of a
N-Back task and both tasks were part of a larger research project where the
subjects also participated in a second session, on a different day, for a
flight simulator task. All subjects received a monetary compensation for their
participation to the two sessions covering also their transportation to the
different facilities. The other components of the larger research project
(i.e. N-Back and flight simulator) are not discussed in this manuscript and
will be published separately.

###  3.2 Task

Figure 1 shows the five different phases of the 3D-MOT task developed with the
Unity engine. (A) presentation phase where 8 yellow spheres are shown in a 3D
volume space for 2 seconds, (B) indexing phase where one, two or three spheres
(targets) change colour (to red) and are highlighted (halo) for 2 seconds, (C)
tracking (or movement) phase where the targets indexed in phase 2 return to
their original colour (yellow) and 1 second later start moving for 8 seconds
crisscrossing and bouncing off of each other and the virtual 3D volume cube
walls, (D) recall phase where the spheres stop moving and the observer is
prompted to identify the spheres originally indexed in phase 2. Each sphere is
labelled with a number between 1 to 8. After identifying the targets, the
observer is asked to provide a confidence level for each answer (either 0%,
25%, 75% or 100% confident). And finally, (E) feedback phase where the correct
targets are clearly identified on the screen. The whole trial takes around 15s
(2s + 2s + 9s + [1-4]s) depending on how long the participant takes to provide
the answers. A video of the task is available online on the repository
provided.

Figure 1: 3D-MOT Task Sequence. (A) All spheres appear on screen. (B) Targets
are highlighted in red for 2 seconds. (3) All the spheres are moving for 8
seconds. (D) Participant must identify the targets and provide a confidence
level. (E) Feedback is provided to the participant showing the correct
answers.

The participant was seated 1.75m and centered from the 65” 3D TV screen
(Panasonic TC-P65VT60) wearing the EEG cap as well as active shutter 3D
glasses (Panasonic TY-ER3D5MA). The 3D-MOT virtual cube was 35 degrees of
visual angle (dva) in size and the spheres were 2.5 dva. The 3D glasses were
carefully slid under the EEG cap in a way to minimally disrupt the EEG signal.
Additional gel was added to some temporal electrodes for some participants to
compensate for the gap created by the 3D glass legs. The participants had a
keyboard on their lap to provide their confidence level after giving the
answers orally to the instructor, seated 2m behind the participant with a
keyboard.

Each participant started with a training block of 20 trials, without the EEG
equipment, to familiarize themselves with the task and for us to obtain an
individualised speed threshold. They were instructed to keep their eyes on the
green fixation dot in the middle of the screen and to let their covert
attention track the targets. An adaptive staircase algorithm modulated the
speed of each new trial based on previous performances to find the individual
speed where the participant gets a performance of 50% (i.e. they get half the
trials right and half wrong) (Faubert and Sidebottom, (2012)). To get a good
trial the participant needs to identify correctly all the targets. For the
training block, only the condition with 3 targets was used. After the 20 trial
training block, the resulting speed was used for the remaining of the
experiment.

The experiment consisted of 4 blocks of 21 trials with 2 conditions: side and
set size. The speed was kept constant, based on the speed obtained at the end
of the training block. In total, 30 trials were presented in the left
hemifield, 10 for each set size (1,2,3), 30 trials were presented in the right
hemifield, 10 for each set size (1,2,3), and 24 trials were not lateralized
and the targets could freely cross from left to right and vice-versa, 8 for
each set size (1,2,3). It is important to note that for lateralized trials
only the targets stayed on one specific hemifield while the other distractors
where moving freely with no restriction. Once the targets stopped moving, a
number between 1 and 8 appeared on each of the spheres and the participants
had to provide their answer by saying the number of the target(s) out loud for
the instructor to enter the answers. After the answers were entered, a visual
cue on top of each of the selected spheres appeared for the participant to
provide a confidence level for each target. The participant used the arrow on
the keyboard to provide the confidence level. Up arrow means 100% confident
(the instruction provided to the participant: ”I was able to track it.”). Down
arrow means 0% (”I lost it, it’s a random guess.”). Right arrow means 75%
confident (”I’m somewhat confident. I think I tracked it but I might have
switch target during an overlap/occlusion.”). Left arrow means 25% confident
(”It’s mostly a guess. I got confused.”). The rational behind this discrete
scoring of confidence was to clearly dissociate between a random guess and a
fully confident answer (0% vs 100%). For the in-between, we decided to avoid
having only one additional option as we felt like most participants might
default to that option whether they are quite confident but do have a small
doubt or if they are mostly guessing it, making this option hard to use for
further analysis. Adding more granularity than four options for the confidence
level would have only added a cognitive load with no additional benefit.
Therefore the compromise with four discrete options was chosen. For the
analysis presented in this manuscript, we regrouped 75% and 100% as being a
confident answer and therefore a good trial is a trial where the participant
identified all the correct targets and indicated a confidence level of 75% or
100%, otherwise it is labeled as a bad trial.

###  3.3 EEG Acquisition

The electroencephalogram (EEG), electrocardiogram (ECG), and electrooculogram
(EOG) were recorded using the Biosemi ActiveTwo system (Biosemi, Amsterdam,
Netherlands) with 71 Ag–Ag/Cl electrodes positioned at 64 standard
International 10/20 System sites (EEG), left and right mastoids for offline
EEG re-reference, 1cm lateral to the external canthi for horizontal EOG
(HEOG), left and right ribs plus right collarbone for ECG. The HEOG was used
for eye movements to confirm that the participant was tracking the targets
with covert and not overt attention (i.e. not moving their eyes).
Electrophysiological signals were digitized at 2048Hz.

###  3.4 EEG Analysis

EEG offline analysis was performed in MNE-Python (Gramfort et al., (2013)) an
open-source Python package for neurophysiological data. Both our EEG data and
code are available online. Here we will breakdown the analysis in two parts,
first in the time domain and second, in the frequency domain to look at the
oscillation variations over time and power spectrum across conditions. Four
conditions were examined across different brain regions: side (left, right),
set size (1, 2, 3), performance (good, bad), 3D-MOT phase (indexing, tracking,
recall). Note that because of low amount of trials with bad performances, we
looked at the data internally but do not draw any conclusion in this paper
about performance. It is important to note that after looking at our results
we realized that we unfortunately had a 150ms offset in our Unity code between
the trigger (i.e. EEG event) and the color change (from yellow to red and red
back to yellow) appearing on screen. The offset is fixed and due to a Unity
fixed update delay we forgot to remove for the color change. This offset does
not apply for when the movement stops and ends. What it means is that the
events at t=0s, and at t=2s, are in fact visible on screen only at t=0.15s and
t=2.15s.

Preprocessing. First, the EEG channels were re-referenced to the left and
right mastoids. Second, independent component analysis (ICA) was used to
remove eye blinks and eye movement artifacts. Third, the EEG data was epoched
in [-1, 15]s windows where t=0s represents the stimuli/trigger of the spheres
being highlighted in red. Fourth, AutoReject (Jas et al., (2017)) was used to
automatically remove bad trials and correct bad channels. After looking at the
clean EEG data and separately analyzing EOG channels for eye movements during
tracking to see if people were really tracking with covert attention instead
of overt attention, 4 subjects were removed and 20 subjects remained for the
analysis.

Time domain. First, the non-lateralized occipital grand average signal was
obtained over O1, O2 and Oz to confirm visual ERPs. Since the 3D-MOT is a
visual task with sudden visual changes between the different phases, a
visual/occipital ERP should be observed accordingly. Second, before looking at
the CDA we seeked to confirm that the brain activity was lateralized for left
vs right trials where targets were presented and moved only in one hemifield.
For the lateralized activity, we looked at different clusters of electrodes,
namely frontal, central, parietal, temporal and occipital. For readability the
electrodes used in each cluster are listed in Table 1 in supplementary
material. For the frontal cluster, all channels with the letter ’F’ were
included. For the central cluster, all the electrodes with the letter ’C’ were
included. For the parietal cluster, all the channels with the letter ’P’ were
included. For the temporal cluster, all the channels with the letter ’T’ were
included. For the occipital cluster, all the channels with the letter ’O’ were
included. For lateralized activity of both left and right trials, we averaged
the left channels (i.e. channels with odd numbers) and subtracted the average
of the right channels resulting in the difference between the two hemispheres.
The rationale behind this analysis is to confirm that indeed we see more
activity in one hemisphere than the other for lateralized trials. The gross
activity observed here should encompass specific neural signatures such as
ERPs and CDA, which are then analyzed in greater detail.

Third, we looked at the effect of the number of targets on the CDA by
averaging of the channels contralateral to the hemifield where the targets
were presented minus the average of the ipsilateral channels. The midline
channels (ending with the letter z) were not included in the CDA nor the
lateralized analysis. The CDA, as defined in the literature, should be
strongest in the parietal region. However, we also calculated the same
activity (i.e. contra minus ipsi) for the different clusters mentioned above.
For both lateralized activity and CDA we used only the trials with a good
performance (i.e. the participant identified all the targets correctly with
high level of confidence) for a total of N=982 trials, an average of 49 trials
per subject. A three-way repeated measures analysis of variance (ANOVA) was
performed with the number of targets (1, 2, and 3), phases (id, tacking, and
recall), and clusters (frontal, central, parietal, temporal, and occipital) as
independent variables and the mean CDA amplitude over a 1s time window as the
dependent variable. For the identification (id) phase, the time window was
from 0.5s to 1.5s, for the tracking we selected the 5s to 6s and for the
recall, we used 11.5s to 12.5s.

Frequency domain. To obtain a more detailed representation of neural
oscillation changes over time, Event-Related Spectrum Perturbation (ERSP)
graphs were used. The time-frequency decomposition was computed using Morlet
wavelets for frequencies between 1 and 50Hz with varying cycles of half the
frequency. The ERSP maps were then obtained by getting the log ratio of the
power relative to the baseline power. For the time-frequency analysis, the
baseline was selected as -1s to 0s prior to the targets being colored in red
(t=0.15s). Instead of using raw power, the log ratio has the advantage of
normalizing the power across participants. To investigate the potential role
of different brain regions we used the midline channels from frontal to
occipital to give a representation in space of the time-frequency activity.
For this analysis we included only the trials with a good performance. With
the same time-frequency decomposition, we analyzed a 1s window of average
power as a dependent variable using a repeated measure three-way ANOVA, with
Phase (Identification [0.3, 1.3]s, Tracking [5, 6]s, and Recall [11.5,
12.5]s), Set Size (1, 2, and 3), and Frequency Band (delta [1-3]Hz, theta
[4-7]Hz, alpha [8-12]Hz, beta [13-30]Hz, gamma [31-50]Hz) as independent
variables. The time windows across the phases were selected to capture mostly
top-down cognitive processes.

##  4 Results

###  4.1 Time Domain

Non-lateralized occipital activity in O1, O2 and Oz (see Figure 2) confirmed
the visual ERPs when drastic visual changes occured, such as the target
spheres changing color from yellow to red at t=0.15s, then reverting back to
yellow at t=2.15s, start moving at t=3s and stop moving at t=11s. Noteworthy,
the occipital ERP of the initial color change from yellow to red is
significantly stronger than when the targets revert back to yellow (t-test on
200ms window; p $<$ 0.001) and the strongest ERP occurs when the targets start
moving. On the other hand, the ERP elicited when spheres stop moving isn’t as
sharp as the other ones, most likely due to the time it takes for individuals
to realize that the spheres have indeed stopped moving. Moreover, we can also
see a sustained cognitive activity following the ERPs at t=0.15s and t=11s,
respectively during identification of targets and recall.

xw

Figure 2: Non-lateralized occipital ERPs (channels: O1, O2, Oz). Grand average
over all participants for all conditions and performances.

As expected, we obtained a clear lateralization of activity in left vs right
trials during both identification and recall phases as shown on Figure 3.
However, we did not observe lateralized activity during tracking despite the
targets staying in only one hemifield for the full duration of the trial.
Zooming in on the identification and recall phases, we observe that the same
brain regions (clusters) are activated the strongest, namely the frontal and
temporal regions, during both identification and recall (see Figure 4). This
could be indicative of similar cognitive functions during identification and
recall phases. Note that automated scales were used to preserve a clear shape
of the signal, however the amplitude, or strength of the signal, varies across
clusters.

Figure 3: Lateralized activity in the frontal region for whole sequence. Left
vs right trials.

Figure 4: Lateralized activity from the different brain regions for
identification (first column) and recall (second column) phases. Left vs right
trials.

Figure 5 shows the CDA amplitude for the frontal cluster for the whole
duration of the trial and on Figure 6 we zoom in on the identification and
recall phases across EEG channel clusters. The contralateral delay activity
observed here has a similar shape as what we see in the literature, peaking
around 500-600ms post-stimuli and slowly decaying for another $\sim$500ms but
we expected the strongest CDA in posterior parietal regions as opposed to
frontal regions (Vogel and Machizawa, (2004); Unsworth et al., (2015)). The
initial three-way repeated measures ANOVA with the number of targets (1, 2,
and 3), phases (id, tacking, and recall), and clusters (frontal, central,
parietal, temporal, occipital) as independent variables and the mean CDA
amplitude as the dependent variable, revealed a significant effect for the
number of targets (F(2,34) = 3.59, p = .039), a strong significant effect for
the phase (F(2,34) = 20.17, p $<$ .0001), a strong significant effect for the
cluster (F(4,68) = 32.85, p $<$ .0001), a near significant effect for the
interaction between the targets and phases (F(4,68) = 2.46, p $=$ .053), a non
significant effect for the interaction between targets and clusters (F(8,136)
= 1.56, p = .14), a strong significant effect for the interaction between
phases and clusters (F(8,136) = 18.3, p $<$ .0001), and a trending but non
significant effect for the interaction between the three (F(16,272) = 1.6, p =
.067). Post hoc one-way ANOVAs looking only at the number of the targets,
independently for each phase and cluster, revealed no significant effect on
the mean CDA amplitude during the identification phase nor during tracking, in
any of the clusters. During recall, however, the effect was significant in the
parietal cluster (F(2,34) = 8.15, p $<$ .005), the temporal cluster (F(2,34) =
7.76, p $<$ .005), the central cluster (F(2,34) = 5.59, p $<$ .01), the
occipital cluster (F(2,34) = 6.93, p $<$ .005) and trending but non
significant in the frontal cluster (F(2,34) = 2.6, p $=$ .088). Note that the
post hoc ANOVA values are provided as is and were not corrected for multiple
comparisons. A total of 15 post hoc ANOVAs were performed (5 clusters x 3
phases). After obtaining and writing the results, We also performed the same
analysis with a 0.5s window to see if it would change the results but it
yielded similar results.

Figure 5: CDA in the frontal region for the full sequence. Trials with 1, 2
and 3 targets.

Figure 6: Lateralized activity from the different brain regions for
identification (first column) and recall (second column) phases. Trials with
1, 2 and 3 targets.

###  4.2 Frequency Domain

The ERSP for the whole trial is shown on Figure 7 which includes 1 second
prior to the targets being identified in red and up to 15s post
identification, with frequencies ranging from 1 to 50Hz. It represents the
grand average across all participants, all conditions, and all channels.
Before breaking down the time-frequency analysis to look at specific elements,
we can easily distinguish the different phases of the MOT task with drastic
changes in neural oscillations across these phases. It is important to note
the range of the color scale of the ERSP maps presented in it section as in
EEG studies blue usually means a desynchronization (i.e. negative value) and
red a synchronization (i.e. positive value). Here, we purposefully used a none
symmetrical color scale to accentuate the differences observed given that most
of the values are negative.

Figure 7: ERSP: Grand Average (GA) across all subjects, all conditions and all
channels. The color represents the log ratio of the power at each instant with
average power of the baseline [-1,0]s for that same frequency, averaged across
subjects.

On the spectral map (Figure 7) we can identify, as expected, a perturbation
near t=0.15s when the spheres turn red, creating a narrow perturbation
corresponding to the ERP in the time domain related to the sudden visual
change. Also visible are perturbations from the ERP near t=2.15s when they
turn back yellow and from the ERP near t=3s when they suddenly start moving.
During tracking, we observe a strong inhibition of lower frequencies until
alpha band ($\sim$8Hz) where the inhibition isn’t as strong, then the
inhibition seems to continue in the $\sim$14-18Hz range after which there is a
change in inhibition intensity for high beta and gamma. More strikingly, we
observe a clear cognitive switch as soon as the spheres stop moving and the
participant is asked to answer (at t=11s). Suddenly, the lower frequencies are
re-activated and alpha slightly reduced. After t=13s, the values can’t really
be interpreted given the variability in speed to answer from trial to trial,
across participants and the number of targets.

Looking at the spatial distribution over the midline channels on Figure 8, we
observe that during tracking (t=[3-11]s), there is a strong delta and theta
inhibition in the frontal regions. During recall (t=[11-13]s), we observe a
sudden re-activation of these lower frequencies in the delta and theta bands
from frontal to occipital regions. During recall, we also observe a strong
activation of high-beta to low-gamma in the occipital region.

To better understand the effect across participants, both the mean and the
median values were calculated for each time-frequency points and what we
observe on Figure 8 is that the mean is not being distorted by outliers and
that the distribution across trials and participants is somewhat symmetrical.
A paired t-test was done for each time-frequency point comparing with the mean
power of the baseline and the time-frequency points with p $>$= .05 were
grayed-out on Figure 9. For readability we show the statistical analysis of
only three channels (Fpz, Cz, Oz) as the six others were showing similar
overall trends. We did not perform any multiple comparison correction, so
given that we have performed tens of thousands of comparisons (i.e. for each
time-frequency point independently) we are exposed to familywise type 1 error,
meaning that the graphs presented here with the statistical masks are most
likely showing more significant effects then there really are. Performing
multiple comparison corrections like false discovery rate (FDR) on such a high
amount of comparisons would obviously result in the opposite and hide all
effects (type 2 error). Different approaches have been suggested to deal with
ERSP significance such as cluster-level statistical permutation tests,
however, given the mean and median of the log ratio with the baseline being
both similar and biologically sound in time and frequency, correcting for the
comparisons would most likely not invalidate the general trends we are
observing here. Also noteworthy, as we can see on the ERSP graphs, the edge
between alpha and theta bands around 8Hz and the edge between alpha and beta
bands around 12Hz are blurry and comes out as non-statistically significant
(vs baseline). This is most likely due to the individual differences in
frequency bands (Haegens et al., (2014)).

Figure 8: ERSP: Midline electrodes, frontal to occipital. The color represents
the log ratio of the power at each instant with average power of the baseline
[-1,0]s for that same frequency, averaged across subjects. Figure 9: ERSP:
Fpz, Cz, and Oz electrodes. The color represents the log ratio of the power at
each instant with average power of the baseline [-1,0]s for that same
frequency. The first column is the mean across subjects. The second column is
the median across subjects. The third column is the mean across subjects with
a gray mask where the p-value of a t-test $>$=.05 (i.e. gray means not
significantly different than baseline).

The analysis for the effect of the number of targets showed a significant
difference only during recall. ERSP values for 1, 2 and 3 targets were
obtained and paired t-tests were performed for condition 1 vs 3 targets on the
whole trial for each time-frequency point. The statistical mask shown on the
right on Figure 10 grays out the points for which the p-value was higher or
equal to 0.05, and therefore, not significantly different from 1 to 3 targets.
The mask is plotted on top of the resulting ERSP map obtained by subtracting
the ERSP values of 1 target to the ERSP values of 3 targets to highlight the
difference in power between the two. The statistical tests for 1 vs 2 and 2 vs
3 targets were not performed because we assume some sort of linearity between
1, 2 and 3 targets and therefore their effect would be somewhere between what
we observe for 1 and 3 targets. For readability only the frontal and parietal
clusters are displayed. Frontal because that’s where we observed the strongest
CDA in the time domain and parietal because that’s where we initially expected
the strongest difference.

Figure 10: ERSP: Frontal and parietal regions for 1, 2, and 3 targets. The
color represents the log ratio of the power at each instant with average power
of the baseline [-1,0]s for that same frequency. The first column is the mean
across subjects. The last column is the ERSP with 3 targets minus the ERSP
with 1 target with a gray mask where the p-value of a t-test between the two
$>$=.05 (i.e. gray means no significant difference between set size of 1
target vs 3 targets.)

Finally, as expected given the results from the ERSP maps showed and explained
above, the repeated measure three-way ANOVA on raw power revealed a
significant effect for Phase (F(2,36) = 12.13, p $<$ .0001), for Frequency
Band (F(4,72) = 50.4, p $<$ .0001), and smaller yet significant effect for Set
Size (F(2,36) = 4.6, p = .016). The interaction between Phase and Frequency
Band was significant (F(8,144) = 11.58, p $<$ .0001), the interaction between
Phase and Set Size was significant (F(4,72) = 6.36, p $<$ .0005), the
interaction between Frequency Band and Set Size was significant (F(8,144) =
5.24, p $<$ .0001) and finally, the interaction between Phase, Set Size and
Frequency Band was also significant (F(16,288) = 6.71, p $<$ .0005). The
statistical analysis was performed on both the raw power (results mentioned
above) and the power ratio (ERSP maps on figures). We reported the results
from the raw power as most studies use the raw power directly. Using the power
ratio instead of raw power provides a more normalized value across subjects,
and we obtained similar results. The differences include the Set Size effect
that came out slightly over the p = .05 significance threshold, with (F(2,38)
= 2.92, p = .066), the interaction between Set Size and Phase was trending
towards significance (F(4,76) = 2.46, p = .0521) and the interaction between
Set Size, Phase and Frequency Band was significant (F(16,304) = 2.88, p $<$
.0005).

##  5 Discussion

Several studies have been published on EEG activity during MOT tasks (e.g.
Drew and Vogel, (2008); Bland et al., (2020); Merkel et al., (2022)), usually
focusing on either the indexing or tracking phase but very little, if at all,
on the recall phase. Here, we took a more holistic view to look at
electrophysiological changes over the whole sequence (i.e. trial).

CDA. During the identification and tracking phases, we did observe lateralized
activity as expected. However, the highest amplitude were seen in frontal and
temporal regions and not in the parietal region as expected from the recent
CDA literature. Also, the amplitude in the parietal region did not increase
with the number of targets during identification and tracking which is in
contradiction with the recent CDA literature (Vogel and Machizawa, (2004);
Drew and Vogel, (2008); Unsworth et al., (2015)) and our own CDA review paper
(Roy and Faubert, (2022)). Moreover, during recall, we actually observed the
opposite effect where the amplitude in certain regions was actually higher
with less targets and the activity lasted longer in time with more targets.
Given the clear lateralized ERPs and activity observed in the results (e.g.
Figure 3) the design of the task clearly had laterialized targets. Therefore,
to explain our results being different from what we expected based on the
literature, we rule out a bad task design and point out four ways in which our
3D-MOT task is different than most VWM tasks used to study CDA. First, as
opposed to change detection tasks where the participant has to remember all
the items at once, here the participants could have recalled the items one by
one, sustaining a longer CDA over time. Second, most VWM tasks in CDA studies
have very short trials of about 1 to 2 seconds long. Third, in most CDA
related tasks, the objects are temporarily removed from the screen forcing the
participant to have an internal representation of the objects. Here the
objects are always visible for the participant and perhaps engage a different
cognitive strategy. Forth, most tasks are done in 2D without stereoscopy.
These confounding differences could make the participant use a different
cognitive strategy relying more on attention mechanisms than working memory
during tracking.

In order to better understand the results we obtained, two things have to be
disentangled. First, is WM required for the 3D-MOT task and if so, are the
targets held in memory for the whole duration of tracking or only held in
memory during indexing and/or recall? Second, is the lateralized activity
observed here the same as the CDA referred to in the literature and believed
to play a role in WM, or is it a different, yet lateralized, activity? If the
answer is yes to both, as we initially hypothesized, we should have observed a
CDA during tracking and the CDA amplitude should have varied in amplitude
based on the number of targets. Our results diverge from Drew and Vogel,
(2008) where they showed a clear CDA with different amplitudes during tracking
in a different and shorter 2D-MOT task. As we can see in our results, the
activity is clearly lateralized and has a CDA-like shape, in both the
identification and recall phases but not during tracking which leads us to
believe that there might be a cognitive switch between memory and attention,
where the targets might not be held in working memory during tracking but
rather only tracked with attention mechanisms. During recall, there is no
doubt that the participant has to leverage working memory to provide the
answers and even more so, the confidence level for each target. Interestingly,
during the identification phase for the lateralized trials, the activity is
nicely symmetrical early post-stimulus and then a change occurs and the
following 500ms are not as symmetrical. We hypothesize that the period between
100-400ms engages more bottom-up brain mechanisms as a response to the stimuli
and the following 500-600ms engages more top-down mechanisms to which there
seems to be a difference between left and right hemispheres.

Eye movements. In order to rule out the possibility that remaining eye
movement artifacts be the main driver of the frontal EEG activity observed in
the results, we calculated the correlation of the EOG channel pair with the
lateralized signal obtained from the frontal cluster. The Pearson product-
moment correlation coefficients was calculated for each participant and then
averaged to obtain the group level correlation coefficient. A weak to moderate
correlation is expected at the minimum, as nearby electrophysiological
channels share information. For the EOG channels, we obtained a correlation
coefficient of 0.765 which, when compared with the other channel pairs (F5-F6
r=0.875; F7-F8 r=0.845; AF7-AF8 r=0.823; F3-F4 r=0.798; FT7-FT8 r=0.788;
AF3-AF4 r=0.769; F1-F2 r=0.739; Fp1-Fp2 r=0.581), is among the weakest
correlation with the lateralized CDA-like activity presented in the results.
This shows that the signal obtained isn’t driven by eye movements but rather
cognitive processes.

Frequency bands. During tracking, there is a strong inhibition of both delta
and theta frequencies, followed by a significant reactivation of these same
frequencies during recall. For such a strong switch between inhibition and
activation, we ruled out the hypothesis that it could be muscle activity or
artifacts based on releasing the tension after being focused during tracking,
because if that was the case, we would see the opposite effect on Figure 10
and the activity would spread longer with only 1 target as they would start
releasing the tension and moving their body faster. However here we see a
stronger but shorter activity in delta and theta during recall for 1 targets,
less strong and slightly longer for 2 targets, and the weakest power but
longer spread over time. This is aligned with the CDA plots in the time domain
(see Figure 6). We also ruled out the motor and speech brain-related activity
from providing the answers, as such activity would be more localized and not
clearly visible on all clusters as seen here.

The activation of lower frequencies (delta and theta) during recall is
distributed spatially but the strong inhibition during tracking is frontal. We
hypothesize that such inhibition might be coming from a top-down cognitive
mechanism to ignore task distractors and prevent them from being encoded
during tracking. Also, the 3D-MOT task requires the participant’s full
attention, because many occlusions and contacts between the objects (targets
and distractors alike) are happening and one tiny lapse in attention can make
the participant fail the trial. Therefore, it is of upmost importance for our
top-down mechanisms to protect our attention from task-related, environment-
related as well as internal distractions. Theta has been linked to memory,
cognitive control and attention systems (Başar et al., (2001); Karakaş,
(2020); Eisma et al., (2021)) and as observed here, has a key role in the MOT
task during tracking and recall. Theta and gamma phase coupling has also been
accumulating supportive evidence for playing a key role in visual processes
involving working memory (Sauseng et al., (2009); Köster et al., (2014);
Lundqvist et al., (2016)). This theta-gamma coupling might explain why during
recall we observe also a strong gamma activation at the same time as the theta
activation, strongest in the occipital region. We haven’t done any phase
coupling nor connectivity analysis to confirm the link between these
frequencies but it is in our future plans to look at phase coupling, source
localization, and connectivity.

As for delta, the literature for its role in cognitive functions, aside from
sleep studies, isn’t as extensive as for the other frequency bands but it has
been linked to similar attention mechanisms than theta (Güntekin and Başar,
(2016)). Alpha, one of the most prominent rhythm in the human brain, also
plays a key role in attention, especially for inhibition of distractors. A
previous study providing online feedback on a 3D-MOT task based on real time
alpha peak frequency has helped improve performance on the task (Parsons and
Faubert, (2021)). While it is clear that alpha is playing a role, it remains
unclear exactly how, as studies have shown both an increase (Wutz et al.,
(2020)) as well as a decrease (Merkel et al., (2022)) in alpha during object
tracking. Based on the previous evidences, looking at the grand average over
the whole trial and then averaging all the trials together might be the wrong
way of looking at alpha as it might be involved in a more granular fashion
during a trial. Calculating the grand average might hide the subtle within-
trial changes of alpha. On the ERPS graphs, we see that power changes are
clearly happening around, and at, alpha frequencies however the effect isn’t
as strong as some other frequencies.

Given the clear EEG pattern we observed in the frequency domain for the whole
trial ($\sim$15s), we wanted to compare with another MOT task and run a
similar frequency analysis to see if we’d see the same pattern of activity.
Thanks to Nicholas S. Bland and colleagues, who shared their EEG data from
their 2020 study (Bland et al., (2020)), we were able to generate an ERSP
graph and we found a striking resemblance. Bland and colleagues used a 2D-MOT
(our task is a 3D-MOT), their stimuli were 2D circles with no filling (i.e.
rings), their trials had either 2 or 4 targets presented either between-
hemifield moving freely left and right but not crossing the middle part
vertically or within-hemifield moving freely up and down but not crossing the
middle part horizontally. At the start of the trial they presented all the
circles in white, then highlighted the targets in blue for 2 seconds (like our
task) after which they reverted back to white for 500ms (we used 1s in our
task) before all objects start moving for 8 seconds (like our task). Their
recall phase was slightly different as their participants had to click on the
targets with a mouse and then received the visual feedback with the correct
answers for 1.5s. In our task the participant gave their answer verbally
before entering their confidence level with the keyboard and then received
visual feedback with the correct answers for 2s. They recorded EEG activity
with a 64 channels BrainCap (BrainProducts) device. On Figure 11, we see the
ERSP grand average across all conditions, all channels and all participants of
our study on the right (as presented on Figure 7 before) and Bland et al.,
2020 on the left. We used all the trials (N=192) for all the participants
(N=41). The spatial distribution on the midline was also similar to ours,
showing a stronger inhibition of delta and theta in the frontal regions during
tracking. Their inhibition of lower frequencies (compared to baseline) was
stronger than our results so we slightly adjusted the color scale to keep a
smooth range.

Figure 11: ERSP: Bland, 2022 vs Roy, 2022.

The main limitation of our study is the number of trials per condition.
According to a recent study by William X. Q. Ngiam and colleagues on the
statistical power to detect set-size effects in contralateral delay activity,
it requires between 30 to 50 clean trials with a sample of 25 subjects to
achieve approximately 80% statistical power on detecting the presence of the
CDA. In our study, we kept 20 subjects with an average of 45 clean
(lateralized) trials after removing bad trials during preprocessing. As
mentioned earlier, this experiment was part of a larger study and we
unfortunately had a time limit with the participants for this task as they
were doing another cognitively demanding task right after. Hence the limited
amount of trials. In the larger study, we train a machine learning classifier
on 1s windows of EEG activity and therefore, the number of trials doesn’t
matter as much as the overall recording time.

In order to keep the participants engaged and energized (knowing they had
another task after), we asked them to give the answers orally as opposed to
entering them with a keyboard. Therefore, the instructor, despite entering the
answers really fast on the numpad, induced an external timing factor during
the recall phase. Finally, given the interesting results we observed during
the recall phase, having more granular information during that phase such as
the time of entry of each answer for each participant would have allowed us to
investigate deeper the neural activity during that phase.

In conclusion, we believe that we offered a more holistic view of the neural
substrates during a 3D-MOT task looking at the activity across the three
different phases (Identification/Indexing, Tracking, Recall/Answer) both in
the time and frequency domains. We also analysed the raw EEG data of another
MOT study to compare our findings and observed similar overall trends,
solidifying our findings. This study sheds more light at what seems to be some
sort of a handoff between focused attention and working memory processes
during tracking and recall and how the delta and theta bands in the frontal
regions play a key role in the 3D-MOT task as they are being toggled like an
on/off switch across phases.

Both the data and the code is available online:
https://github.com/royyannick/3DMOT_EEG

##  6 Funding

This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC-RDC) (reference number: RDPJ 514052-17) and an NSERC
Discovery fund.

## References

  * Alvarez and Franconeri, (2007) Alvarez, G. A. and Franconeri, S. L. (2007).  How many objects can you track?: Evidence for a resource-limited attentive tracking mechanism.  Journal of vision, 7(13):14–14.
  * Başar et al., (2001) Başar, E., Başar-Eroglu, C., Karakaş, S., and Schürmann, M. (2001).  Gamma, alpha, delta, and theta oscillations govern cognitive processes.  International journal of psychophysiology, 39(2-3):241–248.
  * Benoit et al., (2020) Benoit, J. J., Roudaia, E., Johnson, T., Love, T., and Faubert, J. (2020).  The neuropsychological profile of professional action video game players.  PeerJ, 8:e10211.
  * Bland et al., (2020) Bland, N. S., Mattingley, J. B., and Sale, M. V. (2020).  Gamma coherence mediates interhemispheric integration during multiple object tracking.  Journal of Neurophysiology, 123(5):1630–1644.
  * Cavanagh, (2011) Cavanagh, P. (2011).  Visual cognition.  Vision research, 51(13):1538–1551.
  * Cavanagh and Alvarez, (2005) Cavanagh, P. and Alvarez, G. A. (2005).  Tracking multiple targets with multifocal attention.  Trends in cognitive sciences, 9(7):349–354.
  * Chen et al., (2013) Chen, W.-Y., Howe, P. D., and Holcombe, A. O. (2013).  Resource demands of object tracking and differential allocation of the resource.  Attention, Perception, & Psychophysics, 75(4):710–725.
  * Cowan, (2001) Cowan, N. (2001).  The magical number 4 in short-term memory: A reconsideration of mental storage capacity.  Behavioral and brain sciences, 24(1):87–114.
  * Dendorfer et al., (2020) Dendorfer, P., Rezatofighi, H., Milan, A., Shi, J., Cremers, D., Reid, I., Roth, S., Schindler, K., and Leal-Taixé, L. (2020).  Mot20: A benchmark for multi object tracking in crowded scenes.  arXiv preprint arXiv:2003.09003.
  * Drew et al., (2011) Drew, T., Horowitz, T. S., Wolfe, J. M., and Vogel, E. K. (2011).  Delineating the neural signatures of tracking spatial position and working memory during attentive tracking.  Journal of Neuroscience, 31(2):659–668.
  * Drew and Vogel, (2008) Drew, T. and Vogel, E. K. (2008).  Neural measures of individual differences in selecting and tracking multiple moving objects.  Journal of Neuroscience, 28(16):4183–4191.
  * Eisma et al., (2021) Eisma, J., Rawls, E., Long, S., Mach, R., and Lamm, C. (2021).  Frontal midline theta differentiates separate cognitive control strategies while still generalizing the need for cognitive control.  Scientific Reports, 11(1):1–14.
  * Faubert, (2013) Faubert, J. (2013).  Professional athletes have extraordinary skills for rapidly learning complex and neutral dynamic visual scenes.  Scientific reports, 3(1):1–3.
  * Faubert and Sidebottom, (2012) Faubert, J. and Sidebottom, L. (2012).  Perceptual-cognitive training of athletes.  Journal of Clinical Sport Psychology, 6(1):85–102.
  * Gramfort et al., (2013) Gramfort, A., Luessi, M., Larson, E., Engemann, D. A., Strohmeier, D., Brodbeck, C., Goj, R., Jas, M., Brooks, T., Parkkonen, L., et al. (2013).  Meg and eeg data analysis with mne-python.  Frontiers in neuroscience, 7:267.
  * Güntekin and Başar, (2016) Güntekin, B. and Başar, E. (2016).  Review of evoked and event-related delta responses in the human brain.  International Journal of Psychophysiology, 103:43–52.
  * Haegens et al., (2014) Haegens, S., Cousijn, H., Wallis, G., Harrison, P. J., and Nobre, A. C. (2014).  Inter-and intra-individual variability in alpha peak frequency.  Neuroimage, 92:46–55.
  * Hoke et al., (2017) Hoke, J., Reuter, C., Romeas, T., Montariol, M., Schnell, T., and Faubert, J. (2017).  Perceptual-cognitive & physiological assessment of training effectiveness.  In Interservice/Industry Training, Simulation, and Education Conference, Orlando, FL.
  * Iordanescu et al., (2009) Iordanescu, L., Grabowecky, M., and Suzuki, S. (2009).  Demand-based dynamic distribution of attention and monitoring of velocities during multiple-object tracking.  Journal of vision, 9(4):1–1.
  * Jas et al., (2017) Jas, M., Engemann, D. A., Bekhti, Y., Raimondo, F., and Gramfort, A. (2017).  Autoreject: Automated artifact rejection for meg and eeg data.  NeuroImage, 159:417–429.
  * Karakaş, (2020) Karakaş, S. (2020).  A review of theta oscillation and its functional correlates.  International Journal of Psychophysiology, 157:82–99.
  * Köster et al., (2014) Köster, M., Friese, U., Schöne, B., Trujillo-Barreto, N., and Gruber, T. (2014).  Theta–gamma coupling during episodic retrieval in the human eeg.  Brain research, 1577:57–68.
  * Lundqvist et al., (2016) Lundqvist, M., Rose, J., Herman, P., Brincat, S. L., Buschman, T. J., and Miller, E. K. (2016).  Gamma and beta bursts underlie working memory.  Neuron, 90(1):152–164.
  * Luong et al., (2015) Luong, M.-T., Pham, H., and Manning, C. D. (2015).  Effective approaches to attention-based neural machine translation.  arXiv preprint arXiv:1508.04025.
  * Luria et al., (2016) Luria, R., Balaban, H., Awh, E., and Vogel, E. K. (2016).  The contralateral delay activity as a neural measure of visual working memory.  Neuroscience & Biobehavioral Reviews, 62:100–108.
  * Mangine et al., (2014) Mangine, G. T., Hoffman, J. R., Wells, A. J., Gonzalez, A. M., Rogowski, J. P., Townsend, J. R., Jajtner, A. R., Beyer, K. S., Bohner, J. D., Pruna, G. J., et al. (2014).  Visual tracking speed is related to basketball-specific measures of performance in nba players.  The Journal of Strength & Conditioning Research, 28(9):2406–2414.
  * Merkel et al., (2015) Merkel, C., Hopf, J.-M., Heinze, H.-J., and Schoenfeld, M. A. (2015).  Neural correlates of multiple object tracking strategies.  Neuroimage, 118:63–73.
  * Merkel et al., (2022) Merkel, C., Hopf, J.-M., and Schoenfeld, M. A. (2022).  Electrophysiological hallmarks of location-based and object-based visual multiple objects tracking.  European Journal of Neuroscience, 55(5):1200–1214.
  * Milan et al., (2016) Milan, A., Leal-Taixé, L., Reid, I., Roth, S., and Schindler, K. (2016).  Mot16: A benchmark for multi-object tracking.  arXiv preprint arXiv:1603.00831.
  * Oksama and Hyönä, (2016) Oksama, L. and Hyönä, J. (2016).  Position tracking and identity tracking are separate systems: Evidence from eye movements.  Cognition, 146:393–409.
  * Park et al., (2021) Park, Y., Dang, L. M., Lee, S., Han, D., and Moon, H. (2021).  Multiple object tracking in deep learning approaches: A survey.  Electronics, 10(19):2406.
  * Parsons and Faubert, (2021) Parsons, B. and Faubert, J. (2021).  Enhancing learning in a perceptual-cognitive training paradigm using eeg-neurofeedback.  Scientific Reports, 11(1):1–10.
  * Parsons et al., (2016) Parsons, B., Magill, T., Boucher, A., Zhang, M., Zogbo, K., Bérubé, S., Scheffer, O., Beauregard, M., and Faubert, J. (2016).  Enhancing cognitive function using perceptual-cognitive training.  Clinical EEG and neuroscience, 47(1):37–47.
  * Pylyshyn, (1989) Pylyshyn, Z. (1989).  The role of location indexes in spatial perception: A sketch of the finst spatial-index model.  Cognition, 32(1):65–97.
  * Pylyshyn and Storm, (1988) Pylyshyn, Z. W. and Storm, R. W. (1988).  Tracking multiple independent targets: Evidence for a parallel tracking mechanism.  Spatial vision, 3(3):179–197.
  * Rajalingham et al., (2018) Rajalingham, R., Issa, E. B., Bashivan, P., Kar, K., Schmidt, K., and DiCarlo, J. J. (2018).  Large-scale, high-resolution comparison of the core visual object recognition behavior of humans, monkeys, and state-of-the-art deep artificial neural networks.  Journal of Neuroscience, 38(33):7255–7269.
  * Romeas et al., (2016) Romeas, T., Guldner, A., and Faubert, J. (2016).  3d-multiple object tracking training task improves passing decision-making accuracy in soccer players.  Psychology of Sport and Exercise, 22:1–9.
  * Roy and Faubert, (2022) Roy, Y. and Faubert, J. (2022).  Is the contralateral delay activity (cda) a robust neural correlate for visual working memory (vwm) tasks? a reproducibility study.  arXiv preprint.
  * Sauseng et al., (2009) Sauseng, P., Klimesch, W., Heise, K. F., Gruber, W. R., Holz, E., Karim, A. A., Glennon, M., Gerloff, C., Birbaumer, N., and Hummel, F. C. (2009).  Brain oscillatory substrates of visual short-term memory capacity.  Current biology, 19(21):1846–1852.
  * Shim et al., (2008) Shim, W. M., Alvarez, G. A., and Jiang, Y. V. (2008).  Spatial separation between targets constrains maintenance of attention on multiple objects.  Psychonomic bulletin & review, 15(2):390–397.
  * Trick and Pylyshyn, (1993) Trick, L. M. and Pylyshyn, Z. W. (1993).  What enumeration studies can show us about spatial attention: evidence for limited capacity preattentive processing.  Journal of Experimental Psychology: Human Perception and Performance, 19(2):331.
  * Tullo et al., (2018) Tullo, D., Faubert, J., and Bertone, A. (2018).  Examining the benefits of training attention with multiple object-tracking for individuals diagnosed with a neurodevelopmental condition: A cross-over, cognitive training study.  Journal of Vision, 18(10):1021–1021.
  * Unsworth et al., (2015) Unsworth, N., Fukuda, K., Awh, E., and Vogel, E. K. (2015).  Working memory delay activity predicts individual differences in cognitive abilities.  Journal of Cognitive Neuroscience, 27(5):853–865.
  * Vartanian et al., (2016) Vartanian, O., Coady, L., and Blackler, K. (2016).  3d multiple object tracking boosts working memory span: Implications for cognitive training in military populations.  Military Psychology, 28(5):353–360.
  * Vogel and Machizawa, (2004) Vogel, E. K. and Machizawa, M. G. (2004).  Neural activity predicts individual differences in visual working memory capacity.  Nature, 428(6984):748–751.
  * Wutz et al., (2020) Wutz, A., Zazio, A., and Weisz, N. (2020).  Oscillatory bursts in parietal cortex reflect dynamic attention between multiple objects and ensembles.  Journal of Neuroscience, 40(36):6927–6937.
  * Yantis, (1992) Yantis, S. (1992).  Multielement visual tracking: Attention and perceptual organization.  Cognitive psychology, 24(3):295–340.

##  7 Supplementary Material

Cluster | Channels
---|---
Frontal | Fp1, AF7, AF3, F1, F3, F5, F7, FT7
| Fp2, AF8, AF4, F2, F4, F6, F8, FT8
| Fpz, AFz, Fz
Central | C1, C3, C5, CP1, CP3, CP5, FC1, FC3, FC5
| C2, C4, C6, CP2, CP4, CP6, FC2, FC4, FC6
| Cz CPz
Temporal | FT7, T7, TP7
| FT8, T8, TP8
Parietal | TP7, CP1, CP3, CP5, P1, P3, P5, P7, P9, PO3, PO7
| TP8, CP2, CP4, CP6, P2, P4, P6, P8, P10, PO4, PO8
| POz, Pz, CPz
Occipital | O1
| O2
| Oz, POz
Table 1: Electrode Clusters


Spin $1/2$ Particle Dynamics Near Reissner-Nordström Black Holes: Extremal and
Non-Extremal Cases

Özlem Yeşiltaş

$\ddagger$ Gazi University, Faculty of Science, Physics Department, 06500,
Teknikokullar, Ankara, TURKEY, e-mail<EMAIL_ADDRESS>

###### Abstract

In this work, the fermion behaviour near the Reissner-Nordstrom black holes in
quantum mechanics is studied and exact solutions of the Dirac equation for
Reissner-Nordström Black Holes are obtained in terms of special functions. We
have obtained different vector potentials which satisfy the solutions.

keyword: Dirac equation, curved space-time, black holes

PACS: 03.65.Fd, 03.65.Ge, 95.30 Sf

##  1 Introduction

Black holes have their roots in 1915, shortly after Albert Einstein published
his theory of General Relativity that is first modern solution of general
relativity that would characterize a black hole was found by Karl
Schwarzschild in 1916 [1]. In later times, some observations have confirmed
black holes as actual astrophysical objects. After the work by Hawking in 1972
[2], entitled with Black Holes in General Relativity, an avalanche of interest
in black holes has begun. Meeting of two strong theories as quantum mechanics
and general relativity results in conflicts such as the structure of
elementary particles and the particles have to be represented as an extended
field model for compatibility with the stress-energy tensor of Einstein field
equations. By the way, Recently, the demand for the relationships between
black holes and fundamental particles has been constantly increasing and new
information has been obtained. For instance, the Compton wavelength is equal
to the wavelength of a photon whose energy is the same as the mass of that
particle. If we consider black holes as particles, Compton wavelength
$\lambda$ of the black hole as determined by the distant observer using the
asymptotic mass $M$; $\lambda=\frac{\hbar}{cM_{Pl}/2}=2R_{S}$. The diameter of
the black hole is $2R_{S}$, this length becomes identical to the Compton
wavelength. Because for all types of black holes( neutral, charged or
rotating), the horizon mass is always equal to twice the irreducible mass
observed at infinity according to the the horizon mass theorem [3]. If the
horizon mass of a black hole is taken to be the Planck mass $M_{Pl}$, then a
distant observer will find the black hole to have an asymptotic mass
$M_{Pl}/2$ and therefore it is resulting that the Schwarzschild radius is
equal to the Planck length. In case of Kerr–Newman black hole solutions, the
radius of the Kerr singular ring corresponds to the reduced Compton wave
length of the particle [4]. Besides the compatible research efforts, there is
another field of interest which is known as bound state solutions for the
particle states near black holes. Moreover, the electron levels can be located
not only beyond the black hole but also under the Cauchy horizon [5]. If the
charge of the black hole is less than its mass, then it’s geometry has an
outer and inner horizons. The Reissner–Nordström (RN) metric emerged
hypothetically as the static solution of Einstein’s field equations, including
Maxwell’s equations where the time component of the vector potential doesn’t
vanish. It corresponds to a charged but non-rotating spherical gravitational
field of mass. Among the different studies; the level shift in RN geometry is
studied and gravitational shift of the $1S_{1/2}$ state was found as
$10^{-36}Hz$ [6], non-existence time-periodic solutions of the Dirac equation
[7], the strong cosmic censorship for the Dirac field in the higher
dimensional RN-de Sitter black hole [8], stationary solutions of the Dirac
equation in the RN gravitational field [9] and other interesting studies [10],
[11], [12], [13]. The object of the present paper is to propose different
solutions of the equation Dirac equation in the metric of the charged RN black
hole.

##  2 Dirac Equation and Spherical Geometry

A spherically symmetric metric with a radial dependent function $\lambda(r)$
has a form of [14]

$ds^{2}=-\lambda(r)dt^{2}+\frac{1}{\lambda(r)}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}.$ (1)

Here, the function $\lambda(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$ and
$c=G=\hbar=1$, $M$ is the mass of the black hole, $Q$ is the charge, (1) is
known as RN [9]. In case of $|Q|<M$, $\lambda$ has two roots,
$r=r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$. For $|Q|>M$, the metric describes a bare
singularity without an event horizon and $|Q|=M$ corresponds to an extremal
black hole. The Dirac equation written in a pseudo-Riemannian space is written
as

$i\gamma^{\mu}(\partial_{\mu}+iqA_{\mu}+\Gamma_{\mu}-m)\psi=0$ (2)

where $A_{\mu}$ is the electromagnetic four potential, $m$ is the electron
mass, $\Gamma_{\mu}$ is the spin connection and $e^{\mu}_{(a)}$ are the
tetrads. The spin connection can be given by

$\Gamma_{\mu}=\frac{1}{4}\gamma^{(a)}\gamma^{(b)}e_{(a)}^{\nu}e_{(b)\nu;\mu}$
(3)

and $\gamma^{\mu}=e_{(a)}^{\mu}\gamma^{(a)}$. If we use the tetrad which is
given below

$e^{(a)}_{\mu}=diag\left(\sqrt{\lambda(r)},\frac{1}{\sqrt{\lambda(r)}},r,r\sin\theta\right),$
(4)

one can give the Dirac equation which reduces into

$\left(\gamma^{(0)}\frac{1}{\sqrt{\lambda(r)}}\left(\partial_{t}+iqA(r)\right)+\gamma^{(1)}\sqrt{\lambda(r)}\left(\partial_{r}+\frac{1}{r}+\frac{\lambda^{\prime}(r)}{4\lambda(r)}\right)+\gamma^{(2)}\frac{1}{r}\left(\partial_{\theta}+\frac{\cot\theta}{2}\right)+\gamma^{(3)}\frac{1}{r\sin\theta}\partial_{\phi}-m\right)\psi=0.$
(5)

where the gauge potential is $A_{\mu}=(e\varphi(r),0,0,0)$, $q=e$ is the
particle charge (for an electron). If we follow [9] and [15], we take

$\psi(t,r,\theta,\phi)=\frac{e^{-iEt}\bar{\psi}}{\lambda(r)^{1/4}r\sqrt{\sin\theta}}$
(6)

and defining the first order, linear differential operator K as

$\textbf{K}=\left(\gamma^{2}\partial_{\theta}+\frac{\gamma^{3}}{\sin\theta}\partial_{\phi}\right)\gamma^{(0)}\gamma^{(1)}$
(7)

and $\textbf{K}\bar{\psi}=k\bar{\psi}$, $k=0,\pm 1,\pm 2,...$ Taking
$\bar{\psi}$ as separated into the radial and angular parts,

$\bar{\psi}=Y(\theta,\phi)\left(\begin{array}[]{c}\phi_{1}(r)\\\
i\phi_{2}(r)\\\ \end{array}\right),$ (8)

as a result, one can obtain a radial system as

$\left(\sqrt{\lambda(r)}\frac{d}{dr}-\frac{k}{r}\right)\phi_{1}(r)+\left(\frac{E+e\varphi(r)}{\sqrt{\lambda(r)}}+m\right)\phi_{2}(r)=0,$
(9)
$\left(\sqrt{\lambda(r)}\frac{d}{dr}+\frac{k}{r}\right)\phi_{2}(r)-\left(\frac{E+e\varphi(r)}{\sqrt{\lambda(r)}}-m\right)\phi_{1}(r)=0,$
(10)

where we note that the normalization of the radial part is equal to $1$. The
system of (9) and (10) lead to

$\displaystyle-\lambda(r)\frac{d^{2}\phi_{1}(r)}{dr^{2}}+\left(\lambda(r)\frac{U^{\prime}(r)}{U(r)}-\frac{\lambda^{\prime}(r)}{2}\right)\frac{d\phi_{1}}{dr}+\left(\frac{k^{2}}{r^{2}}-\frac{k\sqrt{\lambda(r)}}{r^{2}}+2mU(r)-U(r)^{2}-\frac{k\sqrt{\lambda(r)}U^{\prime}(r)}{rU(r)}\right)\phi_{1}(r)$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle-\lambda(r)\frac{d^{2}\phi_{2}(r)}{dr^{2}}+\left(\lambda(r)\frac{U^{\prime}(r)}{U(r)-2m}-\frac{\lambda^{\prime}(r)}{2}\right)\frac{d\phi_{2}}{dr}+\left(\frac{k^{2}}{r^{2}}+\frac{k\sqrt{\lambda(r)}}{r^{2}}+2mU(r)-U(r)^{2}+\frac{k\sqrt{\lambda(r)}U^{\prime}(r)}{r(U(r)-2m)}\right)\phi_{2}(r)$
$\displaystyle=$ $\displaystyle 0,$

where

$U(r)=\frac{E+e\varphi(r)}{\sqrt{\lambda(r)}}+m.$ (11)

Let us use the transformation below,

$\displaystyle\phi_{i}(r)$ $\displaystyle=$ $\displaystyle
e^{\int^{r}\mu(z)dz}\bar{\phi_{i}}(r),~{}~{}~{}~{}i=1,2$ (12)
$\displaystyle\mu(r)$ $\displaystyle=$
$\displaystyle\frac{2\lambda(r)U^{\prime}(r)+U(r)\lambda^{\prime}(r)}{4U(r)\lambda(r)}.$
(13)

Under this map, we get

$\begin{split}-\lambda(r)\bar{\phi^{\prime\prime}_{1}}(r)-\lambda^{\prime}(r)\bar{\phi^{\prime}_{1}}(r)+(\frac{k^{2}}{r^{2}}+2mU(r)-U(r)^{2}-\frac{k\sqrt{\lambda(r)}}{r^{2}}-\frac{k\sqrt{\lambda(r)}U^{\prime}(r)}{rU(r)}+\frac{3\lambda(r)U^{\prime}(r)^{2}}{4U(r)^{2}}-\frac{U^{\prime}(r)\lambda^{\prime}(r)}{4U(r)}+\\\
\frac{\lambda^{\prime}(r)^{2}}{16\lambda(r)}-\frac{\lambda
U^{\prime\prime}(r)}{2U(r)}-\frac{\lambda^{\prime\prime}(r)}{4})\bar{\phi_{1}}(r)=0,\end{split}$
(14)
$\begin{split}-\lambda(r)\bar{\phi^{\prime\prime}_{2}}(r)-\lambda^{\prime}(r)\bar{\phi^{\prime}_{2}}(r)+(\frac{k^{2}}{r^{2}}+2mU(r)-U(r)^{2}-\frac{k\sqrt{\lambda(r)}}{r^{2}}+\frac{k\sqrt{\lambda(r)}U^{\prime}(r)}{r(U(r)-2m)}+\frac{3\lambda
U^{\prime 2}}{4(U(r)-2m)^{2}}-\frac{U^{\prime}\lambda^{\prime}}{4(U(r)-2m)}\\\
-\frac{\lambda(r)U^{\prime\prime}(r)}{2(U(r)-2m)}+\frac{\lambda^{\prime}(r)^{2}}{16\lambda(r)}-\frac{\lambda^{\prime\prime}(r)}{4})\bar{\phi_{2}}(r)=0.\end{split}$
(15)

It can be noted that (15) can be obtained from (14) using the replace
$m\rightarrow-m$ and that’s why we only include the solutions of the system
(14).

##  3 Extremal and Non-Extremal Cases

In the form of a one parameter family obeyed by the extreme RN black holes we
have $M=Q$ which means that the black hole is like a black body with zero
temperature. Because there is no surface gravity and the Hawking temperature
is equal to zero. $M^{2}<Q^{2}$ stands for the naked singularity where we have
two imaginary roots at $r=0$ and no solution to the Einstein equations. A
regular black hole with two horizons correspond to $M^{2}>Q^{2}$ where we have
two real roots, $r_{+}>r_{-}>0$ and one can construct the Penrose diagrams for
the regions at $r>r_{+}$, $r_{+}>r>r_{-}$ and $r_{-}>r>0$. In this study we
mainly investigate the Dirac equation solutions for only extreme and non-
extremal cases for obtaining solutions in terms of special functions.

###  3.1 non-extremal black hole case: $|Q|<M$

Once we use an approach for a non-extremal black hole, we have $Q\neq 0$,
$\lambda\rightarrow\frac{Q^{2}}{r^{2}}$, $k=0$ and $\varphi(r)=\frac{c}{r}$,
our system (14) has an asymptotic solution when $r\rightarrow 0$ which is

$\bar{\phi}_{1}(r)=C_{1}\sqrt{\frac{2}{(ce+mQ)r}}K\left(2,\sqrt{\frac{2Er}{ce+mQ}}\right),$
(16)

where $K(a,b)$ is the modified Bessel function of the first kind. For the case
$k\neq 0$, we get

$-\chi^{\prime\prime}(r)+\left(\frac{k^{2}}{Q^{2}}+\frac{3}{4r^{2}}-\frac{kQ}{r}+\frac{(2cemQ-c^{2}e^{2})r^{2}}{Q^{4}}\right)\chi(r)=0,$
(17)

where $\chi(r)=\frac{\bar{\phi}_{1}(r)}{r}$. For $c=\frac{2mQ}{e}$, we obtain

$\chi(r)=C_{1}~{}WhittakerM\left(\frac{1}{2},1,\frac{2kr}{Q}\right)+C_{2}~{}WhittakerW\left(\frac{1}{2},1,\frac{2kr}{Q}\right).$
(18)

On the other hand, we may seek for different solutions of (14). For the values
of $\lambda=\frac{Q^{2}}{r^{2}}$, we get

$\begin{split}-\phi^{\prime\prime}_{1}(r)+(\frac{2(E+e\varphi(r))+\frac{mQ}{r}+er\varphi^{\prime}(r)}{mQ+(E+e\varphi(r))r})\phi^{\prime}_{1}(r)+(\frac{k^{2}}{Q^{2}}-\frac{k}{Qr}+\frac{2mr^{2}(m+\frac{r(E+e\varphi(r))}{Q})}{Q^{2}}-\frac{r^{2}(m+\frac{r(E+e\varphi(r))}{Q})^{2}}{Q^{2}}-\\\
\frac{k(E+e\varphi(r)+er\varphi^{\prime}(r))}{Q^{2}(m+\frac{r(E+e\varphi(r))}{Q})})\phi_{1}(r)=0.\end{split}$
(19)

Now, we focus on (19) through the below cases.
_Case I: The solutions in terms of Airy functions: energy dependent potential
parameters _

In order to terminate the coefficient of the first order differential term in
(19), we may write

$\frac{2(E+e\varphi(r))+\frac{mQ}{r}+er\varphi^{\prime}(r)}{mQ+(E+e\varphi(r))r}=0,$
(20)

and we find $\varphi(r)$ as

$\varphi(r)=\frac{C_{1}}{r^{2}}-\frac{mQ}{e}\frac{1}{r}-\frac{E}{e}$ (21)

where $C_{1}$ is the integration constant. Then,(19) becomes

$-\phi^{\prime\prime}_{1}(r)+\left(\frac{k^{2}}{Q^{2}}-\frac{e^{2}C^{2}_{1}}{Q^{4}}+\frac{2emC_{1}}{Q^{3}}r\right)\phi_{1}(r)=0$
(22)

If we take $C_{1}=E$, $y=\alpha r$,
$\epsilon^{2}=\frac{k^{2}Q^{2}-e^{2}E^{2}}{Q^{4}\alpha^{2}}$,
$\alpha=\frac{(2emE)^{1/3}}{Q}$ and $z=y+\epsilon^{2}$, then,

$-\phi^{\prime\prime}_{1}(z)+z\phi_{1}(z)=0$ (23)

which is Airy differential equation and solutions are given by

$\phi_{1}(z)=aAi(z)+bBi(z).$ (24)

Here, $Bi(z)$ diverges for the larger values of $z$, then we use
$\phi_{1}(z)=aAi(z)$. The boundary conditions imply $\phi_{1}|_{{}_{z=0}}=0$,
thus, $z=0=y+\epsilon^{2}$ and

Table 1: Zeros of Airy Function 1 | -2.33810
---|---
2 | -4.08794
3 | -5.52055
4 | -6.78670
5 | -7.94413
6 | -9.02265
7 | -10.04017

boundary condition is satisfied if $\epsilon^{2}_{1}=2.33810$ that implies

$E_{1}=\pm\frac{kQ}{e\sqrt{1+\frac{9.3524m^{2}}{Q^{2}}}}.$ (25)

Hence, other values for the energy can be easily calculated using the table
above.
_Case II: The solutions in terms of hypergeometric functions_ In this case we
can take the coefficient of the first order derivative in (19) as different
from zero and give,

$\frac{2(E+e\varphi(r))+\frac{mQ}{r}+er\varphi^{\prime}(r)}{mQ+(E+e\varphi(r))r}=\frac{1}{r},$
(26)

and we may arrive at $\varphi(r)$ as

$\varphi(r)=-\frac{E}{e}+\frac{C_{1}}{r}$ (27)

where $C_{1}$ is the integration constant. Now consider the transformation
$\phi_{1}(r)=y(r)\exp\mu(r)\phi_{1}(r)$, then we get

$-\phi^{\prime\prime}_{1}(r)+\left(\frac{1}{r}-\frac{2y^{\prime}(r)}{y(r)}-2\mu^{\prime}(r)\right)\phi^{\prime}_{1}(r)+(\frac{k^{2}}{Q^{2}}-\frac{k}{Qr}+\frac{m^{2}Q^{2}-e^{2}C^{2}_{1}}{Q^{4}}r^{2}+\frac{y^{\prime}(r)}{ry(r)}+\frac{\mu^{\prime}(r)}{r}-\frac{2y^{\prime}(r)\mu^{\prime}(r)}{y(r)}-\mu^{\prime
2}(r)-\frac{y^{\prime\prime}(r)}{y(r)}-\mu^{\prime\prime}(r))\phi_{1}(r)=0$
(28)

Using the suggestion for the coefficient of the first order derivative

$\frac{1}{r}-\frac{2y^{\prime}}{y}-2\mu^{\prime}=-(b_{1}+\frac{b_{2}}{r})$
(29)

which gives

$y(r)=r^{\frac{1-b_{2}}{2}}e^{-(\frac{b_{1}r}{2}+\mu(r))},$ (30)

then we get

$-r\phi^{\prime\prime}_{1}(r)+\left(b_{2}+b_{1}r\right)\phi^{\prime}_{1}(r)+\left(-\frac{b_{1}b_{2}}{2}-\frac{k}{Q}+\frac{(1-b_{2})(3+b_{2})}{4r}+(-\frac{b^{2}_{1}}{4}+\frac{k^{2}}{Q^{2}})r+(\frac{m^{2}}{Q^{2}}-\frac{e^{2}C^{2}_{1}}{Q^{4}})r^{3}\right)\phi_{1}(r)=0.$
(31)

In order to transform (31) into the confluent hypergeometric differential
equation, we choose the parameters as given below

$b_{1}=1,~{}~{}b_{2}=-3,~{}~{}k=\pm\frac{Q}{2},~{}~{}C_{1}=\pm\frac{Q^{2}m}{e}$
(32)

that leads to get

$\phi^{\prime\prime}_{1}(r)+(3-r)\phi^{\prime}_{1}(r)+(\frac{b_{1}b_{2}}{2}+\frac{k}{Q})\phi_{1}(r)=0$
(33)

where $\frac{b_{1}b_{2}}{2}+\frac{k}{Q}=-1$ or
$\frac{b_{1}b_{2}}{2}+\frac{k}{Q}=-2$. The solutions are given as

$\phi_{1}(r)\sim
r^{2}\exp(-\frac{r}{2})_{1}F_{1}(\frac{b_{1}b_{2}}{2}+\frac{k}{Q},3,r).$ (34)

_Case 3: The solutions in terms of Whittaker functions_ In this section we
will use the exact definition for the $\lambda(r)$ which is

$\lambda(r)=\frac{Q^{2}}{r^{2}}.$ (35)

Using (35) and the transformation below,

$\phi_{1}(r)=r^{a}e^{C_{1}\int^{r}\mu(r^{\prime})dr^{\prime}}\chi_{1}(r)$ (36)

$a,C_{2}$ are real constants and we obtain

$\begin{split}r\chi_{1}^{\prime\prime}(r)+(-1+2a+2C_{1}r\mu(r)-\frac{rU^{\prime}(r)}{U(r)})\chi^{\prime}_{1}(r)+(k/Q+\frac{a(a-2)}{r}-\frac{k^{2}r}{Q^{2}}+\\\
r^{3}(U(r)^{2}-2mU(r))+C_{1}\mu(r)(2a-1)+C^{2}_{1}r\mu(r)^{2}-\frac{aU^{\prime}}{U}+\frac{krU^{\prime}}{QU}-\frac{C_{1}r\mu(r)U^{\prime}}{U}+C_{1}r\mu^{\prime}(r))\chi_{1}(r)=0.\end{split}$
(37)

If we choose

$\displaystyle\mu(r)$ $\displaystyle=$
$\displaystyle-\frac{1}{2r},~{}~{}U(r)=\frac{C_{1}}{r^{2}},~{}~{}a=0$ (38)

(37) turns into

$r^{2}\chi_{1}^{\prime\prime}(r)+\left(\frac{1}{4}+\frac{C^{2}_{1}}{Q^{2}}-\frac{k}{Q}r-\left(\frac{k^{2}}{Q^{2}}+\frac{2C_{1}m}{Q^{2}}\right)r^{2}\right)\chi_{1}(r)=0.$
(39)

The solutions of (39) are given by

$\chi_{1}(r)=AM_{\alpha,\beta}\left(\frac{2\sqrt{k^{2}+2C_{1}m}r}{Q}\right)+BW_{\alpha,\beta}\left(\frac{2\sqrt{k^{2}+2C_{1}m}r}{Q}\right)$
(40)

where $\alpha=-\frac{k}{2\sqrt{k^{2}+2C_{1}m}},~{}~{}\beta=\frac{iC_{1}}{Q}$,
$M_{\alpha,\beta}(z),W_{\alpha,\beta}(z)$ are the Whittaker functions [16].
$\nu$ is the quantum number, then,

$-\frac{k}{\sqrt{k^{2}+2C_{1}m}}=\nu,~{}~{}\nu=1,2,...$ (41)

and if we choose the constant $C_{1}$ as an energy parameter of the system
$C_{1}=E^{2}$

$E_{\nu}=\pm k\sqrt{\frac{1-4\nu^{2}}{8m\nu^{2}}}.$ (42)

We also remind that the vector potential component which is energy dependent
becomes

$\varphi(r)=-\frac{E}{e}+\frac{C_{1}Q}{er^{3}}-\frac{mQ}{er}.$ (43)

###  3.2 Extremal black hole case: $|Q|=M$

_Case 1:_
We can consider that case $M=|Q|$ as the extremal black hole. We take the
function $\lambda(r)$ as

$\lambda(r)=(1-\frac{M}{r})^{2}.$ (44)

We continue with (14) multiplied by $\frac{r^{2}}{M-r}$, then, it becomes

$-(M-r)\phi^{\prime}_{1}(r)+\left(\frac{M}{r}+\frac{(M-r)U^{\prime}(r)}{U(r)}\right)\phi^{\prime}_{1}(r)+\left(\frac{k}{r}+\frac{k^{2}+r^{2}(2m-U(r))U(r)}{M-r}+\frac{kU^{\prime}(r)}{U(r)}\right)\phi_{1}(r)=0.$
(45)

Substituting $U(r)=\frac{C_{1}}{r}$ in (45) gives

$-\phi_{1}^{\prime\prime}(r)+\left(-2C_{1}m+\frac{C^{2}_{1}-k^{2}-2C_{1}mM}{r-M}\right)\phi_{1}(r)=0$
(46)

and $\phi_{1}(r)$ are found to be as

$\begin{split}\phi_{1}(r)=NJ\left(-2\sqrt{-C^{2}_{1}+k^{2}+2C_{1}mM},~{}~{}2\sqrt{2C_{1}m}(M^{2}-2Mr+r^{2})^{1/4}\right)\Gamma\left(1-2\sqrt{-C^{2}_{1}+k^{2}+2C_{1}mM}\right)\end{split}$
(47)

where $J_{n}(2\sqrt{2C_{1}m}(M^{2}-2Mr+r^{2})^{1/4})$ are the Bessel
polynomials. If we take the parameter $C_{1}$ as energy $C_{1}=E$, then,

$-2\sqrt{-C^{2}_{1}+k^{2}+2C_{1}mM}=n=-2\sqrt{-E^{2}+k^{2}+2EmM},$ (48)

and the energy is obtained as

$E_{n}=\frac{1}{2}(2mM\pm\sqrt{4k^{2}+4m^{2}M^{2}-n^{2}}).$ (49)

In this case, vector potential component can be given by

$\varphi(r)=-\frac{m+E}{e}-\frac{C_{1}M}{er^{2}}+\frac{C_{1}+mM}{er}.$ (50)

_Case 2: Solutions in terms of Heun polynomials_
We substitute $r=\frac{M}{z}$ in (14), then,

$\begin{split}-\frac{z^{4}(z-1)^{2}}{M^{2}}\phi^{\prime\prime}_{1}(z)+\left(-\frac{(z-1)z^{3}(3z-2)}{M^{2}}+\frac{(z-1)^{2}z^{4}U^{\prime}(z)}{M^{2}U(z)}\right)\phi^{\prime}_{1}(z)+\\\
(\frac{kz^{2}(-1+k+z)}{M^{2}}+2mU(z)-U(z)^{2}-\frac{k(z-1)z^{3}U^{\prime}(z)}{M^{2}U(z)})\phi_{1}(z)=0.\end{split}$
(51)

After simplification of (51), we use $\phi_{1}(z)=e^{\int\mu(z)dz}Y(z)$ and
get

$\begin{split}Y^{\prime\prime}(z)+\left(\frac{1}{z-1}+\frac{2}{z}+2\mu(z)-\frac{U^{\prime}(z)}{U(z)}\right)Y^{\prime}(z)+(\frac{k-k^{2}-kz}{(z-1)^{2}z^{2}}-\frac{2mM^{2}U(z)-M^{2}U(z)^{2}}{(z-1)^{2}z^{4}}+\\\
\frac{\mu}{z-1}+\frac{2\mu}{z}+\mu(z)^{2}+\frac{kU^{\prime}(z)}{z(z-1)U(z)}-\frac{\mu(z)U^{\prime}(z)}{U(z)}+\mu^{\prime}(z))Y(z)=0.\end{split}$
(52)

In order to transform (52) into a Heun equation, we can use the value of
$\mu(z)$ given below

$\mu(z)=\frac{\epsilon}{z-a}+\frac{1}{2}\frac{2-\gamma+z(\gamma-3+\delta)}{z(z-1)},$
(53)

where $\epsilon,\delta,\gamma$ and $a$ are real parameters, then we obtain

$\begin{split}Y^{\prime\prime}(z)+(\frac{\delta}{z-1}+\frac{\gamma}{z}+\frac{\epsilon}{z-a})Y^{\prime}(z)+(-\frac{2mM^{2}U(z)}{z^{4}(z-1)^{2}}+\frac{M^{2}U(z)^{2}}{z^{4}(z-1)^{2}}+\frac{(-2+2k+3z)U^{\prime}(z)}{2z(z-1)U(z)}-\frac{3U^{\prime}(z)^{2}}{4U(z)^{2}}+\frac{U^{\prime\prime}(z)}{2U(z)}+\\\
\frac{1}{4(a-z)^{2}(z-1)^{2}z^{2}}(4a^{2}k(1-k))+a^{2}\gamma(\gamma-2)+z^{4}(\gamma-3+\delta+\epsilon)(1+\gamma+\delta+\epsilon)+\\\
z(-8ak-4a^{2}k+8ak^{2}-2a^{2}(-2+\gamma(-2+\gamma+\delta))-2a\gamma(-2+\gamma+\epsilon))+\epsilon(2\gamma+\delta)+\\\
z^{3}(4-4k-2(\gamma+\epsilon)(-2+\gamma+\delta+\epsilon)-2a(-3+\gamma^{2}+\delta(-2+\delta+\epsilon)+\gamma(-2+2\delta+\epsilon))))Y(z)=0.\end{split}$
(54)

Considering the Heun differential equation given in the literare [17], [18]:

$w^{\prime\prime}(z)+(\frac{\gamma}{z}+\frac{\delta}{z-1}+\frac{\epsilon}{z-a})w^{\prime}(z)+\frac{\alpha\beta
z-q}{z(z-1)(z-a)}w(z)=0.$ (55)

Our task is to adapt (54) into (55) using an appropriate $U(z)$ trial function
and parameters. Let us use a constant $U(z)$ as

$U(z)=2m$ (56)

and this choice makes the vanishing terms including the black hole mass $M$ in
(54). From $U(z)=2m$ in (11), we obtain

$\varphi(z)=-\frac{E-m+mz}{e}.$ (57)

Then, we can match the coefficient of derivative-free term $Y(z)$ in (54)
using the relation below:

$\begin{split}(\frac{(-2+2k+3z)U^{\prime}(z)}{2z(z-1)U(z)}-\frac{3U^{\prime}(z)^{2}}{4U(z)^{2}}+\frac{U^{\prime\prime}(z)}{2U(z)}+\\\
\frac{1}{4(a-z)^{2}(z-1)^{2}z^{2}}(4a^{2}k(1-k))+a^{2}\gamma(\gamma-2)+z^{4}(\gamma-3+\delta+\epsilon)(1+\gamma+\delta+\epsilon)+\\\
z(-8ak-4a^{2}k+8ak^{2}-2a^{2}(-2+\gamma(-2+\gamma+\delta))-2a\gamma(-2+\gamma+\epsilon))+\epsilon(2\gamma+\delta)+\\\
z^{3}(4-4k-2(\gamma+\epsilon)(-2+\gamma+\delta+\epsilon)-2a(-3+\gamma^{2}+\delta(-2+\delta+\epsilon)+\gamma(-2+2\delta+\epsilon)))){\color[rgb]{1.00,0.00,0.00}-}{\color[rgb]{1.00,0.00,0.00}\frac{\alpha\beta
z-q}{z(z-1)(z-a)}}=0.\end{split}$ (58)

After collecting the terms with respect to the powers of $z^{i},i=0,1,2,...$,
one can find the parameters as

$\displaystyle q$ $\displaystyle=$ $\displaystyle 2(1-k-2ka+2k^{2}a)$ (59)
$\displaystyle\delta$ $\displaystyle=$ $\displaystyle 1-2k,~{}~{}\epsilon=2$
(60) $\displaystyle\gamma\ $ $\displaystyle=$
$\displaystyle-2(k-1),~{}~{}\alpha=\frac{3+4k(k-2)}{\beta},~{}\beta=1.$ (61)

So, (54) turns into

$Y^{\prime\prime}(z)+\left(\frac{1-2k}{z-1}-\frac{2(1-k)}{z}+\frac{2}{z-a}\right)Y^{\prime}(z)+(\frac{(3+4k(k-2)z)-2+2k+4ka-4k^{2}a}{z(z-1)(z-1+k)})Y(z)=0.$
(62)

Again, reminding Heun’s equation (55) where $z=0,1,a,\infty$ are the regular
singular points satisfying the Fuchsian condition
$\epsilon=\alpha+\beta-\gamma-\delta+1$. Matching (55) and (62) gives us
(59)-(61). On the other hand, our system (62) is two -parameters equation. In
[19], Nikiforov-Uvarov method is extended and applied to the general Heun
differential equation. According to the classification of the Heun equation
given in [19], our system in (62) belongs to class-$V$ and using the results
of [19], we get

$Y_{1}(z)=z^{0}(z-1)^{0}(z-a)^{1-\epsilon}p(_{z}),$ (63)

where $p(z)$ is a polynomial which can be found as

$p(z)=a_{1}(z-2k)z^{2k-1}+a_{2}(z-1)^{2k},$ (64)

where $a_{1},a_{2}$ are constants. Using Fuschian condition, we obtain two
roots for the value of $k$:

$k=0,~{}~{}k=1.$ (65)

_Case 3: Solutions in terms of Laguerre functions _
If we apply the following transformation in (14) as

$\displaystyle\bar{\phi}_{1}(r)$ $\displaystyle=$
$\displaystyle\frac{e^{-r/2}r^{3/2}}{r-M}Y(r)$ (66) $\displaystyle U(r)$
$\displaystyle=$ $\displaystyle C_{2}=const$ (67)

where $\lambda(r)=(1-\frac{M}{r})^{2}$, we get

$\begin{split}rY^{\prime\prime}(r)+(1-r)Y^{\prime}(r)+\left(\frac{1}{2}(-1+4C^{2}_{2}m-8C_{2}mM)-\frac{3}{4r}+\frac{1}{4}(1+4C^{2}_{2}-8C_{2}m)r\right)Y(r)=0.\end{split}$
(68)

And $Y(r)$ can be found as

$Y(r)\sim
C\exp\left[\left(\frac{1}{2}-i\sqrt{C_{2}}\sqrt{C_{2}-2m}\right)r\right]r^{\frac{\sqrt{3}}{2}}L_{n}\left(2i\sqrt{C_{2}}\sqrt{C_{2}-2m}r\right)$
(69)

where

$n=-1-\sqrt{3}-2i\sqrt{C_{2}}\sqrt{C_{2}-2m},$ (70)

where $L_{a}(br)$ are the Laguerre polynomials. Then, $\bar{\phi}_{1}(r)$
becomes

$\bar{\phi}_{1}(r)=C\frac{1}{r-M}\exp\left[\left(-i\sqrt{C_{2}}\sqrt{C_{2}-2m}\right)r\right]r^{\frac{\sqrt{3}+3}{2}}L_{n}\left(2i\sqrt{C_{2}}\sqrt{C_{2}-2m}r\right)$
(71)

where $C$ is the normalization constant. To obtain the real values of $n$, we
can arrange $C_{2}$ as :

$\sqrt{C_{2}}\sqrt{C_{2}-2m}=-iC_{3},~{}\Rightarrow~{}~{}C_{2}=\frac{1}{2}(2m\pm\sqrt{C^{2}_{3}+4m^{2}}).$
(72)

By the way, the power of the exponential function should vanish at infinity,
then, one can take $C_{2}$ as

$C_{2}=\frac{1}{2}\left(2m+\sqrt{C^{2}_{3}+4m^{2}}\right).$ (73)

One can find $\varphi(r)$ vector potential component as

$\varphi(r)=\frac{1}{e}\left(m(1-\frac{M}{r})-E\right).$ (74)

##  4 Conclusions

The real eigenvalues and the corresponding spinor solutions are obtained for
the Dirac equation for Reissner-Nordström Black Holes. In our study, we take
more general vector potential in the non-extremal and extremal black hole
cases while the Coulombic $\sim\frac{1}{r}$ potentials were studied in the
previous works [9]. Our study may bring a new point of view to the bound state
solutions for the partcile dynamics near Reissner-Nordström Black Holes. In
this paper, we have also observed that the energy dependent potentials [20],
[21] which may appear in the bound state solutions. We have obtained the bound
state solutions in terms of Airy, Kummer confluent hypergeometric and
Whittaker functions in the case of non-extremal black holes while we have
obtained those in terms of Bessel, Heun and Laguerre functions in the extremal
case.

## References

  * [1] K. Schwarzschild, ”Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften. 7: 189–196.
  * [2] S. Hawking, Commun. Math. Phys. 25 152 1972.
  * [3] Y. K. Ha, Int. J. of Mod. Phys.D 14(12) 2219 2005.
  * [4] A. Burinskii, Int. J. of Mod. Phys. A 29 1450133 2014.
  * [5] R. Emparan and M. Tomašević, J. of High En. Phys. 2020 38 2020.
  * [6] C. L. Pekeris and K. Frankowski, PNAS April 1 1986 83 (7) 1978-1982; https://doi.org/10.1073/pnas.83.7.1978.
  * [7] F. Finster, J. Smoller and Shing-Tung Yau, J.Math.Phys. 41 2173 2000.
  * [8] X. Liu et al, JHEP 10 186 2019.
  * [9] V. I. Dokuchaev and Yu. N. Eroshenko, J. of Exp. and Theo. Phys. 117(1) 72 2013.
  * [10] M. Soffel, B. Muller and W. Greiner, J. Phys. A 10 551-561 1977.
  * [11] A. Zecca, Adv. Stud. in Theo. Phys. 11 297 2017.
  * [12] Y. Lyu and S.-F. Sun, Int. Jour. Theor. Phys., 52 827-836 2013.
  * [13] J. M. Toledo and V. B. Bezerra, Int. J. of Mod. Phys. D 28 1950023 2019 .
  * [14] Chiang-Mei Chen et al, Class. Quantum Grav. 32 195003 2015.
  * [15] D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. 29 465 1957.
  * [16] Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.
  * [17] M. Hortaçsu, An. Meth. for High En. Phys., 8621573 2018.
  * [18] F. Caruso, J. Martins, V. Oguri, Ann. Phys. 347 130 2014.
  * [19] H. Karayer, D. Demirhan and F. Büyükılıç, J. Math. Phys. 56 063504 2015.
  * [20] J. Formanek, R. J. Lombard, J. Mares, Czech. J. of Phys. 54 289 2004.
  * [21] Miloslav Znojil, Hynek Bila, Vit Jakubsky, Czechoslovak J. Phys. 54 1143 2004.