We fit a generalized linear model in which time-varying state variables pupil size, p(t), and task engagement, b(t), were used to re-weight each unit’s mean evoked response to each noise stimulus, r0t, and spontaneous rate, s0, to generate a prediction of the single-trial spike rate, rfull(t), at each point in time. The model included multiplicative gain parameters (g) and DC offset parameters (d) to capture both types of modulation. We refer to this model as the full model:

To account for nonlinear threshold and saturation of state effects, the summed state signal was passed through a sigmoid nonlinearity, Fd or Fg, before scaling response baseline or gain, respectively (difference of exponentials) (Thorson et al., 2015). With additional constant terms g0 and d0, this model required a total of six free parameters. For comparison, we calculated a state-independent model here referred to as the null model, in which the state variable regressors were shuffled in time. Because shuffling removes any possible correlation between state and neural activity, gain and offset parameters are reduced to d0=g0=1 and dp=db=gp=gb=0, effectively reducing the model to

In practice, fitting to the shuffled data produces parameter values slightly different from zero, and controls for noise in the regression procedure.

We also considered two partial models, one to predict responses based on pupil size only, rpt, and the other to predict responses based on behavior only, rbt, in which the other regressor was shuffled in time. Thus, the pupil-only model accounted only for effects of pupil size,

and the task-only model accounted only for task engagement,

These models tested the effects of a single state variable while ignoring the other.

By comparing performance of the full model to each partial model, we determined the unique contribution of each state variable to the neuron’s activity. We used a 20-fold, nested cross-validation procedure to evaluate model performance, which permitted using the full data set for both model fitting and validation without introducing bias from over-fitting. The model was fit to 95% of the data and used to predict the remaining 5%. Fit and test data were taken from interleaved trials. This procedure was repeated 20 times with nonoverlapping test sets, so that the final result was a prediction of the entire response. Model performance was then quantified by the fraction of variance explained, that is, the squared correlation coefficient, r2, between the predicted and actual time-varying response. Variance uniquely explained by single state variables was calculated as the difference between r2 for the full model and for the partial model in which the relevant variable was shuffled in time.

When comparing pupil and neural data, a 750 ms offset was applied to pupil trace to account for the lagged relationship between changes in pupil size and neural activity in A1 (McGinley et al., 2015).

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