  One major challenge for living cells is the measurement and prediction of
signals corrupted by noise. In general, cells need to make decisions based on
their compressed representation of noisy, time-varying signals. Strategies for
signal noise mitigation are often tackled using Wiener filtering theory, but
this theory cannot account for systems that have limited resources and hence
must compress the signal. To study how accurately linear systems can predict
noisy, time-varying signals in the presence of a compression constraint, we
extend the information bottleneck method. We show that the optimal integration
kernel reduces to the Wiener filter in the absence of a compression constraint.
This kernel combines a delta function at short times and an exponential
function that decays on a timescale that sets the integration time. Moreover,
there exists an optimal integration time, which arises from a trade-off between
time averaging signal noise and dynamical error. As the level of compression is
increased, time averaging becomes harder, and as a result the optimal
integration time decreases and the delta peak increases. We compare the
behaviour of the optimal system with that of a canonical motif in cell
signalling, the push-pull network, finding that the system reacts to signal
noise and compression in a similar way.
