Abstract

Latent linear dynamical systems with generalised-linear observation models arise in a variety of applications, for instance when modelling the spiking activity of populations of neurons. Here, we show how spectral learning methods (usually called subspace identification in this context) for linear systems with linear-Gaussian observations can be extended to estimate the parameters of a generalised-linear dynamical system model despite a non-linear and non-Gaussian observation process. We use this approach to obtain estimates of parameters for a dynamical model of neural population data, where the observed spike-counts are Poisson-distributed with log-rates determined by the latent dynamical process, possibly driven by external inputs. We show that the extended subspace identification algorithm is consistent and accurately recovers the correct parameters on large simulated data sets with a single calculation, avoiding the costly iterative computation of approximate expectation-maximisation (EM). Even on smaller data sets, it provides an effective initialisation for EM, avoiding local optima and speeding convergence. These benefits are shown to extend to real neural data.