# A Pilot Study on the Comparison of Prefrontal Cortex Activities of Robotic
Therapies on Elderly with Mild Cognitive Impairment

King Tai Henry Au-Yeung, William Wai Lam Chan, Kwan Yin Brian Chan, Hongjie
Jiang∗, and Junpei Zhong∗ K.T. AY, W. C, K.C, and J.Z are with the Department
of Rehabilitation Sciences, The Hong Kong Polytechnic University, Hong Kong.
K.T. AY, W. C, K.C are in Master in Occupational Therapy programme, The Hong
Kong Polytechnic University, Hong Kong. K.T. AY is the first author of the
work.H.J is with School of Shien-Ming Wu Intelligent Engineering, South China
University of Technology, Guangzhou China.
∗J. Zhong and H. Jiang are Corresponding authors:
Junpei Zhong<EMAIL_ADDRESS>Hongjie Jiang<EMAIL_ADDRESS>
###### Abstract

Demographic shifts have led to an increase in mild cognitive impairment (MCI),
and this study investigates the effects of cognitive training (CT) and
reminiscence therapy (RT) conducted by humans or socially assistive robots
(SARs) on prefrontal cortex activation in elderly individuals with MCI, aiming
to determine the most effective therapy-modality combination for promoting
cognitive function. This pilot study employs a randomized control trial (RCT)
design. Additionally, the study explores the efficacy of Reminiscence Therapy
(RT) in comparison to Cognitive Training (CT). Eight MCI subjects, with a mean
age of 70.125 years, were randomly assigned to “human-led” or “SAR-led”
groups. Utilizing Functional Near-infrared Spectroscopy (fNIRS) to measure
oxy-hemoglobin concentration changes in the dorsolateral prefrontal cortex
(DLPFC), the study found no significant differences in the effects of human-
led and SAR-led cognitive training on DLPFC activation. However, distinct
patterns emerged in memory encoding and retrieval phases between RT and CT,
shedding light on the impacts of these interventions on brain activation in
the context of MCI.

##  I Introduction

Demographic shifts are occurring in some countries due to the aging of
populations, leading to a notable increase in mild cognitive impairment (MCI)
and its progression to dementia. The MCI manifests as a gradual decline in
cognitive abilities and dementia, encompassing complex attention, executive
function, learning and memory, language, perceptual-motor skills, and social
cognition. The Diagnostic and Statistical Manual of Mental Disorders, Fifth
Edition (DSM-5) [1], introduces a unique category for social cognition within
neurological cognitive illness (NCD), impacting interpersonal dynamics and
elevating the risk of dementia through social isolation (National Academies of
Sciences).

It has been shown that individuals with MCI can benefit significantly from
social interaction [2, 3, 4], fostering positive attitudes and counteracting
cognitive decline. The severity of cognitive impairment is linked to the
intensity of negative experiences, while positive moods associated with social
engagement enhance cognitive function and reduce the likelihood of dementia
development [5]. Chronic depression, prevalent among elderly individuals with
MCI, poses a risk to cognitive abilities, emphasizing the connection of mental
state and cognitive health [6].

Occupational therapists typically administer therapies for individuals with
Mild Cognitive Impairment (MCI). One such example is cognitive-oriented
therapy, which includes cognitive training (CT), cognitive stimulation
(encompassing reality orientation), and cognitive rehabilitation. Recent
evidence underscores the efficacy of cognitive-oriented therapy in enhancing
cognitive functions among individuals with MCI [7]. Notably, both CT and
reality orientation exhibit positive cognitive effects [8]. CT is postulated
to confer benefits by fostering skill improvement through repetitive practice
[8], emphasizing the retention and enhancement of existing cognitive abilities
to enable patients to adeptly manage contemporary tasks [9].

Alternatively, Reminiscence Therapy (RT) concentrates on offering MCI patients
materials associated with their past to evoke recollections of prior
experiences [10]. The theoretical underpinning posits that RT can facilitate
the retrieval of long-term memories, consequently leading to improved
cognition, enhanced quality of life, and behavioral modifications [11].
Various objects, such as old books and pictures, capable of eliciting memories
related to individuals or events from the patients’ past life are employed
[9]. While some studies suggest that RT may enhance cognitive function in
dementia patients[10], not all studies concur with these findings.

Despite the effectiveness of multiple interventions in MCI, they are
impractical due to scarce human resources [12]. Therefore, cost-effective
technological solutions, such as tablets, gaming consoles and robots, have
been adopted [13]. Socially Assistive Robots (SARs) have been developed to
facilitate more comfortable interactions, providing non-verbal and emotional
interactions that could motivate and engage patients in cognitive tasks[14].
The introduction of social assistive robots (SARs) offers a potential positive
impact on mood, similar to interactions with real individuals. Some studies
combining cognitive treatments with SARs have shown promising results in
dementia [15]. The NAO robot, due to its low cost and wide-ranging
capabilities, has been commonly used in human-robot studies [16]. Clinically,
NAO-led computerized CT was found beneficial to MCI [17]. Engaging with SARs,
such as NAO, can activate the prefrontal cortex, potentially uplifting the
mood of seniors with MCI, akin to conversations with humans [18]. In this
context, SARs function as companions, employing both verbal and non-verbal
communication and imitating human actions through embodied structures. Joint
action robots, simulating genuine interaction, utilize verbal dialogue, facial
expressions, and gestures to activate cognitive functions, with respondents
expressing preferences for specific interaction sides, enhancing the perceived
realism of engagement with robots [19].

In this project, we would like to measure the effects of RT and CT in the
activation of prefrontal cortex, using both human- and robot-led
interventions. Recruiting MCI elderly as subjects, to the best of our
knowledge, this is the first study will answer the following questions:

  * •

What are the effects of using SARs (i.e. the NAO robot) to conduct RT to MCI,
compared with human-led interventions;

  * •

What are the differences of effects on prefrontal cortex activities of using
CT and RT to MCI;

  * •

What is the best combination of “therapy (CT/RT) x modality (robot-/ human-
led)” in terms of its activation on prefrontal cortex activities.

##  II Literature Review

###  II-A Neurophysiological Deficits with MCI and the Mechanisms for
Interventions

Short-term memory, attention, and orientation are the most reliable predictors
in screening for MCI [20]. They are also predictors for early-onset AD, which
is associated with poor prognosis[21].

The physiological underpinnings of MCI and AD are the medial temporal lobe,
temporal lobe glucose metabolism, and episodic memory function for orientation
[21], and the prefrontal cortex (PFC) for short-term memory and attention[22].
Lateral PFC is critical in attention and memory[23], while the dorsolateral
and ventrolateral prefrontal cortices (DLPFC and VLPFC) are responsible for
working memory selection, monitoring and manipulation [24], as well as working
memory encoding and retrieval [25] respectively. DLPFC and VLPFC have close
functional relationships when memory recall tasks are elicited [24], [26].
Recent research found that the DLPFC-VLPFC network relationship was
emotionally-mediated, suggesting that memory encoding and retrieval are
influenced by mood [27]. Further, recent studies have shown hypoperfusion of
oxygenated blood in the DLPFC during memory tasks (e.g. word retrieval) in
elders with MCI when compared to the well elderly counterparts[28]. On the
other hand, recent research states that orientation in space, time, and person
are related to the default mode network (DMN) including the medial frontal
cortex, thus indicating that frontal lobe lesion is associated with
orientation dysfunction [29].

Thus, it could be deduced that the frontal lobe is one of the most susceptible
regions to early-stage neurocognitive deficits. The key to preventing
cognitive decline may lie in frontal lobe activation.

The finding postulates an anatomic-functional deduction that any treatments
with stimuli that activate the frontal lobe (hence exercising orientation,
short-term memory, and attention) can theoretically prevent neurocognitive
decline, regardless of RT or CT. Thus, frontal lobe activation measurements
were justified to indicate treatment effectiveness. As this study focused on
the memory function, DLPFC was selected as the main region of interest (ROI).

###  II-B Social Robots for MCI

A comprehensive review of studies investigating the use of robots in therapy
for individuals with Mild Cognitive Impairment (MCI) reveals diverse
interventions across various countries. Notable robots include Kompai, Paro,
TIAGo Iron, and NAO, each employed in distinct experimental setups.

Studies, such as those by Agrigoroaie et al. [30] and Chen et al. [31],
highlight individual and group interventions utilizing robots like Kompai and
Paro in elderly care facilities and long-term care centers. Notably, the
duration of interventions ranges from 2 days to 8 weeks.

Reminiscence Therapy (RT) is explored in studies like Granata et al. [32],
where robots like Kompaï engage individuals with materials evoking past
memories. This approach aims to improve cognition, quality of life, and
behavior.

Additionally, investigations by Park et al. [33] and Pino & Palestra [34]
utilize humanoid robots, Sil-bot and NAO, in group interventions at community
centers. The studies emphasize the significance of diagnostic criteria,
including the Mini-Mental State Exam (MMSE), Petersen guidelines, and
cognitive function assessments.

These robotic developments as well as studies with MCI in these studies
contribute valuable insights into the potential of robotic interventions for
cognitive therapy in MCI.

###  II-C Using Functional Near Infrared Spectroscopy for Subjective
Measurement

In prior HRI investigations involving individuals with MCI, efficacy
assessment commonly relied on subjective measures such as ratings,
observations, and feedback. However, objective assessments at the
neurophysiological level have been lacking. To address this gap, Functional
Near Infrared Spectroscopy (fNIRS) should be employed to non-invasively
measure brain activity during interventions [35, 36]. The fNIRS, capable of
tracking changes in oxyhemoglobin concentration as an indicator of regional
brain activity, provides a real-time, direct, and cost-effective approach for
evaluating the performance of CT and RT with MCI.

In our study, the choice of fNIRS over functional assessments like the
Montreal Cognitive Assessment (MoCA) was justified due to the heterogeneous
etiologies of MCI, making the anti-Alzheimer’s disease (AD) deterioration
effects of treatments challenging to discern within functional assessments.
Additionally, improvements in cognition among MCI elders may not manifest
overtly in functional assessments for a relatively prolonged period.

Moreover, fNIRS has demonstrated effectiveness in detecting cognitive
activation in both robot- and human-led interventions, particularly in tasks
involving memory training (i.e. CT and RT) and human-robot interactions.
Furthermore, this study seeks to explore whether CT and RT using SARs can
yield comparable benefits to human-led interventions, an area that has not
been previously explored. Given inconclusive findings in prior RT studies,
largely attributed to methodological flaws, this research aims to refine
methodologies for a more conclusive evaluation of RT’s effects in comparison
to CT, both led by SARs and humans.

##  III Methodology

###  III-A Experimental Setting

The “HK-MoCA 5-minutes” [37] was used to screen whether a subject has MCI. It
is of high correlation with HK-MoCA original version ($r=0.87;p<0.001$). A
total of $8$ subjects were recruited by signing up voluntarily from elderly
homes and public posters. Inclusion criteria: (1)Adult aged 50 or above,
(2)Mild Cognitive Impairment (MCI), (3)have lived in Hong Kong for at least 25
years, and (4) able to comprehend Cantonese. Subjects not previously diagnosed
were screened by Montreal Cognitive Assessment-5 minutes (MoCA-5), and
required to score between the 2nd and 16th percentile, considering age and
educational level. Exclusion criteria were: (1)medical history of frontal lobe
lesions, (2)auditory defects, (3)medical diagnosis of Parkinson’s Disease,
Schizophrenia, Attention Deficit Hyperactivity Disorder (ADHD), (4)have
behavioral or emotional symptoms. All these conditions would have potential
implications for cognitive impairments, including attention and memory span
[38, 39].

After satisfying the inclusion/exclusion criteria, all qualified participants
were assigned a random code. The subjects, assessment testers, and treatment
experimenters were different persons, and thus were blinded to the treatment
grouping. Each participant was given HK$100 as an incentive after having
completed all the procedures, to reduce attrition.

All the participants were divided into $2$ groups. The experimental groups is
summarized in Tab. I. To test question 1, the recruited subjects were divided
into 2 groups randomly: the “robot-led” group and the “human-led” group. To
test question 2, the subjects were then randomly divided into two groups. They
were given 2 types of stimuli (verbal stories) with a clear indication of time
sense to simulate CT (present information) or RT (past information): (1)
Recent stories happening in 2023 (present-time sense), which is labeled as
“present’; and the questions for recent stories focus on verbal memory recall
of the story contents, simulating cognitive training (CT); (2) Past stories
involving elements more than 20 years ago (past-time sense), which is labeled
as “past”; and the questions for past stories focus on sharing of subject’s
personal experiences during the past decade depicted by the stories,
simulating reminiscence therapy (RT). To counterbalance the order effect of
stimuli presentation, each group was equally divided randomly into 2
subgroups, with either the sequence of administering “CT (present) first, RT
(past) second” (Group 1a, 2a) or “RT (past) first, CT (present) second” (Group
1b, 2b).

The experimental procedure was summarized in Figs 1 and 2. One intervention
session was provided to each participant. In the session, fNIRS recordings
were done during the story telling (T1-T4), participants were given $1$ minute
to rest to establish the signal baseline (T0). After that, each subject was
given a “Story” interaction, Story 1 (T1), either by the NAO robot or human.
Within each “Story” (Fig 2), participants were first given the first part of
verbal stimuli (“encoding”) lasting for $40$ seconds, followed by $2$
questions (“retrieval”) lasting for a total of $40$ seconds. Then, the second
part of the verbal stimulus was given for $40$ seconds and was followed by
another $2$ questions. Stories $2$ to $4$ (T2-T4) were then given sequentially
following the format of T1. After each interaction, participants were given a
1-minute break.

Figure 1: Study Procedure

Figure 2: Procedure of the “Story” session

Procedure of each story sessions (T1-T4). “Stimuli” refers to the verbal
stimuli either given by a NAO robot or by a human. “Q1” to “Q4” are “Q & A”
for the participants, each lasting for $20$ seconds.

TABLE I: Experimental Groups | Robot-led | Human-led
---|---|---
Group Number | 1a | 1b | 2a | 2b
Story 1 |  |  |  |
Story 2 | Present | Past | Present | Past
Story 3 |  |  |  |
Story 4 | Past | Present | Past | Present

###  III-B Experimental stimuli standardization

The stimuli employed in CT and RT necessitate standardization due to
variations in their respective protocols. In alignment with the recommendation
proposed by Cotelli et al. [40], we implemented a standardized protocol,
utilizing nearly identical storylines for CT and RT interventions. This
approach aimed to eliminate potential confounding factors that might obscure
the discernment of CT/RT’s beneficial effects.

Our focus centered on introducing a singular distinction among stimuli,
specifically the temporal orientation towards either the present or past. This
deliberate manipulation sought to simulate the differential approaches of CT
and RT for the purpose of standardization.

The experimental stimuli comprised succinct verbal narratives in Cantonese,
addressing commonplace aspects of life and familiar objects prevalent in Hong
Kong. Two distinct topics, namely “Leisure” and “Living”, were chosen.
Standardization ensured that each stimulus consisted of $132$ words, delivered
in a neutral Cantonese tone, and maintained an equal duration for a fair
comparative analysis. The language used in the narratives was drawn from
Lexical Lists for Chinese Learning in Hong Kong, ensuring the
comprehensibility of stimuli for participants with potentially lower
educational levels. This meticulous approach ensured that the stimuli remained
accessible and understandable to all study participants.

###  III-C Outcome measures

Outcome measures were assessed starting from set up (T0) until the end of the
experiment (T4).

####  III-C1 fNIRS Measurement and Removal of Confounding Noises

We utilized a portable fNIRS device, NIRSIT (OBELAB, South Korea), specialized
in measuring hemodynamic responses in the prefrontal cortex [41]. The device
is equipped with 32 detection sensors sampling at a rate of 8.138 Hz,
generating a maximum of 204 channels.

To enhance data quality, several measures were implemented to minimize
confounding signals or ”noise.” Following best-practice recommendations by
Yücel et al. [42], the following steps were taken: (1) setting the Signal-to-
Noise Ratio (SNR) threshold to 20 or above to eliminate cardiac power noises,
(2) mitigating motion artifacts by instructing subjects to minimize movements
and speak only when necessary, and (3) employing the ”ABC-A” experimental
paradigm to balance out confounding systemic signals, such as high blood
pressure. Subjects were placed in rooms with minimal distraction to further
ensure data accuracy. Brain activities in the frontal lobe were subsequently
measured using the established fNIRS setup. The overall setup of the Robot-led
intervention was shown in Fig. 3.

Figure 3: Robot-led Intervention with the fNIRS Measurement

####  III-C2 Acceptability of SAR

Additionally, the Robotic Social Attributes Scale (RoSAS) developed by
Carpinella et al.[43] was used to measure the perception of NAO by the
participants. The RoSAS has been validated empirically and uses 18 items to
measure the social acceptability of robots regardless of the context and robot
type.

###  III-D Data Analysis

####  III-D1 Data Processing

The fNIRS data was processed using the NIRSIT analysis tool (v2.1, OBELAB).
The raw data was converted to oxyhemoglobin (HbO2) and deoxyhemoglobin (HbR)
concentration changes by using the modified Beer-Lambert Law [44]. A cut-off
frequency of $0.005-0.1$ Hz was applied to eliminate ambient and physiological
noises [45]. To minimize signal drifting, the average value from the 10s
period before each task was used as the baseline for the block average.
Channel signals were averaged based on the corresponding brain areas for each
subject 4. The averaged data for each individual were categorized into
“encoding” phases (each lasts for 40 seconds) and “retrieval” phases (each
lasts for 40 seconds) based on the tasks given as Fig.2 showed. These data
were further grouped into “robot-led” group, “human-led” group, “CT” segment
and “RT” segment for comparison.

Figure 4: Grouping of the NIRSIT Channels based on the Brodmann Areas

####  III-D2 Statistical Analysis

Baseline characteristics and channel data were compared using independent-
samples t-test, paired t-test, and chi-square test. A probability of $p<0.05$
was considered statistically significant. For research question 1,
independent-samples t-tests were performed: (1) between the “robot” and
“human” groups, (2) between “robot-CT” and “human-CT” groups, (3) between
“robot-RT” and “human-RT” groups. For research question 2, paired t-tests are
performed between: (1) “CT” and “RT” interventions within subjects, (2) “CT-
robot” and “RT-robot” interventions within subjects, and (3) “CT-human” and
“RT-human” intervention within subjects. In each comparison, data was also
finely analyzed in two different phases: “encoding” and “retrieval” phases.
Statistical analyses were performed using SPSS.

##  IV Results

###  IV-A Baseline

Baseline characteristics of the subjects are listed in Tab.II. No significant
difference was found in terms of age ($t(6)=0.068,p>0.05$), gender
($\chi^{2}(1)=0.533,p>0.05$), and MoCA-5-minutes baseline scores
($t(6)=-0.208,p>0.05$) between “robot-led” and “human-led” groups. There was
no attrition in the study.

TABLE II: Baseline Characteristics of Subjects | Robot-led | Human-led |  $t(6)$ or $\chi^{2}(1)$ | $p$
---|---|---|---|---
Age | $70.2500\pm 6.02080$ | $70.0000\pm 4.24264$ | $0.08$ | $0.948$
Gender (Male) | $2(50\%)$ | $1(25\%)$ | $0.533$ | $0.465$
MoCA Score | $19.0000\pm 2.61406$ | $19.3750\pm 2.49583$ | $-0.208$ | $0.842$

###  IV-B Between-group comparison between “robot-” and “human-led”
interventions

HbO2 changes from the baseline of the DLPFC in “robot-” and “human-led” groups
were shown in Tab.III. During the encoding and retrieval phase, there was no
significant difference between “robot-led” and “human-led” groups (encoding
phase: right: $t(6)=1.284,p=0.247$; left: $t(6)=1.341,p=0.229$)(retrieval
phase: right: $t(6)=0.017,p=0.987$; left : $t(6)=-0.206,p=0.843$). Both groups
gave negative HbO2 changes across left and right cortices during the encoding
phase, while in the retrieval phase, both groups gave positive HbO2 changes.

TABLE III: Comparison of HbO2 changes in the DLPFC during robot- and human-led Intervention Comparison |  Condition |  dPFC, right |  dPFC, left
---|---|---|---
Robot- & Human-led |  Encoding phase (all inclusive) |  $NAO=-0.00000925\pm 0.00031734;Human=-0.00028580\pm 0.000291340,t=1.284,p=0.247$ |  $NAO=-0.00008150\pm 0.00017997;Human=-0.00038297\pm 0.00041221,t=1.341,p=0.229$
Retrieval phase (all inclusive) |  $NAO=0.00008987\pm 0.00015995;Human=0.00008840\pm 0.00007983,t=0.017,p=0.987$ |  $NAO=-0.00008227\pm 0.00011394;Human=0.00011452\pm 0.000291100,t=-0.206,p=0.843$
CT, encoding phase |  $NAO=-0.00005543\pm 0.00025933;Human=-0.00023994\pm 0.00037567,t=0.808,p=0.450$ |  $NAO=-0.00004328\pm 0.00022114;Human=0.00000276\pm 0.00039315,t=-0.204,p=0.845$
RT, encoding phase |  $NAO=0.00003687\pm 0.00037755;Human=-0.00033185\pm 0.00022452,t=1.679,p=0.144$ |  $NAO=-0.00011968\pm 0.00024224;Human=-0.00076874\pm 0.00046551,t=2.474,p=0.048^{*}$
CT, retrieval phase |  $NAO=0.00006849\pm 0.00021184;Human=0.00001517\pm 0.00030186,t=0.289,p=0.782$ |  $NAO=0.00002122\pm 0.00016662;Human=-0.00005532\pm 0.00037510,t=0.373,p=0.722$
RT, retrieval phase |  $NAO=0.00011141\pm 0.00015298;Human=0.00016193\pm 0.00017686,t=-0.432,p=0.681$ |  $NAO=0.00014311\pm 0.00006981;Human=0.00028435\pm 0.00028277,t=-0.970,p=0.370$

###  IV-C Comparison between HbO2 changes in CT and RT

HbO2 changes from baseline in the DLPFC during CT and RT stories were compared
using paired t-test in Tab.IV. During both encoding and retrieval phases, a
significant difference was found between CT and RT in the left DLPFC
(Encoding: $t(7)=2.668,p=0.032$; Retrieval: $t(7)=-2.592,p=0.036$). In the
encoding phase, both CT and RT gave negative HbO2 values, with RT giving more
negative HbO2 change. During the retrieval phase, RT gave positive HbO2
changes, in contrast to the negative change in CT. Comparing CT and RT with
the consideration of intervention medium (robot or human), the only
statistically significant difference was found in human-led group x encoding
phase in left DLPFC, where CT gave significantly higher HbO2 change (though
close to zero) than the negative change in RT ($t(7)=6.146,p=0.009$).

TABLE IV: Comparison of subject HbO2 changes in the DLPFC during CT and RT Comparison |  Condition |  dPFC, right |  dPFC, left
---|---|---|---
Both groups |  Encoding phase |  $CT=-0.00014768\pm 0.00031469,RT=-0.00014748\pm 0.00034863,t=-0.003,p=0.998$ |  $CT=-0.00002025\pm 0.00029632,RT=-0.00044421\pm 0.00048825,t=2.668,p=0.032*$
Retrieval phase |  $CT=-0.00004183\pm 0.000243093,RT=-0.00013667\pm 0.00015545,t=-0.803,p=0.448$ |  $CT=-0.00001705\pm 0.00027179,RT=0.00021373\pm 0.00020508,t=-2.592,p=0.036*$
Robot-led |  Encoding phase |  $CT=-0.00005543\pm 0.00025933,RT=-0.00003687\pm 0.00037755,t=-1.397,p=0.258$ |  $CT=-0.00004328\pm 0.00022114,RT=-0.00011968\pm 0.00024224,t=-0.521,p=0.638$
Retrieval phase |  $CT=0.00006849\pm 0.00021184,RT=0.000111417\pm 0.00015298,t=-0.465,p=0.673$ |  $CT=0.00002122\pm 0.00016662,RT=0.00014311\pm 0.00006981,t=-2.097,p=0.127$
Human-led |  Encoding phase |  $CT=-0.00023994\pm 0.00037567,RT=-0.00033185\pm 0.00022452,t=0.884,p=0.442$ |  $CT=-0.00000276\pm 0.00039315,RT=-0.00076874\pm 0.00046551,t=6.146,p=0.009**$
Retrieval phase |  $CT=0.00001517\pm 0.00030186,RT=0.00016193\pm 0.00017686,t=-0.627,p=0.575$ |  $CT=-0.00005532\pm 0.00037510,RT=0.00028435\pm 0.00028277,t=-2.118,p=0.124$

###  IV-D Acceptability of SAR

Using the SAR survey, most subjects from the robot-led group expressed
positive opinions towards the NAO robot (83.3% agree with “competence”, 54.2%
agree with “warmth”). Only a small proportion of subjects expressed negative
sentiment towards the SAR (4.2% agree with “discomfort”) (Fig. 5.

Figure 5: Summary of the RoSAS

##  V Discussion

###  V-A Comparative Analysis of the Impact of Human- and Robot-Led
Interventions on MCI

The alterations in HbO2 concentration within the DLPFC during both robot-led
and human-led interventions did not exhibit significant differences. This
suggests a similarity in DLPFC activation between the two intervention
modalities, indicating that robot-led interventions may evoke brain activation
patterns comparable to those observed in traditional human-led interventions.
Recent findings supporting the positive impact of NAO robot-led cognitive
training further underscore this notion [16]. This study pioneers the
comparison of such effects with therapies conducted by human practitioners,
suggesting that robot-led interventions could serve as a viable alternative in
therapy sessions for elders with MCI.

In the context of the memory retrieval phase, both robot-led and human-led
groups displayed increased DLPFC activation, as evidenced by positive HbO2
changes. This aligns with contemporary research highlighting PFC activation in
MCI subjects during memory retrieval phase [46, 47, 48], albeit some results
of these studies showed the PFC activation levels in MCI during memory
retrieval phase were actually lower than those of the cognitively-healthy
elders. Consequently, memory retrieval tasks mediated by both humans and
robots may induce similar PFC activation patterns.

Contrastingly, during the memory encoding phase in both robot and human
groups, the DLPFC exhibited reduced activation, i.e. deactivation, compared to
the resting state. This contradicts established studies suggesting PFC
activation during memory encoding tasks [47, 48]. However, some studies have
reported PFC hypoperfusion in MCI patients during memory tasks, where the PFC
is less activated compared to those with normal cognitive function [28]. It
has also been observed that lateral prefrontal and parietal brain regions are
activated during tasks involving memory and executive function [28, 49]. Bokde
et al. [49] proposed that this phenomenon could be attributed to limited brain
resources in MCI, leading to an impaired ability to compensate for age-
associated hypoperfusion in the PFC. Hence, a similar explanation could be
posited for the reduced DLPFC activation in this study: during memory encoding
tasks, brain regions other than the PFC might have been activated, diverting
oxygenated blood supply away from the PFC. Considering the limited pool of
brain resources in elders with MCI, this might result in diminished
oxygenation in the PFC.

##  VI Comparative Analysis: Reminiscence Therapy (RT) vs. Cognitive Training
(CT)

###  VI-A Encoding Phase

As two of the most widely utilized therapeutic approaches for Mild Cognitive
Impairment (MCI), RT and CT exhibit three significant differences during the
encoding phase.

The first crucial distinction is evident in the HbO2 concentration changes in
the left DLPFC. Both RT and CT display negative values during this phase,
signifying deactivation. However, RT results in more negative values than CT,
indicating a greater deactivation of the left DLPFC when processing past-
information in RT compared to present-information in CT.

Our study pioneers the investigation of the therapeutic effects of RT on MCI
elders using fNIRS brain activation data. Unlike prior research that primarily
focused on cognitively intact elders’ memory retrieval phase, our finer
analysis delves into both memory encoding and retrieval phases. Notably,
studies before 2023 relied on post-intervention outcome measurements, such as
functional, cognitive, or mood assessments, yielding inconclusive results
[50].

One plausible explanation for DLPFC deactivation during verbal memory encoding
aligns with Bokde et al.’s postulation that MCI elders experience age-related
hypoperfusion in the PFC [49]. The cognitive demands of the encoding process,
perceived as cognitively demanding by the subjects, may lead to DLPFC
hypoperfusion as limited cognitive resources are directed to other brain
regions.

These findings find support in other fNIRS/fMRI studies exploring verbal
memory encoding and neuro-correlates. Emch et al. [51] revealed that high
attentional control in verbal memory encoding led to holistic cerebral blood
flow distribution to the PFC, hippocampus, parietal cortex, and cerebellum. In
elders with MCI, facing cognitively demanding verbal memory encoding triggered
blood flow to the entire PFC and other structures, outside the left DLPFC,
explaining its hypoperfusion.

Further, RT encoding caused even more hypoperfusion than CT encoding, and
previous fNIRS/fMRI studies provide insights:

  1. 1.

Carelli & Olsson [52] demonstrated that present time-sense activates superior
medial frontal cortex (BA32), while past time-sense activates medial frontal
cortex (BA10). As the left DLPFC is located in BA33-35, closer to present-time
activated cortex, CT induces less hypoperfusion compared to RT, where blood
flow must travel a longer distance to activate BA10.

  2. 2.

Skye et al. [53] emphasized the involvement of the time-orientation network
(precuneus, medial temporal lobes, and occipitotemporal cortex) in time
orientation, integral for MCI elders. RT’s longer time-distance than CT in the
universal time frame drains more blood to the whole time-orientation network
from the left DLPFC, causing increased hypoperfusion.

  3. 3.

Ferreri et al. [54, 55] attributed older adults’ DLPFC reduced activation to
facilitating semantic background in stimuli, for example background music-
playing instead provided a richer context to lessen DLPFC verbal memory
encoding demand and hence reduced activation, compared to silence. In our
study, RT’s story content and obsolete lifestyles related to the ”1970s” are
highly time-specific and contrast distinctively to subjects’ present-day
lives, compared to CT’s less time-specific modern lifestyles as described in
”2023”. Therefore, ”1970s” time-specific semantic background could benefit
MCIs’ verbal memory encoding, by providing a helpful, highly distinctive time-
context and therefore less demand on DLPFC, leading to increased
hypoperfusion.

###  VI-B Memory Encoding Phase - Human-led Group

The second significant difference emerges during the memory encoding phase in
the human-led group. HbO2 concentration changes in the left DLPFC show a
significant difference between CT and RT. RT results in negative values,
indicating deactivation, while CT shows slightly positive values. This
suggests more prominent left DLPFC deactivation when processing past-
information in RT, particularly in the human-led group.

The explanation builds upon previous paragraphs, with additional
considerations for the human-led group. In this context, MCI subjects,
compared to the robot-led group, manage RT past-information stimuli from human
speakers, demanding attention to facial features. Bardon et al. [56] noted the
activation of ”face neurons” in the inferior temporal cortex, further draining
limited cognitive resources from the left DLPFC in MCI elders. With MCI’s
inherent limited attentional resources to verbal memory encoding and RT’s past
time-sense, the simultaneous tri-functional demand exacerbates left DLPFC
hypoperfusion during RT encoding in the human-led group.

###  VI-C Retrieval Phase

The final significant difference surfaces in HbO2 concentration changes in the
left DLPFC between CT and RT during the retrieval phase. RT yields positive
values, indicating activation, while CT shows negative values. This suggests
left DLPFC activation during sharing familiar past in RT, contrasting with
deactivation during recalling recently-heard story-stimuli in front of
experimenters.

The positive effect of RT during retrieval is echoed by Kobayashi et al. [57],
linking frontal lobe activation to positive mood induction during RT memory
retrieval tasks. This emphasizes the importance of considering RT as a mood-
oriented intervention, with the DLPFC-VLPFC network being emotionally-mediated
in verbal memory tasks [27].

CT’s brain deactivation during retrieval may be explained by cognitive
demands, as discussed earlier. Notably, CT shows positive values in the robot-
led group but negative values in the human-led group during the retrieval
phase. This discrepancy may be attributed to elders’ self-ego or anxiety
issues about performance in front of humans, causing brain deactivation.
Socially-assistive robots may offer advantages in CT for elders with MCI,
alleviating performance anxiety [58].

###  VI-D Limitations and Future Directions

This study has several limitations. Firstly, the sample size is small due to
practical and study design concerns. The study faced constraints related to
budget limitations and time constraints in recruiting subjects. Additionally,
there was a limited availability of subjects diagnosed with MCI in the
community. On the positive side, the study was conducted on a pilot basis,
focusing on providing preliminary data for a larger-scale study and checking
the feasibility of the proposed protocol. However, the small sample size may
introduce bias, and any conclusions derived from statistical analysis should
be approached with caution.

Furthermore, the absence of a control group in the study means that the
effectiveness of the intervention compared to non-intervention could not be
determined.

Future studies should prioritize a larger sample size to enhance the
reliability and generalizability of the findings. Including control groups
would provide valuable insights into the comparative effectiveness of
interventions for elders with MCI. The experimental protocol can be refined by
employing clearer instructions to minimize confusion among subjects and
adopting more flexible protocols. Measures such as using chairs with headrests
can be implemented to mitigate motion artifacts.

Exploring future directions, the application of SARs in CT and RT warrants
investigation. Subsequent studies could delve into the brain activation
patterns during the memory encoding and retrieval phases. Given the recorded
reduced activation of the PFC during memory encoding, a larger sample size in
future studies can validate and elucidate the observed brain activation
patterns. Moreover, as RT has demonstrated comparable effects to CT,
especially in therapies involving the sharing of past experiences, future
research should explore the influence of subject engagement levels and anxiety
factors. This avenue presents a promising direction for further research to
clarify the inconclusive effects noted in previous studies with methodological
flaws. It is anticipated that future research, incorporating larger sample
sizes and methodological refinements, will contribute to a more robust
affirmation of the therapeutic effects of RT.

##  VII Conclusion

In conclusion, robot-led interventions demonstrate comparable effectiveness to
human-led interventions in addressing cognitive challenges among elders with
Mild Cognitive Impairment (MCI). The positive reception of Socially Assistive
Robots (SARs) by the subjects suggests their potential as a novel therapeutic
approach in MCI rehabilitation.

However, Cognitive Training (CT) and Reminiscence Therapy (RT) exhibit
distinct effects on brain activation. Specifically, the act of listening to
information during the memory encoding phase may lead to Prefrontal Cortex
(PFC) deactivation, indicating a cognitive demand among elders with MCI.
Notably, listening to past information (RT) appears to impose a greater
cognitive demand compared to listening to present information (CT). This
pattern is evident in the human-led group but not observed in the robot-led
group. Interestingly, discussing past information during RT prompts more PFC
activation than recalling present information, suggesting that RT may offer
advantages over CT when encouraging elders with MCI to share their familiar
past stories. The underlying mechanisms behind these findings were explored,
and the study proposes several clinical implications and avenues for future
research.

## Acknowledgment

The authors would like to thank Professor Sam Chan, Professor Stella Cheng,
Ms. Vera Lam and Ms. Anita Ngan, The Hong Kong Polytechnic University, for
their occupational therapy teaching about dementia and ageing as well as
countless support throughout this study.

Mr. Au-Yeung King Tai, the first author, would also like to thank Mr. Johnny
Lam, The Hong Kong Polytechnic University, for his advice as the author’s
academic advisor.

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# On natural densities of sets of some type integers

Dmitry I. Khomovsky  Lomonosov Moscow State University, Moscow, Russia. Email:
<EMAIL_ADDRESS>
###### Abstract

Let $a_{0}=b_{0}=0$ and $0<a_{1}\leq b_{1}<a_{2}\leq b_{2}<\ldots\leq b_{n}$
be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_{j},b_{j}]\right)$ be the
number of integers between $1$ and $x$ such that all exponents in their prime
factorization are in $\bigcup_{j=1}^{n}[a_{j},b_{j}]$. The following formula
holds:

$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_{j},b_{j}]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$

In this paper, we prove this result and then generalize it.

##  1 Introduction

In [1], Rényi and Turán proved the following theorem.

######  Theorem 1.

Let $\Delta$ be a function on natural numbers such that $\Delta(1)=0$ and for
a positive integer $u=\prod_{i=1}^{r}{p_{i}}^{\alpha_{i}}>1$ the following
holds: $\Delta(u)=\sum_{i=1}^{r}(\alpha_{i}-1).$ Then, for any integer $k\geq
0$, there exists a density $d_{k}$ of the set of integers $y$ such that
$\Delta(y)=k$, and the following identity holds:

$\sum\limits_{k=0}^{\infty}d_{k}z^{k}=\prod\limits_{p}\left(1-\frac{1}{p}\right)\left(1+\frac{1}{p-z}\right),$
(1)

where $|z|<2.$

It should be noted that if we put $z=0$, then we get the density of square-
free numbers. We will show that other densities of subsets of integers can be
obtained by analogy. To do this, we need to modify the function $\Delta$ in a
certain way and carry out the calculations that were used in the paper [1].

We need the following classical result [2].

######  Theorem 2.

Let $f$ be a multiplicative arithmetic function such that the series
$\sum_{n=1}^{\infty}f(n)$ is absolutely convergent, then

$\sum_{n=1}^{\infty}f(n)=\prod\limits_{p}\left(1+\sum\limits_{i=1}^{\infty}f(p^{i})\right).$
(2)

######  Corollary 1.

Let $g$ be a function on natural numbers such that $g(1)=0$ and for a positive
integer $u=\prod_{i=1}^{r}{p_{i}}^{\alpha_{i}}>1$ the following holds:
$g(u)=\sum_{i=1}^{r}h(\alpha_{i})$, where $h\colon\mathbb{Z}_{+}\to\\{0,1\\}$
is an arbitrary function. Let $z$ and $s$ be complex numbers such that the
Dirichlet series $\sum_{n=1}^{\infty}{z^{g(n)}}/{n^{s}}$ is absolutely
convergent, then

$\sum\limits_{n=1}^{\infty}\frac{z^{g(n)}}{n^{s}}=\prod_{p}\left(1+\sum_{i=1}^{\infty}\frac{z^{h(i)}}{(p^{s})^{i}}\right).$
(3)

##  2 Main theorems

The following result was obtained and proved by Shevelev, see [3, 4, 5]. Note
also that it is easy to obtain using the theorem $1$ presented by Tóth in [6].
We give one more proof.

######  Theorem 3.

Let $a_{0}=b_{0}=0$ and $0<a_{1}\leq b_{1}<a_{2}\leq b_{2}<\ldots\leq b_{n}$
be integers. Let $Q\left(x;\bigcup_{j=1}^{n}[a_{j},b_{j}]\right)$ be the
number of integers between $1$ and $x$ such that all exponents in their prime
factorization are in $\bigcup_{j=1}^{n}[a_{j},b_{j}]$. The following formula
holds:

$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_{j},b_{j}]\right)}{x}}=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right).$
(4)

###### Proof.

Let $g$ be a function that satisfies all requirements of Corollary $1$. Let
the function $h$ be such that if $\alpha\in\bigcup_{j=1}^{n}[a_{j},b_{j}]$,
then $h(\alpha)=0$, else $h(\alpha)=1.$ Consider the Dirichlet series

$\delta(s,z)=\sum\limits_{n=1}^{\infty}\frac{z^{g(n)}}{n^{s}}.$ (5)

Using Corollary $1$ we get

$\delta(s,z)=\zeta(s)\prod_{p}\left(1-\frac{1}{p}\right)\left(1+\sum_{i\in
I}\frac{1}{(p^{s})^{i}}+\sum_{i\notin I}\frac{z}{(p^{s})^{i}}\right),$ (6)

where $I=\bigcup_{j=1}^{n}[a_{j},b_{j}]$. Using the reasoning presented in [1]
and performing the necessary calculations, we get

$\lim\limits_{m\to\infty}\frac{1}{m}\sum\limits_{n=1}^{m}z^{g(n)}=\sum\limits_{k=0}^{\infty}d_{k}z^{k}=\prod_{p}\left(1-\frac{1}{p}\right)\left(1+\sum_{i\in
I}\frac{1}{p^{i}}+\sum_{i\notin I}\frac{z}{p^{i}}\right),\text{ where }|z|\leq
1.$ (7)

Here, $d_{k}$ is the natural density of the subset of integers $y$ such that
$g(y)=k.$ Thus

$d_{0}=\prod_{p}\left(1-\frac{1}{p}\right)\left(1+\sum_{i\in
I}\frac{1}{p^{i}}\right)=\prod\limits_{p}\sum\limits_{i=0}^{n}\left(\frac{1}{p^{a_{i}}}-\frac{1}{p^{b_{i}+1}}\right),\,\,\,a_{0}=b_{0}=0.$
(8)

It is clear that

$\lim_{x\to\infty}{\frac{Q\left(x;\bigcup_{j=1}^{n}[a_{j},b_{j}]\right)}{x}}=d_{0}.$
(9)

∎

Below we demonstrate examples of using Theorem $3$. Some of them are well-
known.

$\displaystyle\lim_{x\to\infty}{\frac{Q\left(x;[1,k]\right)}{x}}=$
$\displaystyle\prod_{p}\left(1-\frac{1}{p^{k+1}}\right)=\frac{1}{\zeta(k+1)},$
(10)
$\displaystyle\lim_{x\to\infty}{\frac{Q\left(x;[1,1]\cup[k,\infty]\right)}{x}}=$
$\displaystyle\prod_{p}\left(1-\frac{1}{p^{2}}+\frac{1}{p^{k}}\right),$ (11)
$\displaystyle\lim_{x\to\infty}{\frac{Q(x;[1,k-1]\cup[k+1,\infty])}{x}}=$
$\displaystyle\prod\limits_{p}\left(1-\frac{1}{p^{k}}+\frac{1}{p^{k+1}}\right),$
(12)
$\displaystyle\lim_{x\to\infty}{\frac{Q(x;\bigcup_{j=0}^{\infty}[2j+1,2j+1])}{x}}=$
$\displaystyle\prod\limits_{p}\left(1-\frac{1}{p(p+1)}\right),$ (13)
$\displaystyle\lim_{x\to\infty}{\frac{Q(x;\bigcup_{j=0}^{\infty}[\ell j+1,\ell
j+1])}{x}}=$
$\displaystyle\prod\limits_{p}\left(1-\frac{p^{\ell-1}-1}{p(p^{\ell}-1)}\right),\,\,\,\ell\in\mathbb{Z}_{+}.$
(14)

The formula $(\ref{f})$ gives the density of exponentially odd numbers, see
the sequence $A065463$ in Sloane’s Online Encyclopedia of Integer Sequences
[13]. Other interesting results can be found in [7, 8, 9, 10, 11, 12].

Let us consider the generalized theorem.

######  Theorem 4.

Let $g$ be a function on natural numbers such that $g(1)=0$ and for a positive
integer $u=\prod_{i=1}^{r}{p_{i}}^{\alpha_{i}}>1$ the following holds:
$g(u)=\sum_{i=1}^{r}h(\alpha_{i},p_{i})$, where
$h\colon\mathbb{Z}_{+}\times\mathbb{P}\to\\{0,1\\}$ is an arbitrary function.
Let $Q\left(x;h\right)$ be the number of integers $n$ between $1$ and $x$ such
that $g(n)=0$. The following formula holds:

$\lim_{x\to\infty}{\frac{Q(x;h)}{x}}=\prod\limits_{p}\left(1-\frac{p-1}{p}\sum\limits_{i=1}^{\infty}\frac{h(i,p)}{p^{i}}\right).$
(15)

###### Proof.

Similarly to the proof of the previous theorem, we have

$\delta(s,z)=\zeta(s)\prod_{p}\left(1-\frac{1}{p}\right)\left(1+\sum_{i=1}^{\infty}\frac{z^{h(i,p)}}{(p^{s})^{i}}\right).$
(16)

Since $\lim\limits_{z\to 0}z^{h(i,p)}=1-h(i,p)$, we get

$d_{0}=\prod\limits_{p}\left(1-\frac{p-1}{p}\sum\limits_{i=1}^{\infty}\frac{h(i,p)}{p^{i}}\right).$
(17)

Finally we conclude that $\lim\limits_{x\to\infty}{\frac{Q(x;h)}{x}}=d_{0}.$ ∎

This theorem is a powerful tool for obtaining densities of various subsets of
integers. Let us show it.

Example $1$. Suppose that we need to obtain the natural density of numbers
$n=\prod_{i=1}^{r}{p_{i}}^{\alpha_{i}}$ such that $p_{i}>q$ and $\alpha_{i}<k$
for all $1\leq i\leq r,$ here $q$ is some prime. In other words, these are
$k$th-power-free numbers for which $\gcd(n,q\\#)=1$. We need to use the
following function:

$h(\alpha,p)=\begin{cases}1&\mbox{if }p\leq q\\\ 1&\mbox{if }p>q\text{ and
}\alpha\geq k\\\ 0&\mbox{if }p>q\text{ and }\alpha<k.\end{cases}$ (18)

By using $(\ref{PP})$ we get

$\lim_{x\to\infty}{\frac{Q(x;h)}{x}}=\frac{\prod\limits_{p\leq
q}\left(1-\frac{1}{p}\right)}{\zeta(k)\prod\limits_{p\leq
q}\left(1-\frac{1}{p^{k}}\right)}.$ (19)

Example $2$. Let $S=\\{q_{1},q_{2},\ldots,q_{m}\\}$ be the set of arbitrary
distinct primes. Consider $k$th-power-free numbers $n$ such that
$\gcd(n,q_{i})=1$ for all $1\leq i\leq m$. To find the density we have to use

$h(\alpha,p)=\begin{cases}1&\mbox{if }p\in S\\\ 1&\mbox{if }\alpha\geq k\\\
0&\mbox{if }p\notin S\text{ and }\alpha<k.\end{cases}$ (20)

Then

$\lim_{x\to\infty}{\frac{Q(x;h)}{x}}=\frac{1}{\zeta(k)}\prod\limits_{q\in
S}\frac{q^{k}-q^{k-1}}{q^{k}-1}.$ (21)

Example $3$. Let us consider the numbers
$n=\prod_{i=1}^{r}{p_{i}}^{\alpha_{i}}$ for which one of exponents
$\alpha_{i}$ can be any number, and the rest are less than $k$. We denote the
density of such numbers by $\rho$. First, we need to obtain a density of
numbers for which the prime factorization can have any exponent for prime $p$,
and the other prime exponents less than $k$. With the help of $(\ref{PP})$ we
conclude that this density is

$\rho_{p}=\frac{1}{\zeta(k)}\frac{p^{k}}{p^{k}-1}.$ (22)

Finally we have

$\rho=\frac{1}{\zeta(k)}+\sum\limits_{p\in\mathbb{P}}\left(\rho_{p}-\frac{1}{\zeta(k)}\right)=\frac{1}{\zeta(k)}\left(1+\sum\limits_{p\in\mathbb{P}}\frac{1}{p^{k}-1}\right).$
(23)

Acknowledgments. The author is grateful to the participants of the internet
forum https://dxdy.ru for their fruitful discussions.

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