  Neural encoding studies explore the relationships between measurements of
neural activity and measurements of a behavior that is viewed as a response to
that activity. The coupling between neural and behavioral measurements is
typically imperfect and difficult to measure.To enhance our ability to
understand neural encoding relationships, we propose that a behavioral
measurement may be decomposable as a sum of two latent components, such that
the direct neural influence and prediction is primarily localized to the
component which encodes temporal dependence. For this purpose, we propose to
use a non-separable Kronecker sum covariance model to characterize the
behavioral data as the sum of terms with exclusively trial-wise, and
exclusively temporal dependencies. We then utilize a corrected form of Lasso
regression in combination with the nodewise regression approach for estimating
the conditional independence relationships between and among variables for each
component of the behavioral data, where normality is necessarily assumed. We
provide the rate of convergence for estimating the precision matrices
associated with the temporal as well as spatial components in the Kronecker sum
model. We illustrate our methods and theory using simulated data, and data from
a neural encoding study of hawkmoth flight; we demonstrate that the neural
encoding signal for hawkmoth wing strokes is primarily localized to a latent
component with temporal dependence, which is partially obscured by a second
component with trial-wise dependencies.
