Broadly speaking, descriptive statistical models aim to identify a probability distribution that captures the statistical properties of an observed neural dataset, while remaining agnostic about the mechanisms that gave rise to it. Two classes of modeling approaches that have generated a large following in the literature are descriptive statistical models and mechanistic models. To describe this activity, modelers have at their disposal a wide array of tools that give rise to different forms of low-dimensional activity (Cunningham & Yu, 2014). Recent work on large-scale neural population recordings has suggested that neural activity is often confined to a low-dimensional space, with fewer dimensions than the number of neurons in a population (Churchland, Byron, Sahani, & Shenoy, 2007 Gao & Ganguli, 2015 Gallego, Perich, Miller, & Solla, 2017 Saxena & Cunningham, 2019 Jazayeri & Ostojic, 2021). A surprising consequence of our results is that a partially observed RNN is better represented by an LDS model than by an RNN consisting of only observed units. Conversely, we show that linear RNNs can be mapped onto LDS models, with latent dimensionality at most twice the rank of the RNN. When can one model class be converted to the other, and vice versa? We show that latent LDS models can only be converted to RNNs in specific limit cases, due to the non-Markovian property of latent LDS models. Here we examine the precise relationship between latent LDS models and linear low-rank RNNs.

Although these two modeling approaches have strong similarities, they arise in different contexts and tend to have different domains of application. A second approach involves low-rank recurrent neural networks (RNNs), in which population activity arises directly from a low-dimensional projection of past activity. One approach involves latent linear dynamical system (LDS) models, in which population activity is described by a projection of low-dimensional latent variables with linear dynamics. However, there are a variety of different approaches for modeling low-dimensional neural population activity. A large body of work has suggested that neural populations exhibit low-dimensional dynamics during behavior.

Activity 3.4 linear dimensions