Variance as a signature of neural computations during decision making

Abstract

Traditionally, insights into neural computation have been furnished by averaged firing rates from many stimulus repetitions or trials. We pursue an analysis of neural response variance to unveil neural computations that cannot be discerned from measures of average firing rate. We analyzed single-neuron recordings from the lateral intraparietal area (LIP), during a perceptual decision-making task. Spike count variance was divided into two components using the law of total variance for doubly stochastic processes: (1) variance of counts that would be produced by a stochastic point process with a given rate, and loosely (2) the variance of the rates that would produce those counts (i.e., "conditional expectation"). The variance and correlation of the conditional expectation exposed several neural mechanisms: mixtures of firing rate states preceding the decision, accumulation of stochastic "evidence" during decision formation, and a stereotyped response at decision end. These analyses help to differentiate among several alternative decision-making models.

Figure 1. Overview of the task and neural responses

Monkeys decided the net direction of motion in dynamic random dot displays and indicated their choices by making an eye movement to a peripheral choice target. Analyses of neural data focus on three epochs during the trials. Examples show subsets of data presented in subsequent figures. Left: Pre-decision epoch. Responses are aligned in time to the onset of choice targets (red circles in cartoons, above). Mean firing rates are from all 2-choice trials (16,444 trials). Mean rates are calculated from spikes counted in 60 ms bins (counting windows). Curves are running means; error bars are SEM from non-overlapping 60 ms windows (most are too small to be visible). Middle: Early decision formation. Responses are aligned to the onset of random dot motion. All 2-choice trials where motion was in the T_in direction are included (9,654 trials). Inset: responses grouped by motion strength (color, labels). Trials contribute to the averages up to 340 ms after motion onset or 100 ms before saccade initiation, whichever occurs first. Arrow indicates beginning of decision related activity, approximately 190 ms after motion onset. Right: End of decision. Responses are aligned to the initiation of the saccadic eye movement response. Averages reflect correct T_in choices only. All motion strengths are included (7008 trials).

Figure 2. Examples of doubly stochastic point processes

a–e, Each process is characterized by a rate function that may vary from trial to trial and a random point process that realizes that rate. Both sources of variability contribute to total spike count variance. For each process, theoretical rate functions are shown with simulations of a nonstationary Poisson point process. Mean spike rate and spike count variance are calculated in non-overlapping windows using the same method as for analysis of data in subsequent figures (60 ms; 20,000 simulated trials). Ten random spike trains are shown in the rasters below the panels. a. Constant rate without trial-to-trial variation. Spike count variability arises only from the stochastic point process (PPV), hence VarCE=0. b. Constant rate with trial-to-trial variation. A random value perturbs each rate function for the duration of the trial. Gray traces: examples of rate functions used to generate spikes. Total variance is comprised of PPV and VarCE. c. Same as b but with time varying rates. d. Same as c except that a new random perturbation is sampled every 10ms. e. Drift-diffusion. Rate is the sum of a deterministic “drift” function (same linear rise as in b and c) plus the cumulative sum of independent, random values drawn from a Normal distribution (mean=0). Individual rate traces resemble 1-dimensional Brownian motion (with drift). f. VarCE for the five examples. The VarCE captures the portion of total variance owing to variation in the rate functions across trials. Thick dashed lines show theoretical values $({\sigma }_{\langle N\rangle }^2)$ of VarCE for doubly stochastic Poisson point processes. Thin solid lines show VarCE estimates using the algorithm applied to the simulated spike trains $(s_{\langle N\rangle }^2)$. Counting window = 60 ms. Line color corresponds to the colors used in a–e.

Figure 3. VarCE in the pre-decision epoch depends on number of alternatives

Responses are aligned to the onset of the choice targets as in Fig. 1 (left panel). Mean and VarCE are calculated from spikes counted in a 60 ms sliding window. Top, mean firing rates for 2- and 4-choice conditions. Points include values from 70 neurons (2-choice, 16,444 trials; 4-choice, 32,882 trials). Error bars are SEM for non-overlapping windows (most are too small to be visible). Bottom, VarCE. Curves show the VarCE from the same 2- and 4-choice trials. Data from 70 neurons were combined using residual deviations from the means, respecting neuron identity. Error bars are SE (bootstrap; see Methods).

Figure 4. VarCE and CorCE during decision formation support a diffusion-like process

Responses are aligned to the onset of stimulus motion (vertical lines), as in Fig. 1 (middle panel). a–c. Spike rates and count variance are derived from 60 ms counting windows. Top row: Mean firing rates. Error bars are SEM for nonoverlapping 60 ms bins. Bottom row: VarCE computed from the residual deviations from means, respecting neuron identity, motion strength, direction and number of choices. Error bars are SE (bootstrap); many are too small to be visible. a. 0% motion strength only (8,815 trials). Both panels: Arrow marks the time that mean responses begin to diverge as function of stimulus direction and motion strength, as indicated in Figure 1 (v). b. All motion strengths (50,326 trials). c. Comparison of 2- and 4-choice tasks; all motion strengths (16,444 and 33,882 trials, for 2- and 4-choice, respectively). d. Correlation of conditional expectations (CorCE) of counts as a function of time separation during decision formation. The matrix of CorCE values is displayed as a heat map (color bar to right; centers of the 60 ms counting windows are indicated by the axes). Data are from all trials, as in b, using the same residual deviations. e. Decay of correlation in time. Graph shows CorCE of the first time bin with each subsequent time bin (top row of the matrix in d).

Figure 5. VarCE declines at the end of decision formation

Responses are aligned to the time of saccade initiation (vertical lines), as in Fig. 1 (right panel). Top row: Mean firing rates. Error bars are SEM. Bottom row: VarCE computed from the residual deviations. Error bars are SE (bootstrap). a. The decline in VarCE was faster and more pronounced for T_in choices (13,788 trials) than for T_out choices (13,724 trials). b. The decline in VarCE for T_in choices was apparent for strong and weak motion (2,815 and 2,838 trials; see Supplementary Figure 4 for comparison of all motion strengths). c. The decline in VarCE for T_in choices was apparent for 2- and 4-choice tasks (7235 and 6553 trials, respectively). Error trials for nonzero motion strengths are excluded in all panels.

Figure 6. Analysis of VarCE and CorCE in candidate models of decision-making

Columns show analyses of simulated neural responses from five mechanisms in the epoch of early decision formation (corresponding to the epoch beginning ~190 ms after motion onset in the LIP data). a–e: Average firing rates (5,000 simulated trials). Gray traces: 10 randomly chosen trials. f–j: VarCE. k–o: CorCE matrices displayed as heat maps (scale bar near panel k). Same conventions as Fig. 4d, except for 190 ms delay of start time in LIP data. p–t: CorCE between first and subsequent time bins. Black: CorCE from the top row of the corresponding CorCE matrix in panels k–o. Blue: CorCE for the LIP data (same as Fig. 4e).

Figure 7. Use of VarCE to evaluate a mixture-of-states model

Top, A mixture of low and high firing rate states can reproduce the mean firing rates from LIP in 2-choice (blue) and 4-choice (red) tasks. The analysis is for 0% coherent motion trials that culminate in a T_in choice. Black curves reconstruct average firing rates by mixing counts drawn from the start and end of the decisions (see Methods). Distributions of counts comprising these start and end sets were established separately for each neuron. Abscissae show time relative to the beginning of decision related activity in LIP, 190 ms after motion onset. Bottom, VarCE from the mixture model was estimated using a bootstrap procedure (600 samples). Each sample is a reconstruction of the mean firing rate, using the same number of trials as in the data. Black traces: average of the 600 VarCE values at each time point. Gray traces: individual sample reconstructions. VarCE from the data (blue and red curves) is outside the range associated with the mixture hypothesis. a. 2-choice. b. 4-choice. Note: VarCE associated with T_in choices is expected to rise initially, but then decrease owing to the exclusion of diffusion paths that lead to the alternative choices. This is evident in the 2-choice data (left) and would be evident in the 4-choice data (right) at later times (not shown).