  We consider the problem of estimating the sparse time-varying parameter
vectors of a point process model in an online fashion, where the observations
and inputs respectively consist of binary and continuous time series. We
construct a novel objective function by incorporating a forgetting factor
mechanism into the point process log-likelihood to enforce adaptivity and
employ $\ell_1$-regularization to capture the sparsity. We provide a rigorous
analysis of the maximizers of the objective function, which extends the
guarantees of compressed sensing to our setting. We construct two recursive
filters for online estimation of the parameter vectors based on proximal
optimization techniques, as well as a novel filter for recursive computation of
statistical confidence regions. Simulation studies reveal that our algorithms
outperform several existing point process filters in terms of trackability,
goodness-of-fit and mean square error. We finally apply our filtering
algorithms to experimentally recorded spiking data from the ferret primary
auditory cortex during attentive behavior in a click rate discrimination task.
Our analysis provides new insights into the time-course of the spectrotemporal
receptive field plasticity of the auditory neurons.
