Learning to Estimate Dynamical State with Probabilistic Population Codes.
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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.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.
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PubMed Central - PubMed
Affiliation: Center for Integrative Neuroscience, University of California, San Francisco, San Francisco, California, United States of America.
ABSTRACT
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. No MeSH data available. Related in: MedlinePlus |
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Mentions: Optimal (or nearly optimal) position estimation for these dynamical systems requires tracking velocity and position, so we plot receptive fields (RFs) in position-velocity space. Now, for oscillatory dynamics, high speeds rarely co-occur with positions far from zero (equilibrium), which leaves the “corners” of such RFs empty. This obscures the pattern of RFs and the corresponding state-estimation scheme learned by the rEFH. Therefore, for simplicity, we present results from a network trained on a third dynamical model (“no-spring”): uncontrolled, and with no spring force (see Methods). (Similar results, albeit less clean, are observed in the corresponding analyses for oscillatory dynamics; see S3 Text in the supporting material.) In Fig 5A, the position-velocity receptive fields are plotted for all 225 hidden units of this rEFH, arranged in a 15 × 15 grid. The ordinate of each subsquare corresponds to position (increasing from top to bottom), and the abscissa to velocity (increasing from left to right). The large majority of receptive fields are negatively sloped “stripes” in this space. Interestingly, they resemble in this the receptive fields of neurons in MSTd of a rhesus macaque trained to track moving stimuli [22]—although in that work there are positively-sloped stripes as well. |
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Affiliation: Center for Integrative Neuroscience, University of California, San Francisco, San Francisco, California, United States of America.
No MeSH data available.