Interpretable deep learning in single-cell omics

Manoj M Wagle1,2,3,+, Siqu Long1,3,+, Carissa Chen1,3, Chunlei Liu1,3, Pengyi
Yang1,2,3,4,*

1 Computational Systems Biology Unit, Children’s Medical Research Institute,
Faculty of Medicine and Health, The University of Sydney, Westmead, NSW 2145,
Australia
2 School of Mathematics and Statistics, Faculty of Science, The University of
Sydney, Camperdown, NSW 2006, Australia
3 Sydney Precision Data Science Centre, The University of Sydney, Camperdown,
NSW 2006, Australia
4 Charles Perkins Centre, The University of Sydney, Camperdown, NSW 2006,
Australia
\+ Equal contribution

* Correspondence<EMAIL_ADDRESS>
## Abstract

Recent developments in single-cell omics technologies have enabled the
quantification of molecular profiles in individual cells at an unparalleled
resolution. Deep learning, a rapidly evolving sub-field of machine learning,
has instilled a significant interest in single-cell omics research due to its
remarkable success in analysing heterogeneous high-dimensional single-cell
omics data. Nevertheless, the inherent multi-layer nonlinear architecture of
deep learning models often makes them ‘black boxes’ as the reasoning behind
predictions is often unknown and not transparent to the user. This has
stimulated an increasing body of research for addressing the lack of
interpretability in deep learning models, especially in single-cell omics data
analyses, where the identification and understanding of molecular regulators
are crucial for interpreting model predictions and directing downstream
experimental validations. In this work, we introduce the basics of single-cell
omics technologies and the concept of interpretable deep learning. This is
followed by a review of the recent interpretable deep learning models applied
to various single-cell omics research. Lastly, we highlight the current
limitations and discuss potential future directions. We anticipate this review
to bring together the single-cell and machine learning research communities to
foster future development and application of interpretable deep learning in
single-cell omics research.

## Introduction

The advances in high-throughput omics technologies have transformed our
ability to probe molecular programs at a large scale, providing insight into
the complex mechanisms underlying various biological systems and diseases.
Until recently, most early omics technologies have been typically applied to
profile a population of cells, known as ‘bulk’ profiling [19], where the
heterogeneity of cells and cell types are masked by the average signal across
the cell population [59]. Recent establishments of technologies such as
single-cell RNA-sequencing (scRNA-seq) [53, 47] and single-cell assay for
transposase-accessible chromatin by sequencing (scATAC-seq) [5] enables the
dissection of cellular composition and heterogeneity at the single-cell level
based on their gene expression and chromatin accessibility profiles. The
latest advancements in single-cell omics technologies towards multimodality
have further made it possible to obtain multimodal measurements simultaneously
from the same cell in a single experiment [4]. These new developments in the
single-cell omics field hold great promise for unlocking genetic information
at an unparalleled resolution for understanding multi-layered molecular
networks that underlie a broad range of cellular processes and diseases [3,
29].

Deep learning, a rapidly evolving sub-field of machine learning, has gained
considerable attention in the single-cell community for its capability to deal
with heterogeneous, sparse, noisy, and high-dimensional single-cell omics data
and versatility in handling a wide range of applications [37]. For example,
deep learning models have been demonstrated to excel in tasks such as
dimension reduction, batch effect removal, data imputation, cell type
annotation, and inferring cellular trajectories [31, 55, 2, 64, 30].
Nevertheless, deep learning models are well known for their lack of
interpretability [57]. That is, predictions made by these models are often
hard to interpret, especially towards understanding the underlying molecular
mechanisms that drive cellular processes and phenotype. To this end, improving
model interpretability has attracted increasing attention, in particular, in
applications such as identifying molecular regulators and reconstructing
molecular networks [24, 8, 15, 34].

In this work, we review the basics of single-cell omics technologies and key
principles behind interpretable deep learning. We next summarise the latest
development of interpretable deep learning models specifically tailored to
single-cell omics research, providing a global view of the current
applications in the main interpretable deep learning taxonomy. Finally, we
discuss the challenges and potential future directions in this burgeoning
field. We hope that this review will shed light on the current state of the
field and guide researchers toward making deep learning both robust and
reliable for single-cell research.

## Fundamentals of single-cell omics and interpretable deep learning

### The advent of single-cell omics technologies

The establishment of scRNA-seq technologies that enable transcriptomic
profiling at single-cell resolution (Fig. 1a) has revolutionised biomedical
research and has since emerged as a powerful tool for dissecting cellular
composition [53]. With its potential to reveal variability in cell-to-cell
gene expression at an unparalleled accuracy, the use of scRNA-seq has led to
ground-breaking discoveries that are unattainable from bulk data, such as cell
type annotation for model organisms [25, 12] and cell lineage tracing during
development and disease progression [27, 58]. The development of single-cell
techniques that profile other modalities (e.g., scATAC-seq (Buenrostro et al.,
2015) and single-cell bisulfite sequencing [scBS-seq] of methylomes [49]), and
their combination with other single-cell techniques such as cytometry [50]
have led to the generation of additional data modalities and together can
provide a more holistic view of the multi-layered molecular programs in single
cells (Fig. 1b,c). Nevertheless, these ‘unimodal’ single-cell omics
technologies are often independently applied, and each molecular attribute is
profiled separately, creating significant difficulties in data modality
integration from such ‘unpaired’ data.

The recent advance of single-cell omics technologies towards multimodality
alleviates the difficulties in integrating data modalities from unpaired data
generated by unimodal technologies by measuring multiple data modalities from
each single cell [4]. Thus, the multimodal single-cell omics data generated
from such ‘paired’ experiments provide a true single-cell view across multiple
molecular attributes. For example, cellular indexing of transcriptomes and
epitopes by sequencing (CITE-Seq) [51], simultaneous high-throughput ATAC and
RNA expression with sequencing (SHARE-seq) [38], and simultaneous single-cell
methylome and transcriptome sequencing (scMT-seq) [22] each capture a
different combination of two modalities in a single cell (Fig. 1d), and
techniques such as TEA-seq [52] and single-cell nucleosome, methylation and
transcription sequencing (scNMT-seq) [11] can capture a combination of three
modalities in a single cell. A recent review has summarised a comprehensive
list of multimodal single-cell omics technologies [56], and the integrative
analyses of such data are poised to revolutionise molecular and cellular
biology by transforming our understanding of molecular regulators and networks
that underlie a broad range of cellular processes and diseases.

Figure 1: Summary illustration of single-cell omics. (a) A schematic of single
cells from different complex tissues/organs. (b) Molecular attributes in a
single cell and their corresponding modalities in single-cell omics. (c, d)
Example unimodal (c) and multimodal (d) single-cell omics technologies.

### Interpretability of deep learning models

While the term ‘model interpretability’ still lacks a universal consensus of
definition in the machine learning community [32], it is broadly considered to
be the ability of the model to generate, explain, and present in human
understandable terms its decision-making process or insights of data [14, 40,
1]. Deep learning models, while incredibly successful in their application to
various domains, are typically considered as ‘black boxes’ for the lack of
interpretability [45] due to their complex multi-layer non-linear
architecture, activation functions, and potentially a large number of
parameters [66]. Such opacity poses significant challenges in establishing
trust and ensuring accurate validation, especially in domains such as
molecular biology and biomedicine, where the reasoning behind predictions is
crucial for our understanding and subsequent applications [41].

Adopting popular taxonomies in machine learning [14, 40, 1], the
interpretability of deep learning models can be largely categorised into (i)
intrinsic, whether prior knowledge or interpretable designs are incorporated
into the neural network structures, and post-hoc, where additional analyses
are performed to extract interpretable knowledge from the trained neural
networks [40], and (ii) model-specific, where the interpretability is tailor-
made for a specific neural network design, and model-agnostic, where the
techniques could be used across different neural network architectures. For
example, post-hoc feature attribution techniques such as Shapley value
estimation [35] and LIME [44] are frequently used for identifying important
learning features from trained neural networks and are generally considered to
be model-agnostic interpretations as they are applicable to various neural
network architectures. In contrast, the design of specific model architectures
for interpretable learning, such as the use of transformer networks with
attention layers, is intrinsic and specific to the model [7]. In the next
section, we will review the current works of interpretable deep learning
applications to single-cell omics research in light of these overarching
categories.

## Harnessing the power of interpretable deep learning for single-cell omics
research

The application of deep learning models to single-cell omics data analysis has
been met with remarkable success owing to their capability to deal with
challenging data characteristics (e.g. heterogeneity, sparsity, noise, high-
dimensionality) and versatility in handling a wide range of applications in
single-cell omics research. Nonetheless, deep learning models often lack
interpretability, complicating efforts towards understanding the underlying
molecular mechanisms that drive cellular processes and phenotype. In
recognition of this limitation, increasing research has been directed to
interpretable deep learning in single-cell omics. This section summarises the
latest developments in this fast-moving field based on the biological
applications of model interpretability, such as identifying cell identity
genes and molecular features (Fig. 2a), discovering gene sets (Fig. 2b) or
gene programs (iii) (Fig. 2c) that underlie cell types, and inferring
molecular networks (Fig. 2d) among other applications.

Figure 2: Schematic of example interpretable deep learning models applied to
single-cell omics. (a) Using post-hoc feature attribution techniques to
identify cell identity genes that distinguish cell types [24]. Colours denote
the estimated importance of input features (b) Designing an intrinsic and
model-specific attention layer to detect gene sets for annotating cell types
[8]. Colours denote the estimated importance of latent features (c)
Incorporating prior knowledge to design embeddings for detecting activated
gene programs underlie cell types [34]. Colours denote different gene
programs. (d) Using prior knowledge to design neural network architectures for
modelling molecular networks [15]. Colours denote different molecular species.

### Identifying cell identity genes and molecular features from unimodal data

The identification of genes and other molecular features that underlie cell
identity and can discriminate cell types is an essential task in single-cell
omics data analysis [63]. scDeepFeatures exemplifies the use of several post-
hoc approaches for identifying cell identity genes from scRNA-seq data, where
a simple multilayer perceptron (MLP)-based neural network is trained to
classify cell types, and subsequently various feature attribution techniques
(e.g. LIME, feature ablation, occlusion, DeepLIFT) are applied to identify
genes that discriminate cell types [24]. Similarly, Hu et al. proposed a post-
hoc and model-agnostic data permutation approach to extract surface proteins
from cytometry data using a convolutional neural network [23]. Alternatively,
scMGCA is a post-hoc and model-specific approach where a graph convolutional
autoencoder is first used to learn an embeddings-by-cells matrix from the
input data of a normalised count matrix and a cell graph, and a post-hoc
embedding analysis procedure is used to identify key cell identity genes that
separate cell types and are functionally enriched in the Gene Ontology (GO)
analysis [65]. scETM serves as a good example for intrinsic and model-specific
approaches and uses a variational autoencoder and a linear decoder to
factorise the input data into a tri-factor of cells-by-topics, topics-by-
embeddings, and embeddings-by-genes matrices. It allows the incorporation of
prior pathway information and together enables the identification of
interpretable gene markers and cellular signatures when integrating multiple
scRNA-seq datasets [62]. scBERT is another intrinsic and model-specific
approach that uses a transformer with an attention layer to encode the pre-
training data into an embeddings-by-cells matrix and leverage this embedding
matrix for subsequent cell-type annotation of future datasets. Cell-type
discriminative genes and their long-range dependencies are captured by the
attention layer in the model [62]. Finally, siVAE uses cell-wise and gene-wise
variational autoencoders to learn interpretable embeddings linked to important
genes and co-expression networks [10].

### Detecting genes and molecular features from multimodal data

Similar to gene selection from unimodal scRNA-seq data, various methods have
also been developed for identifying other molecular features from multimodal
single-cell omics data. To this end, several popular approaches make use of
autoencoders where multiple data modalities are integrated through embedding
learning. For example, Matilda uses a multi-task learning framework of a
variational autoencoder and a classification head to classify cell types and a
post-hoc feature attribution procedure to identify molecular features from
multimodal single-cell omics data that contribute to the classification of
each cell type [33]. Similarly, UnitedNet uses a dual autoencoder framework
for integrating data modalities and subsequently identifying important
molecular features using a post-hoc feature attribution technique of Shapley
value estimation [54]. totalVI also learns from multiple data modalities using
a variational autoencoder and offers a post-hoc archetypal analysis to
interpret each latent dimension by relating them to the molecular features in
the input data [16]. Lastly, scMM builds a mixture-of-experts of variational
autoencoders for integrating data modalities and then performs a post-hoc step
to traverse latent dimensions for identifying molecular features strongly
associated with each latent dimension [39]. These four methods can be
generally considered model-agnostic given that the post-hoc approaches they
used for identifying molecular features, albeit different, are largely model-
independent.

### Discovering gene sets and programs that govern cell identity and states

Another essential task that extends to gene identification is to discover gene
sets or gene programs that jointly govern cell identity, cell states, and
cellular processes. scDeepSort represents one of the first methods that use
GNNExplainer, a post-hoc and model-agnostic approach, for selecting gene sets
that are predictive of cell types from a weighted graph neural network trained
on scRNA-seq data [48]. Compared to post-hoc and model-agnostic approaches,
methods that rely on intrinsic and model-specific mechanisms have enjoyed more
popularity. For example, scCapsNet uses a capsule network model for
interpretable learning and captures gene sets that are predictive to cell
types from scRNA-seq data [60]. TOSICA uses a multi-head self-attention
network to incorporate prior biological knowledge for identifying gene sets
that belong to pathways or regulons for cell type annotation using scRNA-seq
data [8]. Alternatively, VEGA utilises prior knowledge of gene modules for
designing an interpretable latent space in variational autoencoders and
subsequently detecting active modules from models trained on scRNA-seq data
[46]. Likewise, ExpiMap incorporates prior knowledge of gene programs into a
sparsely connected variational autoencoder during pre-training on large scRNA-
seq reference atlases and subsequently identifies gene programs that are
associated with cell types and cell states in query datasets [34]. Finally,
pmVAE trains multiple variational autoencoders, each incorporating prior
information of a pathway module for detecting biological effects such as cell
stimulation from scRNA-seq data [18].

### Inferring molecular networks from single-cell omics data

Building on the concept of gene sets and pathways, the next related task is to
infer molecular networks, such as gene regulatory networks (GRNs), or capture
regulatory relationships among transcription factors (TFs) and their target
genes [3, 29]. One of the first methods towards achieving this aim is the
knowledge-primed neural network (KPNN), an intrinsic and model-specific
approach that explores the design of a sparsely connected neural network based
on prior knowledge of genome-wide regulatory networks and subsequently trains
the model using scRNA-seq data to learn regulatory strengths as weights of
network edges between TFs and their target genes [15]. Alternatively,
scGeneRAI attempts to infer GRNs by predicting the expression of a gene from a
set of other genes using scRNA-seq data and layer-wise relevance propagation
(LRP), a post-hoc and model-specific feature attribution technique [28].
Similar to scGeneRAI, STGRNS aims to reconstruct GRNs by predicting TF
expressions using gene sets but relies on an intrinsic and model-specific
approach where a transformer network with a multi-head attention layer is
trained on scRNA-seq data [61]. More recently, methods such as DeepMAPS have
been developed for intrinsic and model-specific interpretable learning from
multimodal single-cell omics data [36]. In particular, DeepMAPS uses a graph
autoencoder for integrating data modalities and then inserts the trained graph
autoencoder into a heterogeneous graph transformer with mutual attention
layers for inferring cell-type-specific GRNs.

### Predicting transcription factor binding sites and sequence motifs

In GRNs, TFs regulate their target genes through binding to cis-regulatory
elements (CREs) that contain specific sequence motifs. The prediction of TF
binding sites and sequence motifs therefore goes further than inferring GRNs
solely based on gene expression and deepens our understanding of underlying
mechanisms of molecular network regulation. Several studies have demonstrated
the utility of convolutional neural networks (CNNs) for addressing this task.
In particular, ExplainNN predicts TF binding sites and sequence motifs through
learning convolutional filters on DNA sequences. The model predictions are
validated using scATAC-seq data and curations in databases such as JASPAR
[42]. IscPNAM is another CNN-based method for predicting TF binding sites but
integrates DNA sequences with bulk data (e.g. ATAC-seq data) for their
prediction and generates interpretations from additional attention modules in
the network [17]. Similar to ExplainNN, IscPNAM also uses scATAC-seq data for
its prediction evaluation. Lastly, scover also uses CNN for interpretable
learning of sequence motifs using convolutional filters. While scATAC-seq data
are used for validation purposes in ExplainNN and IscPNAM, they are used for
training CNNs in scover for discovering regulatory motifs that are cell-type-
specific and reside in distal CREs and therefore can benefit from the cell-
type-specific information captured by single-cell omics data [20]. Although
IscPNAM uses model intrinsic attention modules for interpretations, all three
methods rely on intrinsic and model-specific mechanisms for interpretable
learning.

### Other applications

The above applications of interpretable deep learning mostly centre around
some closely related tasks of identifying genes and gene sets and inferring
their regulatory relationships from unimodal and multimodal single-cell omics
data. Beyond these, a few studies have explored other potential applications.
Examples include TAPE and UCDBase for bulk transcriptomic data deconvolution
using scRNA-seq data. Specifically, TAPE implements a training stage for an
autoencoder and an adaptive learning stage to extract interpretable
information from the trained model, including cell-type-specific signature
matrix and gene expression profiles, and predicted cell-type composition in
the bulk data [9]. On the other hand, UCDBase first pre-trains a densely-
connected neural network model using large-scale scRNA-seq atlases and
transfers the model for bulk data deconvolution. The model interpretations are
generated using a post-hoc model-specific feature attribution technique of
integrated gradients [6]. Another example is PAUSE which models transcriptomic
variation by using a biologically constrained autoencoder to attribute
variations in scRNA-seq data to pathway modules [26].

## Current challenges and future opportunities

Interpretable deep learning has made a significant impact on the single-cell
omics research field. However, the current application of interpretable deep
learning techniques to single-cell omics research is still limited to a few
related tasks of identifying genes and programs and inferring molecular
networks they form. Other common tasks, such as trajectory inferences and
cell-cell interactions, are crucial in single-cell omics data analysis [21]
but remain less explored in the context of interpretable deep learning. We
anticipate that future method development and application will investigate the
potential utility of interpretable deep learning in addressing these tasks.

Besides multimodality, the recent development of single-cell omics research is
increasingly towards multi-sample and multi-condition. While methods such as
ExpiMap have been designed for data generated from different samples and with
various perturbations and conditions in mind [34], there is still a lack of
application of interpretable deep learning models to specifically address
these emerging data structures that go beyond typical tasks of cell type
annotation, cell identity gene selection, and GRN inference from a normal
sample without experimental perturbations. Given the increasing application of
single-cell omics techniques to studying human diseases and drug
perturbations, we expect to see new methods developed to extract interpretable
information from these more complex data structures.

Another aspect of single-cell omics technological advancements is in the field
of spatial transcriptomics [43]. Few interpretable deep learning methods have
been specifically designed to take advantage of the extra spatial information
besides the gene expression profile of cells. Yet, such information can be
valuable for studying cell-cell communications and cellular microenvironments
that drive normal development and diseases such as cancer. Therefore,
developing deep learning models to integrate spatial information and other
omic data modalities in single cells for interpretable learning may lead to a
better understanding of developmental processes and improved treatments of
cancer.

It is important to note that annotating interpretable deep learning models
reviewed in this work into the four overarching categories is useful for their
summarisation. However, this is only intended to serve as a conceptual
framework for ease of understanding the main strategies used in each method.
Some methods use multiple interpretable learning techniques that can fit into
more than one category, while others develop new strategies that may not
precisely fit into any of these categories. Furthermore, there are additional
taxonomy strategies, such as classifying model interpretability to be global
and local [1], but are excluded due to their less utility in summarising
methods reviewed in this work.

Related to the above, the concept and definition of interpretability are fast-
evolving. For example, methods that use simpler neural network architectures
and perform linear transformation can be viewed as more interpretable. Besides
improving model interpretability, methods that generate biologically
meaningful results can also be viewed as more interpretable. For instance, in
scvis, increased interpretability is defined as generating embeddings that
better preserve the local and global neighbour structures in the original
high-dimensional scRNA-seq data [13]. While these definitions of
interpretability are beyond the scope of this review, they are nonetheless
important aspects that will contribute to the future development of
interpretable deep learning in the single-cell omics research field.

## Conclusions

Deep learning models, previously viewed as ‘black boxes’ for their lack of
interpretability, have become increasingly interpretable due to the recent
progress made in interpretable deep learning. This has stimulated a growing
interest in using interpretable deep learning techniques for single-cell omics
research, as the ability to identify and understand molecular regulators and
networks is critical for guiding downstream experimental validations. Here, we
briefly introduce the key concepts in single-cell omics technologies and
interpretable deep learning techniques and then review the recent advancement
in the development and application of interpretable deep learning models to
various single-cell omics data analysis tasks. We discuss current challenges
and opportunities and hope this review will catalyse this multidisciplinary
research field for future development and application of interpretable deep
learning to accelerate single-cell omics research.

### Acknowledgements

P.Y. is supported by a National Health and Medical Research Council
Investigator Grant (1173469). The authors thank the useful feedback from the
members of the Computational Systems Biology Unit.

### Author contributions

P.Y. conceptualised this work and supervised all authors to review the
literature, and write and edit the manuscript.

### Competing interests

The authors declare no competing interests.

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11institutetext: Koushik Chatterjee 22institutetext: Institute for Research in
Electronics and Applied Physics, University of Maryland, 8279 Paint Branch
Drive, College Park, MD 20742, USA, 22email<EMAIL_ADDRESS>33institutetext:
Matthew Liska 44institutetext: Center for Relativistic Astrophysics, Georgia
Institute of Technology, Howey Physics Bldg, 837 State St NW, Atlanta, GA
30332, USA, 44email<EMAIL_ADDRESS>55institutetext: Alexander
Tchekhovskoy 66institutetext: Center for Interdisciplinary Exploration &
Research in Astrophysics (CIERA), Physics & Astronomy, Northwestern
University, Evanston, IL 60202, USA, 66email<EMAIL_ADDRESS>77institutetext: Sera Markoff 88institutetext: Anton Pannekoek Institute for
Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands, 88email<EMAIL_ADDRESS>99institutetext: Ramesh Narayan
1010institutetext: Black Hole Initiative at Harvard University, 20 Garden
Street, Cambridge, MA 02138, USA, 1010email<EMAIL_ADDRESS>
# Extreme magnetic fields around black holes

Koushik Chatterjee0000-0002-2825-3590
Matthew Liska0000-0003-4475-9345
Alexander Tchekhovskoy0000-0002-9182-2047
Sera Markoff0000-0001-9564-0876  and
Ramesh Narayan0000-0002-1919-2730

###### Abstract

Recent results of the event horizon-scale images of M87* and Sagittarius A*
from the Event Horizon Telescope Collaboration show that strong magnetic
fields are likely present around the central black holes (BHs) in these
sources. Magnetically arrested disks (MADs), the end stage of magnetic flux
saturation around BHs, are especially rich in horizon-scale physics due to the
presence of powerful jets and magnetic flux eruptions that provide significant
feedback on the accretion mechanism. Here, we present an overview of our
current knowledge about the magnetic field evolution in numerical simulations
of accreting BHs, focusing on jet launching, black hole-interstellar medium
feedback, and black hole imaging of MADs. We find that misaligned MAD
accretion flows seemingly exhibit jet ejection cycles that could produce
flaring states in radio-quiet active galactic nuclei. Further, we show that
advances in horizon-scale interferometric telescopes could identify disk
misalignment by imaging the disk-jet connection region.

##  1 Introduction

Figure 1: (Left) Magnetised gas turbulence triggers particle acceleration that
leads to simultaneous moderate nIR/X-ray flares, but cannot explain the most
energetic flares Chatterjee2021 . We show a multi-wavelength SED (solid and
dashed lines) of Sagittarius A* from a GRMHD simulation of a weakly magnetised
accretion flow around a spinning BH assuming synchrotron emission from both
thermal and non-thermal (i.e. accelerated) electron populations. Sgr A* data
is shown in black (see Chatterjee2021  for more details). Note that we ignore
synchrotron self-Compton calculations in this SED. (Right) Magnetically
arrested disks (MADs) exhibit magnetic explosions that heat up gas (red) in
the disk (green/blue), a mechanism favourable for producing extreme high-
energy flares.

While the idea of magnetically dominated accretion flows has been around for
at least 50 years BK1974 , this mode of accretion, dubbed magnetically
arrested disks (MADs), truly became a subject of interest over the past 2
decades or so Igumenshchev2003 ; Narayan2003 . Part of the interest stemmed
from advances in computational development of 3D general relativistic
magnetohydrodynamic (GRMHD) codes that enabled self-consistent simulations of
black hole (BH) accretion. On the observational front, the multitude of
Sagittarius A* (or Sgr A*) X-ray flares captured by the Chandra observatory
Baganoff2001 ; Nowak2012 ; Haggard2019  provided indications of particle
acceleration, possibly driven by reconnection of magnetic fieldlines close to
the BH Markoff2001 ; Yuan2003 . Figure 1 shows that our standard picture of
magnetorotational instability (MRI)-driven accretion produces X-ray and near-
infrared (nIR) flares too weak to explain some of Sgr A*’s brightest flares
Do2019 ; Haggard2019 , suggesting the need for stronger magnetic fields. More
recently, the polarised horizon-scale EHT images of M87 and Sgr A* also favour
strong vertical magnetic fields EHTpaperVIIIM872021 ; EHTpaperVIIISgrA2024 .
Indeed, the MAD state could be a natural outcome of starved BHs (i.e.,
accreting at highly sub-Eddington rates), which is the case for both Sgr A*
and M87. There is seemingly an abundant supply of poloidal magnetic field
either by advecting $\mu$Gauss-level magnetic fields from the interstellar
medium (ISM) Narayan2003 ; Haverkorn2013  or from magnetised stellar winds (as
thought to occur in Sgr A*; Ressler2020 ), or by in-situ generation of
poloidal flux via dynamo processes in the disk Liska2020 , favouring the
emergence of the MAD state. Motivated by these advances in theory, numerics
and observations, we discuss the many features of MAD accretion flows and
their outflows, with a focus on recent results of misaligned MAD flows from
Chatterjee et al. Chatterjee2023tiltedMADs .

##  2 Revisiting magnetically arrested accretion flows

In this section, we introduce some salient features of the MAD state. We
divide our discussion into two parts: time-averaged and time-dependent
disk/jet properties. The former category of analysis allows us to understand
broad features of MADs and cross-compare between multiple simulations, while
the latter involves tracking the gas and magnetic field dynamics over time
with the aim of studying the evolution of magnetic explosions often seen in
MADs, and their effect on jet morphology. For our discussion, we make use of
results from MAD simulations, performed using the GRMHD code H-AMR Liska2022 ,
from a number of published papers Chatterjee2020 ; Chatterjee2022 ;
Chatterjee2023tiltedMADs . These results are largely consistent with
simulations from other codes.

We focus on three quantities: the accretion rate $\dot{M}=-\iint\rho
u^{r}dA_{\theta\varphi}$, the poloidal magnetic flux $\Phi_{\rm
BH}=(1/2)\iint|B^{r}|dA_{\theta\varphi}$ calculated at the event horizon, and
the total outflow power $P=\dot{M}c^{2}-\dot{E}$. For these quantities, we
require the gas density $\rho$, the radial component of the gas 4-velocity
$u^{r}$, the radial component of the 3-magnetic field vector $B^{r}$ and the
total energy flux in the radial direction $\dot{E}=\iint
T^{r}_{t}dA_{\theta\varphi}$, defined using the stress-energy tensor
$T^{\mu}_{\nu}$ and the area element
$dA_{\theta\varphi}=\sqrt{-det(g_{\mu\nu})}d\theta d\varphi$. We assume
natural units $G=c=1$, which implies that our characteristic length-scale, the
gravitational radius $r_{\rm g}=GM/c^{2}$, and the corresponding light-
crossing time-scale $t_{\rm g}=r_{\rm g}/c$, are normalised to the BH mass
$M$. For our simulations, we can set $M=1$. Further, we can normalise the
magnetic flux and the outflow power with the time-averaged accretion rate
$\langle\dot{M}\rangle$, giving us the dimensionless magnetic flux $\phi_{\rm
BH}=\Phi_{\rm BH}/\sqrt{\langle\dot{M}\rangle}$ and the outflow efficiency
$\eta=P/\langle\dot{M}\rangle$ (note that $r_{\rm g}$ and the speed of light
$c$ are unity according to our units). We refer the reader to Porth et al.
Porth2019  for more details about these quantities.

Figure 2: (Left) We show the accretion-rate normalised magnetic flux at the BH
horizon for GRMHD simulations of MADs around a variety of BH spins
Chatterjee2022 . We also show the $1\sigma$ standard deviation for each model
to indicate the amount of variability in $\phi_{\rm BH}$. The dashed line
shows a cubic fit to the numerical data, using the scipy function curvefit.
(Right) We show the corresponding outflow efficiency $\eta$ along with the
$1\sigma$ standard deviation for each model. The dashed line shows the
expected Blandford-Znajek (BZ) jet efficiency $\eta_{\rm BZ}\propto\phi_{\rm
BH,fit}^{2}\Omega_{\rm H}^{2}$ BZ1977 ; Tchekhovskoy2012 ; Chatterjee2023JP ,
where $\Omega_{\rm H}=a/2r_{\rm H}$ is the horizon angular frequency and
$r_{\rm H}=(1+\sqrt{1-a^{2}})r_{\rm g}$ is the event horizon radius. We find
that for BHs with a large prograde spin, (1) the BZ power matches the
simulation results as jets dominate the total outflow power, and (2) both
$\phi_{\rm BH}$ and $\eta$ exhibit the largest variability among the models,
presumably due to powerful and frequent magnetic flux eruptions.

###  2.1 Time-averaged properties of MADs: magnetic flux saturation and
relativistic jets

MADs deal with magnetically-saturated BH environments where forces due to
magnetic fields are roughly in radial balance with the ram pressure from the
accreting gas. It thus follows that such a system should exhibit the maximum
horizon-scale poloidal magnetic flux possible given sufficiently large supply
from the disk. Any amount of extra magnetic flux accumulated near the horizon
results in expulsion of magnetic fluxtubes, or “magnetic flux eruptions”,
creating temporal oscillations of the (largely radial) magnetic flux
$\phi_{\rm BH}$ at the BH horizon. We will discuss these eruptions in the next
subsection. Figure 2 shows the time-averaged $\phi_{\rm BH}$ as a function of
BH spin, calculated over $5000\,t_{\rm g}$ using H-AMR simulations
EHTpaperVSgrA2022 ; Chatterjee2022 . We see that $\phi_{\rm BH}$ has a cubic
dependence of the BH spin parameter $a$: $\phi_{\rm
BH,fit}=41.84+26.06a-12.68a^{2}-14.23a^{3}$, which is roughly consistent with
previous work using GRMHD simulations Tchekhovskoy2012 ; Narayan2022 ;
Chatterjee2023tiltedMADs . While this cubic dependence on spin has not yet
been worked out from first principles, it is reasonable to assume that the BH
spin should strongly affect the MAD magnetic flux since (1) the magnitude of
$\phi_{\rm BH}$ depends on the horizon area (which, in turn, is spin-
dependent), and (2) the accretion disk close to the BH is subject to BH frame-
dragging. For example, Narayan et al. Narayan2022  showed that the disk scale-
height appears to be closely tied to the BH spin. Additionally, for BHs
accreting at or close to their Eddington rate, radiation physics would also
strongly affect disk structure Ricarte2023 ; Liska2024 ; Lowell2024 .

In MADs, the strong magnetic fields near the BH not only modulate disk
dynamics, but also power highly efficient relativistic jets. GRMHD simulations
show that, in most cases, there is always an outflow of energy (and sometimes
mass) in the form of jets and disk winds, provided that the BH environment is
sparse enough to allow jet propagation. Figure 2 shows one of the most
interesting properties of MADs: rapidly spinning prograde BHs can launch jets
with efficiencies $\eta$ exceeding 100% in the MAD state, first shown by
Tchekhovskoy et al. Tchekhovskoy2011 , which provides direct evidence of spin
energy extraction from BHs, a direct consequence of the Blandford-Znajek (BZ)
mechanism BZ1977 . Indeed, it has been shown that jets from MADs spindown
their BHs, with the equilibrium BH spin close to zero Tchekhovskoy2012 ;
Chatterjee2022 ; Narayan2022 ; Ricarte2023 ; Lowell2024 , which could have
exciting implications for spindown of BHs accreting at super-Eddington rates,
e.g., in gamma-ray bursts and ultra-luminous X-ray binaries Narayan2017 .

###  2.2 How MADs evolve over time: the role of magnetic flux eruptions

Figure 3: (Left) We show the x-y and x-z planes of two representative
snapshots of a MAD flow around a Schwarzschild BH from Chatterjee et al.
Chatterjee2022 , tracking the movement of a magnetic fluxtube as it propagates
into the disk. From the figures, we see that a bundle of fieldlines get
separated from the jet and form a fluxtube. The accretion flow gradually
shears out the fluxtube over time, giving rise to Rayleigh-Taylor-like (RT)
and Kelvin-Helmholtz (KH) instabilities at the fluxtube boundary that could
explain nIR flares seen in Sgr A* Zhdankin2023 . These fluxtubes push out disk
material, powering short-term winds, thus influencing angular momentum
transport Chatterjee2022 . (Right) Time evolution of $\phi_{\rm BH}$ and
$\eta$ for the $a=0.94$ simulation from Fig. 2. We also indicate the time-
averaged values from Fig. 2 with dashed lines. Flux eruptions repeatedly take
away magnetic flux from the BH magnetosphere when magnetic forces overcome the
accreting gas.

Numerical simulations show that the BH magnetosphere can continue accumulating
magnetic flux well past the time-averaged $\phi_{\rm BH}$ shown in Fig. 2,
slowly forming a current sheet layer, typically in the equatorial plane. At
the point of magnetic flux over-saturation, magnetic reconnection occurs close
to the BH, and nearly-cylindrical fluxtubes filled with vertical magnetic
fields are ejected out into the disk. Vital questions about the ejection
mechanism remain due to the complex 3D evolution of the reconnection event,
such as: (1) Is the formation of the current sheet only due to reconnection of
the azimuthal magnetic field? (2) Does reconnection of the radial magnetic
field trigger the eruption event, or does the azimuthal field also play a
role? (3) What determines the limited azimuthal range covered by the expelled
fluxtubes? (4) When are fluxtubes ejected and at what frequency in time? (5)
Does the disk structure and dynamics also affect the ejection mechanism apart
from shearing out the fluxtube once ejected into the disk?

Figure 3 shows an example of how an ejected fluxtube propagates through the
accretion disk in a GRMHD simulation. Additionally we show the time-evolution
of the magnetic flux $\phi_{\rm BH}$ and the outflow efficiency $\eta$. We
clearly see temporal oscillations in the time-series plots where there is a
slow rise as magnetic flux is advected along with the gas towards the BH,
thereby strengthening the outflow, and a fast decay due to fluxtube ejections.
High resolution ideal and resistive GRMHD as well as GR-Particle-in-Cell
simulations are actively investigating the ejection mechanism with a
particular focus on the decay rate of the magnetic flux and its relation to
the local reconnection rate Bransgrove2021 ; Ripperda2022 ; Galishnikova2023 .
Further, once ejected, magnetic flux eruptions push against gas in the disk
and can power disk winds, effectively regulating the disk angular momentum
Chatterjee2022 ; Manikantan2024 , possibly alongside Maxwell stresses due to
large-scale poloidal fields Narayan2012 . In summary, recent advances in
simulations present an exciting opportunity to study magnetic flux eruptions
in great detail, which could lead to new insights in the triggering, frequency
and energetics of flares in accreting BHs (e.g., see Fig. 1).

##  3 Jet - ISM collisions in MAD accretion flows

Figure 4: (Top) Chatterjee et al. Chatterjee2023tiltedMADs  finds that a
small-scale BH feedback loop can form for highly misaligned MAD accretion
flows. As the newly formed jet strengthens over time due to the accumulation
of magnetic flux at the horizon. When the jet begins to align with the BH
spin, it collides with the accreting gas, disrupting the supply of magnetic
flux to the horizon, leading to jet quenching. Once the jet subsides, new
magnetic flux can be advected and brought to the BH, thereby restarting jet
launching, and completing the feedback loop. In this particular simulation,
the timescale for the feedback loop is around $5\times 10^{4}t_{\rm g}$, or
around 3 days to 8 years for a $10^{6-9}M_{\odot}$ BH, which presents an
interesting possibility for flaring states in radio-quiet AGN and quiescent
X-ray Binaries. (Bottom left) Vertical cross-section plot of gas density plot
of the same simulation as above showing an extremely warped jet (blue)
launched out of a spinning BH which is accreting from a $90^{\circ}$
misaligned disk (red). (Bottom right) The distorted jet inflates shocked
cavities in the ISM. The shocked gas shows relativistic temperatures ($T>1$ in
relativistic units) suggesting significant non-thermal X-ray flaring. The jet
also shows limb-brightened features as magnetic fields become squeezed between
the jet and the ISM, which triggers reconnection and heats the gas. See this
link for a movie of this simulation: https://youtu.be/mTLbZgd5CtY.

Jets travelling through dense ISM environments, such as that found in quasi-
spherical accretion flows, could be subject to large-scale kink instabilities,
resulting in jet collapse. Gas infall in the polar regions of the BH could
also inhibit jet production. Thus jet propagation is intimately connected to
both the intrinsic jet power via the BZ process as well as environment
properties. Powerful jets can burrow through the environment, creating a
cocoon of magnetised gas that forms backflows and helps in jet collimation
Rohoza2024 . For jets with smaller $\phi_{\rm BH}$, the jets can quickly lose
stability and collapse, at times creating shocked cavities. Recent advances in
computational resources have enabled GRMHD simulators to expand to larger
length-scales and experiment with different types of accretion flows in order
to self-consistently capture the complex dynamics of jet-ISM interactions
Ressler2020 ; Lalakos2022 ; Cho2023 ; Kaaz2023 ; Ressler2023 ; Rohoza2024 .

Figure 4 shows an interesting case when the incoming gas and magnetic fields
are so misaligned from the BH spin vector such that the jet in the MAD state
collides directly with the disk bulk, turns almost $90^{\circ}$ degrees and
eventually falls apart Chatterjee2023tiltedMADs . In this scenario, the jets
actively regulate the advection of magnetic flux onto the BH horizon in a
manner similar to flux eruptions (but with a lot more power!), creating a
feedback loop as a result. The figure shows a plethora of interesting features
such as jet remnants, magnetised filaments, cavities during the “quenched jet
phase” and a twisted or bent jet during the “jetted MAD phase”. The gas
temperature ($T$) plot in Fig. 4 shows that such systems could exhibit
relativistically-hot gas that can produce X-ray emission alongside small-scale
radio jets. We require more in-depth analysis of the radiative emission of
these systems to see if such feedback loops can explain radio and X-ray
flaring in radio-quiet AGN as an alternative mechanism to binary BH-disk
interactions. See Chatterjee2023tiltedMADs  for details on the simulations and
jet/disk evolution for such systems.

##  4 Outlook

Figure 5: Misaligned accretion onto spinning BHs result in bent jets that do
not follow the BH spin direction. Misalignment significantly affects the
evolution of the horizon-scale structure in these systems, altering the BH
shadow size and shape as well as the extended jet structure. We show a
snapshot of raytraced images for disks with misalignment angles of
$0^{\circ}$, $30^{\circ}$ and $60^{\circ}$ Chatterjee2020 , measured at large
distances, and their corresponding synthetic reconstructions at 230GHz (using
the eht-imaging toolkit Chael2019ehtim ) with the 2017 EHT array
EHTpaperIM872019  and the future ngEHT array Raymond2021 ; Johnson2023 ,
assuming the M87 BH mass and distance. The 2017 EHT array fails to properly
capture the differences between these disks, demonstrating the need to resolve
the disk-jet connection region by including more short-baseline arrays.

Magnetically arrested disks present a uniquely rich laboratory for exploring
magnetised plasmas in extreme conditions, from the micro-scale physics of
reconnection, hotspots and shocks, to the meso-scales of BH accretion, disk
warping in misaligned inflows and magnetic flux eruptions, and all the way to
the large-scales of twisted and kink-unstable jets, magnetised backflows and
cocoons. Figure 5 shows one such example. We show raytraced horizon-scale
images of 3 misaligned near-MAD models at sub-millimetre wavelengths (using
the GR-radiative transfer code BHOSS Younsi2020 ) and their reconstructions
assuming the telescope array properties of the 2017 EHT array EHTpaperIM872019
, and a future large interferometric array (the next-generation EHT, or ngEHT
Raymond2021 ; Johnson2023 ), including only thermal noise for simplicity. We
find that even after accounting for the systematic errors in imaging, it is
possible to capture 3 primary features with the ngEHT array that indicate
misaligned accretion: (1) the shape of the BH shadow region (i.e., the
brightness depression in the center), given our knowledge of the source
inclination, (2) the ring asymmetry, and (3) the warped disk-jet just outside
the ring. We note that while these reconstructions are quite simplistic and
more careful work needs to be done (much like that seen in the ngEHT Analysis
Challenge papers; Roelofs2023 ; Chatterjee2023ngEHT ), it is encouraging that
observing the horizon-scale accretion flow with such high resolution is
possible either with ground-based arrays and potentially future space
missions.

Capturing the tell-tale signs of magnetic flux eruptions would be crucial to
identify the MAD state. The equatorial plane reconnection event that triggers
the eruption as well as the disk-fluxtube shearing/RT/KH instabilities could
produce high-energy flares Ripperda2022 , while the expulsion of accretion
material from the BH magnetosphere could produce a dip in the submillimetre
flux Gelles2022  in AGNs like M87 and Sgr A* in addition to affecting
polarimetric features in the horizon-scale image Jia2023  as well as the jet
boundary Davelaar2023 . Further, the development of the MAD state in
misaligned accretion often forces the inner part of the disk to “magneto-spin
align” with the BH spin McKinney2013 ; Chatterjee2023tiltedMADs , which have
crucial consequences for interpreting the hard state of X-ray Binaries
Fragile2023 , where some sources show quasi-periodic oscillations in the X-ray
lightcurve. It is possible that flux eruptions may explain some quasi-periodic
activity such as in AGNs Cui2023 , but power-spectral-density calculations of
the MAD $\dot{M}$ time-series have not yet revealed promising results,
primarily showing red noise. Evidently further investigations of the MAD state
in standard torii need to be done, such as the dependence of accretion
properties on simulation grid resolutions White2019 ; SosapantaSalas2024 ,
similar to analysis done for weakly-magnetised, MRI-dominated accretion flows
Porth2019 . Similarly, understanding the ejection mechanism of fluxtubes holds
important consequences for the interpretation of high-energy flares. Guided by
the plethora of multi-wavelength observations of accreting BHs and jets, the
future looks bright for numerical simulations dealing with accreting BHs with
plenty of exciting scientific achievements to look forward to.

###### Acknowledgements.

KC thanks Alex Raymond and Daniel Palumbo for their help with scripts to
generate and reconstruct synthetic data from GRMHD simulations using eht-
imaging. KC and RN were supported in part by grants from the Gordon and Betty
Moore Foundation and the John Templeton Foundation to the Black Hole
Initiative at Harvard University, and by NSF award OISE-1743747. ML was
supported by the John Harvard, ITC and NASA Hubble Fellowship Program
fellowships. AT acknowledges support from the NSF AST-2009884 and NASA
80NSSC21K1746 grants. AT was supported by BSF grant 2020747 and NSF grants
AST-2107839, AST-1815304, AST-1911080, AST-2206471, OAC-2031997.This research
was enabled by support provided by an INCITE program award PHY129, using
resources from the Oak Ridge Leadership Computing Facility, Summit, which is a
US Department of Energy office of Science User Facility supported under
contract DE-AC05- 00OR22725, as well as Calcul Quebec
(http://www.calculquebec.ca) and Compute Canada (http://www.computecanada.ca).
This work has made use of NASA’s Astrophysics Data System (ADS).

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