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Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory

Abstract

Prefrontal persistent activity during the delay of spatial working memory tasks is thought to maintain spatial location in memory. A 'bump attractor' computational model can account for this physiology and its relationship to behavior. However, direct experimental evidence linking parameters of prefrontal firing to the memory report in individual trials is lacking, and, to date, no demonstration exists that bump attractor dynamics underlies spatial working memory. We analyzed monkey data and found model-derived predictive relationships between the variability of prefrontal activity in the delay and the fine details of recalled spatial location, as evident in trial-to-trial imprecise oculomotor responses. Our results support a diffusing bump representation for spatial working memory instantiated in persistent prefrontal activity. These findings reinforce persistent activity as a basis for spatial working memory, provide evidence for a continuous prefrontal representation of memorized space and offer experimental support for bump attractor dynamics mediating cognitive tasks in the cortex.

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Figure 1: Behavioral and neural fingerprints of spatial working memory.
Figure 2: Bump attractor dynamics during the delay can explain behavioral inaccuracies.
Figure 3: Tuning curves computed from clockwise and counterclockwise behavioral trials show model-predicted shift in the delay period.
Figure 4: Responses to flank stimuli (1 and 2 locations from preferred) become correlated with upcoming behavior during the delay.
Figure 5: As predicted by the bump attractor model, neuronal variability correlates with upcoming behavioral responses in a cue-dependent way: maximally for flank stimuli and earlier in the delay for cues closer to the neuron's preferred location.
Figure 6: Fano factors follow the predictions of the bump attractor model.
Figure 7: Noise correlations between pairs of neurons depend on the stimulus as predicted by the bump attractor model.
Figure 8: Comparison of three alternative memory representations in three mechanistic models: a bump attractor model maintained by continuous topographic recurrent excitation (left), a discrete attractor model with eight populations (middle) and a non-attractor decaying bump model sustained by a slow intrinsic current (right).

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Acknowledgements

We thank M. Goldman, J. de la Rocha and A. Roxin for fruitful discussions. We wish to acknowledge the late P. Goldman-Rakic, in whose laboratory experimental data were collected. This work was funded by the Ministry of Economy and Competitiveness of Spain, the European Regional Development Fund (Ref: BFU2009-09537) and by the National Science Foundation (NSF PHY11-25915 and DMS-0847749). K.W. acknowledges funding from the German Research Foundation (research fellowship Wi 3767/1-1). D.Q.N. was supported by the Generalitat de Catalunya (PIV-DGR 2010 program). C.C. received support from the Tab Williams Family Endowment. The work was carried out at the Esther Koplowitz Centre and partly at the Kavli Institute for Theoretical Physics, and at the Centre de Recerca Matemàtica.

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Authors

Contributions

A.C. conceived and designed the study. C.C. collected the experimental data. K.W. and A.C. analyzed the data. D.Q.N. and A.C. implemented the computational models. A.C., C.C., D.Q.N. and K.W. wrote the manuscript.

Corresponding author

Correspondence to Albert Compte.

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The authors declare no competing financial interests.

Integrated supplementary information

Supplementary Figure 1 Schematic illustration of the first prediction from the bump attractor model regarding the relation between single-neuron delay tuning curves and the direction of small behavioral imprecisions.

(a) Scheme of 2 possible delay population activity profiles (red and blue lines) in response to the repeated presentation of a 90° cue. Repeated presentation of any given cue results in variable behavior, with saccades directed CW (red) and CCW (blue) from the correct cue location. Neural responses are marked for one single neuron in the network (circles, preferred cue 144°). (b) Same as a for a different cue, presented at 180°. Again two network responses for two different trials are plotted, which give rise to CW (red) and CCW (blue) behavioral imprecisions. Responses from the same neuron with preference at 144° are indicated by filled circles. (c) Tuning curves computed for the single neuron indicated in a and b, separately recorded from trials with CCW and CW behavior. The tuning curve is constructed by plotting the neuron's response to different stimuli locations (x-axis). Here panels a and b contribute two points to each tuning curve in panel c, the rest of the points are obtained by presenting all other possible cues to the network (not shown). Notice that CW tuning curves are displaced CCW relative to CCW tuning curves: the tuning bias defined as the distance from the CCW to the CW tuning curve centers is predicted to be positive.

Supplementary Figure 2 Comparison of delay tuning properties of the databases of tuned and non-tuned neurons.

(a) Mean tuning for tuned neurons (solid) is higher than for non-tuned neurons (dashed). Note that the alignment to preferred locations induces a spurious tuning by itself. (b) Histograms of tuning strength T for tuned (black) and non-tuned neurons (gray) differ significantly in their medians (triangles, 0.0832 vs. 0.1757, P < 1e–6, Wilcoxon test) but have a significant overlap. (c) Neurons with similar tuning strength T in the two databases (tuned, black and non-tuned, gray) have lower mean firing rate for non-tuned neurons, thus explaining non-significant tuning in cells with high tuning strength T in the non-tuned neuron database. Low, medium and high tuning was defined using tertiles obtained from the tuned neuron database and splitting the non-tuned database using the same intervals (n = 353, 111 and 59 non-tuned neurons for low, medium and high tuning, respectively).

Supplementary Figure 3 Late-delay Fano factor difference between accurate and inaccurate trials for flank cues increases for more extreme behavioral deviations.

As the fraction of trials used to build accurate and inaccurate response classes is reduced (i.e. the two classes differ more in their mean response accuracy) the difference between Fano Factors computed from neural responses at the end of the delay (last 500 ms) for flank cue stimuli in each of these classes of trials grows parametrically. Significance was assessed with a permutation test (P < 0.05).

Supplementary Figure 4 Validation of ANOVA models.

Graphical representation of the ANOVA residuals corresponding to the analyses in Fig. 6a-d and Fig. 7e-h. Data deviated mildly from the normality assumption due to long tails (panels a, e) and some outlier points (panel e), and from the homoscedasticity assumption in panels b, d, g, h (Levene's test P < 0.05). Note that despite this significant heteroscedasticity, there is not more than a factor 3 between the various dispersions. Heteroscedasticity is the most worrisome aspect of the data, and it is particularly severe in the case of unequal sample sizes. In our case the sample size corresponding to the different monkeys (d) and the different neuronal pair tunings (h) were different. However, the cases with the smaller sample size had smaller variance and this condition is known to reduce the power of the test rather than inflate the false positive rate. Thus, for both ANOVA tests, non-normality and heteroscedasticity do not question the rejection of the hypotheses, since the corresponding corrections are unlikely to cancel the strong significance of the effects found (P = 0.0013 for the interaction cue × accuracy in Fig. 6 and panels a-d, and P < 0.001 for main effect of selectivity in Fig. 7 and panels e-h).

Supplementary Figure 5 Coding models that match experimental data quantitatively.

The models in Fig 8 were in qualitative agreement with experimental behavioral and neural data, but the quantitative match of neuronal firing rates was limited by the network implementation. To evaluate quantitative predictions that took into account the specific firing rates and heterogeneity present in the data (Fig. 1), we tested two coding models for inaccurate memory-guided behavioral responses: the bump attractor model and the decaying bump model. (a) In the bump attractor model, dispersion of behavioral responses is due to noise-induced diffusion of the location of a rigid bump representation. In different trials (red and blue) in response to the same cue the bump diffuses during the delay to different locations (lower panel), giving rise to different read-outs and inaccurate, off-target behavioral responses. (b) In the decaying bump model, the bump is established at cue presentation, but begins to decay once the cue is removed. During the delay, the coding slowly decays away, with different decay rates in different trials. The read-out is inaccurate at the end of the delay due to the decreased selectivity. (c-d) Generation of multiple firing rate trials (8 equidistant cues, 10 trials per cue) and surrogate Poisson spike trains for n = 200 neurons from the models in corresponding panels a and b above generates behavioral and neural data in quantitative agreement with the experimental data (Fig. 1). Behavioral responses from computational models (panel c, left, and panel d, right) are represented as in Fig. 1b. The memory dependence of these behavioral inaccuracies is revealed by comparing with the models' behavioral responses for a brief 0.5 s delay (black dashed curve). Average tuning curves of model neurons (panel c, right and panel d, left) in the delay period computed as in Fig. 1d.

Supplementary Figure 6 Model-derived surrogate data supports a bump attractor representation, not a decaying bump representation.

(a,b) Tuning curve bias analysis (Fig. 3c) performed on surrogate data from the diffusing and decaying bump models (Supplementary Fig. 5), respectively. For the bump attractor, but not the decaying bump model, the tuning curve bias computed from trials with CW versus CCW behaviors (Fig. 3) becomes increasingly positive through the delay. (c,d) Rate-behavior correlation analysis (Fig. 4c) applied to the surrogate data of Supplementary Fig. 5. Only for the bump attractor model spike counts in response to flank stimuli correlated with behavioral deviations in the direction of the neuron's preferred cue as delay progressed. (e,f) For surrogate data generated from the bump attractor model, but not the decaying bump model, Fano Factors computed separately for trials when a flank stimulus was presented are larger when considering inaccurate trials with large behavioral deviations (solid line) as compared to accurate trials (dashed line) (c.f., Fig. 6d). This is shown for average activity in the last 500 ms of delay period, when considering cues presented at different points of the neuronal tuning curve. (g,h) Only surrogate data derived from the bump attractor model mimics the experimental results (Fig. 7e,f) that neuron pair noise correlation is negative for those pairs with dissimilar tuning, when responding to a middle flank stimulus (In-flank condition). Solid lines correspond to labels “Peak,” “Flanks,” and “Tails” as in Fig. 7e. Dashed lines correspond to labels “In-flank,” “Peaks,” and “Out-flanks” as in Fig. 7f.

Supplementary information

Supplementary Text and Figures

Supplementary Figures 1–6 (PDF 652 kb)

Supplementary Code 1

Matlab code to run the bump attractor network model of Fig. 8. The model is described in Online Methods generically, and this Matlab code specifies all the parameters necessary to run the simulation. (TXT 4 kb)

Supplementary Code 2

Matlab code to run the discrete attractor network model of Fig. 8. The model is described in Online Methods generically, and this Matlab code specifies all the parameters necessary to run the simulation. (TXT 4 kb)

Supplementary Code 3

Matlab code to run the decaying bump network model of Fig. 8. The model is described in Online Methods generically, and this Matlab code specifies all the parameters necessary to run the simulation. (TXT 4 kb)

Video demonstrating network activity and behavioral read-out in the course of one simulation of the bump attractor network model of Fig. 8.

The upper left panel shows the temporal progress along the task timeline (F=fixation, C=cue, D=delay, R=response) with a red time-stamp marker. The upper right panel shows in black the simulated visual scene: fixating cross and square visual cue, together with the instantaneous network readout during the delay in red. The lower panel shows the activity of excitatory neurons in the network. The 8 possible locations of cue stimuli are indicated on the lower panel with gray dots. When external stimuli are applied to the network, these are indicated with a thick black line on the location of corresponding neurons on the x-axis. This occurs in the cue period of the trial (C in upper panel), and is implemented as a depolarizing current to excitatory neurons around the cue location at θ= 0°), and in the response period (R in upper panel), where a hyperpolarizing current is injected to all excitatory neurons in the network. The location encoded instantaneously in the excitatory population activity is read out continuously with a population vector algorithm (Online Methods and Supplementary Code 1) and it is indicated with a black triangle in the lower panel and with a red radial line in the upper right panel. Parameters for this simulation are as detailed in Supplementary Code 1. (MOV 6610 kb)

Video demonstrating network activity and behavioral read-out in the course of one simulation of the discrete attractor network model of Fig. 8.

Same layout and symbols as for Supplementary Video 1. Parameters for this simulation are as detailed in Supplementary Code 2. (MOV 7248 kb)

Video demonstrating network activity and behavioral read-out in the course of one simulation of the decaying bump network model of Fig. 8.

Same layout and symbols as for Supplementary Video 1. Parameters for this simulation are as detailed in Supplementary Code 3. (MOV 7060 kb)

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Wimmer, K., Nykamp, D., Constantinidis, C. et al. Bump attractor dynamics in prefrontal cortex explains behavioral precision in spatial working memory. Nat Neurosci 17, 431–439 (2014). https://doi.org/10.1038/nn.3645

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