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# Sources of noise during accumulation of evidence in unrestrained and voluntarily head-restrained rats.

^{1,}

^{2}, Constantinople CM

^{1,}

^{2}, Erlich JC

^{3}, Tank DW

^{1,}

^{2,}

^{4}, Brody CD

^{1,}

^{2,}

^{5}.

### Author information

- 1
- Princeton Neuroscience Institute, Princeton University, Princeton, United States.
- 2
- Department of Molecular Biology, Princeton University, Princeton, United States.
- 3
- NYU-ECNU Institute of Brain and Cognitive Science, New York University Shanghai, Shanghai, China.
- 4
- Bezos Center for Neural Circuit Dynamics, Princeton University, Princeton, United States.
- 5
- Howard Hughes Medical Institute, Princeton University, Princeton, United States.

### Abstract

Decision-making behavior is often characterized by substantial variability, but its source remains unclear. We developed a visual accumulation of evidence task designed to quantify sources of noise and to be performed during voluntary head restraint, enabling cellular resolution imaging in future studies. Rats accumulated discrete numbers of flashes presented to the left and right visual hemifields and indicated the side that had the greater number of flashes. Using a signal-detection theory-based model, we found that the standard deviation in their internal estimate of flash number scaled linearly with the number of flashes. This indicates a major source of noise that, surprisingly, is not consistent with the widely used 'drift-diffusion modeling' (DDM) approach but is instead closely related to proposed models of numerical cognition and counting. We speculate that this form of noise could be important in accumulation of evidence tasks generally.

#### KEYWORDS:

accumulation of evidence; decision-making; drift diffusion model; head restraint; neuroscience; rat; signal detection theory

- PMID:
- 26673896
- PMCID:
- PMC4749559
- DOI:
- 10.7554/eLife.11308

- [Indexed for MEDLINE]

**B**) Timing of task events in an example trial. Rat was presented with two left flashes and one right flash and correctly oriented to the left side poke after an auditory cue. (C) Behavioral performance of data pooled from all unrestrained rats across the set of all stimuli. Color indicates the percentage of trials in which the subjects chose the right sideport. (D) Psychophysical performance on the accumulation task pooled from all unrestrained rats.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.003

**A**) A histogram of the number of left flashes presented on all trials included in this paper (data is pooled across all unrestrained rats). (

**B**) Histogram of the number of right flashes presented on all trials. (

**C**) Histogram of the difference in flash number on all trials. (

**D**) Histogram of the total number of flashes presented on each trial. (

**E**) Histogram of post-stimulus delay durations (seconds). Delay durations on some trials were greater than 6 s, but those represented a small portion of the dataset. (

**F**) Distribution of inter-flash intervals (seconds). (

**G**) The distribution of delay durations from panel E, replotted from 0–2 s. (

**H**) The distribution of inter-flash intervals from panel F, replotted from 0–2 s. (

**I**) Percent correct for each of the 1956 behavioral sessions included in this paper. All sessions with more than 100 trials in the final training stage were included. (

**J**) Number of trials completed for all behavioral sessions included in this paper.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.004

**DOI:**http://dx.doi.org/10.7554/eLife.11308.005

**A**) For trials of fixed duration and fixed difference in number of flashes (Δ

**-**flashes, colored lines), behavioral performance decreased with more flashes. (

**B**) Reverse correlation analysis indicating the relative contribution of flashes occurring at different times in the trial to the subject’s behavioral choice. Each point on the upper line represents the probability that on trials in which the subject poked to the right, there was an extra right flash in each time bin. The lower lines represent the same analysis for trials in which the subject poked left. The flatness of the lines suggests that rats use early, middle and late flashes equally to guide their decision. Lines and error bars represent mean and standard error across rats. This result suggests a long time constant of accumulation. (

**C**) Changes in behavioral performance (% correct) as a function of the number of total flashes presented. Data points indicate behavioral performance relative to the average performance (Δ Performance) across trials with identical differences in flash number (|#R-#L| = ΔF) but with varied total flash number (#R+#L=F). Red lines are regression lines to those points, weighted by the number of behavioral trials contributing to each point. (

**D**) Changes in behavioral performance as a function of the trial duration. Performance across all delay durations was computed for each unique combination of total number of flashes and difference in the number of flashes. For each unique combination of flash number and difference, performance relative to that average performance (Δ Performance) was computed for different trial durations binned in 50 ms bins (black circles). Red lines are regression lines to those points, weighted by the number of behavioral trials contributing to each point.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.006

**A**) Schematic of the accumulation model of used here to compare the contribution of flash- and time-associated noise to behavioral variability. At each moment in time the model represents the accumulated evidence as a decision variable,

*a*(

*t*)(black line). Colored arrows indicate the timing of left (blue) and right (red) flashes. σ

_{s}

^{2}parameterizes the noise added with each flash, σ

_{a}

^{2}parameterizes the noise added at each time point, τ (or 1/λ) parameterizes the memory time constant of

*a(*. λ<0 suggests that the memory decays to

*t*)*a*= 0 with time, λ>0 suggests that the magnitude of

*a(*increases over time.

*t*)*B*parameterizes sticky bounds: if

*a(*ever reaches +/-

*t*)*B,*integration stops and the animal is committed to that decision (go right/go left, respectively). There are a few other terms in the model that are not represented in the schematic. A bias term represents an offset of

*a(t)*at the beginning of each trial. A lapse rate parameterizes the percent of trials on which the animal behaves randomly. φ and τ

_{φ}parameterize sensory adaptation dynamics. After a flash, φ is a constant that scales the effect of the flash; it recovers to an unadapted/facilitated magnitude with time constant τ

_{φ}. φ >1 indicates that successive flashes facilitate; φ <1 indicates that they depress. Fits to those parameters are not shown here, because we found that the inter-flash-intervals presented here were sufficiently long to minimize any adaptation/facilitation effects of subsequent flashes. Light gray lines indicate alternative runs of the model on the same trial. (B) Model fits of flash-associated noise (σ

_{s}

^{2}; blue circles) and noise associated with time (σ

_{a}

^{2}; red circles). To evaluate these parameters in comparable units, σ

_{s}

^{2}was divided by the average number of flashes per second for each rat. For each subject, noise associated with time (σ

_{a}

^{2}) is close to zero, whereas noise correlated with flashes (σ

_{s}

^{2}) is predominant, consistent with previous studies (; ). (C) The drift in the accumulator’s memory is parameterized by λ. A leaky integrator would have negative values of λ, an impulsive integrator would have positive values. The time constant of the integrator, τ, is 1/λ. For many, but not all, of the rats, the time constant is close to or greater than 2 s ( λ <= 0.5). (D) The bound value for all except two rats is larger than the maximum number of flashes (on one side) that each rat experienced. This suggests that rats accumulated/used all of the flashes to inform their decision. (E) The bias term for the each rat. (F) The lapse rate term for each rat, represented as the percent of trials in which animals behaved randomly. (G) The number of behavioral trials that were used to fit the model parameters for each rat, represented on a logarithmic scale.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.007

**DOI:**http://dx.doi.org/10.7554/eLife.11308.008

**DOI:**http://dx.doi.org/10.7554/eLife.11308.009

**DOI:**http://dx.doi.org/10.7554/eLife.11308.010

**) Schematic of the model used to determine the standard deviation (**

**A***σ*) of the subjects’ estimate of flash number. Left panel indicates the stimuli from an example trial in which four flashes were presented to the left side (green) and six flashes were presented to the right (orange). The model assumes that on any given trial, a subjects’ estimate of flash number on each side is a continuous random variable drawn from a Gaussian distribution whose mean is the number of flashes on that side (

*), and whose variance is a free parameter () (*

**n***middle panel*). The choice on each trial is determined by comparing these two random variables. Errors occur when the difference of the two random variables (greater magnitude – lesser magnitude) is less than zero. The variance for each Gaussian representing a given flash number ( … ) was fit to the behavioral data (

*right panel*) using maximum likelihood estimation. (

**) Model fits of the standard deviations ( …) in the rats’ estimate for different numbers of flashes. Error bars indicate the 95% confidence intervals for the mean based on one thousand-fold resampled data. Note the deviation from pure linear dependence of the parameters on**

**B***n*for

*n*<2. (

**) Comparison of the behavioral data (**

**C***left panel*) with the predictions of the model (

*right panel*) based on the values calculated as shown in . Color indicates the percentage of trials on which the subject responded correctly. (

**) Comparison of psychometric performance of the rats (data, green triangles) and model prediction of performance (model, blue squares). (**

**D****) Three models that predict how the standard deviation (σ) of the numerical estimate scales with the number of flashes. Scalar variability predicts that σ scales linearly with the number of flashes (SV, yellow). Subitizing predicts that σ is zero until a limit (3 or 4), and then follows scalar variability prediction (black dashed). The drift diffusion models predict that the variance of the estimate scales linearly, and σ scales with the square root of the number of flashes (LV, blue). Purple triangles are model estimates of σ, replotted from . Each model was fit using linear regression to the model estimate of σ, weighted by the number of data points contributing to each triangle. Additionally, all models were constrained to intersect with the origin. (**

**E****) Goodness of fit of the shown in to subitizing (SUB+SV, black), the drift diffusion model (LV, blue) and scalar variability (SV, yellow) using least squares regression. Analysis indicates that the data is best fit by a scalar variability model. Error bars represent the 95% confidence intervals based on fits derived from a thousand-fold resampling of the data.**

**F****DOI:**http://dx.doi.org/10.7554/eLife.11308.011

**DOI:**http://dx.doi.org/10.7554/eLife.11308.012

**DOI:**http://dx.doi.org/10.7554/eLife.11308.013

**DOI:**http://dx.doi.org/10.7554/eLife.11308.014

**DOI:**http://dx.doi.org/10.7554/eLife.11308.015

^{2}values for scalar variability (SV) and linear variance (LV) () for a thousand bootstrapped samples. The gray line is a histogram of the null hypothesis distribution for a thousand bootstrapped samples. The red line indicates the p-value for the permutation test: the area under the gray line at the value corresponding to the observed difference in the distribution of r

^{2}values (i.e. the mean of the black distribution).

**DOI:**http://dx.doi.org/10.7554/eLife.11308.016

**DOI:**http://dx.doi.org/10.7554/eLife.11308.017

_{0}+kN, where N is the number of clicks after the simultaneous click. Color indicates trial difficulty, parameterized by γ, the log of the ratio of the rates of the clicks on each side. Across all animals, performance increases as a function of cue duration for a fixed γ (colored dots). Whereas pure scalar variability predicts flat psychometric function (see ), inclusion of an offset term captures increased performance with cue duration.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.018

**) General forms of the signal detection theory-based models that were compared assumed either single or dual accumulators. Each model determines which choice to make on each trial, by randomly selecting a value (**

**A***) from a Gaussian with mean equal to the difference in total right minus left flashes, and standard deviation . If*

**a***a*>0, the model decides “Right”, and if

*a*<0 the model decides “Left”. Noise, parameterized as the standard deviation of the distributions of flash number or flash difference (), enters the accumulation process differently in each model. Single accumulator models assume that noise depends only on the difference in the number of flashes (R-L) while dual accumulator models assume that noise depends on the total number of flashes (R+L). (

**) For each class of models (single vs. dual accumulators), we implemented versions that assumed scalar variability or linear variance and that assumed a static or a sequential sampling process. For the static model, equivalent to our original signal detection theory-based model, noise scales with the number of flashes seen at the end of the trial. For the sequential sampling model, noise is added to the subject’s estimate with each flash, and the magnitude of the noise depends on the value of the accumulator at the time of the flash (**

**B***a(*). (

**t**)**) Cladogram indicating the details of each of the eight models (a-h). (**

**C****) Comparison of the likelihood of the model given the data for each of the eight model versions. A bootstrapping procedure was used, in which behavioral trials were resampled (with replacement) and each model was fit to each resample. Error bars indicate the 95**

**D**^{th}percentiles of the model likelihoods using the best-fit parameters for all resamples. The number of parameters is equal across all models, allowing direct comparison of likelihoods. (

**) The permutation test used to assess statistical significance. Black line indicates the distribution of the differences of likelihoods between the two models with the largest likelihood, scalar variability, dual accumulator static version (**

**E****b**) and linear variance, dual accumulator, static version (

**d**), across all resamples. Gray line indicates the distribution under the null hypothesis (see Materials and methods: Model comparison). Model b has a significantly larger likelihood than model d and all other models (p<<.001; see Materials and methods: Model comparison).

**DOI:**http://dx.doi.org/10.7554/eLife.11308.019

**b**) and the seven other model versions, described in , for each rat. Error bars indicate the 95

^{th}percentiles of the model likelihoods using the best-fit parameters for all resamples. The number of parameters were equal across all models, allowing direct comparison of likelihoods. A bootstrapping procedure was used, in which behavioral trials were resampled (with replacement) and each model was fit to each resample. Permutation test revealed that in 6/7 rats the scalar variability, dual accumulator static sampling model (model b) had a significantly greater likelihood than all other models (a,c-h) (p<0.001). For rat S142 model b was not significantly greater than other models (p>0.001) except model (

**f**) (p<0.001). Note that rat S142 may not have performed enough trials (n<2000) to accurately fit the model.

**DOI:**http://dx.doi.org/10.7554/eLife.11308.020

**DOI:**http://dx.doi.org/10.7554/eLife.11308.021

**) Distribution of flash durations. In a subset of experiments, flash durations were drawn from a Gaussian distribution with a mean of 10 ms. As a consequence, on some trials in which the same number of flashes were presented to both sides (#R=#L), there was a greater overall flash duration on one side (up to 50 ms) (**

**A****) Schematic of the timing and duration of left (blue) and right (red) flashes on an example trial in which #R=#L, but there was greater overall flash duration from the right LED. (**

**B****) Percent of trials on which the animal went right as a function of difference in right-left LED duration (specifically (durR-durL)/(durR+durL)) across all combinations of left and right flash number. Because LED duration is correlated with number of flashes on most trials, this looks very similar to performance as a function of flash number. Yellow line is the regression line to the data. (**

**C****) Percent of trials on which the animal went right as a function of difference in flash number ((#R-#L)/(#R+#L)). Green line is the regression line to the data. (**

**D****) Behavioral choice as a function of the difference in the overall flash duration ((durR-durL)/(durR+durL)) on trials with equal numbers of flashes on both sides (#R=#L). Black circles indicate the percent of trials in which animals went right on trials of different overall flash durations. Error bars indicate 95% confidence intervals for the mean assuming a binomial distribution. Solid gray line is the regression line to the data, whose slope is displayed in panel F. (**

**E****) Bar plots indicate the slope of the regression lines in panels C and D for trials with equal numbers of flashes. Error bars are standard error of the coefficient (slope) estimates. These represent different models of what the slope of the regression line to the data would be if the animal were integrating flash duration (‘Duration model’, yellow) or flash number (‘Flash model’, green). The slope of the regression line fit to the data (‘Data’) is significantly different from the line relating the LED duration to choice, but not significantly different from zero. This suggests that rats integrate flash number.**

**F****DOI:**http://dx.doi.org/10.7554/eLife.11308.022