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Peiran Gao (Stanford University)
In many experiments, neuroscientists tightly control behavior, record many trials, and obtain trial-averaged firing rates from hundreds of neurons in circuits containing millions of behaviorally relevant neurons. Dimensionality reduction has often shown that such datasets are strikingly simple; they can be described using a much smaller number of dimensions (principal components (PCs)) than the number of recorded neurons, and the resulting projections onto these components yield a remarkably insightful dynamical portrait of circuit computation.
This ubiquitous simplicity raises several profound and timely conceptual questions. What is the origin of this simplicity and its implications for the complexity of brain dynamics? Would neuronal datasets become more complex if we recorded more neurons? How and when can we trust dynamical portraits obtained under a vastly under-sampled measurement regimes, where only hundreds of neurons are recorded from circuits containing millions of neurons? We present a theory that answers these questions, and test it using data from monkeys performing a reaching task.
We prove that the measured dimensionality, the participation ratio of the PCA spectrum of a dataset, is upper bounded by the dimensionality of a stationary and factored approximation to its task correlation matrix. Conceptually, this upper bound has a natural interpretation as a quantitative measure of the data's neural task complexity, and provides a general analytic framework to ascertain whether neural dimensionality is constrained by task complexity or intrinsic brain dynamics. Interestingly, the dimensionality of motor cortical data is close to this bound, indicating neural activity is as complex as possible, given task and smoothness constraints.
Also, by connecting the act of recording random subsets of neurons to the theory of random projections, we further describe sufficient conditions for the recovery of low dimensional dynamical portraits in datasets collected under the vastly under-sampled regime. The theory further suggests a favorable logarithmic scaling of the requisite number of recorded neurons with respect to our definition of neural task complexity, which we verify in motor cortical data. Our framework yields conceptual insights into how we can reliably recover dynamical portraits in the under-sampled measurement regimes, and quantitative guidelines for the design of future experiments.