Stable encoding of spatial structure by a recurrent neural network in the presence of noisy data

We develop conditions that guarantee stable encoding of spatial structure by a recurrent neural network in the presence of random perturbations in the data. A condition on the max-norm of the connection matrix and the derivative of the sigmoid guarantees BIBO statistical stability of a time-invariant input for a general network. For stimuli that fall into the linear region of the sigmoid we show that a condition on the eigenvalues of the connection matrix results in stable steady state in the mean and covariance. For the case of a homogeneous symmetric neural network, of which lateral inhibitory cortical networks are an example, we provide expressions for the mean steady state, the stationary co-variance and the eigenvalues. We also present a condition that guarantees that the network will reduce the variance of the input upon reaching stationarity. A general sufficient condition assuring the linear results for the symmetric, homogeneous network is given