High-dimensional neuronal activity from low-dimensional latent dynamics: a solvable model
Valentin Schmutz, Ali Haydaroglu, Shuqi Wang, Yixiao Feng, Matteo Carandini, Kenneth D Harris
Computation in recurrent networks of neurons has been hypothesized to occur at the level of low-dimensional latent dynamics, both in artificial systems and in the brain. This hypothesis seems at odds with evidence from large-scale neuronal recordings in mice showing that neuronal population activity is high-dimensional. To demonstrate that low-dimensional latent dynamics and high-dimensional activity can be two sides of the same coin, we present an analytically solvable recurrent neural network (RNN) model whose dynamics can be exactly reduced to a low-dimensional dynamical system, but generates high-dimensional activity with a power-law covariance eigenspectrum. This raises the question: Do low-dimensional latents explain the high-dimensional activity observed in mouse visual cortex? Spectral theory tells us that the covariance eigenspectrum alone does not allow us to recover the dimensionality of the latents, which can be low or high, when neurons are nonlinear. To address this indeterminacy, we develop Neural Cross-Encoder (NCE), an interpretable, nonlinear latent variable modeling method for calcium recordings, and find that high-dimensional neuronal responses to drifting gratings and spontaneous activity in visual cortex can be reduced to low-dimensional latents, while the responses to natural images cannot. Together, our results show that taking into account neuronal nonlinearities suffices to reconcile apparently conflicting views on the dimensionality of neural activity in computational and systems neuroscience.