Analytic model of response statistics in noisy neural populations with divisive normalization

Divisive normalization is an essential neural computation that is commonly used in models of visual processing. However, normalization models rarely incorporate neural noise. This is not realistic. Here, we expand the classic model of normalization to include the effects of noise. The classic model is characterized by a noiseless unnormalized drive (e.g., linear-receptive-field outputs) that is divided by a scalar normalization signal. The normalization signal is, itself, a function of the unnormalized drives (e.g. L2-norm). The result is a normalized population response. We expand this classic model by incorporating noise into the unnormalized drive. We model the drive as multivariate Gaussian, entailing that noise affects both the numerator (drive) and the denominator (normalization signal) of the normalization equation. We derive analytic formulas for the mean and covariance of the normalized responses, given the unnormalized-drive statistics. The formulas can incorporate arbitrary noise correlations and different types of biologically-plausible normalization (e.g. broadband, feature-specific). The formulas are also differentiable, making them suitable for optimization routines used in computational models, and in fitting neural data. The behavior of the model is consistent with classic noiseless models (e.g. responses saturate at high contrasts), but non-obvious behaviors are also predicted. First, normalization transforms independent drive-noise into noise correlations in the normalized responses. Second, these normalization-induced response noise correlations are stimulus-dependent. Third, increasing noise in fixed-mean unnormalized drives reduces mean normalized responses. Our model includes elements (e.g. large neural populations, dependence of normalization on noisy population drive) that are missing from other attempts to analytically model the interaction of normalization and noise. The formulas derived here describe in a principled way the consequences of normalization on noisy visual processing, and constitute a tool for quantitatively modeling the behaviors of real neural systems that have not previously been analytically linked to normalization.