#  Suppression without Thawing:
Constraining Structure Formation and Dark Energy with Galaxy Clustering

Shi-Fan Chen<EMAIL_ADDRESS>School of Natural Sciences, Institute for
Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA  Mikhail M. Ivanov
<EMAIL_ADDRESS>Center for Theoretical Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA  Oliver H. E. Philcox<EMAIL_ADDRESS>Simons Society of Fellows, Simons Foundation, New York, NY 10010, USA  Center
for Theoretical Physics, Columbia University, New York, NY 10027, USA  Lukas
Wenzl<EMAIL_ADDRESS>Department of Astronomy, Cornell University, Ithaca,
NY, 14853, USA

###### Abstract

We present a new perturbative full-shape analysis of BOSS galaxy clustering
data, including the full combination of the galaxy power spectrum and
bispectrum multipoles, baryon acoustic oscillations, and cross-correlations
with the gravitational lensing of cosmic microwave background measured from
Planck. Assuming the $\Lambda$CDM model, we constrain the matter density
fraction $\Omega_{m}=0.3154\pm 0.0089$, the Hubble constant $H_{0}=68.34\pm
0.77\,\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, and the mass fluctuation
amplitude $\sigma_{8}=0.686\pm 0.027$ (equivalent to $S_{8}=0.704\pm 0.031$).
Cosmic structure at low redshifts appears suppressed with respect to the
Planck $\Lambda$CDM concordance model at $4.5\sigma$. We explore whether this
tension can be explained by the recent DESI preference for dynamical dark
energy (DDE): the BOSS data combine with DESI BAO and PantheonPlus supernovae
competitively compared to the CMB, yielding no preference for DDE, but the
same $\sim 10\%$ suppression of structure, with dark energy being consistent
with a cosmological constant at 68% CL. Our results suggest that either the
data contains residual systematics, or more model-building efforts may be
required to restore cosmological concordance.

††preprint: MIT-CTP/5731

Introduction. — Observational and theoretical efforts over the last three
decades have led to the establishment of the standard model of cosmology:
$\Lambda$CDM. This model can successfully fit a wide range of cosmological
data, in particular the various correlators of cosmological fluctuations
traced by the cosmic microwave background (CMB) anisotropies and large-scale
structure of the Universe (e.g., Aghanim  _et al._ , 2020; Alam  _et al._ ,
2021).

Despite its phenomenological successes, the $\Lambda$CDM model suffers from
significant theoretical questions. Many of its core ingredients, such as
cosmic inflation, dark matter, and dark energy are, at best, highly exotic.
The latter is particularly puzzling from the theoretical viewpoint. The
simplest explanation for dark energy is the famous cosmological constant,
which gives rise to the naturalness paradox that shatters the fundamental
pillars of physics: symmetry-based selection rules and dimensional analysis
Weinberg (1989); Burgess (2015). Whilst anthropic Weinberg (1987) and
landscape Polchinski (2006) explanations are possible, the cosmological
constant problem still poses a formidable conceptual challenge in fundamental
physics. This challenge is particularly relevant given the possible ($\gtrsim
2.5\sigma$) evidence for dynamical dark energy (DDE, also known as
$w_{0}w_{a}$CDM) recently reported by the Dark Energy Survey Instrument (DESI)
collaboration DESI Collaboration  _et al._ (2024a, b); Adame  _et al._ (2024).

In addition to DDE, the data contain other anomalies whose presence could
signal the breakdown of cosmological concordance. The most prominent is the
Hubble tension, i.e. the apparent disagreement between the direct and indirect
measurements of the Hubble constant $H_{0}$, a proxy for the age of the
Universe Riess  _et al._ (2022). Another important anomaly is the disagreement
of the direct and indirect probes of the growth of structure encoded by the
mass fluctuation amplitude $\sigma_{8}$, or the related structure growth
parameters $S_{8}\equiv(\Omega_{m}/0.3)^{1/2}\sigma_{8}$ and $f\sigma_{8}$
(where $f$ is the redshift-dependent logarithmic growth factor) Di Valentino
_et al._ (2021); Nguyen  _et al._ (2023). This discrepancy is observed in
multiple independent low-redshift datasets Abdalla  _et al._ (2022) (a
selection of which are shown in Fig. 1): cluster counts (e.g., Ade  _et al._ ,
2016; Bolliet  _et al._ , 2020), weak lensing measurements (e.g., Asgari  _et
al._ , 2021; Abbott  _et al._ , 2022), CMB lensing cross-correlations (e.g.,
Marques  _et al._ , 2024; White  _et al._ , 2022), and galaxy clustering in
redshift space (e.g., Nguyen  _et al._ , 2023; Ivanov  _et al._ , 2023;
Philcox and Ivanov, 2022; Chen  _et al._ , 2022a, b; Ivanov, 2021), though
there exist some outliers (Beutler  _et al._ , 2017; Horowitz and Seljak,
2017; Kobayashi  _et al._ , 2022; Dalal  _et al._ , 2023; Miyatake  _et al._ ,
2023; Yu  _et al._ , 2023; D’Amico  _et al._ , 2024; Farren  _et al._ , 2024;
Shaikh  _et al._ , 2024). In general relativity the expansion history and
growth of structure are intricately related through the equations of motion,
and the accumulation of cosmological tensions raise a natural question: do
they all point to a particular new physics model in a correlated fashion? This
Letter addresses this question focusing on the case of DDE and the
$\sigma_{8}$ tension.

Figure 1: Left: A comparison on various $S_{8}$ results available in the
literature (Chen  _et al._ , 2022; Farren  _et al._ , 2024; Abbott  _et al._ ,
2022; Busch  _et al._ , 2022; Madhavacheril  _et al._ , 2024; Tristram  _et
al._ , 2024), with our measurement (bottom) including galaxy two- and three-
point information ($G^{2}$ and $G^{3}$), cross-correlation with lensing
($G\kappa$), and BAO. Right: Dependence of our results on analysis choices
including choice of galaxy split, dataset, and maximum cross-correlation
scale.

We present an independent reanalysis of the galaxy clustering data from the
Baryon acoustic Oscillation Spectroscopic Survey (BOSS) in combination with
Planck CMB lensing, in an attempt to link the possible $\sigma_{8}$ tension
found in these data with the hints of DDE reported by DESI. Using
methodologies developed in previous works (Philcox and Ivanov, 2022; Ivanov
_et al._ , 2023, 2020a, 2020b, 2022a; Philcox  _et al._ , 2020), we measure
the BOSS three-dimensional redshift-space power spectrum and bispectrum
multipoles, and post-reconstructed BAO data. We also include the angular cross
correlation between the BOSS galaxies and Planck CMB lensing following Wenzl
_et al._ (2024a) (see also Pullen  _et al._ (2016); Singh  _et al._ (2017);
Doux  _et al._ (2018); Singh  _et al._ (2019); Darwish  _et al._ (2021); Chen
_et al._ (2022); Wenzl  _et al._ (2024a, b)). For the first time, we
consistently analyze all of these observables within the effective field
theory (EFT)-based full-shape (FS) framework Ivanov  _et al._ (2020a); D’Amico
_et al._ (2020); Chen  _et al._ (2022b). Our first important result is that,
when combined with a BBN prior on the baryon density, this dataset yields a
measurement of matter clustering amplitude $\sigma_{8}$ discrepant with the
Planck concordance value at the $4.5\sigma$ level (cf. Fig. 1). This
represents the strongest evidence for the $\sigma_{8}$ tension from the BOSS
dataset to date.

In the second part of this Letter, we study whether the $\sigma_{8}$ tension
can be explained by the DDE model suggested by the combination
DESI$+$Supernovae (SNe) and CMB. Specifically, we analyze a combination of the
BOSS data described above (both FS and BAO), including the galaxy-CMB lensing
correlations, the DESI BAO data at redshift $z>0.8$, and PantheonPlus SNe data
assuming the DDE model. We find that DDE does not restore concordance between
galaxy clustering data and the primary CMB. The optimal values of $\sigma_{8}$
in our DDE analysis is still in a $4.5\sigma$ tension with Planck, though the
$H_{0}$ is consistent with Planck, but not with the value implied by the
Cepheid-calibrated distance ladder Riess  _et al._ (2022) (though in better
agreement with (Freedman and Madore, 2022)). In addition, our dataset
constrains DDE competitively compared to DESI BAO but does not display any
evidence for DDE. The combination of the above results suggests that internal
tensions between the datasets seem to pull cosmological parameters in
directions uncorrelated with each other; this could motivate more efforts from
the model building perspective, as well as searches for systematic effects in
the full combination of the large-scale structure data.

Data. — Our primary dataset is the clustering of galaxies from the twelfth
data release of the BOSS survey Reid  _et al._ (2016); Dawson  _et al._
(2013). These galaxies are observed in both the northern (NGC) and southern
(SGC) galactic caps and are composed of the LOWZ and CMASS samples, each of
which are restricted to the redshift ranges $0.15<z<0.43$ and $0.43<z<0.70$ in
order to avoid overlap.111Unlike our previous works (e.g., Philcox and Ivanov,
2022), we split the sample by their physical type rather than imposing a
redshift-cut (e.g., Beutler  _et al._ , 2017). Combining both galactic caps,
the LOWZ and CMASS catalogs cover $8,579$ and $9,493$ deg2 with $361,762$ and
$777,202$ galaxies, respectively. The complete DR12 catalogs also contain
galaxies in two chunks LOWZE2 and LOWZE3 selected using different criteria
than the main LOWZ sample. These are often combined with the main samples in
order to maximize the survey volume, but, since the different selections imply
different galaxy properties, we will instead omit them in this work (this
choice was made also in pre-DR16 BOSS analyses (e.g., Gil-Marín  _et al._ ,
2016), leading to a smaller area in the LOWZ sample compared to CMASS).

To characterize the clustering of the above galaxy samples, we utilize the
power spectrum and bispectrum statistics, measured using the window-free
estimators derived in (Philcox, 2021a, b; Ivanov  _et al._ , 2023) (now
implemented in the PolyBin3D code (Philcox and Flöss, 2024)). We include the
standard systematic and FKP weights constructed by BOSS Reid  _et al._ (2016),
which imply that the power spectrum probes clustering at an effective redshift
$z_{\rm eff}=\int dV\,z\bar{n}^{2}(z)/\int dV\bar{n}^{2}(z)$, equal to $0.316$
($0.555$) for the LOWZ (CMASS) sample, where $\bar{n}$ is the weighted galaxy
number density. The same weights applied to the bispectrum would result in a
different effective redshift (instead weighted by $\bar{n}^{3}$); to
ameliorate this, we an additional redshift weight
$w_{B}(z)\equiv\bar{n}(z)^{-1/3}$ when computing the bispectrum. We
additionally include the real-space power spectrum proxy $Q_{0}$ (equal to the
power spectrum perpendicular to the line of sight) is estimated from the
redshift space multipoles Ivanov  _et al._ (2022a).

In addition to the power spectrum and bispectrum, we also include post-
reconstruction BAO measurements from the BOSS galaxies. Since this signal does
not depend strongly on galaxy properties, we will use measurements of the BAO
scale from the combined BOSS sample covering the full survey area and redshift
range, including the LOWZE2 and LOWZE3 samples omitted in the above full-shape
analysis (specifically, those from (Philcox  _et al._ , 2020)). This combined
sample is split into non-overlapping redshift bins $0.2<z<0.5$ (z1) and
$0.5<z<0.75$ (z3) chunks following Alam  _et al._ (2017); Beutler  _et al._
(2017) with effective redshifts of $z_{\rm eff}=0.38$ and $0.59$,
respectively. We compute the total covariance of the above measurements using
measurements from the 2048 public MultiDark Patchy mocks Kitaura  _et al._
(2016); Rodríguez-Torres  _et al._ (2016).

Figure 2: Constraints on cosmological parameters from our baseline dataset
(BOSS two- and three-point galaxy clustering correlations, cross correlations
of galaxies and CMB lensing and BAO; BOSS $G^{2}G^{3}G\kappa$+BAO) on the
standard $\Lambda$CDM model (red), as well as those on the dynamical dark
energy (DDE) model from the baseline data in addition to DESI BAO (at $z>0.8$)
and PantheonPlus supernovae (blue). Dashed lines mark the $\Lambda$CDM values
of the dark energy equation of state parameters $w_{0}$ and $w_{a}$,
corresponding to the cosmological constant.

In order to measure the lensing cross correlation with galaxies we use
publicly-available CMB lensing maps reconstructed from Planck data.
Specifically, we use the PR4 map introduced in Carron  _et al._ (2022), which
uses the updated NPIPE pipeline and slightly more data than previous releases,
leading to a $20\%$ improvement in signal-to-noise compared to PR3. We compute
the cross-correlations of the lensing convergence $\kappa$ with the LOWZ and
CMASS galaxies using the NaMaster algorithm Alonso  _et al._ (2019) adopting
the same numerical choices (including filters and apodization) as described in
Wenzl  _et al._ (2024a)—to which we direct the interested reader for further
details including extensive systemaics tests—except that we split the cross
correlations according to galactic cap. In particular we use NaMaster to
compute the bandpower window $M_{L\ell}$ relating the observed bandpowers in
bin $L$ to the unbinned theory $\hat{C}_{L}=M_{L\ell}C_{\ell}$, as well as an
analytic (Gaussian) covariance matrix using the theory predictions for the
measured $C_{\ell}$’s. We treat this covariance independently from that
derived from the three-dimensional clustering of galaxies since the mode
overlap is negligible Taylor and Markovič (2022). Similarly to the bispectrum,
we re-weight the galaxies when computing the cross correlation $C^{\kappa
g}_{\ell}$ by the ratio of the galaxy and CMB lensing kernels in order to
homogenize the effective redshifts probed with the galaxy power spectrum Chen
_et al._ (2022).

Finally, in addition to BOSS-volume data, we will supplement our analysis with
constraints on cosmological expansion at lower and higher redshfts obtained
from Supernovae Type Ia (SNIa) and external BAO data. For the former we adopt
the PantheonPlus dataset Scolnic  _et al._ (2022); Riess  _et al._ (2022)
(which constrains the redshift-dependence of luminosity distances) and BAO
likelihoods for the 6-degree Field Galaxy Redshift Survey (6dFGS) Beutler  _et
al._ (2011) and the Main Galaxy Sample (MGS) in SDSS DR7 Ross  _et al._
(2015), as implemented in MontePython Audren  _et al._ (2013); Brinckmann and
Lesgourgues (2019). For the latter, we use all galaxy BAO measurements from
DESI with $z>0.8$ DESI Collaboration  _et al._ (2024a) as well as the DESI
Ly$\alpha$ measurement DESI Collaboration  _et al._ (2024b). We also adopt the
BBN baryon density constraint from the primordial deuterium abundance
$\Omega_{b}h^{2}=0.02268\pm 0.00038$ Cooke  _et al._ (2018); Ivanov  _et al._
(2020a), and fix222Alternatively, one can free $n_{s}$ in the fit or use the
Harrison-Zeldovich theoretical value $n_{s}=1$. These choices have a marginal
impact on our results. the spectral tilt to the Planck best-fit value
$n_{s}=0.9649$.

Theoretical model. — We begin with a short overview of the theory model used
for galaxy clustering. At the background level, we adopt either a baseline
$\Lambda$CDM model, or the popular dynamical dark energy (DDE) extension,
which is parametrized according to the equation of state
$w(a)=w_{0}+w_{a}(1-a)$ Fang  _et al._ (2008). We compute predictions for the
redshift-space power spectrum and bispectrum, as well as the real-space
matter-galaxy cross power spectrum, using the effective field theory of large
scale structure (EFT) Baumann  _et al._ (2012); Carrasco  _et al._ (2012);
Ivanov (2023) as implemented in the CLASS-PT code Chudaykin  _et al._ (2020)
in a modification to the public
likelihoods.333https://github.com/oliverphilcox/full_shape_likelihoods Our
modeling follows the the conventions of Chudaykin  _et al._ (2020, 2021a);
Philcox and Ivanov (2022), to which we refer the reader for further details.
Briefly, the galaxy power spectrum is computed to one-loop in perturbation
theory while the bispectra are computed using the same bias parameters up to
quadratic order. The lensing cross correlation probes the real-space cross
power spectrum of galaxies with matter $P_{mg}$ perpendicular to the line of
sight; we evaluate $P_{mg}$ to one-loop order including an additional
counterterm for the matter field. In all cases the effects of long-wavelength
displacements on the BAO wiggles in the linear power spectrum are resummed
following Blas  _et al._ (2016a, b); Vlah  _et al._ (2016); Ivanov and
Sibiryakov (2018); Chen  _et al._ (2024) (see also Senatore and Zaldarriaga
(2015); Baldauf  _et al._ (2015)).

To make contact with observations, we rescale the wavenumbers in the redshift-
space power spectrum and bispectrum to correct for the mismatch of the true
cosmology and the fiducial cosmology (with $\Omega_{m}=0.31$) used to convert
the angles and redshifts in the galaxy catalogs to rectilinear coordinates.
Following this conversion, the power spectrum and bispectrum are then
converted into the measured multipoles by integrating over the requisite
angles Scoccimarro (2015); Scoccimarro  _et al._ (1999) and combined into the
measured $k$-bins, including weights to correct for discreteness effects as
described in Ivanov  _et al._ (2023). No such conversions are needed for the
angular power spectrum multipoles which are given in the Limber approximation
by Limber (1953); LoVerde and Afshordi (2008)

$\displaystyle C^{\kappa
g}_{\ell}=\int\frac{d\chi}{\chi^{2}}\Big{(}W^{\kappa}(\chi)W^{g}(\chi)\
P_{mg}\big{(}k=\frac{\ell+\frac{1}{2}}{\chi},z(\chi)\big{)}$
$\displaystyle+W^{\kappa}(\chi)W^{\mu}(\chi)(2\alpha-1)P_{mm}\big{(}k=\frac{\ell+\frac{1}{2}}{\chi},z(\chi)\big{)}\Big{)},$

where $\chi$ is comoving distance and the CMB lensing and galaxy density
kernels are given by

$W^{\kappa}(\chi)=\frac{3}{2}H_{0}^{2}\Omega_{m}(1+z)\frac{\chi(\chi_{\ast}-\chi)}{\chi_{\ast}},\
W^{g}(\chi)=\frac{dN}{d\chi}.$

We evaluate $P_{mg}$ at the effective redshift rather than parameterizing its
redshift evolution since the galaxy redshift distribution is very narrow.
Furthermore, $W^{\mu}$ is the galaxy lensing kernel and $\alpha$ is the
magnification bias—the latter contribution was studied extensively in Wenzl
_et al._ (2024c) and we use the values of $\alpha$ measured for LOWZ and CMASS
therein. Unlike the $\kappa-g$ term, the magnification bias contribution
probes the matter power spectrum to non-linear scales, though its support at
the smallest scale is curtailed since $W^{\kappa,\mu}$ fall to zero at short
distances. As this term gives only a small contribution, but one dependent on
non-perturbative physics, we model it via the one-loop $P_{mm}$ EFT prediction
supplemented with a phenomenological “resummed” version of the counterterm
whose parameters were fitted from HMcode Mead  _et al._ (2015). We stress that
the choice of non-linear corrections for the magnification bias has a
negligible impact on final results.

Our baseline analysis of BOSS (which we dub $G^{2}G^{3}G\kappa$+BAO) uses
redshift-space scale cuts $k_{\rm max}^{P_{\ell}}=0.2\,h{\text{Mpc}}^{-1}$ and
$k_{\rm max}^{B_{\ell}}=0.08\,h{\text{Mpc}}^{-1}$, and real-space scale cuts
for $Q_{0}$ and $P_{mg}$ at $k_{\rm max}^{\rm real}=0.4\,h{\text{Mpc}}^{-1}$
(corresponding to angular cuts $\ell_{\rm
max}^{\texttt{LOWZ},\texttt{CMASS}}=400,600$) validated in Ivanov  _et al._
(2020a); Nishimichi  _et al._ (2020); Chudaykin  _et al._ (2021a); Schmittfull
_et al._ (2021); Ivanov  _et al._ (2022b, a); Philcox and Ivanov (2022); Chen
_et al._ (2022); Ivanov  _et al._ (2023); Krause  _et al._ (2024).

Parametrization and priors. — In this Letter, we follow the EFT
parametrization in Ivanov  _et al._ (2020a); Philcox and Ivanov (2022); Ivanov
_et al._ (2023); Krause  _et al._ (2024). Briefly, galaxy clustering is
described by one linear, two quadratic, and one cubic bias parameters, along
with three counterterms and three stochastic terms up to $k^{2}$ in scale
dependence. In addition to these contributions we include a next-order finger-
of-god (FoG Jackson (1972)) term $k^{4}\mu^{4}P_{\rm lin}$ to account for the
effect leading to larger dynamical non-linearities than other effects
Chudaykin  _et al._ (2020). The tree-level bispectrum is described by these
bias parameters up to quadratic order and two additional stochastic terms
associated with the non-Gaussianity and density-dependence of short modes and,
like the power spectrum, a phenomenological FoG term. The real-space
clustering of matter requires an additional real space counterterm
$P_{mg}^{\rm c.t.}=2b_{1}c_{0}k^{2}P_{\rm lin}(k)$, for which we use the
Gaussian prior $\mathcal{N}(0,10^{2})$ in units $[h^{-1}{\text{Mpc}}]^{2}$,
resulting in 14 free parameters per sample.444Part of this counterterm also
enters the galaxy power spectrum, but the counterterm combinations that appear
in $P_{0,2,4}$ are linearly independent from that of $P_{gm}$, leading to four
parameters for four independent spectra.

Results. — We start by analyzing the BOSS FS and BAO data within the baseline
$\Lambda$CDM model. Our results are displayed in Fig. 2 and Tab. 1. We find
that the optimal values of cosmological parameters are consistent with the
Planck baseline CMB values Aghanim  _et al._ (2020), except for $\sigma_{8}$,
which shows a $4.5\sigma$ disagreement. The tension with the ACT CMB lensing
results Madhavacheril  _et al._ (2024) has a similar strength: 4.3$\sigma$.
For the $S_{8}$ parameter, the discrepancy is somewhat weaker, $3.8\sigma$,
though our results appear in agreement with weak lensing measurements by DES
Abbott  _et al._ (2022), KiDS Asgari  _et al._ (2021), and HSC Dalal  _et al._
(2023); Miyatake  _et al._ (2023), see Fig. 1.

Parameter |  $\Lambda$CDM | DDE
---|---|---
$\Omega_{m}$ | $0.3155~{}(0.3103)_{-0.0090}^{+0.0087}$ |  | $0.3223~{}(0.3232)_{-0.0081}^{+0.0079}$ |
$H_{0}$ | $68.54~{}(68.54)_{-0.81}^{+0.77}$ |  | $68.19~{}(67.47)^{+0.89}_{-0.89}$ |
$\sigma_{8}$ | $0.687~{}(0.704)_{-0.028}^{+0.026}$ |  | $0.686~{}(0.691)_{-0.028}^{+0.026}$ |
$S_{8}$ | $0.704~{}(0.716)^{+0.031}_{-0.031}$ |  | $0.711~{}(0.717)^{+0.029}_{-0.029}$ |
$w_{0}$ | $-1$ |  | $-0.907~{}(-0.971)_{-0.073}^{+0.072}$ |
$w_{a}$ | $0$ |  | $-0.49~{}(-0.05)^{+0.42}_{-0.34}$ |
Table 1: Cosmological parameters and their 68% confidence limits (with best-
fit values shown in parentheses), from the full BOSS dataset under the
$\Lambda$CDM model (left) and from BOSS+DESI+SNe under the DDE model (right).

Notably, the tension remains for different subsets of the underlying data (see
Fig. 1). Our tests include: (a) adopting a more conservative choice of
$\ell_{\rm max}=350$ for the lensing data (which yields
$\sigma_{8}=0.698_{-0.03}^{+0.03}$); (b) fitting only the galaxy power
spectrum multipoles using the Lagrangian EFT with the velocileptors
code555https://github.com/sfschen/velocileptors/ Chen  _et al._ (2020, 2021,
2022b); Maus  _et al._ (2024a) ($\sigma_{8}=0.731^{+0.048}_{-0.048}$); (c)
fitting only the multipoles plus CMB-lensing cross correlation
($\sigma_{8}=0.708^{+0.037}_{-0.037}$) as in ref. Chen  _et al._
(2022a)666https://github.com/sfschen/BOSSxPlanck; (d) removing the galaxy-CMB
lensing cross correlation ($\sigma_{8}=0.709_{-0.033}^{+0.031}$); (e)
analyzing the z1/z3 split of the BOSS data including LOWZE2/LOWZE3 samples
omitted in our main FS analysis ($\sigma_{8}=0.722_{-0.03}^{+0.03}$). We find
consistent results in all cases (Fig. 1).

Another important observation is that the preference for low $\sigma_{8}$ is
not a prior effect (see e.g. Ivanov  _et al._ (2020a); Chudaykin  _et al._
(2021b); Philcox and Ivanov (2022) for related studies), as previous studies
have shown that the preference for a low $\sigma_{8}$ value in the BOSS data
is present at the level of the raw $\chi^{2}$ statistic Chen  _et al._
(2022b); Philcox and Ivanov (2022). The tension also remains when more
informative reasonably narrow priors are applied. It will be interesting to
see if informative simulation-based priors Ivanov  _et al._ (2024a) can
sharpen our constraints further.

Secondly, we analyze the full combination of BOSS clustering, DESI BAO, and
SNe data assuming the DDE model. Our results are shown in Fig. 2 and Tab. 1.
The inferred $H_{0}$ value is consistent with previous CMB and LSS
measurements based on the $\Lambda$CDM model, confirming the standard lore
that DDE cannot resolve the Hubble tension (Di Valentino  _et al._ , 2021).
Turning to the $\sigma_{8}$ tension, we find nearly identical constraints in
the DDE model as in $\Lambda$CDM, implying that DDE cannot resolve the
$\sigma_{8}$ discrepancy in the BOSS galaxy clustering data. Finally, we
observe that the $w_{0}-w_{a}$ posterior is consistent with the $\Lambda$CDM
values $(-1,0)$ within 68% CL.

Notably, the FS data provides an independent channel to extract $\omega_{m}$
(and $\Omega_{m}$), relevant for the DDE constraints Ivanov  _et al._ (2020a);
Chudaykin  _et al._ (2021b). Our analysis suggests that the inclusion of this
information leads to a non-detection of DDE, compared to the weak preference
found in Appendix A of (Adame  _et al._ , 2024), which used only the BAO data
from BOSS/eBOSS. The BOSS FS data delivers constraints on the matter density
whose precision rivals that of the Planck CMB, which when combined with DESI
and PantheonPlus, cf. Adame  _et al._ (2024), i.e. BOSS FS can competitively
replace the CMB in breaking degeneracies inherent in BAO data. This remains
true if we vary the spectral tilt $n_{s}$ in the fit; as such, our results do
not require any input from the CMB.777In this case, we find
$\sigma_{8}=0.655~{}(0.683)_{-0.034}^{+0.031}$,
$w_{0}=-0.887~{}(-0.875)_{-0.08}^{+0.075}$,
$w_{a}=-0.80~{}(-0.65)_{-0.42}^{+0.52}$.

Conclusions. — In this Letter, we have presented a novel analysis of public
galaxy clustering and CMB lensing data from Planck and BOSS, representing the
most complete combination of galaxy correlators yet performed. Our results
suggest a value of the mass fluctuation amplitude in tension with the best-fit
$\Lambda$CDM value predicted by Planck CMB anisotropies, which cannot be
accounted for by dynamical dark energy (DDE); furthermore, the combination of
BOSS with BAO data from DESI and expansion data from supernovae does not yield
any evidence for DDE. Our results have several important implications.

From the phenomenological perspective, it would be interesting to build a
model that can resolve this $\sigma_{8}$ tension. There exist many proposals
that can readily produce some suppression on small scales, such as massive
neutrinos Lesgourgues and Pastor (2006); Ivanov  _et al._ (2020b), ultralight
axions Laguë  _et al._ (2022); Rogers  _et al._ (2023), light but massive
relics Xu  _et al._ (2022), baryon-dark matter scattering He  _et al._ (2023),
dark sector interactions Rubira  _et al._ (2023); Nunes  _et al._ (2022);
Joseph  _et al._ (2023), and beyond; however, it is unclear whether these can
account for the part of the suppression in BOSS that is present on large
scales $k\lesssim 0.1\,h{\text{Mpc}}^{-1}$ Chen  _et al._ (2022b); Philcox and
Ivanov (2022). In addition, such a model should keep cosmic structure at
$z\gtrsim 1$ unsuppressed, as suggested by eBOSS quasar Neveux  _et al._
(2020); Chudaykin and Ivanov (2023) and Lyman-$\alpha$ data Chabanier  _et
al._ (2019); Ivanov  _et al._ (2024b).

The $\sigma_{8}$ tension is generated by data that effectively measure the
cross-correlation between the galaxy field and a probe of matter, through
either CMB lensing or redshift-space distortions (which probe the velocity
field): $\sigma_{8}$ is extracted from the ratio between the relevant cross-
(i.e. the quadrupole and $P_{gm}$) and auto- galaxy power spectrum (i.e. the
monopole). An additive foreground, arising for example due to contaminants in
the photometric selection of target galaxies, would enhance the auto-spectrum
but cancel in both cross-correlations, leading to smaller ratios between them
and consequently lower $\sigma_{8}$. If such a systematic correction is
present, it will affect these two seemingly independent measurements of
$\sigma_{8}$ in a correlated fashion. That said, the addition of the
bispectrum, which instead probes structure growth through cancelling quadratic
and linear terms in the redshift-space galaxy density, is also found to reduce
the measured $\sigma_{8}$, though we caution that the impact of foreground
systematics on the 3-point function is less well-explored. In any case, we
expect the coming generation of galaxy surveys, which feature more robust data
and updated treatments of foreground systematics, to shed light on this issue.

Finally, it will be important to understand if the tensions in the expansion
history, particularly due to deviations away from a cosmological constant,
that appear from different combinations of the LSS data from BOSS and DESI are
physical, and if there is a new physics model that can account for them in a
consistent manner. Such effects, as well as further examination of the
low-$\sigma_{8}$ discrepancy, will certainly be illuminated with future full-
shape analyses of galaxy clustering data from DESI Maus  _et al._ (2024b) and
beyond.

Acknowledgements. — We are grateful to Emanuele Castorina, Guido D’Amico,
Julien Lesgourgues, John Peacock, Douglas Scott, Martin White, Matias
Zaldarriaga, and Pierre Zhang for insightful discussions. SC acknowledges the
support of the National Science Foundation at the Institute for Advanced
Study. OHEP is a Junior Fellow of the Simons Society of Fellows. We thank the
City of Edinburgh for their kaleidoscopic selection of mashed grains.

## References

  * Aghanim  _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Alam  _et al._ (2021) S. Alam _et al._ (eBOSS), Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Weinberg (1989) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
  * Burgess (2015) C. P. Burgess, in  _100e Ecole d’Ete de Physique: Post-Planck Cosmology_ (2015) pp. 149–197, arXiv:1309.4133 [hep-th] .
  * Weinberg (1987) S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987).
  * Polchinski (2006) J. Polchinski, in  _23rd Solvay Conference in Physics: The Quantum Structure of Space and Time_ (2006) pp. 216–236, arXiv:hep-th/0603249 .
  * DESI Collaboration  _et al._ (2024a) DESI Collaboration, A. G. Adame, J. Aguilar, S. Ahlen, S. Alam, D. M. Alexander, M. Alvarez, O. Alves, A. Anand, U. Andrade, _et al._ , arXiv e-prints , arXiv:2404.03000 (2024a), arXiv:2404.03000 [astro-ph.CO] .
  * DESI Collaboration  _et al._ (2024b) DESI Collaboration, A. G. Adame, J. Aguilar, S. Ahlen, S. Alam, D. M. Alexander, M. Alvarez, O. Alves, A. Anand, U. Andrade, _et al._ , arXiv e-prints , arXiv:2404.03001 (2024b), arXiv:2404.03001 [astro-ph.CO] .
  * Adame  _et al._ (2024) A. G. Adame _et al._ (DESI), (2024), arXiv:2404.03002 [astro-ph.CO] .
  * Riess  _et al._ (2022) A. G. Riess _et al._ , Astrophys. J. Lett. 934, L7 (2022), arXiv:2112.04510 [astro-ph.CO] .
  * Di Valentino  _et al._ (2021) E. Di Valentino _et al._ , Astropart. Phys. 131, 102604 (2021), arXiv:2008.11285 [astro-ph.CO] .
  * Nguyen  _et al._ (2023) N.-M. Nguyen, D. Huterer, and Y. Wen, Phys. Rev. Lett. 131, 111001 (2023), arXiv:2302.01331 [astro-ph.CO] .
  * Abdalla  _et al._ (2022) E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Ade  _et al._ (2016) P. A. R. Ade _et al._ (Planck), Astron. Astrophys. 594, A24 (2016), arXiv:1502.01597 [astro-ph.CO] .
  * Bolliet  _et al._ (2020) B. Bolliet, T. Brinckmann, J. Chluba, and J. Lesgourgues, Mon. Not. Roy. Astron. Soc. 497, 1332 (2020), arXiv:1906.10359 [astro-ph.CO] .
  * Asgari  _et al._ (2021) M. Asgari _et al._ (KiDS), Astron. Astrophys. 645, A104 (2021), arXiv:2007.15633 [astro-ph.CO] .
  * Abbott  _et al._ (2022) T. M. C. Abbott _et al._ (DES), Phys. Rev. D 105, 023520 (2022), arXiv:2105.13549 [astro-ph.CO] .
  * Marques  _et al._ (2024) G. A. Marques _et al._ (ACT, DES), JCAP 01, 033 (2024), arXiv:2306.17268 [astro-ph.CO] .
  * White  _et al._ (2022) M. White _et al._ , JCAP 02, 007 (2022), arXiv:2111.09898 [astro-ph.CO] .
  * Ivanov  _et al._ (2023) M. M. Ivanov, O. H. E. Philcox, G. Cabass, T. Nishimichi, M. Simonović, and M. Zaldarriaga, Phys. Rev. D 107, 083515 (2023), arXiv:2302.04414 [astro-ph.CO] .
  * Philcox and Ivanov (2022) O. H. E. Philcox and M. M. Ivanov, Phys. Rev. D 105, 043517 (2022), arXiv:2112.04515 [astro-ph.CO] .
  * Chen  _et al._ (2022a) S.-F. Chen, M. White, J. DeRose, and N. Kokron, JCAP 07, 041 (2022a), arXiv:2204.10392 [astro-ph.CO] .
  * Chen  _et al._ (2022b) S.-F. Chen, Z. Vlah, and M. White, JCAP 02, 008 (2022b), arXiv:2110.05530 [astro-ph.CO] .
  * Ivanov (2021) M. M. Ivanov, Phys. Rev. D 104, 103514 (2021), arXiv:2106.12580 [astro-ph.CO] .
  * Beutler  _et al._ (2017) F. Beutler _et al._ (BOSS), Mon. Not. Roy. Astron. Soc. 466, 2242 (2017), arXiv:1607.03150 [astro-ph.CO] .
  * Horowitz and Seljak (2017) B. Horowitz and U. Seljak, Mon. Not. Roy. Astron. Soc. 469, 394 (2017), arXiv:1609.01850 [astro-ph.CO] .
  * Kobayashi  _et al._ (2022) Y. Kobayashi, T. Nishimichi, M. Takada, and H. Miyatake, Phys. Rev. D 105, 083517 (2022), arXiv:2110.06969 [astro-ph.CO] .
  * Dalal  _et al._ (2023) R. Dalal, X. Li, A. Nicola, J. Zuntz, M. A. Strauss, S. Sugiyama, T. Zhang, M. M. Rau, R. Mandelbaum, M. Takada, _et al._ , Phys. Rev. D 108, 123519 (2023), arXiv:2304.00701 [astro-ph.CO] .
  * Miyatake  _et al._ (2023) H. Miyatake, S. Sugiyama, M. Takada, T. Nishimichi, X. Li, M. Shirasaki, S. More, Y. Kobayashi, A. J. Nishizawa, M. M. Rau, _et al._ , Phys. Rev. D 108, 123517 (2023), arXiv:2304.00704 [astro-ph.CO] .
  * Yu  _et al._ (2023) B. Yu, U. Seljak, Y. Li, and S. Singh, JCAP 04, 057 (2023), arXiv:2211.16794 [astro-ph.CO] .
  * D’Amico  _et al._ (2024) G. D’Amico, Y. Donath, M. Lewandowski, L. Senatore, and P. Zhang, JCAP 05, 059 (2024), arXiv:2206.08327 [astro-ph.CO] .
  * Farren  _et al._ (2024) G. S. Farren _et al._ (ACT), Astrophys. J. 966, 157 (2024), arXiv:2309.05659 [astro-ph.CO] .
  * Shaikh  _et al._ (2024) S. Shaikh _et al._ (ACT, DES), Mon. Not. Roy. Astron. Soc. 528, 2112 (2024), arXiv:2309.04412 [astro-ph.CO] .
  * Chen  _et al._ (2022) S.-F. Chen, M. White, J. DeRose, and N. Kokron, JCAP 2022, 041 (2022), arXiv:2204.10392 [astro-ph.CO] .
  * Busch  _et al._ (2022) J. L. v. d. Busch _et al._ , Astron. Astrophys. 664, A170 (2022), arXiv:2204.02396 [astro-ph.CO] .
  * Madhavacheril  _et al._ (2024) M. S. Madhavacheril _et al._ (ACT), Astrophys. J. 962, 113 (2024), arXiv:2304.05203 [astro-ph.CO] .
  * Tristram  _et al._ (2024) M. Tristram _et al._ , Astron. Astrophys. 682, A37 (2024), arXiv:2309.10034 [astro-ph.CO] .
  * Ivanov  _et al._ (2020a) M. M. Ivanov, M. Simonović, and M. Zaldarriaga, JCAP 05, 042 (2020a), arXiv:1909.05277 [astro-ph.CO] .
  * Ivanov  _et al._ (2020b) M. M. Ivanov, M. Simonović, and M. Zaldarriaga, Phys. Rev. D 101, 083504 (2020b), arXiv:1912.08208 [astro-ph.CO] .
  * Ivanov  _et al._ (2022a) M. M. Ivanov, O. H. E. Philcox, M. Simonović, M. Zaldarriaga, T. Nischimichi, and M. Takada, Phys. Rev. D 105, 043531 (2022a), arXiv:2110.00006 [astro-ph.CO] .
  * Philcox  _et al._ (2020) O. H. E. Philcox, M. M. Ivanov, M. Simonović, and M. Zaldarriaga, JCAP 2020, 032 (2020), arXiv:2002.04035 [astro-ph.CO] .
  * Wenzl  _et al._ (2024a) L. Wenzl, R. Bean, S.-F. Chen, G. S. Farren, M. S. Madhavacheril, G. A. Marques, F. J. Qu, N. Sehgal, B. D. Sherwin, _et al._ , Phys. Rev. D 109, 083540 (2024a), arXiv:2401.12971 [astro-ph.CO] .
  * Pullen  _et al._ (2016) A. R. Pullen, S. Alam, S. He, and S. Ho, MNRAS 460, 4098 (2016), arXiv:1511.04457 .
  * Singh  _et al._ (2017) S. Singh, R. Mandelbaum, and J. R. Brownstein, MNRAS 464, 2120 (2017), arXiv:1606.08841 .
  * Doux  _et al._ (2018) C. Doux, M. Penna-Lima, S. D. P. Vitenti, J. Tréguer, E. Aubourg, and K. Ganga, MNRAS 480, 5386 (2018), arXiv:1706.04583 [astro-ph.CO] .
  * Singh  _et al._ (2019) S. Singh, S. Alam, R. Mandelbaum, U. Seljak, S. Rodriguez-Torres, and S. Ho, MNRAS 482, 785 (2019), arXiv:1803.08915 [astro-ph.CO] .
  * Darwish  _et al._ (2021) O. Darwish, M. S. Madhavacheril, B. D. Sherwin, S. Aiola, N. Battaglia, J. A. Beall, D. T. Becker, J. R. Bond, E. Calabrese, S. K. Choi, _et al._ , MNRAS 500, 2250 (2021), arXiv:2004.01139 [astro-ph.CO] .
  * Wenzl  _et al._ (2024b) L. Wenzl, R. An, N. Battaglia, R. Bean, E. Calabrese, S.-F. Chen, S. K. Choi, O. Darwish, J. Dunkley, G. S. Farren, _et al._ , arXiv e-prints , arXiv:2405.12795 (2024b), arXiv:2405.12795 [astro-ph.CO] .
  * D’Amico  _et al._ (2020) G. D’Amico, J. Gleyzes, N. Kokron, K. Markovic, L. Senatore, P. Zhang, F. Beutler, and H. Gil-Marín, JCAP 05, 005 (2020), arXiv:1909.05271 [astro-ph.CO] .
  * Freedman and Madore (2022) W. L. Freedman and B. F. Madore, IAU Symp. 376, 1 (2022), arXiv:2308.02474 [astro-ph.CO] .
  * Reid  _et al._ (2016) B. Reid, S. Ho, N. Padmanabhan, W. J. Percival, J. Tinker, R. Tojeiro, M. White, D. J. Eisenstein, C. Maraston, A. J. Ross, _et al._ , MNRAS 455, 1553 (2016), arXiv:1509.06529 [astro-ph.CO] .
  * Dawson  _et al._ (2013) K. S. Dawson _et al._ (BOSS), Astron. J. 145, 10 (2013), arXiv:1208.0022 [astro-ph.CO] .
  * Gil-Marín  _et al._ (2016) H. Gil-Marín, W. J. Percival, J. R. Brownstein, C.-H. Chuang, J. N. Grieb, S. Ho, F.-S. Kitaura, C. Maraston, F. Prada, S. Rodríguez-Torres, _et al._ , MNRAS 460, 4188 (2016), arXiv:1509.06386 [astro-ph.CO] .
  * Philcox (2021a) O. H. E. Philcox, Phys. Rev. D 103, 103504 (2021a), arXiv:2012.09389 [astro-ph.CO] .
  * Philcox (2021b) O. H. E. Philcox, Phys. Rev. D 104, 123529 (2021b), arXiv:2107.06287 [astro-ph.CO] .
  * Philcox and Flöss (2024) O. H. E. Philcox and T. Flöss, (2024), arXiv:2404.07249 [astro-ph.CO] .
  * Alam  _et al._ (2017) S. Alam _et al._ (BOSS), Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO] .
  * Kitaura  _et al._ (2016) F.-S. Kitaura _et al._ , Mon. Not. Roy. Astron. Soc. 456, 4156 (2016), arXiv:1509.06400 [astro-ph.CO] .
  * Rodríguez-Torres  _et al._ (2016) S. A. Rodríguez-Torres _et al._ , Mon. Not. Roy. Astron. Soc. 460, 1173 (2016), arXiv:1509.06404 [astro-ph.CO] .
  * Carron  _et al._ (2022) J. Carron, M. Mirmelstein, and A. Lewis, JCAP 2022, 039 (2022), arXiv:2206.07773 [astro-ph.CO] .
  * Alonso  _et al._ (2019) D. Alonso, J. Sanchez, A. Slosar, and LSST Dark Energy Science Collaboration, MNRAS 484, 4127 (2019), arXiv:1809.09603 [astro-ph.CO] .
  * Taylor and Markovič (2022) P. L. Taylor and K. Markovič, Phys. Rev. D 106, 063536 (2022), arXiv:2205.14167 [astro-ph.CO] .
  * Scolnic  _et al._ (2022) D. Scolnic, D. Brout, A. Carr, A. G. Riess, T. M. Davis, A. Dwomoh, D. O. Jones, N. Ali, P. Charvu, R. Chen, _et al._ , ApJ 938, 113 (2022), arXiv:2112.03863 [astro-ph.CO] .
  * Riess  _et al._ (2022) A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, _et al._ , ApJ 934, L7 (2022), arXiv:2112.04510 [astro-ph.CO] .
  * Beutler  _et al._ (2011) F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, MNRAS 416, 3017 (2011), arXiv:1106.3366 [astro-ph.CO] .
  * Ross  _et al._ (2015) A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden, and M. Manera, MNRAS 449, 835 (2015), arXiv:1409.3242 [astro-ph.CO] .
  * Audren  _et al._ (2013) B. Audren, J. Lesgourgues, K. Benabed, and S. Prunet, JCAP 02, 001 (2013), arXiv:1210.7183 [astro-ph.CO] .
  * Brinckmann and Lesgourgues (2019) T. Brinckmann and J. Lesgourgues, Phys. Dark Univ. 24, 100260 (2019), arXiv:1804.07261 [astro-ph.CO] .
  * Cooke  _et al._ (2018) R. J. Cooke, M. Pettini, and C. C. Steidel, Astrophys. J. 855, 102 (2018), arXiv:1710.11129 [astro-ph.CO] .
  * Fang  _et al._ (2008) W. Fang, W. Hu, and A. Lewis, Phys. Rev. D 78, 087303 (2008), arXiv:0808.3125 [astro-ph] .
  * Baumann  _et al._ (2012) D. Baumann, A. Nicolis, L. Senatore, and M. Zaldarriaga, JCAP 07, 051 (2012), arXiv:1004.2488 [astro-ph.CO] .
  * Carrasco  _et al._ (2012) J. J. M. Carrasco, M. P. Hertzberg, and L. Senatore, JHEP 09, 082 (2012), arXiv:1206.2926 [astro-ph.CO] .
  * Ivanov (2023) M. M. Ivanov, “Effective Field Theory for Large-Scale Structure,” (2023) arXiv:2212.08488 [astro-ph.CO] .
  * Chudaykin  _et al._ (2020) A. Chudaykin, M. M. Ivanov, O. H. E. Philcox, and M. Simonović, Phys. Rev. D 102, 063533 (2020), arXiv:2004.10607 [astro-ph.CO] .
  * Chudaykin  _et al._ (2021a) A. Chudaykin, M. M. Ivanov, and M. Simonović, Phys. Rev. D 103, 043525 (2021a), arXiv:2009.10724 [astro-ph.CO] .
  * Philcox and Ivanov (2022) O. H. E. Philcox and M. M. Ivanov, Phys. Rev. D 105, 043517 (2022), arXiv:2112.04515 [astro-ph.CO] .
  * Blas  _et al._ (2016a) D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov, JCAP 07, 052 (2016a), arXiv:1512.05807 [astro-ph.CO] .
  * Blas  _et al._ (2016b) D. Blas, M. Garny, M. M. Ivanov, and S. Sibiryakov, JCAP 07, 028 (2016b), arXiv:1605.02149 [astro-ph.CO] .
  * Vlah  _et al._ (2016) Z. Vlah, U. Seljak, M. Yat Chu, and Y. Feng, JCAP 2016, 057 (2016), arXiv:1509.02120 [astro-ph.CO] .
  * Ivanov and Sibiryakov (2018) M. M. Ivanov and S. Sibiryakov, JCAP 07, 053 (2018), arXiv:1804.05080 [astro-ph.CO] .
  * Chen  _et al._ (2024) S.-F. Chen, Z. Vlah, and M. White, arXiv e-prints , arXiv:2406.00103 (2024), arXiv:2406.00103 [astro-ph.CO] .
  * Senatore and Zaldarriaga (2015) L. Senatore and M. Zaldarriaga, JCAP 02, 013 (2015), arXiv:1404.5954 [astro-ph.CO] .
  * Baldauf  _et al._ (2015) T. Baldauf, M. Mirbabayi, M. Simonović, and M. Zaldarriaga, Phys. Rev. D 92, 043514 (2015), arXiv:1504.04366 [astro-ph.CO] .
  * Scoccimarro (2015) R. Scoccimarro, Phys. Rev. D 92, 083532 (2015), arXiv:1506.02729 [astro-ph.CO] .
  * Scoccimarro  _et al._ (1999) R. Scoccimarro, H. M. P. Couchman, and J. A. Frieman, Astrophys. J. 517, 531 (1999), arXiv:astro-ph/9808305 .
  * Limber (1953) D. N. Limber, ApJ 117, 134 (1953).
  * LoVerde and Afshordi (2008) M. LoVerde and N. Afshordi, Phys. Rev. D 78, 123506 (2008), arXiv:0809.5112 [astro-ph] .
  * Wenzl  _et al._ (2024c) L. Wenzl, S.-F. Chen, and R. Bean, MNRAS 527, 1760 (2024c), arXiv:2308.05892 [astro-ph.CO] .
  * Mead  _et al._ (2015) A. Mead, J. Peacock, C. Heymans, S. Joudaki, and A. Heavens, Mon. Not. Roy. Astron. Soc. 454, 1958 (2015), arXiv:1505.07833 [astro-ph.CO] .
  * Nishimichi  _et al._ (2020) T. Nishimichi, G. D’Amico, M. M. Ivanov, L. Senatore, M. Simonović, M. Takada, M. Zaldarriaga, and P. Zhang, Phys. Rev. D 102, 123541 (2020), arXiv:2003.08277 [astro-ph.CO] .
  * Schmittfull  _et al._ (2021) M. Schmittfull, M. Simonović, M. M. Ivanov, O. H. E. Philcox, and M. Zaldarriaga, JCAP 05, 059 (2021), arXiv:2012.03334 [astro-ph.CO] .
  * Ivanov  _et al._ (2022b) M. M. Ivanov, O. H. E. Philcox, T. Nishimichi, M. Simonović, M. Takada, and M. Zaldarriaga, Phys. Rev. D 105, 063512 (2022b), arXiv:2110.10161 [astro-ph.CO] .
  * Krause  _et al._ (2024) E. Krause _et al._ (Beyond-2pt), (2024), arXiv:2405.02252 [astro-ph.CO] .
  * Jackson (1972) J. C. Jackson, Mon. Not. Roy. Astron. Soc. 156, 1P (1972), arXiv:0810.3908 [astro-ph] .
  * Chen  _et al._ (2020) S.-F. Chen, Z. Vlah, and M. White, JCAP 2020, 062 (2020), arXiv:2005.00523 [astro-ph.CO] .
  * Chen  _et al._ (2021) S.-F. Chen, Z. Vlah, E. Castorina, and M. White, JCAP 2021, 100 (2021), arXiv:2012.04636 [astro-ph.CO] .
  * Maus  _et al._ (2024a) M. Maus, S. Chen, M. White, J. Aguilar, S. Ahlen, A. Aviles, S. Brieden, D. Brooks, T. Claybaugh, S. Cole, _et al._ , arXiv e-prints , arXiv:2404.07312 (2024a), arXiv:2404.07312 [astro-ph.CO] .
  * Chudaykin  _et al._ (2021b) A. Chudaykin, K. Dolgikh, and M. M. Ivanov, Phys. Rev. D 103, 023507 (2021b), arXiv:2009.10106 [astro-ph.CO] .
  * Ivanov  _et al._ (2024a) M. M. Ivanov, C. Cuesta-Lazaro, S. Mishra-Sharma, A. Obuljen, and M. W. Toomey, (2024a), arXiv:2402.13310 [astro-ph.CO] .
  * Lesgourgues and Pastor (2006) J. Lesgourgues and S. Pastor, Phys. Rept. 429, 307 (2006), arXiv:astro-ph/0603494 .
  * Laguë  _et al._ (2022) A. Laguë, J. R. Bond, R. Hložek, K. K. Rogers, D. J. E. Marsh, and D. Grin, JCAP 01, 049 (2022), arXiv:2104.07802 [astro-ph.CO] .
  * Rogers  _et al._ (2023) K. K. Rogers, R. Hložek, A. Laguë, M. M. Ivanov, O. H. E. Philcox, G. Cabass, K. Akitsu, and D. J. E. Marsh, JCAP 06, 023 (2023), arXiv:2301.08361 [astro-ph.CO] .
  * Xu  _et al._ (2022) W. L. Xu, J. B. Muñoz, and C. Dvorkin, Phys. Rev. D 105, 095029 (2022), arXiv:2107.09664 [astro-ph.CO] .
  * He  _et al._ (2023) A. He, M. M. Ivanov, R. An, and V. Gluscevic, Astrophys. J. Lett. 954, L8 (2023), arXiv:2301.08260 [astro-ph.CO] .
  * Rubira  _et al._ (2023) H. Rubira, A. Mazoun, and M. Garny, JCAP 01, 034 (2023), arXiv:2209.03974 [astro-ph.CO] .
  * Nunes  _et al._ (2022) R. C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino, and O. Mena, Phys. Rev. D 105, 123506 (2022), arXiv:2203.08093 [astro-ph.CO] .
  * Joseph  _et al._ (2023) M. Joseph, D. Aloni, M. Schmaltz, E. N. Sivarajan, and N. Weiner, Phys. Rev. D 108, 023520 (2023), arXiv:2207.03500 [astro-ph.CO] .
  * Neveux  _et al._ (2020) R. Neveux _et al._ (eBOSS), Mon. Not. Roy. Astron. Soc. 499, 210 (2020), arXiv:2007.08999 [astro-ph.CO] .
  * Chudaykin and Ivanov (2023) A. Chudaykin and M. M. Ivanov, Phys. Rev. D 107, 043518 (2023), arXiv:2210.17044 [astro-ph.CO] .
  * Chabanier  _et al._ (2019) S. Chabanier _et al._ (eBOSS), JCAP 07, 017 (2019), arXiv:1812.03554 [astro-ph.CO] .
  * Ivanov  _et al._ (2024b) M. M. Ivanov, M. W. Toomey, and N. G. Karaçaylı, (2024b), arXiv:2405.13208 [astro-ph.CO] .
  * Maus  _et al._ (2024b) M. Maus, Y. Lai, H. E. Noriega, S. Ramirez-Solano, A. Aviles, S. Chen, S. Fromenteau, H. Gil-Marín, C. Howlett, M. Vargas-Magaña, _et al._ , arXiv e-prints , arXiv:2404.07272 (2024b), arXiv:2404.07272 [astro-ph.CO] .


# Chimera States and Seizures in a Mouse Neuronal Model

Henry M. Mitchell  Peter Sheridan Dodds  J. Matthew Mahoney  Christopher M.
Danforth

###### Abstract

Chimera states—the coexistence of synchrony and asynchrony in a nonlocally-
coupled network of identical oscillators—are often used as a model framework
for epileptic seizures. Here, we explore the dynamics of chimera states in a
network of modified Hindmarsh-Rose neurons, configured to reflect the graph of
the mesoscale mouse connectome. Our model produces superficially epileptiform
activity converging on persistent chimera states in a large region of a two-
parameter space governing connections (a) between subcortices within a cortex
and (b) between cortices. Our findings contribute to a growing body of
literature suggesting mathematical models can qualitatively reproduce
epileptic seizure dynamics.

##  1 Introduction

###  1.1 Chimera States

A well-studied example of complex behavior arising from simple mechanisms is
the coexistence of synchrony and asynchrony within a system of identical
coupled oscillators, a phenomenon known as a chimera state [Kuramoto &
Battogtokh(2002), Abrams & Strogatz(2004)]. The existence of these chimera
states is surprising, as they represent asymmetry within symmetric systems.
One of the simplest systems which produces chimera states is the Abrams model
which consists of two populations of identical oscillators with a stronger
coupling strength within the populations than between them [Abrams _et
al._(2008)Abrams, Mirollo, Strogatz & Wiley]. The system is described by the
following:

$\derivative{\theta_{i}^{\sigma}}{t}=\omega+\sum_{\sigma^{\prime}=1}^{2}\frac{K_{\sigma\sigma^{\prime}}}{N_{\sigma^{\prime}}}\sum_{j=1}^{N_{\sigma^{\prime}}}\sin(\theta_{j}^{\sigma^{\prime}}-\theta_{i}^{\sigma}-\alpha),$
(1)

where

$K=\bmqty{undef}\mbox{\quad and\quad}\sigma\in\Bqty{1,2}.$

In this model, $\mu$ represents the intra-population strength, and $\nu$
represents the inter-population strength, with $\mu>\nu$. Time can be scaled
such that $\mu+\nu=1$. If $\mu-\nu$ is not too large, and $\alpha$ is not too
much less than $\frac{\pi}{2}$, then this system can produce chimera states.
Fig. 1 shows a simulation of the Abrams model on two populations of 128
oscillators.

Figure 1: A simulation of the Abrams model for two populations of 128
oscillators. We employed a 4th-order Runge-Kutta solver
($\differential{t}=0.01$, $t_{\text{max}}=1000$). A. Time series of the
simulation for $t\in\pqty{800,1000}$. B. Snapshot at $t\approx 800$.

An analogous system has recently been analyzed in the physical world [Martens
_et al._(2013)Martens, Thutupalli, Fourriere & Hallatschek]. Two swinging
platforms were coupled together with springs of variable spring constant
$\kappa$, and 15 metronomes—all tuned to the same frequency—were placed on
each platform. The metronomes on the same platform are coupled through the
motion of the swing, which heavily influences the motion of the metronomes,
represented in the Abrams model by $\mu$. The metronomes on opposite platforms
are coupled through the springs, which is a much weaker interaction,
represented in the Abrams model by $\nu$. For a wide range of values of
$\kappa$, all of the metronomes on one platform would synchronize, while the
metronomes on other platform would remain asynchronous.

While chimera states may present themselves obviously when observed in a plot
or the physical world, they can be harder to pin down analytically. In order
to do so, we will investigate a system of $M$ communities of nonlocally-
coupled oscillators, and we sample their phases at times
$t\in\bqty{1,\ldots,T}$. A useful pair of measures for detecting the presence
of a chimera state are the chimera-like index $\chi$ and the metastability
index $m$ [Shanahan(2010), Hizanidis _et al._(2016)Hizanidis, Kouvaris,
Zamora-López, Díaz-Guilera & Antonopoulos]. To develop these two measures, we
start with the order parameter
$r_{c}(t)=\absolutevalue{\expectationvalue{e^{i\phi_{k}(t)}}_{k\in C}}$, where
$\phi_{k}$ is the phase of oscillator $k$, and $\expectationvalue{f}_{k\in C}$
is the average of $f$ over all $k$ in community $C$. The order parameter $r$
indicates the instantaneous synchrony of a community (how similar the phases
of the oscillators are to the others in $C$), and not its overall coherence
(how similar the trajectories of the oscillators are). From this, we define
the two measures:

$\displaystyle\chi$
$\displaystyle=7\times\expectationvalue{\sigma_{\text{chi}}}_{T},$ (2)
$\displaystyle m$
$\displaystyle=12\times\expectationvalue{\sigma_{\text{met}}}_{C},$ (3)

where

$\sigma_{\text{chi}}(t)=\frac{1}{M-1}\sum_{c\in
C}\pqty{r_{c}(t)-\expectationvalue{r_{c}}_{C}}^{2},$ (4)

and

$\sigma_{\text{met}}(c)=\frac{1}{T-1}\sum_{t\leq
T}\pqty{r_{c}(t)-\expectationvalue{r_{c}}_{T}}^{2}.$ (5)

To put this into words, the chimera-like index $\chi$ is the average over time
of the variance of the order parameter across communities, while the
metastability index $m$ is the average across communities of the variance of
the order parameter within a given community over time,

The normalization constants follow from the indices' maximum possible values
[Shanahan(2010)]. If a community spends equal time in a maximally chimeric
state and a minimally chimeric state, then its chimera-like index will be at
its maximum111While it is possible for half of a system’s communities to be
synchronous and the other half asynchronous for all times (resulting in a
chimera-like index of $\frac{2}{7}$), this is transient due to the effects of
metastability [Shanahan(2010)]. Therefore, we will ignore this case.:
$\chi_{\text{max}}=\frac{1}{7}$. If a community $c$ spends equal time in all
stages of synchronization (i.e., the phase parameter of $c$ is uniformly
distributed), then $\sigma_{\text{met}}(c)$ is at its maximum, which is the
variance of the uniform distribution: $m_{\text{max}}=\frac{1}{12}$.

###  1.2 Seizures

Chimera states have been observed in many other systems, whether they be
purely mathematical, biological, electrical, or mechanical [Shanahan(2010),
Abrams & Strogatz(2004), Andrzejak _et al._(2016)Andrzejak, Rummel, Mormann &
Schindler, Hizanidis _et al._(2016)Hizanidis, Kouvaris, Zamora-López, Díaz-
Guilera & Antonopoulos, Kuramoto & Battogtokh(2002), Martens _et
al._(2013)Martens, Thutupalli, Fourriere & Hallatschek, Panaggio &
Abrams(2015), Santos _et al._(2015)Santos, Szezech, Batista, Caldas, Viana &
Lopes, Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista,
Viana & Kurths, Kruk _et al._(2018)Kruk, Maistrenko & Koeppl, Xie _et
al._(2014)Xie, Knobloch & Kao]. One of the most common ways that chimera
states are discussed is in regards to seizures.

As is often the case with emergent phenomena, it is wildly impractical to
simulate the collective behavior of a brain by simulating its constituent
neurons. Since the human brain has approximately $10^{11}$ neurons with
$10^{14}$ synapses, direct simulation is too computationally intensive. To
better understand the dynamics of large portions of the brain, many
researchers have turned to the techniques of thermal and statistical physics
[Breakspear(2017)], resulting in neural field models and neural mass networks.
The first treats the brain as a continuous sheet of cortex, within which
activity obeys wave equations. The second represents the brain as a discrete
graph of cortices, or a network of coupled oscillators. The network used for
the coupling of the oscillators is determined by the brain's connectivity
matrix, or connectome. An example of a neural mass network model is the
modified Hindmarsh-Rose model [Eqs. 6, 7 and 9], which we discuss later.

One of the benefits of a neural mass network model is that its outputs are
similar to those of an electroencephalograph, or EEG. The EEG is a device used
to record the electrical activity of the brain. Electrodes are placed in
specific areas on the scalp, and then changes in voltage are measured from
neural masses beneath the skull. Much of the signal is distorted and
attenuated by the bone and tissue between the brain and the electrodes, which
act like resistors and capacitors. This means that, while the membrane voltage
of the neuron changes by millivolts, the EEG reads a signal in the microvolt
scale [Westbrook(2013)]. The EEG also has relatively low spatial and temporal
resolution (16 electrodes for the whole brain, and a sampling rate of
$33\text{\,}\mathrm{ms}$). However, when properly treated, neural mass models
make for effective predictors of the output from EEGs [Taylor _et
al._(2012)Taylor, Goodfellow, Wang & Baier, Leistritz _et al._(2007)Leistritz,
Putsche, Schwab, Hesse, Süße, Haueisen & Witte]. This is useful, as EEGs are
the main tool used to detect and categorize seizures.

####  1.2.1 Seizure Ætiology

Researchers define seizures as abnormal, excessive, or overly-synchronized
neural activity [Westbrook(2013), Baier _et al._(2012)Baier, Goodfellow,
Taylor, Wang & Garry]. It is important to distinguish between seizures and
epilepsy, as the two are often conflated. Seizures are an acute event, whereas
epilepsy is a chronic condition of repeated seizures. While classification
schemes vary, all center around the division between generalized and focal
seizures.

The focus of the present work is focal seizures, which start in one part of
the brain (the seizure focus). They are typically preceded by auras such as a
sense of fear, or hearing music, and often manifest as clonic movement of the
extremities. In many cases, they secondarily generalize, spreading to the
entire brain. This can make focal seizures and primary generalized seizures
hard to distinguish, as a focal seizure can generalize rapidly after a brief
aura. This can lead to misdiagnoses and improper treatments.

##  2 Literature Review

Chimera states in brain models have often been linked loosely to
unihemispheric sleep, seizures, and other brain behaviors [Abrams _et
al._(2008)Abrams, Mirollo, Strogatz & Wiley, Panaggio & Abrams(2015), Martens
_et al._(2013)Martens, Thutupalli, Fourriere & Hallatschek, Abrams &
Strogatz(2004), Shanahan(2010), Höhlein _et al._(2019)Höhlein, Kemeth &
Krischer, Bansal _et al._(2019)Bansal, Garcia, Tompson, Verstynen, Vettel &
Muldoon, Chouzouris _et al._(2018)Chouzouris, Omelchenko, Zakharova, Hlinka,
Jiruska & Schöll]. In most cases, these connections are made in off-hand
remarks to introduce the concept of a chimera state, but serious connections
between these phenomena are rarely drawn. However, there are some notable
cases of investigations of chimera states in brains.

A example of a chimera state being investigated in neural models was an
exploration of chimera states on a network of Hindmarsh-Rose neurons (Eqs. 6,
7 and 9) [Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas,
Batista, Viana & Kurths]. This model was simulated on the connectome of a cat.
Parameter space for the two connection strengths $\alpha$ and $\beta$ was
explored. Chimera states are most prevalent for low values of $\beta$, the
inter-cortex connection strength [Santos _et al._(2017)Santos, Szezech,
Borges, Iarosz, Caldas, Batista, Viana & Kurths]. This is unsurprising. If the
inter-cortex connection strength is too high as compared to the intra-cortex
connection strength, the coupling acts globally instead of nonlocally. This
means that each cortex has less holding it together than pulling it apart,
allowing the system to descend into asynchrony.

Further, with increasing input current $I_{j}$ (and increasing noise in the
input current), chimera states give way to incoherence. This also intuitively
makes sense. As the input current increases, its significance relative to the
coupling also increases. Thus, the oscillators have no reason to synchronize.
And, of course, adding noise will simply amplify the effect [Santos _et
al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana & Kurths].

##  3 Methods

###  3.1 Model

The model we used was a modified Hindmarsh-Rose neural model222The
modification is to add in the coupling, turning it into a network model
instead of a single neuron. The brain has different connection strengths and
types within cortices than it does between cortices, which is why the network
is separated into intra- and intercortical connections in Eq. 6. taken from
[Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana
& Kurths].

$\displaystyle\begin{split}\dot{x}_{j}={}&y_{j}-x_{j}^{3}+bx_{j}^{2}+I_{j}-z_{j}\\\
&-\frac{\alpha}{n^{\prime}_{j}}\sum_{k=1}^{N}G^{\prime}_{jk}\Theta_{j}(x_{k})-\frac{\beta}{n^{\prime\prime}_{j}}\sum_{k=1}^{N}G^{\prime\prime}_{jk}\Theta_{j}(x_{k}),\end{split}$
(6) $\displaystyle\dot{y}_{j}={}$ $\displaystyle 1-5x_{j}^{2}-y_{j},$ (7)

and

$\dot{z}_{j}={}\mu\pqty{s\pqty{x_{j}-x_{\text{rest}}}-z_{j}},$ (9)

where

$\Theta_{j}(x_{k})=\frac{x_{j}-x_{\text{rev}}}{1+e^{-\lambda\pqty{x_{k}-\theta}}}$
(10)

is the sigmoidal activation function. This function helps the model better
approximate the behavior of neural masses, as opposed to specific neurons.
Table 1 shows the values and meanings of the symbols in the model.

Symbol | Value |  Meaning
---|---|---
$x_{j}$ | — |  Membrane potential of the $j$th neural mass
$y_{j}$ | — |  Associated with the fast processes
$z_{j}$ | — |  Associated with slow processes
$b$ | 3.2 |  Tunes the spiking frequency
$I_{j}$ | 4.4 |  External input current
$x_{\text{rev}}$ | 2 |  Ambient reversal potential
$\lambda$ | 10 |  Activation function parameter
$\theta$ | -0.25 |  Activation function parameter
$\mu$ | 0.01 |  Time scale for variation of $z$
$s$ | 4 |  Governs adaptation
$x_{\text{rest}}$ | -1.6 |  Resting/equilibrium potential
$\alpha$ | Varied |  Coupling strength within cortices
$n_{j}^{\prime}$ | See Fig. 3 A |  Number of connections within a cortex from the $j$th neuron
$G_{jk}^{\prime}$ | See Fig. 3 C |  Intra-cortical connection strength
$\beta$ | Varied |  Coupling strength between cortices
$n_{j}^{\prime\prime}$ | See Fig. 3 B |  Number of connections between cortices from the $j$th neuron
$G_{jk}^{\prime\prime}$ | See Fig. 3 D |  Inter-cortical connection strength
Table 1: The list of parameters used in modeling the Hindmarsh-Rose network.

The measurable output of an EEG corresponds to the mean of the membrane
potential within a cortex (i.e., the observable values are
$\expectationvalue{x_{j}}_{j\in C}$).

We chose this model due to the intelligibility of its parameters, as well as
its proven ability to exhibit chimera-like behavior as a neural mass model
[Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana
& Kurths]. Additionally, the Hindmarsh-Rose model was not designed to emulate
seizures, which provides further evidence for the assertion that chimeras may
be a universal aspect of brain activity.

It is worth noting that one of the limitations of this model is the changing
nature of intra- and inter-cortical connection strengths (corresponding to
$\alpha$ and $\beta$) in the actual brain. The strengths of connections and
the amounts by which they are amplified vary in time. However, they will be
treated as constant, in order to present a view of parameter space.

Additionally, the model assumes a network of identical nodes. This is not the
case in the brain, as there is heterogeneity of neuronal behavior, even among
neurons of similar types. The model does allow some flexibility for creating
differences between neural masses through the $I_{j}$ external input current
term. Changes to this term have been shown to disrupt chimera states [Santos
_et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana &
Kurths], possibly due to intermediate levels of heterogeneity enabling optimal
information encoding [Baroni & Mazzoni(2014)]. However, a network of identical
nodes provides a good qualitative picture of the parameter space, with room
for refinement by the introduction of noise.

###  3.2 Network

We implemented the model on a mesoscale mouse connectome, which comprises of
213 fine areas, grouped into 13 coarse areas, along with measured connection
strengths between the subcortices [Oh _et al._(2014)Oh, Harris, Ng, Winslow,
Cain, Mihalas, Wang, Lau, Kuan, Henry, Mortrud, Ouellette, Nguyen, Sorensen,
Slaughterbeck, Wakeman, Li, Feng, Ho, Nicholas, Hirokawa, Bohn, Joines, Peng,
Hawrylycz, Phillips, Hohmann, Wohnoutka, Gerfen, Koch, Bernard, Dang, Jones &
Zeng]. We used these coarse areas as the sets $C$ (Eqs. 2,3) for chimera and
metastability analyses. We reduced the connection strengths to those with
sufficient certainty ($p<0.01$), and segmented as follows:

$G_{jk}=\begin{cases}0\text{ if }O_{jk}<10^{-4},\\\ 1\text{ if }10^{-4}\leq
O_{jk}<10^{-2},\\\ 2\text{ if }10^{-2}\leq O_{jk}<1,\\\ 3\text{ if }1\leq
O_{jk},\end{cases}$ (11)

where $O_{jk}$ is the raw connection strength provided by [Oh _et
al._(2014)Oh, Harris, Ng, Winslow, Cain, Mihalas, Wang, Lau, Kuan, Henry,
Mortrud, Ouellette, Nguyen, Sorensen, Slaughterbeck, Wakeman, Li, Feng, Ho,
Nicholas, Hirokawa, Bohn, Joines, Peng, Hawrylycz, Phillips, Hohmann,
Wohnoutka, Gerfen, Koch, Bernard, Dang, Jones & Zeng]. We performed this
simplification to match the analysis of Santos et al. more closely [Santos _et
al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana & Kurths].

We show $G$ in Fig. 2, and break it down into its inter- and intra-connections
in Fig. 3. This brain network is a small-world network [Oh _et al._(2014)Oh,
Harris, Ng, Winslow, Cain, Mihalas, Wang, Lau, Kuan, Henry, Mortrud,
Ouellette, Nguyen, Sorensen, Slaughterbeck, Wakeman, Li, Feng, Ho, Nicholas,
Hirokawa, Bohn, Joines, Peng, Hawrylycz, Phillips, Hohmann, Wohnoutka, Gerfen,
Koch, Bernard, Dang, Jones & Zeng], a graph topology which lends itself well
to the development of chimera states, as it facilitates nonlocal coupling
[Hizanidis _et al._(2016)Hizanidis, Kouvaris, Zamora-López, Díaz-Guilera &
Antonopoulos].

Figure 2: A. A matrix representation of the mouse connectome, with strengths
as defined by Eq. 11. The cortices represented are, left to right (top to
bottom), the striatum, the olfactory areas, the isocortex, the crebellar
cortex, the hippocampal formation, the midbrain, the hypothalamus, the
pallidum, the pons, the medulla, the cortical subplate, the thalamus, and the
cerebellar nuclei. B. An embedding of the graph. Edge colors indicate the
source location.

Another benefit to this network is that it is comparatively accurate and
complete. Given the complexity of brains, creating an accurate structural or
functional connectome is extremely difficult. It has yet to be done to a
large-scale extent in humans, and was only recently done in mice. Moreover, as
mice are common analogues for humans in laboratory settings, the mouse seemed
a fitting ``guinea pig'' for the creation of chimera states.

Figure 3: A breakdown of the network. A. & B. show $n_{j}^{\prime}$ and
$n_{j}^{\prime\prime}$, effectively the number of nonzero elements in the
$j$th row of $G^{\prime}$ and $G^{\prime\prime}$, respectively. C. & D. show
$G^{\prime}$ and $G^{\prime\prime}$, which are $G$ (Fig. 2 A) only within and
between cortices, respectively.  Figure 4: The average connection strengths
for each neuron $j$, within cortices (blue) and between them (orange). A. All
of the subcortices. B. All of the subcortices for which neither intra- nor
inter-cortical average strength was 0.

###  3.3 Implementation

We coded the modified Hindmarsh-Rose model using Python (Python version 3.7.0,
NumPy version 1.15.2, Pandas version 0.23.4, SciPy version 1.1.0), and
integrated using a 4th-order Runge-Kutta with variable step size333Step size
was determined by SciPy’s internal algorithms, but was limited to a maximum of
0.01. $\differential{t}<0.01$. We verified the code by reproducing the results
of [Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista,
Viana & Kurths]. We ran the model for a time period of
$T_{\text{sim}}=\bqty{-1000,5000}$, where only times $T=\bqty{0,4000}$ were
saved. We threw away the times $\bqty{-1000,0}$ to eliminate transients. The
chimeras were extremely unlikely to be eliminated on such a time scale, due to
the size of the network [Wolfrum & Omel’chenko(2011)]. We calculated the times
$\bqty{4000,5000}$ to facilitate analysis of the phase.

We computed the phase of the $j$th neuron in the resulting waveform as

$\phi_{j}(t)=2\pi\times\frac{t-t_{i}}{t_{i+1}-t_{i}},$ (12)

where $t_{i}$ is the time at which the $j$th neuron fires ($x_{j}$ crosses 0
in a positive direction) for the $i$th time444This is a similar measure for
the phase as was used in [Santos _et al._(2017)Santos, Szezech, Borges,
Iarosz, Caldas, Batista, Viana & Kurths], but allows for easier discrimination
between physical and aphysical parameter sets. It is modified to keep
$\phi_{j}\in\pqty{0,2\pi}$ and to eliminate ambiguity about the meanings of
the subscripts. . In order for this calculation to be possible for all values
in $T$, it was necessary to have each neuron fire at least once after $T$ had
finished (i.e., there has to be some $t_{i+1}\notin T$ in order to calculate
the phase for times $t_{i}\leq t\leq t_{\text{max}}=4000$). The calculated
time range went so far beyond $t_{\text{max}}$ so that any extremely slow-
firing neurons were allowed to do so, to ensure that as much of parameter
space was in the physical region (Section 4.2). We then used phase to find the
chimera and metastability indices of the result using Eq. 2 and Eq. 3
respectively.

We repeated this process for various parameter sweeps of $\alpha\times\beta$,
summarized in Table 2. Note that the step in each strength
$i\in\Bqty{\alpha,\beta}$ is $\Delta i=\frac{i_{\text{max}}}{n_{i}-1}$, due to
the fact that the ranges are inclusive of both endpoints.

$\alpha_{\text{max}}$, $\beta_{\text{max}}$ |  $\Delta\alpha$ ($n_{\alpha}$), $\Delta\beta$ ($n_{\beta}$) | Figure
---|---|---
1.6, 0.4 | 0.0203 (80), 0.0211 (20) | —
3.2, 0.8 | 0.0405 (80), 0.0205 (40) | —
0.2, 0.1 | 0.00253 (80), 0.00256 (40) | Fig. 8
0.9, 0.9 | 0.0101 (80), 0.0101 (80) | —
1.0, 1.0 | 0.0101 (100), 0.0101 (100) | Fig. 7
Table 2: The sweeps we used in evaluating the effects of $\alpha$ and $\beta$
on the chimera and metastability indices. All parameter sweeps started at
$(\alpha,\beta)=(0,0)$. We performed the $\bqty{0,1}\times\bqty{0,1}$ sweep 10
times, and averaged the resulting chimera-like indices.

Initial conditions were drawn from uniform distributions of
$x_{j}\in\bqty{-2,2}$, $y_{j}\in\bqty{0,0.2}$, $z_{j}\in\bqty{0,0.2}$. We
performed all simulations on the Vermont Advanced Computing Core, and is
available online555https://github.com/henmitch/chimera-2019.

##  4 Results

We investigate three aspects of the model's output. First, we compare the
output from the model to real-world data, on a qualitative level. We then
discuss the region of parameter space for which the model produces aphysical
results. Finally, we draw connections between chimera states in the model and
their physiological analogues.

###  4.1 Model Quality

It is worthwhile to first discuss the quality of the model used, and its
relationship to reality. Fig. 5 shows several types of behavior one can expect
on an EEG trace.

Figure 5: A typical EEG trace. The first row (``Wild type'') shows a normal
awake adult mouse EEG trace. The other four rows (``Mutant'') show typical
abnormal/epileptiform activity. Taken from [Ljungberg _et al._(2009)Ljungberg,
Sunnen, Lugo, Anderson & D'Arcangelo].

Healthy brain behavior presents as low-amplitude oscillations on an EEG, as
the asynchrony leads the firings of individual neurons to cancel each other
out. Seizures and seizure-like activity present as higher-amplitude
oscillations, as the synchrony decreases the variance between neurons, making
the mean closer to the behavior of each neuron. One of the main challenges of
simulating seizures is that only trained experts can truly identify seizures;
as yet, there is no mathematical technique for identifying seizures
[Westbrook(2013)].

Figure 6: A typical run of the Hindmarsh-Rose simulation. A. The mean membrane
potential within each cortex. B. The phase $\phi$ of the entire timeseries for
a simulation of the Hindmarsh-Rose network.

The highly chimeric portion of the landscape appears to be mostly below the
$\beta=\alpha$ line.

However, one can indicate whether a model resembles epileptiform activity on a
qualitative level. Fig. 6 shows the results of a simulation of the Hindmarsh-
Rose network for $\pqty{\alpha,\beta}=\pqty{0.608,0.267}$. Qualitatively, in
our simulation, the thalamus, the pons, and the striatum look like the wild
type EEG; the cerebellar cortex shows some spiking behavior, as well as some
spike runs; the medulla and the hypothalamus look to be in repetitive spike
runs; and the cortical subplate seems to be exhibiting seizure-like behavior
over some time periods. This shows that the behavior visible on an EEG can be
approximately reproduced in this model.

While obtaining a higher neuronal resolution from the EEG is not possible due
to the nature of the method, we can see the neural dynamics of the model (Fig.
6 B). It is evident that the thalamus, the pons, and the striatum are each
highly asynchronous, which corresponds to their wild-type presentation.

Both presentation methods have their benefits, as Fig. 6 A looks similar to an
EEG (and therefore lends itself well to the comparison), where Fig. 6 B allows
us to view the individual subcortical behaviors leading to the net dynamics.

###  4.2 Aphysical Region

We choose not to include figures for the first two sweeps of Table 2 because a
large portion of parameter space leads to an aphysical model. Specifically,
for certain value pairs of $\pqty{\alpha,\beta}$, certain neurons never fired
(increased past 1). Despite the drastically increased time of evaluation (see
Section 3.3), a vast swath of parameter space gave nonsense results (the white
shown in Fig. 7).

Figure 7:  The chimera-like landscape of parameter space on
$\pqty{\alpha,\beta}\in\pqty{0,1.0}\times\pqty{0,1.0}$. The aphysical region
of the model is shown in white. The black rectangle in the bottom left corner
indicates the region of parameter space shown in Fig. 8. The dashed line has a
slope of $-1$, to serve as a guide for Section 4.3. The chimera-like index
(defined in Eq. 2) is normalized to 1, as usual.

The boundary between physical and aphysical appears to be linear, with a
negative slope. This means that $\alpha$ can range further when $\beta$ is
low, and vice versa. This makes sense, as increased $\alpha$ and $\beta$
influence the model in the same way (increasing the coupling and decreasing
$\dot{x}_{j}$).

Furthermore, the slope of the boundary is greater than $-1$, which means that
$\alpha$ has an greater influence on the physicality of the model. This is
also reasonable, but for slightly less self-evident reasons. To explain why,
we must look specifically at the coupling term from Eq. 6:

$-\frac{\alpha}{n^{\prime}_{j}}\sum_{k=1}^{N}G^{\prime}_{jk}\Theta_{j}\pqty{x_{k}}-\frac{\beta}{n^{\prime\prime}_{j}}\sum_{k=1}^{N}G^{\prime\prime}_{jk}\Theta_{j}\pqty{x_{k}}.$

This coupling will, in fact, be positive, as $\Theta_{j}\pqty{x_{k}}<0$ if
$x_{j}<2$, which is true almost all of the time. This means that, as $\alpha$
and $\beta$ increase, so does the overall coupling strength. So, there is some
threshold $K$ for which the overall coupling is too strong if, for some $j$,

$\frac{\alpha}{n^{\prime}_{j}}\sum_{k=1}^{N}G^{\prime}_{jk}\absolutevalue{\Theta_{j}\pqty{x_{k}}}+\frac{\beta}{n^{\prime\prime}_{j}}\sum_{k=1}^{N}G^{\prime\prime}_{jk}\absolutevalue{\Theta_{j}\pqty{x_{k}}}>K_{j}.$
(13)

In order for $\alpha$ to influence the coupling's proximity to $K$ more than
$\beta$ does, there must exist some $j$ such that
$\frac{1}{n^{\prime}_{j}}\sum
G^{\prime}_{jk}\absolutevalue{\Theta_{j}\pqty{x_{k}}}>\frac{1}{n^{\prime\prime}_{j}}\sum
G^{\prime\prime}_{jk}\absolutevalue{\Theta_{j}\pqty{x_{k}}}$. Seeing as
$\overline{g}^{\prime}_{j}=\frac{1}{n^{\prime}_{j}}\sum G^{\prime}_{jk}$ and
$\overline{g}^{\prime\prime}_{j}=\frac{1}{n^{\prime\prime}_{j}}\sum
G^{\prime\prime}_{jk}$ are the average connection strength within and between
cortices (shown in Fig. 4 A), these are simply a function of the topology of
the graph.

It may look from Fig. 4 A like $\beta$ should have more influence than
$\alpha$, as for most $j$,
$\overline{g}^{\prime\prime}_{j}>\overline{g}^{\prime}_{j}$. However, for most
of those cases,
$\overline{g}^{\prime}_{j_{0}}>\overline{g}^{\prime\prime}_{j_{0}}=0$. This
means that, for those $j_{0}$, $\partialderivative{K_{j_{0}}}{\alpha}=0$. So,
those cases contribute to the value of the threshold, but do not influence the
physicality's dependence on $\alpha$ and $\beta$.

If we remove the $j$ for which
$0\in\Bqty{\overline{g}^{\prime}_{j},\overline{g}^{\prime\prime}_{j}}$, we
find that, on average, $\overline{g}^{\prime}_{j}=2.100$, slightly more than
$\overline{g}^{\prime\prime}_{j}=2.079$ (see Fig. 4 B). This explains the
slope of the boundary between the physical region and the aphysical region.

###  4.3 Chimera states

We show the normalized chimera-like index of the entire physical region in
Fig. 7. Near the maximal edge of the physical region, the highest values of
the chimera index appear to follow a slope of $-1$. It is unsurprising that
chimera states would be prevalent when the coupling is large (out near the
boundary of the aphysical range).

What is surprising, however, is the presence of the chimeric patch in the
bottom left corner of Fig. 7, shown at a higher resolution in Fig. 8.

Figure 8: A. The chimera-like index $\chi$ of runs with
$(\alpha,\beta)\in(0,0.2)\times(0,0.2)$. As before, the chimera-like index is
normalized to $\frac{1}{7}$. Note that the values of the index are much higher
in this patch than in most of the rest of $(\alpha,\beta)\in(0,1)\times(0,1)$
(Fig. 7). B. The variance of the chimera-like index.

Plotting the results of the simulations (Fig. 9), it is evident that this is
not a calculation error, but is an actual feature of the parameter landscape.

Figure 9: A run of the Hindmarsh-Rose simulation in the chimeric island. A.
The mean membrane potential within each cortex. B. The phase $\phi$ of the
entire timeseries for a simulation of the Hindmarsh-Rose network.
Synchronization is most consistently evident in the medulla, the hypothalamus,
and the isocortex.

The highly chimeric patch within the physical portion of the landscape appears
to be mostly below the $\beta=\alpha$ line. This is reasonable, as chimera
states occur when coupling within groups is greater than coupling between
groups. A small portion of the chimeric patch lies above the $\beta=\alpha$
line, likely because the average strength between cortices is greater than the
average strength within cortices (see Fig. 4 A).

The chimera-like index $\chi$ greatly lessens at $\alpha\approx 0.1$. A
possible explanation for this comes from comparing the order of $\dot{x}$
without the coupling terms, and the coupling terms themselves. From our
simulation, we find that $\dot{x}$ without the coupling terms ranges roughly
from -6 to 3. The coupling terms each666Since the same can be said for both
$\alpha$ and $\beta$, we will discuss only $\alpha$, with the understanding
that $\beta$ could be substituted into the proceeding sentences. range from 0
to approximately $30\alpha$. This means that, when $\alpha>0.1$, the coupling
is at least of the same order as the sum of the rest of the terms in the
equation. This leads to a qualitative difference between the two states, which
likely manifests itself as the less-chimeric states.

##  5 Discussion

The logical next step is to compare the simulated EEG traces to actual data
collected from mice, with the aid of an epileptologist. With further work in
this direction, our research could potentially go from a mathematical
curiosity to an applicable therapeutic and diagnostic tool. However, the
initial results are encouraging. The majority of the simulated seizures showed
excessive synchrony in areas such as the hypothalamus and the hippocampal
formation, which are the most common origins of ictal activity [Maguire &
Salpekar(2013), Avoli(2007)]. The exact origin of focal seizures is poorly
understood, and our implementation of the model makes it difficult to narrow
in on the precise onset of chimera states. This could be a future direction
for research in this area.

Finding an instructive phase-space embedding of the Hindmarsh-Rose network
would be challenging (seeing as it is a 639-dimensional system), but would
likely reveal potentially useful insights into the nature of the mechanisms
underlying these systems. The same could be said for Lyapunov analysis, as
well as finding an informative way to create a bifurcation diagram and perform
more in-depth bifurcation analysis.

Another way our work could be extended is by looking at chimera state collapse
and its relationship to secondary seizure generalization. However, it would be
extremely computationally expensive, given the size of the system, and would
therefore require some clever handiwork [Wolfrum & Omel’chenko(2011)].

Future work could also naturally be performed on better, more up-to-date
connectomes [Knox _et al._(2019)Knox, Harris, Graddis, Whitesell, Zeng,
Harris, Shea-Brown & Mihalas].

## Acknowledgments

This work could not have been completed without help from Taras Lakoba and
Sean Flynn. The authors also thank Kameron Harris for helpful comments on the
manuscript.

Additional Figures Fig. 10 shows some additional analyses of the parameter
landscape.

Figure 10: Additional analyses of the parameter landscape. A. The number of
runs (out of 10) which gave physical results. All parameter sets which are not
yellow in A were excluded from the other analyses. B. The variance of the
chimera-like index. C. The metastability index. D. The variance of the
metastability index. E. The mean order parameter over time, averaged across
runs. F. The variance between runs of the mean order parameter.

## References

  * [Abrams _et al._(2008)Abrams, Mirollo, Strogatz & Wiley] Abrams, D. M., Mirollo, R., Strogatz, S. H. & Wiley, D. A. [2008] ``Solvable model for chimera states of coupled oscillators,'' _Physical Review Letters_ 101, doi:10.1103/PhysRevLett.101.084103.
  * [Abrams & Strogatz(2004)] Abrams, D. M. & Strogatz, S. H. [2004] ``Chimera states for coupled oscillators,'' _Physical Review Letters_ 93, doi:10.1103/PhysRevLett.93.174102.
  * [Andrzejak _et al._(2016)Andrzejak, Rummel, Mormann & Schindler] Andrzejak, R. G., Rummel, C., Mormann, F. & Schindler, K. [2016] ``All together now: Analogies between chimera state collapses and epileptic seizures,'' _Scientific Reports_ 6, doi:10.1038/srep23000.
  * [Avoli(2007)] Avoli, M. [2007] ``The epileptic hippocampus revisited: Back to the future,'' _Epilepsy Currents_ 7, 116–118, doi:10.1111/j.1535-7511.2007.00194.x, URL https://doi.org/10.1111/j.1535-7511.2007.00194.x.
  * [Baier _et al._(2012)Baier, Goodfellow, Taylor, Wang & Garry] Baier, G., Goodfellow, M., Taylor, P. N., Wang, Y. & Garry, D. J. [2012] ``The importance of modeling epileptic seizure dynamics as spatio-temporal patterns,'' _Frontiers in Physiology_ 3, doi:10.3389/fphys.2012.00281.
  * [Bansal _et al._(2019)Bansal, Garcia, Tompson, Verstynen, Vettel & Muldoon] Bansal, K., Garcia, J. O., Tompson, S. H., Verstynen, T., Vettel, J. M. & Muldoon, S. F. [2019] ``Cognitive chimera states in human brain networks,'' _Science Advances_ 5, eaau8535.
  * [Baroni & Mazzoni(2014)] Baroni, F. & Mazzoni, A. [2014] ``Heterogeneity of heterogeneities in neuronal networks,'' _Frontiers in Computational Neuroscience_ 8, doi:10.3389/fncom.2014.00161, URL https://doi.org/10.3389/fncom.2014.00161.
  * [Breakspear(2017)] Breakspear, M. [2017] ``Dynamic models of large-scale brain activity,'' _Nature Neuroscience_ 20, 340–352.
  * [Chouzouris _et al._(2018)Chouzouris, Omelchenko, Zakharova, Hlinka, Jiruska & Schöll] Chouzouris, T., Omelchenko, I., Zakharova, A., Hlinka, J., Jiruska, P. & Schöll, E. [2018] ``Chimera states in brain networks: Empirical neural vs. modular fractal connectivity,'' _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 28, doi:10.1063/1.5009812.
  * [Hizanidis _et al._(2016)Hizanidis, Kouvaris, Zamora-López, Díaz-Guilera & Antonopoulos] Hizanidis, J., Kouvaris, N. E., Zamora-López, G., Díaz-Guilera, A. & Antonopoulos, C. G. [2016] ``Chimera-like states in modular neural networks,'' _Scientific Reports_ 6, doi:10.1038/srep19845.
  * [Höhlein _et al._(2019)Höhlein, Kemeth & Krischer] Höhlein, K., Kemeth, F. P. & Krischer, K. [2019] ``Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators,'' _Physical Review E_ 100, doi:10.1103/physreve.100.022217.
  * [Knox _et al._(2019)Knox, Harris, Graddis, Whitesell, Zeng, Harris, Shea-Brown & Mihalas] Knox, J. E., Harris, K. D., Graddis, N., Whitesell, J. D., Zeng, H., Harris, J. A., Shea-Brown, E. & Mihalas, S. [2019] ``High-resolution data-driven model of the mouse connectome,'' _Network Neuroscience_ 3, 217–236.
  * [Kruk _et al._(2018)Kruk, Maistrenko & Koeppl] Kruk, N., Maistrenko, Y. & Koeppl, H. [2018] ``Self-propelled chimeras,'' _Physical Review E_ 98, doi:10.1103/PhysRevE.98.032219.
  * [Kuramoto & Battogtokh(2002)] Kuramoto, Y. & Battogtokh, D. [2002] ``Coexistence of coherence and incoherence in nonlocally coupled phase oscillators,'' _Nonlinear Phenomena in Complex Systems_ 5, 380–385.
  * [Leistritz _et al._(2007)Leistritz, Putsche, Schwab, Hesse, Süße, Haueisen & Witte] Leistritz, L., Putsche, P., Schwab, K., Hesse, W., Süße, T., Haueisen, J. & Witte, H. [2007] ``Coupled oscillators for modeling and analysis of EEG/MEG oscillations,'' _Biomedizinische Technik/Biomedical Engineering_ 52, 83–89.
  * [Ljungberg _et al._(2009)Ljungberg, Sunnen, Lugo, Anderson & D'Arcangelo] Ljungberg, M. C., Sunnen, C. N., Lugo, J. N., Anderson, A. E. & D'Arcangelo, G. [2009] ``Rapamycin suppresses seizures and neuronal hypertrophy in a mouse model of cortical dysplasia,'' _Disease Models & Mechanisms_ 2, 389–398.
  * [Maguire & Salpekar(2013)] Maguire, J. & Salpekar, J. A. [2013] ``Stress, seizures, and hypothalamic–pituitary–adrenal axis targets for the treatment of epilepsy,'' _Epilepsy & Behavior_ 26, 352–362, doi:10.1016/j.yebeh.2012.09.040, URL https://doi.org/10.1016/j.yebeh.2012.09.040.
  * [Martens _et al._(2013)Martens, Thutupalli, Fourriere & Hallatschek] Martens, E. A., Thutupalli, S., Fourriere, A. & Hallatschek, O. [2013] ``Chimera states in mechanical oscillator networks,'' _Proceedings of the National Academy of Sciences_ 110, 10563–10567.
  * [Oh _et al._(2014)Oh, Harris, Ng, Winslow, Cain, Mihalas, Wang, Lau, Kuan, Henry, Mortrud, Ouellette, Nguyen, Sorensen, Slaughterbeck, Wakeman, Li, Feng, Ho, Nicholas, Hirokawa, Bohn, Joines, Peng, Hawrylycz, Phillips, Hohmann, Wohnoutka, Gerfen, Koch, Bernard, Dang, Jones & Zeng] Oh, S. W., Harris, J. A., Ng, L., Winslow, B., Cain, N., Mihalas, S., Wang, Q., Lau, C., Kuan, L., Henry, A. M., Mortrud, M. T., Ouellette, B., Nguyen, T. N., Sorensen, S. A., Slaughterbeck, C. R., Wakeman, W., Li, Y., Feng, D., Ho, A., Nicholas, E., Hirokawa, K. E., Bohn, P., Joines, K. M., Peng, H., Hawrylycz, M. J., Phillips, J. W., Hohmann, J. G., Wohnoutka, P., Gerfen, C. R., Koch, C., Bernard, A., Dang, C., Jones, A. R. & Zeng, H. [2014] ``A mesoscale connectome of the mouse brain,'' _Nature_ 508, 207–214.
  * [Panaggio & Abrams(2015)] Panaggio, M. J. & Abrams, D. M. [2015] ``Chimera states: coexistence of coherence and incoherence in networks of coupled oscillators,'' _Nonlinearity_ 28, R67–R87.
  * [Santos _et al._(2015)Santos, Szezech, Batista, Caldas, Viana & Lopes] Santos, M., Szezech, J., Batista, A., Caldas, I., Viana, R. & Lopes, S. [2015] ``Recurrence quantification analysis of chimera states,'' _Physics Letters A_ 379, 2188–2192.
  * [Santos _et al._(2017)Santos, Szezech, Borges, Iarosz, Caldas, Batista, Viana & Kurths] Santos, M., Szezech, J., Borges, F., Iarosz, K., Caldas, I., Batista, A., Viana, R. & Kurths, J. [2017] ``Chimera-like states in a neuronal network model of the cat brain,'' _Chaos, Solitons & Fractals_ 101, 86–91.
  * [Shanahan(2010)] Shanahan, M. [2010] ``Metastable chimera states in community-structured oscillator networks,'' _Chaos: An Interdisciplinary Journal of Nonlinear Science_ 20, doi:10.1063/1.3305451.
  * [Taylor _et al._(2012)Taylor, Goodfellow, Wang & Baier] Taylor, P. N., Goodfellow, M., Wang, Y. & Baier, G. [2012] ``Towards a large-scale model of patient-specific epileptic spike-wave discharges,'' _Biological Cybernetics_ 107, 83–94.
  * [Westbrook(2013)] Westbrook, G. L. [2013] ``50,'' _Seizures and Epilepsy_ , 5th ed. (McGraw Hill Medical), pp. 1116–1139.
  * [Wolfrum & Omel’chenko(2011)] Wolfrum, M. & Omel'chenko, O. E. [2011] ``Chimera states are chaotic transients,'' _Physical Review E_ 84, doi:10.1103/PhysRevE.84.015201.
  * [Xie _et al._(2014)Xie, Knobloch & Kao] Xie, J., Knobloch, E. & Kao, H.-C. [2014] ``Multicluster and traveling chimera states in nonlocal phase-coupled oscillators,'' _Physical Review E_ 90, doi:10.1103/PhysRevE.90.022919.