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A large body of work has suggested that neural populations exhibitlow-dimensional dynamics during behavior. However, there are a variety ofdifferent approaches for modeling low-dimensional neural population activity.One approach involves latent linear dynamical system (LDS) models, in whichpopulation activity is described by a projection of low-dimensional latentvariables with linear dynamics. A second approach involves low-rank recurrentneural networks (RNNs), in which population activity arises directly from alow-dimensional projection of past activity. Although these two modelingapproaches have strong similarities, they arise in different contexts and tendto have different domains of application. Here we examine the preciserelationship between latent LDS models and linear low-rank RNNs. When can onemodel class be converted to the other, and vice versa? We show that latent LDSmodels can only be converted to RNNs in specific limit cases, due to thenon-Markovian property of latent LDS models. Conversely, we show that linearRNNs can be mapped onto LDS models, with latent dimensionality at most twicethe rank of the RNN.

Probing the relationship between linear dynamical systems and low-rank recurrent neural network models