Modeling stimulus-dependent variability improves decoding of population neural responses

Abstract

Neural responses to repeated presentations of an identical stimulus often show substantial trial-to-trial variability. How the mean firing rate varies in response to different stimuli or during different movements (tuning curves) has been extensively modeled in a wide variety of neural systems. However, the variability of neural responses can also have clear tuning independent of the tuning in the mean firing rate. This suggests that the variability could contain information regarding the stimulus/movement beyond what is encoded in the mean firing rate. Here we demonstrate how taking variability into account can improve neural decoding. In a typical neural coding model spike counts are assumed to be Poisson with the mean response depending on an external variable, such as a stimulus or movement. Bayesian decoding methods then use the probabilities under these Poisson tuning models (the likelihood) to estimate the probability of each stimulus given the spikes on a given trial (the posterior). However, under the Poisson model, spike count variability is always exactly equal to the mean (Fano factor = 1). Here we use two alternative models -the Conway-Maxwell-Poisson (CMP) model and Negative Binomial (NB) model -to more flexibly characterize how neural variability depends on external stimuli. These models both contain the Poisson distribution as a special case but have an additional parameter that allows the variance to be greater than the mean (Fano factor >1) or, for the CMP model, less than the mean (Fano factor <1). We find that neural responses in primary motor (M1), visual (V1), and auditory (A1) cortices have diverse tuning in both their mean firing rates and response variability. Across cortical areas, we find that Bayesian decoders using the CMP or NB models improve stimulus/movement estimation accuracy by 4-12% compared to the Poisson model. Moreover, the uncertainty of the non-Poisson decoders more accurately reflects the magnitude of estimation errors. In addition to tuning curves that reflect average neural responses, stimulus-dependent response variability may be an important aspect of the neural code. Modeling this structure could, potentially, lead to improvements in brain machine interfaces.

A universal probabilistic spike count model reveals ongoing modulation of neural variability

“…The task is to jointly recover the ground truth count likelihoods, their tuning to covariates, and latent trajectories if relevant from activity generated using two synthetic populations. The first population was generated with a parametric heteroscedastic Conway-Maxwell-Poisson (CMP) model [53], which has decoupled mean and variance modulation as well as simultaneously over- and underdispersed activity (Fano factors above and below 1). The second population consists of Poisson neurons tuned to head direction and an additional hidden signal, which gives rise to apparent overdispersion [28] as well as noise correlations when only regressing to observed covariates.…”

“…We compare our universal model to the log Cox Gaussian process or Poisson GP model [33] and the heteroscedastic negative binomial GP (NBh) model which places GP priors on both the rate and shape parameter, a non-parametric extension of [53]. The more flexible CMPh model, analogous to NBh, has difficulty in scaling to large datasets due to the series approximation of the partition function (Appendix B).…”

“…Heteroscedastic count models, characterized by input-dependent noise [58], additionally regress the dispersion parameter of count distributions to covariates [59, 60]. This has shown improvements in stimulus decoding and more calibrated posterior uncertainties [53]. Copula-based models [35, 61] separate marginal distributions of single neurons from the multivariate dependency structure of the population parameterized by the copula family, and thus do not place parametric constraints on single neuron count statistics.…”

“…Related work Neural encoding model provide a statistical description of neural count activity, and typically rely on a parametric count likelihood such as the Poisson [33], negative binomial [30,31] or Conway-Maxwell-Poisson distribution [53]. This choice is independent of the empirical count statistics and is often mismatched to the data.…”

“…We use the cross-correlation between time series x t and y t r xy (∆) = (x t+∆ − x t+∆ )(y t − y t ) σ x σ y (53) which includes the auto-correlation as a special case, e.g. r xx (∆).…”