Population neural recordings with long-range temporal structure are often best understood in terms of a shared underlying low-dimensional dynamical process. Advances in recording technology provide access to an ever larger fraction of the population, but the standard computational approaches available to identify the collective dynamics scale poorly with the size of the dataset. Here we describe a new, scalable approach to discovering the low-dimensional dynamics that underlie simultaneously recorded spike trains from a neural population. Our method is based on recurrent linear models (RLMs), and relates closely to timeseries models based on recurrent neural networks. We formulate RLMs for neural data by generalising the Kalman-filter-based likelihood calculation for latent linear dynamical systems (LDS) models to incorporate a generalised-linear observation process. We show that RLMs describe motor-cortical population data better than either directly-coupled generalised-linear models or latent linear dynamical system models with generalised-linear observations. We also introduce the cascaded linear model (CLM) to capture low-dimensional instantaneous correlations in neural populations. The CLM describes the cortical recordings better than either Ising or Gaussian models and, like the RLM, can be fit exactly and quickly. The CLM can also be seen as a generalization of a low-rank Gaussian model, in this case factor analysis. The computational tractability of the RLM and CLM allow both to scale to very high-dimensional neural data.

Simultaneous recordings of many neurons embedded within a recurrently-connected cortical network may provide concurrent views into the dynamical processes of that network, and thus its computational function. In principle, these dynamics might be identified by purely unsupervised, statistical means. Here, we show that a Hidden Switching Linear Dynamical Systems (HSLDS) model---in which multiple linear dynamical laws approximate a nonlinear and potentially non-stationary dynamical process---is able to distinguish different dynamical regimes within single-trial motor cortical activity associated with the preparation and initiation of hand movements. The regimes are identified without reference to behavioural or experimental epochs, but nonetheless transitions between them correlate strongly with external events whose timing may vary from trial to trial. The HSLDS model also performs better than recent comparable models in predicting the firing rate of an isolated neuron based on the firing rates of others, suggesting that it captures more of the "shared variance" of the data. Thus, the method is able to trace the dynamical processes underlying the coordinated evolution of network activity in a way that appears to reflect its computational role.

This paper presents a neural network model of demyelination of the mouse motor pathways, coupled to a central pattern generation (CPG) model for quadruped walking. Demyelination is the degradation of the myelin layer covering the axons which can be caused by several neurodegenerative autoimmune diseases such as multiple sclerosis. We use this model - to our knowledge first of its kind - to investigate the locomotion deficits that appear following demyelination of axons in the spinal cord. Our model meets several physiological and behavioral results and predicts that whereas locomotion can still occur at high percentages of demyelination damage, the distribution and location of the lesion are the most critical factors for the locomotor performance.