Learning to Estimate Dynamical State with Probabilistic Population Codes

Tracking moving objects, including one’s own body, is a fundamental ability of higher organisms, playing a central role in many perceptual and motor tasks. While it is unknown how the brain learns to follow and predict the dynamics of objects, it is known that this process of state estimation can be learned purely from the statistics of noisy observations. When the dynamics are simply linear with additive Gaussian noise, the optimal solution is the well known Kalman filter (KF), the parameters of which can be learned via latent-variable density estimation (the EM algorithm). The brain does not, however, directly manipulate matrices and vectors, but instead appears to represent probability distributions with the firing rates of population of neurons, “probabilistic population codes.” We show that a recurrent neural network—a modified form of an exponential family harmonium (EFH)—that takes a linear probabilistic population code as input can learn, without supervision, to estimate the state of a linear dynamical system. After observing a series of population responses (spike counts) to the position of a moving object, the network learns to represent the velocity of the object and forms nearly optimal predictions about the position at the next time-step. This result builds on our previous work showing that a similar network can learn to perform multisensory integration and coordinate transformations for static stimuli. The receptive fields of the trained network also make qualitative predictions about the developing and learning brain: tuning gradually emerges for higher-order dynamical states not explicitly present in the inputs, appearing as delayed tuning for the lower-order states.Author Summary: A basic task for animals is to track objects—predators, prey, even their own limbs—as they move through the world. Because the position estimates provided by the senses are not error-free, higher levels of performance can be, and are, achieved when the velocity and acceleration, as well as the position, of the object are taken into account. Likewise, tracking of limbs under voluntary control can be improved by considering the motor command that is (partially) responsible for its trajectory. Engineers have built tools to solve precisely these problems, and even to learn dynamical features of the object to be tracked. How does the brain do it? We show how artificial networks of neurons can learn to solve this task, simply by trying to become good predictive models of their incoming data—as long as some of those data are the activities of the neurons themselves at a fixed time delay, while the remainder (imperfectly) report the current position. The tracking scheme the network learns to use—keeping track of past positions; the corresponding receptive fields; and the manner in which they are learned, provide predictions for brain areas involved in tracking, like the posterior parietal cortex.