The covariance perceptron: A new paradigm for classification and processing of time series in recurrent neuronal networks

Learning in neuronal networks has developed in many directions, in particular to reproduce cognitive tasks like image recognition and speech processing. Implementations have been inspired by stereotypical neuronal responses like tuning curves in the visual system, where, for example, ON/OFF cells fire or not depending on the contrast in their receptive fields. Classical models of neuronal networks therefore map a set of input signals to a set of activity levels in the output of the network. Each category of inputs is thereby predominantly characterized by its mean. In the case of time series, fluctuations around this mean constitute noise in this view. For this paradigm, the high variability exhibited by the cortical activity may thus imply limitations or constraints, which have been discussed for many years. For example, the need for averaging neuronal activity over long periods or large groups of cells to assess a robust mean and to diminish the effect of noise correlations. To reconcile robust computations with variable neuronal activity, we here propose a conceptual change of perspective by employing variability of activity as the basis for stimulus-related information to be learned by neurons, rather than merely being the noise that corrupts the mean signal. In this new paradigm both afferent and recurrent weights in a network are tuned to shape the input-output mapping for covariances, the second-order statistics of the fluctuating activity. When including time lags, covariance patterns define a natural metric for time series that capture their propagating nature. We develop the theory for classification of time series based on their spatio-temporal covariances, which reflect dynamical properties. We demonstrate that recurrent connectivity is able to transform information contained in the temporal structure of the signal into spatial covariances. Finally, we use the MNIST database to show how the covariance perceptron can capture specific second-order statistical patterns generated by moving digits. Author summary The dynamics in cortex is characterized by highly fluctuating activity: Even under the very same experimental conditions the activity typically does not reproduce on the level of individual spikes. Given this variability, how then does the brain realize its quasi-deterministic function? One obvious solution is to compute averages over many cells, assuming that the mean activity, or rate, is actually the decisive signal. Variability across trials of an experiment is thus considered noise. We here explore the opposite view: Can fluctuations be used to actually represent information? And if yes, is there a benefit over a representation using the mean rate? We find that a fluctuation-based scheme is not only powerful in distinguishing signals into several classes, but also that networks can efficiently be trained in the new paradigm. Moreover, we argue why such a scheme of representation is more consistent with known forms of synaptic plasticity than rate-based network dynamics.