11institutetext: Multidisciplinary Brain Research Center, Bar-Ilan University,
Ramat Gan, Israel 22institutetext: Department of Computer Science, University
of Oregon, Eugene, Oregon, USA 33institutetext: Institute of Health and
Neurodevelopment, College of Health and Life Sciences, Aston University,
Birmingham, UK
44institutetext: Department of Mathematics and Statistics, Auburn University,
Alabama, USA
55institutetext: Computational Engineering Division, Lawrence Livermore
National Laboratory, Livermore, California, USA
66institutetext: NPLab, Network Science Institute, Northeastern University
London, London, UK
77institutetext: School of Industrial Engineering, Purdue University, West
Lafayette, Indiana, USA
88institutetext: School of Biomedical Engineering, Purdue University, W.
Lafayette, Indiana, USA
99institutetext: Department of Mathematics, Purdue University, W. Lafayette,
Indiana, USA
1010institutetext: School of Public Health, Yale University, New Heaven,
Connecticut, USA
1111institutetext: Department of Mathematics, U.S. Naval Academy, Annapolis,
Maryland, USA
1212institutetext: Department of Biostatistics, Epidemiology and Informatics,
University of Pennsylvania, Philadelphia, Pennsylvania, USA1212email:
<EMAIL_ADDRESS>
1313institutetext: Co-supervising Authors

# Volume-optimal persistence homological scaffolds of hemodynamic networks
covary with MEG theta-alpha aperiodic dynamics

Nghi Nguyen 11 Tao Hou 22 Enrico Amico 33 Jingyi Zheng 44 Huajun Huang 44 Alan
D. Kaplan 55 Giovanni Petri 66 Joaquín Goñi 7788 Ralph Kaufmann 99 Yize Zhao
1010 Duy Duong-Tran 111112121313 Li Shen 12121313

###### Abstract

Higher-order properties of functional magnetic resonance imaging (fMRI)
induced connectivity have been shown to unravel many exclusive topological and
dynamical insights beyond pairwise interactions. Nonetheless, whether these
fMRI-induced higher-order properties play a role in disentangling other
neuroimaging modalities’ insights remains largely unexplored and poorly
understood. In this work, by analyzing fMRI data from the Human Connectome
Project Young Adult dataset using persistent homology, we discovered that the
volume-optimal persistence homological scaffolds of fMRI-based functional
connectomes exhibited conservative topological reconfigurations from the
resting state to attentional task-positive state. Specifically, while
reflecting the extent to which each cortical region contributed to functional
cycles following different cognitive demands, these reconfigurations were
constrained such that the spatial distribution of cavities in the connectome
is relatively conserved. Most importantly, such level of contributions
covaried with powers of aperiodic activities mostly within the theta-alpha
(4-12 Hz) band measured by magnetoencephalography (MEG). This comprehensive
result suggests that fMRI-induced hemodynamics and MEG theta-alpha aperiodic
activities are governed by the same functional constraints specific to each
cortical morpho-structure. Methodologically, our work paves the way toward an
innovative computing paradigm in multimodal neuroimaging topological learning.
The code for our analyses is provided in
https://github.com/ngcaonghi/scaffold_noise.

##  1 Introduction

Many real-world systems can be characterized not only by graphs, which are
described by nodes and pairwise relations but also by complexes and
hypergraphs, elucidating higher-order relations [16, 15]. Interactions between
these higher-order structures have been demonstrated to elucidate complex
dynamics, such as phase transitions induced by emergent phenomena in complex
networks [4], that could not be fully explained by pairwise interactions
alone. In neuroscience, brain dynamics analysis based on such higher-order
descriptions of neuronal populations has yielded statistically and
topologically significant results [16, 40, 32, 10, 9]. Underlying such
successes is a family of analytical tools commonly referred to as Topological
Data Analysis (TDA) [8, 12], which includes persistent homology (PH) [13, 41].
Using PH on functional connections deriving from functional magnetic resonance
imaging (fMRI), several studies have identified differences between cognitive
abilities [31], brain states [28], and brain regions [24, 9] in terms of
higher-order features (e.g., topological invariants) pertaining to voids
(cavities) and cycles. However, the relationships between these topological
invariants (e.g., cycles) and other neural correlates, such as anatomical
structures and electrophysiological dynamics, remain poorly understood.

Among such neural correlates are cortical aperiodic dynamics, which have
recently gained growing interest due to their implicated roles in cognition
[27], consciousness [33], aging [36], and diseases [25, 29, 30]. Generative
mechanisms of these dynamics, particularly those within the gamma band ($>$ 30
Hz), have been linked to several biological processes [1, 2, 5, 3, 14, 22,
11]. However, aperiodic activities on the level of cortical regions below the
gamma band ($<$ 30 Hz) are significantly less understood.

In this study, leveraging computations in PH, we associate a critical link
between fMRI-induced human brain large-scale functional connectivity cycles
and aperiodic magnetoencephalography (MEG) activities in theta-alpha range
(4-12 Hz). Specifically, we discovered that for each parcellated region in the
cortex, changes in its contribution to maintaining homological cycle
persistence were positively correlated with changes in its theta-alpha
aperiodic power. Comprehensively, our pioneering work paves the way toward a
new, innovative computing paradigm in multimodal neuroimaging topological
learning.

##  2 Methods

###  2.1 Overview of Topological Data Analysis

Topological Data Analysis (TDA) [8, 12] is an emerging field focusing on
understanding the “shape” of data (denoted as $X$) through homology theory
[19]. Classical homology groups [19] capture voids and cycles of topological
spaces in various dimensions $i$ (e.g., connected components ($i=0$), rings
($i=1$), and cavities ($i=2$), and so on). Functional connectomes (FCs,
denoted as $G$) in this paper refer to symmetric matrices, whose entries are
pair-wise functional couplings quantified by Pearson correlations, realized
from $X$. The progression to glean topological information progresses an FC
onto a topological space by realizing its simplicial clique complex
$\Delta(G)$ which models higher order interactions: $G\rightarrow\Delta(G)$.
Defining a threshold $r$, sampled from a set of distinct weight values in FC,
determines a corresponding binarized scaffold $G(r)$; the associated homology
groups $H_{i}(\Delta(G))$ becomes a function of $r$. Scanning $r$ from $0$ to
$1$ homology is born and annihilated. The sequence of these events is
mathematically captured by persistence homology and can be encoded and
visualized in terms of bar codes. The topological space, which is simplicial
in nature, has associated topological invariants, such as the homology
$H_{i}(\Delta(G))$ and Betti numbers $b_{i}$. Topological spaces $\Delta(G)$
take the form of a _filtration_ $\mathcal{F}:K_{0}\subseteq
K_{1}\subseteq\cdots\subseteq K_{m}$, which is a growing sequence of
discretized spaces called _simplicial complexes_. In $\mathcal{F}$, each
simplicial complex $K_{i}$ is a union of building blocks such as vertices,
edges, triangles, and higher dimensional analogs. Given a filtration
$\mathcal{F}$, _persistent homology_ [13] produces a set of intervals (called
the _barcodes_) as a scheme for quantifying the “significance” (e.g.
persistence) of topological features in spaces. In the barcode of
$\mathcal{F}$, each interval registers the birth and death of a homology
feature in the sequence with the longer intervals typically considered to
correspond to more significant features, see [6, 8].

###  2.2 Volume-optimal Persistent Cycles

Homology is the study of classifications of _cycles_ in topological spaces and
features in homology are also represented by the cycles. However, the barcode
of a filtration $\mathcal{F}$ does not provide concrete representative cycles
for the homology features born and died in the sequence. These representatives
for persistent homology, termed as _persistent cycles_ , turned out to reveal
crucial insight into the topological spaces over which filtrations are taken.

###### Definition 1

For an interval $[b,d)$ in the barcode of $\mathcal{F}$, a _persistent cycle_
for $[b,d)$ is a cycle born in $K_{b}$ (existing in $K_{b}$ but not in
$K_{b-1}$) and becoming trivial in $K_{d}$ (is not a boundary in $K_{d-1}$ but
becomes a boundary in $K_{d}$).

Notice that boundaries are special cycles representing the trivial homology
feature (zero). As indicated in several works (e.g., [7, 26]), an interval in
the barcode could have more than one persistent cycles, among which a
canonical choice is to compute an _optimized_ persistent cycle with weights
typically reflecting the intrinsic areas of the cycles. Such weight-optimized
cycles can provide the tightest representations of the holes characterized by
persistent homology. However, Dey et al. [7] showed that computing optimal
persistent cycles is NP-hard in general. In this paper, we utilize a Python
library called HomCloud developed by Obayashi [26] for computing persistent
cycles. HomCloud uses a linear-program-based algorithm for computing optimized
persistent cycles and indeed targets _volume-optimal_ persistent cycles, where
weights of the higher dimensional bounding chains are optimized. In this
paper, we found that the persistent cycles produced by HomCloud typically have
good quality (better than the non-optimized ones produced by the original
persistence algorithm [13]). More importantly, mathematical constructs based
on such cycles can afford additional interpretations about the volume of the
cavities that they wrap around.

###  2.3 HCP data processing

####  2.3.1 Persistence homological scaffolds of resting-state and task-based
hemodynamic networks.

We used resting-state and task-based 3T fMRI data from a subset of the Human
Connectome Project Young Adult dataset [34], preprocessed by the Neuromatch
Academy [18]. Our study focused on left-hand, right-hand, left-foot, and
right-foot motor tasks, and 0-back faces, 0-back tools, 2-back faces, and
2-back tools working memory tasks. To ensure valid extrapolation to MEG neural
correlates, we excluded 20 subjects that underwent the same task conditions
during MEG recording trials. The resulting sample size was N = 319, each of
which represents the BOLD signals of one subject under resting and task
conditions and parcellated according to the Glasser atlas [17].

Figure 1:  Data processing pipelines. I. From fMRI functional connectivity to
persistence centrality. II. From sensor-level MEG signals to Glasser-
parcellated power spectrum decomposition.

Functional connectivity matrices were computed for each subject by computing
Pearson correlation coefficients between BOLD time series of region pairs. To
perform persistent cycle discovery with HomCloud on every connectivity matrix
$G$, we converted each pairwise correlation term $r$ into a pseudo-distance
index $\omega:=1-r$ while excluding negative correlations (figure 1.I.A). For
each pseudo-distance matrix with weights $\\{\omega\\}$, we generated a
Vietoris-Rips filtration $\mathcal{F}$ of the homological group $H_{1}$. Every
complex in $\mathcal{F}$ entailed a birth-death interval $[b,d)$ where
$b,d\in\\{\omega\\}$, a volume-optimal persistent cycle $g_{t}$, and its
persistence $\pi_{g_{t}}=d-b$ (figures 1.I.B and 1.I.C). From this point, we
followed the approach outlined by Petri et al. [28] to calculate the
_persistence homological scaffold network_ $\mathcal{H}^{p}_{G}$ composed of
the volume-optimal cycles (figure 1.I.D). Specifically, for every edge
$e_{i,j}$ between nodes $i$ and $j$ belonging to multiple cycles
$g_{0},g_{1},\cdots,g_{s}$, its weight is $w_{i,j}^{\pi}=\sum_{g_{t}\mid
e_{i,j}\in g_{t}}\pi_{g_{t}}.$ The group-averaged scaffold matrix was
generated for each experimental condition (figure 1.I.E), from which we
calculated the _persistence centrality_ (PC) indices defined by
$PC(i)=\sum_{j}w^{\pi}_{i,j}\left(\sum_{j,k}w^{\pi}_{j,k}\right)^{-1}$ for
each node $i$ (figure 1.I.F). For comparative purposes, we also calculated the
degree centrality (DC) indices from the original functional connectivity
matrix $G$ with edge weights $\\{r\\}$, whereby
$DC(i)=(\sum_{j}r_{i,j})(\sum_{j,k}r_{j,k})^{-1}.$

####  2.3.2 MEG data preprocessing.

We used MEG data from the same HCP dataset under the same task conditions but
with the subjects previously excluded from the fMRI data analysis (n = 20) and
preprocessed as previously detailed by Larson-Prior et al., 2013 [23]. Each
subject’s data comprised 12-second segments of randomly epoched resting-state
data and task-evoked data starting from the first stimulus onset of each task
block. We performed source reconstruction using linearly constrained minimum
variance (LCMV) beamformers [35] on boundary element method (BEM) surfaces
derived from subject anatomy data. The source estimates were then grouped by
labels defined by the Glasser atlas and averaged across sources (figure
1.II.A), resulting in a single time series per epoch for every Glasser region.

####  2.3.3 Power spectrum decomposition.

On each Glasser-parcellated MEG epoch, we computed the power spectral density
using Welch’s method (with 2-second Hann windows and 1-second overlaps) and
performed spectrum decomposition using the irregular-resampling auto-spectral
analysis (IRASA) method developed by Wen & Liu, 2015 [37]. The resulting
aperiodic and periodic spectra spanned a frequency band of 1 to 120 Hz (Figure
1.II.B). Epoch-level spectra were averaged to produce those for the resting
state and every task condition (Figure 1.II.C). To avoid high-frequency
artifacts that might contaminate downstream analyses, we only considered power
in the range of 1 to 90 Hz.

##  3 Result

###  3.1 Directions of change in persistence centralities are consistent
across attentional task conditions

Figure 2:  Resting state-subtracted PC ($\Delta PC$) captures task-evoked
hemodynamic patterns nonspecific to task conditions, whereas resting state-
subtracted DC ($\Delta DC$) is more condition-specific.

In Figure 2.A and 2.D, the spatial distributions of resting-state PC and DC
are portrayed on a Glasser-parcellated surface. Figure 2.B maps $\Delta PC$,
representing the PCs during the motor task and right foot condition subtracted
by the resting-state PCs, on the same brain surface. The mapped surface shows
a coarse-grained global gradient of $\Delta PC$ starting from the deepest
$\Delta PC$ sinks (in blue) in the dorsal areas to the sources (in red) in
ventral regions. Notably, this gradient does not exhibit a hemisphere
asymmetry (figure 2.C), as evidenced by similar distributions of sinks and
sources across hemispheres for each functional network ($p>0.34>0.05$, Mann-
Whitney $U$-test). Figures 2.E and 2.F demonstrate the $\Delta DC$
counterparts of figures 2.B and 2.C, respectively, revealing a different
distribution of sinks and sources that shows hemispheric asymmetry. See
Supplementary information, figures S1 and S2 for a comprehensive comparison of
$PC$, $DC$, $\Delta PC$, and $\Delta DC$ distributions across all task
conditions.

Figure 2.G demonstrates the relationship between resting-state $PC$ (or $DC$)
and $\Delta PC$ (or $\Delta DC$) evoked by each task condition. All $\Delta
PC$ vectors have strongly negative correlations with the resting-state $PC$
($p<<0.05$, Kendall’s $\tau$ for rank association), while their $\Delta DC$
counterparts show weak or non-significant correlations. $\Delta PC$ and
$\Delta DC$ clusters are at least not positively correlated as demonstrated by
the minimum angular difference of $1.7373$ radians, i.e., nearly orthogonal
(visualized by multidimensional scaling, MDS, in figure 2.H). Figure 2.I
sketches the distributions of angular differences within each cluster, showing
that spatial patterns of $PC$ sinks and sources are more consistent across
tasks and conditions than those of $DC$.

From the observations above, we concluded that $\Delta PC$ was more correlated
with changes in cognitive demand than with the nature of the task, while
$\Delta DC$ was task-specific. Spatially, $\Delta PC$ exhibited patterns such
as “sinks” concentrated in the Dorsal-Attention network and “sources” in the
Auditory network and part of the Cingulo-Opercular network previously
identified as the salience network [21]. Such “sink” and “source” regions
indicated the reduction and increase, respectively, of the volumes of
functional connectome cavities incident to those regions. Since the spatial
distributions of $\Delta PC$ “sinks” and “sources” were consistent across task
conditions (Supplementary information, figures S1), we further concluded that
under different cognitive demands, cavities in the functional connectomes were
conserved in space and relatively invariant to changes in cortical
computations.

###  3.2 Persistence centralities and theta-alpha aperiodic band powers change
in the same direction

Figure 3: Regional alterations in bandpower ratio ($\Delta$ BPR) of cortical
aperiodic activity within the theta-alpha range (4 - 12 Hz) exhibit a stronger
alignment with $\Delta PC$ compared to $\Delta DC$.

We observed that $\Delta PC$ coincided with aperiodic dynamic patterns
outlined in previous fMRI-EEG studies [20]. Further corroborating this
correlation, cosine similarities between $\Delta PC$ and changes in band power
ratios ($\Delta BPRs$) within the theta-alpha range were significantly higher
than those between $\Delta BPRs$ and $\Delta DC$.

Figures 3.A and 4.B present Isomap projections of $\Delta PC$ and $\Delta DC$,
respectively, evoked by right-foot cues during the motor task. These allow for
visual comparison with the $\Delta$ BPR of canonical bands (delta: 2-4 Hz,
theta: 4-8 Hz, alpha: 8-12 Hz, beta: 12-25 Hz, gamma: 25-90 Hz), also
projected using Isomap and depicted in figures 3.C to 4.G. $\Delta$ BPR was
also computed for 32 frequency bands whose endpoints were logarithmically
spaced within the range of 2 to 90 Hz. Cosine similarities between these
$\Delta$ BPRs and $\Delta PC$ (orange) or $\Delta DC$ (blue) under the motor,
right-foot condition are shown in Figure 3.H. Absolute cosine similarities
averaged across conditions are shown in Figure 3.I, suggesting that $\Delta
PC$ better predicts $\Delta$ BPRs in the theta-alpha and gamma ranges (p =
0.0156 and p = 0.0078, respectively, cluster-based permutation paired t-test).
Similar analyses for the periodic component resulted in statistically
insignificant correlations between periodic activities and $\Delta PC$ or
$\Delta DC$.

##  4 Discussion

By exploring the relationship between fMRI and MEG data from a novel bi-modal
perspective, this study reveals two previously undocumented phenomena.
Firstly, the persistence centrality (PC) of each cortical structure,
representing its participation in persistent cycles relative to the total
participation, demonstrates predictable changes between brain states induced
by different attentional demands. Specifically, regardless of the task nature,
functional cavities incident to structures with high PCs tend to contract when
attention is required, and vice versa. Secondly, the directions of change in
PC predict those in the power of theta-alpha aperiodic activities on the
brain-wide scale. These observations underscore a relationship between BOLD
signals and electrophysiological activities that cannot be described or
explained only in terms of pairwise interactions captured by node degrees. In
other words, it must rely on the notion of volume-optimal functional cycles.

Since changes in these functional cycles and their associated changes in the
aperiodic spectra are robust across normal brain states, it might be possible
that they emerge from a common constraint regime governing both hemodynamics
and aperiodic electrodynamics. This relatively new notion of a constraint
regime described by cycles suggests new directions toward understanding the
generative mechanisms of BOLD signals and sub-gamma aperiodic spectra. Future
investigations—which might involve connections to subcortical structures such
as the thalamus—can help detail or expand the current models of BOLD
functional connectivity on the systemic macrovascular level. More broadly, by
leveraging topological assumptions that persist throughout different normal
brain states, homological cycles in the brain can provide better guidance for
treatments of outlying brain states indicative of psychological disorders,
neuropathologies, and neurodegeneration [38, 39].

####  4.0.1 Acknowledgements

LS acknowledges financial support from the National Institutes of Health
grants RF1 AG068191, U01 AG066833, and U01 AG068057. DDT acknowledges
financial support from the Office of Naval Research N0001423WX00749. NN
acknowledges financial support from the Erasmus Mundus Joint Master’s
Programme in Brain and Data Science, European Commission. TH acknowledges
financial support from the National Science Foundation grant CCF 2348238. Data
used in this project were obtained from the Human Connectome Project, WU-Minn
HCP data – 1200 subjects (db.humanconnectome.org).

####  4.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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$\displaystyle\|h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\lesssim\,$
$\displaystyle\|h(t,u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}+\|h(t,u_{m,n})-h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}$
$\displaystyle\leq\,$ $\displaystyle\overline{C}\left({\rm
e}^{\frac{-1}{\|u_{m,n}\|_{H^{\sigma_{0}}}}}\right)^{2}+q(\overline{C})\|v\|_{H^{\sigma_{0}}}^{2},\
\ t\in[0,T_{l}\wedge\tau^{m,n}_{R}]\ \ \mathbb{P}-a.s.,$

where $q(\cdot)$ is given in (1.8). As a result, for any fixed $s>3$, by
applying Lemmas 2.5 and 6.1 again, we can pick
$\lambda>2(1+\frac{s-\sigma_{0}-1}{2-\delta})$ to derive

$\displaystyle\|h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\lesssim\,$
$\displaystyle\|u_{m,n}\|^{2\lambda}_{H^{\sigma_{0}}}+\|v\|_{H^{\sigma_{0}}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\left(n^{-s+\sigma_{0}}+n^{\frac{\delta}{2}-1}\right)^{2\lambda}+\|v\|_{H^{\sigma_{0}}}^{2}\lesssim\,n^{2r_{s}}+\|v\|_{H^{\sigma_{0}}}^{2},\
\ t\in[0,T_{l}\wedge\tau^{m,n}_{R}]\ \ \mathbb{P}-a.s.$

Via (6.19), we have

$\displaystyle 2\left|(E,v)_{H^{\sigma_{0}}}\right|\leq
2\|E\|_{H^{\sigma_{0}}}\|v\|_{H^{\sigma_{0}}}\lesssim\|E\|_{H^{\sigma_{0}}}^{2}+\|v\|_{H^{\sigma_{0}}}^{2}\lesssim
n^{2r_{s}}+\|v\|_{H^{\sigma_{0}}}^{2}.$

Using Lemma 2.2, (6.10), (2.7), integration by parts, and the embedding
$H^{s}\hookrightarrow H^{\sigma_{0}}\hookrightarrow W^{1,\infty}$, we obtain
that the almost surely

$\displaystyle\left|\left(\left(\mathcal{H}v\right)\partial_{x}u_{m,n},v\right)_{H^{\sigma_{0}}}\right|\lesssim\,\|\mathcal{H}v\|_{H^{\sigma_{0}}}\|u_{m,n}\|_{H^{\sigma_{0}+1}}\|v\|_{H^{\sigma_{0}}}\lesssim\,\|u_{m,n}\|_{H^{s}}\|v\|_{H^{\sigma_{0}}}^{2}\lesssim\|v\|_{H^{\sigma_{0}}}^{2},\
t\in[0,T_{l}\wedge\tau^{m,n}_{R}],$

and

$\displaystyle\left|\left(\left(\mathcal{H}u^{m,n}\right)\partial_{x}v,v\right)_{H^{\sigma_{0}}}\right|$
$\displaystyle=\,$
$\displaystyle\left|([D^{\sigma_{0}},\mathcal{H}u^{m,n}]\partial_{x}v,D^{\sigma_{0}}v)_{L^{2}}+(\mathcal{H}u^{m,n}D^{\sigma_{0}}\partial_{x}v,D^{\sigma_{0}}v)_{L^{2}}\right|$
$\displaystyle\lesssim\,$
$\displaystyle\left(\|D^{\sigma_{0}}\mathcal{H}u^{m,n}\|_{L^{2}}\|\partial_{x}v\|_{L^{\infty}}+\|\partial_{x}\mathcal{H}u^{m,n}\|_{L^{\infty}}\|D^{\sigma_{0}-1}\partial_{x}v\|_{L^{2}}\right)\|v\|_{H^{\sigma_{0}}}+\|\partial_{x}\mathcal{H}u^{m,n}\|_{L^{\infty}}\|v\|_{H^{\sigma_{0}}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\|\mathcal{H}u^{m,n}\|_{H^{s}}\|v\|_{H^{\sigma_{0}}}^{2}\lesssim\|u^{m,n}\|_{H^{s}}\|v\|_{H^{\sigma_{0}}}^{2}\lesssim\|v\|_{H^{\sigma_{0}}}^{2},\
t\in[0,T_{l}\wedge\tau^{m,n}_{R}].$

To sum up, we obtain that

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|v(t)\|_{H^{\sigma_{0}}}^{2}\lesssim
T_{l}n^{2r_{s}}+\int_{0}^{T_{l}}\mathbb{E}\sup_{t^{\prime}\in[0,t\wedge\tau^{m,n}_{R}]}\|v(t^{\prime})\|_{H^{\sigma_{0}}}^{2}\,{\rm
d}t.$

Using the Grönwall inequality, we obtain (6.21).

Now we prove (6.22). To this end, we first notice that $2s-\sigma_{0}>3$ and
$u_{m,n}$ is the unique solution to (6.5). Then, similar to (3.27), we can use
(6.20) and Assumption (D) to find for each fixed $n\in\mathbb{N}$ that

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u^{m,n}\|_{H^{2s-\sigma_{0}}}^{2}\lesssim\mathbb{E}\|u_{m,n}(0)\|_{H^{2s-\sigma_{0}}}^{2}+\int_{0}^{T_{l}}\mathbb{E}\sup_{t^{\prime}\in[0,t\wedge\tau^{m,n}_{R}]}\|u^{m,n}\|_{H^{2s-\sigma_{0}}}^{2}\,{\rm
d}t.$

Using the Grönwall inequality and Lemmas 6.1 and 2.5, we have

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u^{m,n}\|_{H^{2s-\sigma_{0}}}^{2}\lesssim\mathbb{E}\|u_{m,n}(0)\|_{H^{2s-\sigma_{0}}}^{2}\lesssim(n^{\frac{\delta}{2}-1}+n^{s-\sigma_{0}})^{2}\lesssim
n^{2s-2\sigma_{0}},\ \ n\geq 1.$

Hence, by Lemmas 6.1 and 2.5 again, we arrive at

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u^{m,n}-u_{m,n}\|_{H^{2s-\sigma_{0}}}^{2}$
$\displaystyle\leq\ $ $\displaystyle
2\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u^{m,n}\|_{H^{2s-\sigma_{0}}}^{2}+2\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u_{m,n}\|_{H^{2s-\sigma_{0}}}^{2}\leq\
Cn^{2s-2\sigma_{0}},\ \ n\geq 1.$

Therefore, we complete the proof. ∎

###  6.3. Concluding the proof of Theorem 1.4

To begin with, we have the following property:

######  Lemma 6.4.

Let Assumption (D) hold true. Suppose that for some $R_{0}\gg 1$, the
$R_{0}$-exiting time of the zero solution to (1.2) is strongly stable. Then we
have

$\lim_{n\rightarrow\infty}\tau^{m,n}_{R_{0}}=\infty\ \mathbb{P}-a.s.$ (6.23)

###### Proof.

By Assumption (D), the unique solution with zero initial data to (1.2) is
zero. Now we notice that for all $s^{\prime}<s$,
$\lim_{n\rightarrow\infty}\|u_{m,n}(0)\|_{H^{s^{\prime}}}=\lim_{n\rightarrow\infty}\|u_{m,n}(0)-0\|_{H^{s^{\prime}}}=0$
and the $R_{0}$-exiting time of the zero solution is $\infty$. Then the
assumption that $R_{0}$-exiting time of the zero solution to (1.2) is strongly
stables immediately implies (6.23). ∎

###### Proof of Theorem 1.4.

We only need to show that if the $R_{0}$-exiting time is strongly stable at
the zero solution for some $R_{0}\gg 1$, then $\\{u^{-1,n}\\}$ and
$\\{u^{1,n}\\}$ are two sequences of pathwise solutions such that (1.19),
(1.20), (1.21) and (1.22) are satisfied.

For each $n>1$ and for fixed $R_{0}\gg 1$, Lemmas 6.1, 2.5 and (6.20) give
$\mathbb{P}\\{\tau^{m,n}_{R_{0}}>0\\}=1$, and Lemma 6.4 implies (1.19). Then,
it follows from Theorem 1.1 and (6.20) that $u^{m,n}\in
C([0,\tau^{m,n}_{R_{0}}];H^{s})$ $\mathbb{P}-a.s.$ and (1.20) holds true.
Next, we check (1.21). By interpolation, we have

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|_{H^{s}}$
$\displaystyle\lesssim\,$
$\displaystyle\left(\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|_{H^{\sigma_{0}}}\right)^{\frac{1}{2}}\left(\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|_{H^{2s-\sigma_{0}}}\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\,$
$\displaystyle\left(\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|^{2}_{H^{\sigma_{0}}}\right)^{\frac{1}{4}}\left(\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|^{2}_{H^{2s-\sigma_{0}}}\right)^{\frac{1}{4}}.$

Combining Lemma 6.3 and the above estimate yields

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|_{H^{s}}\lesssim\,$
$\displaystyle n^{\frac{1}{4}\cdot 2r_{s}}\cdot
n^{\frac{1}{4}\cdot(2s-2\sigma_{0})}=n^{r^{\prime}_{s}},$ (6.24)

where $r_{s}$ is defined by (6.14) and

$0>r^{\prime}_{s}=r_{s}\cdot\frac{1}{2}+(s-\sigma_{0})\cdot\frac{1}{2}=\frac{\delta-1}{2}.$

Since $r^{\prime}_{s}<0$, we can deduce that

$\displaystyle\lim_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R_{0}}]}\|u_{m,n}-u^{m,n}\|_{H^{s}}=0.$
(6.25)

Since $\delta<1$, we have

$\displaystyle\|u^{-1,n}(0)-u^{1,n}(0)\|_{H^{s}}^{2}=\,$
$\displaystyle\|u_{-1,n}(0)-u_{1,n}(0)\|_{H^{s}}^{2}$ $\displaystyle=\,$
$\displaystyle\left\|2\mathcal{H}\left(n^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right)\right\|_{H^{s}}^{2}\lesssim\left\|n^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right\|_{H^{s}}^{2}\lesssim
n^{\frac{\delta}{2}-1}\|\tilde{\phi}\|_{H^{s}}\rightarrow 0,{~{}\rm
as~{}}n\rightarrow\infty,$

which implies that (1.21) holds true.

Now we prove (1.22). Let $T_{l}>0$ be given in Lemma 6.1. We can infer from
(6.25) that

$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u^{-1,n}(t)-u^{1,n}(t)\|_{H^{s}}$
$\displaystyle\gtrsim\,$
$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u_{-1,n}(t)-u_{1,n}(t)\|_{H^{s}}$
$\displaystyle-\lim_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u_{-1,n}(t)-u^{-1,n}(t)\|_{H^{s}}$
$\displaystyle-\lim_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u_{1,n}(t)-u^{1,n}(t)\|_{H^{s}}$
$\displaystyle\gtrsim\,$
$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u_{-1,n}(t)-u_{1,n}(t)\|_{H^{s}}.$

Then it follows from the construction of $u_{m,n}$, Lemmas 6.1, 2.5 and 6.4,
and Fatou’s lemma that

$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\|u_{-1,n}(t)-u_{1,n}(t)\|_{H^{s}}$
$\displaystyle=\,$
$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}\left\|-2n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx)\sin(t)+[u_{l,-1,n}(t)-u_{l,1,n}(t)]\right\|_{H^{s}}$
$\displaystyle\gtrsim\,$
$\displaystyle\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{-1,n}_{R_{0}}\wedge\tau^{1,n}_{R_{0}}]}n^{-\frac{\delta}{2}-s}\Big{\|}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx)\Big{\|}_{H^{s}}|\sin
t|-\liminf_{n\rightarrow\infty}n^{\frac{\delta}{2}-1}$
$\displaystyle\gtrsim\,$ $\displaystyle\sup_{t\in[0,T_{l}]}|\sin t|,$

which is (1.22). The proof is therefore completed. ∎

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