Linear Discriminant Analysis and Noise Correlations in Neuronal Activity

Abstract

The effects of noise correlations on neuronal stimulus discrimination have been the subject of sustained debate. Both experimental and computational work suggest beneficial and detrimental contributions of noise correlations. The aim of this study is to develop an analytically tractable model of stimulus discrimination that reveals the conditions leading to improved or impaired performance from model parameters and levels of noise correlation. We begin with a mean firing rate integrator model as an approximation of underlying spiking activity in neuronal circuits. We consider two independent units receiving constant input and time fluctuating noise whose correlation across units can be tuned independently of firing rate. We implement a perceptron-like readout with Fisher Linear Discriminant Analysis (LDA). We exploit its closed form solution to find explicit expressions for discrimination error as a function of network parameters (leak, shared inputs, and noise gain) as well as the strength of noise correlation. First, we derive equations for discrimination error as a function of noise correlation. We find that four qualitatively different sets of results exist, based on the ratios of the difference of means and variance of the distributions of neural activity. From network parameters, we find the conditions for which an increase in noise correlation can lead to monotonic decrease or monotonic increase of error, as well as conditions for which error evolves non-monotonically as a function of correlations. These results provide a potential explanation for previously reported contradictory effects of noise correlation. Second, we expand on the dependency of the quantitative behaviour of the error curve on the tuning of specific subsets of network parameters. Particularly, when the noise gain of a pair of units is increased, the error rate as a function of noise correlation increases multiplicatively. However, when the noise gain of a single unit is increased, under certain conditions, the effect of noise can be beneficial to stimulus discrimination. In sum, we present a framework of analysis that explains a series of non-trivial properties of neuronal discrimination via a simple linear classifier. We show explicitly how different configurations of parameters can lead to drastically different conclusions on the impact of noise correlations. These effects shed light on abundant experimental and computational results reporting conflicting effects of noise correlations. The derived analyses rely on few assumptions and may therefore be applicable to a broad class of neural models whose activity can be approximated by a multivariate distribution.