Noise in neural populations accounts for errors in working memory

Abstract

Errors in short-term memory increase with the quantity of information stored, limiting the complexity of cognition and behavior. In visual memory, attempts to account for errors in terms of allocation of a limited pool of working memory resources have met with some success, but the biological basis for this cognitive architecture is unclear. An alternative perspective attributes recall errors to noise in tuned populations of neurons that encode stimulus features in spiking activity. I show that errors associated with decreasing signal strength in probabilistically spiking neurons reproduce the pattern of failures in human recall under increasing memory load. In particular, deviations from the normal distribution that are characteristic of working memory errors and have been attributed previously to guesses or variability in precision are shown to arise as a natural consequence of decoding populations of tuned neurons. Observers possess fine control over memory representations and prioritize accurate storage of behaviorally relevant information, at a cost to lower priority stimuli. I show that changing the input drive to neurons encoding a prioritized stimulus biases population activity in a manner that reproduces this empirical tradeoff in memory precision. In a task in which predictive cues indicate stimuli most probable for test, human observers use the cues in an optimal manner to maximize performance, within the constraints imposed by neural noise.

Keywords: Poisson noise; divisive normalization; neural gain; population coding; resource; short term memory.

Figure 1.

The population coding model. a, Stimulus orientations were encoded in the activity of idealized neurons with preferred orientations evenly distributed on the circle and bell-shaped tuning functions (width ω). Each of N stimuli was encoded by an independent subpopulation of M neurons. Divisive normalization operated across the whole population, scaling population activity to a level determined by the gain constant, γ. (Note that the 180° range of orientations is represented here by the circular parameter space −π to π). b, Each neuron generated spikes according to a Poisson process, with mean firing rate determined by the normalized input of a neuron. Subsequent recall was modeled as ML decoding of the spiking activity of the subpopulation of neurons corresponding to a probed stimulus over a fixed time window. c, Simulations showed that error in the recalled orientation depended on gain (summed output) of the decoded subpopulation, which declined with increasing N as a result of divisive normalization. At high gains, errors had an approximately normal distribution (e.g., blue curves). As gain decreased (magenta to red curves), variability increased and error distributions deviated from normality. Note that error distributions shown are normalized by peak probability to better illustrate distribution shape. d, Variance of simulated errors under the population coding model as a function of gain and tuning width. The lowest variances (blue) were obtained with high gains and narrow tuning functions. The range of variances typically observed in human recall corresponds approximately to the yellow band. e, Kurtosis of errors. Circular kurtosis approached 0 at the highest gains, indicating that errors had an approximately circular normal distribution. Kurtosis also approached 0 at the lowest gains, as errors approached a uniform distribution on the circle. Positive kurtosis (hot colors) was observed at intermediate gains for all tuning widths, indicating deviations from circular normality. f, Exponent of a power law relating gain to error variance (estimated from the change of variance resulting from halving the gain, equivalent to doubling the number of items in memory). At high gains, the exponent approached −1, indicating variance was inversely proportional to gain (Seung and Sompolinsky, 1993). However, at intermediate gains, the exponent became more strongly negative (less than −1; green and blue regions) for all tuning widths.

Figure 2.

Human recall errors and model fits. a, Black symbols show distribution of recall errors made by a representative human observer for displays of one to eight orientation stimuli. Red curves show error distributions generated by the population coding model for ML parameters γ and ω. Note that the model reproduced the changes in error distribution with array size (left to right) despite model parameters remaining fixed. b, ML values of population gain γ and tuning width ω for eight subjects. Open circle corresponds to subject shown in a. c, Black symbols show mean distribution of recall errors for the group (error bars indicate ±1 SE). Red curves show mean error distributions for the population coding model with ML parameters. d, Variance of error in recalled orientation for human observers (black symbols) and for the population coding model with ML parameters (red curve, dashed lines indicate ±1 SE). Variance has an approximately power-law relationship with set size (appearing linear on the log–log plot). e, Deviation from the circular normal distribution. Black symbols plot mean discrepancy between subject error frequencies shown in c and circular normal (Von Mises) distributions matched in variance. Red curves plot equivalent deviations for the population coding model with ML parameters. f, Kurtosis of recall errors for subjects (black symbols) and the population coding model (red curve, dashed lines indicate ±1 SE). The circular normal distribution has kurtosis around 0.

Figure 3.

Recall with an informative cue. a, Black symbols show mean distribution of recall errors for displays of two to eight orientations in Experiment 2, in which one stimulus, indicated by a cue, was more likely to be selected for test. Overall error variability increased with array size (top, left to right). When separating trials according to cue validity, recall was found to be consistently less variable for the cued item (middle) than for uncued items (bottom). In the population coding model, this corresponds to an increased weighting of activity related to the cued stimulus. Colored curves show mean error distributions generated by the model with ML parameters. Note that only the weighting of activity to stimuli differed across array sizes (left to right), with total population activity and tuning width remaining constant for each subject. b, Variance of error for cued items (red) and uncued items (green), for human observers (symbols) and the population coding model with ML parameters (solid lines; dashed lines indicate ±1 SE). Differences in variance between cued and uncued items are accounted for in the model by differential weighting of activity. c, Kurtosis of errors for cued and uncued items. d, Comparison of weighting factors estimated from subject errors on the cued task (Experiment 2, white bars) and optimal weightings that would be expected to minimize total error variance (blue bars, based on ML parameters γ and ω obtained from the uncued task, Experiment 1). Optimal behavior predicts that weighting of activity in favor of the cued item should increase with each increase in display size (blue bars, left to right), and this predicted pattern was observed in the weighting factors estimated from subject errors (white bars).

Figure 4.

Effects of variations in tuning and noise correlations on decoding errors at low gain. a, Simulated error distributions (middle, normalized by peak probability) and deviations of errors from the circular normal distribution (bottom) based on decoding a homogeneous population of neurons with broad tuning (top; ω = 0.5). b, Simulation results as in a but with narrower tuning curves (top; ω = 0.2). Note that decoding precision is increased compared with a more broadly tuned population with the same gain (compare curves of same color in a and b). c, As in b but with the addition of a baseline level of activity (illustrated top; f₍₀₎ = 0.25). For a specified gain, the presence of baseline activity decreases decoding precision. However, the pattern of deviations from normality is unaffected (bottom; see also Fig. 5). d, Decoding errors obtained by simulation of neurons with randomly distributed preferred orientations and heterogeneous tuning curves (examples shown at the top; amplitude, a = 1 ± 0.5; width, ω = 0.2 ± 0.1; baseline, f₍₀₎ = 0.25 ± 0.125). Heterogeneity had minimal impact on error distributions or deviations from normality, except to add noise at very low gains. e, As in d but with cosine instead of bell-shaped tuning curves. f, Decoding errors obtained from simulated populations with short-range pairwise correlations in spiking activity (illustrated top; c₀ = 0.25, other parameters as in b). For a specified gain, decoding precision was decreased compared with uncorrelated activity. Unlike the other manipulations shown here (a–e), the impact of short-range correlations depended strongly on population size (solid lines, M = 100 neurons; dashed lines, M = 1000 neurons). However, the introduction of correlations had minimal impact on the pattern of deviations from normality (bottom).

Figure 5.

Effects of baseline activity on ML parameters and decoding errors. a, Population gain (ML values for subjects in Experiment 1) as a function of baseline activation of the neural population (shaded area indicates ±1 SE). Note that the population gain that best fits the data rises steeply as the proportion of activity attributable to baseline increases. b, ML values of the tuning width ω as a function of baseline activation. c, Error distributions corresponding to ML parameters of the model with different baseline activation levels (1–8 items). It is not possible to reliably distinguish on the basis of error distributions between decoding of a population with low activity of which little or none is attributable to baseline (e.g., black, blue curves) and a population with substantially higher activity of which a much larger proportion is attributable to baseline (e.g., green, red curves). d, SNR per neuron as a function of baseline activation, calculated from ML parameters. SNR shown is for a population encoding a single stimulus. The SNR corresponding to observed error distributions is approximately independent of the baseline activity level.

Figure 6.

Comparison with resource-based models of working memory errors. a, Errors under the slots + averaging model are drawn from a finite mixture of normal distributions of different widths, corresponding to different numbers of slots allocated to the probed item, and plotted here as a cumulative probability over precision (based on ML parameters for representative subject; Fig. 2a). Different set sizes are indicated by different colors (overlying lines are slightly shifted horizontally for ease of viewing). Note that, for large set sizes, there is a non-zero probability that an item receives zero slots, in which case errors are drawn from a distribution with zero precision, i.e., the uniform distribution. b, Error distributions corresponding to the slots + averaging model parameters illustrated in a. c, Variance of error in recalled orientation for human observers (black symbols) and for the slots + averaging model with ML parameters (red curve, dashed lines indicate ±1 SE). d, Kurtosis of recall errors for subjects (black symbols) and the slots + averaging model (red curve). e–h, The variable precision model. Errors are drawn from an infinite mixture of normal distributions of different widths, shown here as a cumulative probability over precision (e). Corresponding error distributions are shown in f; variance and kurtosis corresponding to ML parameters in g and h. i–l, In the population coding model, the precision of recall is correlated with the total number of spikes contributing to the estimate. The cumulative probability and corresponding precision of different spike counts is shown in i, for the model with zero baseline activity. Error distributions are shown in j. Note that, for large set sizes, there is a non-zero probability of zero spikes occurring within the decoding window, resulting in an estimate with zero precision, i.e., uniformly distributed. The error distributions (k) corresponding to a particular precision in i are approximately normally distributed: deviations from circular normal are plotted in l. m–p, Corresponding results for the population coding model incorporating baseline activity (example shown, 50% of peak). Spike counts corresponding to a specified precision (m) are substantially higher than in the zero baseline model, and there is negligible probability of zero spikes; errors corresponding to a specified spike count (o) deviate strongly from normal (p); the resulting error distributions (n) are indistinguishable from the zero baseline case (j).