  We present a new notion $R_\ell$ of higher-order common information, which
quantifies the information that $\ell\geq 2$ arbitrarily distributed random
variables have in common. We provide analytical lower bounds on $R_3$ and $R_4$
for jointly Gaussian distributed sources and provide computable lower bounds
for $R_\ell$ for any $\ell$ and any sources. We also provide a practical method
to estimate the lower bounds on, e.g., real-world time-series data. As an
example, we consider EEG data acquired in a setup with competing acoustic
stimuli. We demonstrate that $R_3$ has descriptive properties that is not in
$R_2$. Moreover, we observe a linear relationship between the amount of common
information $R_3$ communicated from the acoustic stimuli and to the brain and
the corresponding cortical activity in terms of neural tracking of the
envelopes of the stimuli.
