Fig 1.
Ca2+ imaging from hippocampal neurons during linear-track exploration yields temporally sparse neuronal activity.
(A) Ca2+ imaging in the hippocampus of freely behaving mice as they run back and forth on a linear track. (B) Ten example contours of detected cells overlaid on the projection of all the detected cells in a given imaging session. (C) The fluorescence traces (colors) extracted for the example cells shown in B and their estimated spike trains (black). (D-F) The distribution of the average firing rates across all positions (D), of the firing rates in the most active position (E), and of the total number of active time bins in a given session (F) for hippocampal place cells. Distributions in D-F are for 44,981 place cells during running periods, pooled from N = 9 mice (1,717–9,906 cells per mouse) recorded over 8 sessions in each of two different environments per mouse. We focused only on sufficiently active cells (≥ 5 active time bins in a given session). Cells were analyzed independently, without tracking them across sessions.
Fig 2.
Limited sample sizes and temporally sparse neuronal activity positively bias the naïve calculation of information content.
(A) Top: Neuronal activity (red) of four example cells overlaid on the mouse trajectory (cyan) during rightward running epochs in the linear track. Inset, the full mouse trajectory during 200 seconds in the linear track, with the rightward running epochs marked in cyan. Middle: Tuning curves of the same example cells, and their naïve SI expressed in bit/spike. Bottom: The naïve SI (blue) and distribution of shuffle SI (black) of the same cells. Note that the naïve SI of a cell with significant place tuning and high firing rates (cell 2) is lower than that of a cell without significant place tuning and with low firing rates (cell 4). (B) Naïve SI (mean ± SD) as a function of the sample duration for real (blue) and shuffled (black) data. Inset, entropy of the mouse position (mean ± SD) as a function of sample duration. (C-D) The naïve SI (mean ± SD) as a function of the average firing rate (C) or number of active time bins (D) for real (blue) and shuffled (black) data. Data in B-D were averaged across N = 9 mice. For each mouse, SI was averaged across the last four imaging sessions in each of the two environments when they are familiar (a total of 740–4,003 place cells per mouse).
Fig 3.
Correcting the upward bias in naïve SI using the SSR method.
Validation based on data from simulated place cells with similar tuning properties to those observed in the real hippocampal data, corresponding to 20 minutes of free exploration in the linear track. (A) Demonstration of the shuffle reduction (SR) method. Estimated SI as a function of sample duration for the naïve calculation (blue), shuffle (black), SR (red), and the true SI (cyan). (B) Demonstration of the scaled shuffle reduction (SSR) method. Estimated SI as a function of sample duration for the naïve calculation (blue), shuffle (black), SSR (magenta), and the true SI (cyan). The SSR estimation is obtained by assuming a fixed ratio between the bias in the naïve SI (biasn, blue) and the bias in the shuffle SI (biass, black) for two different sample durations, and using this bias ratio to subtract from the naïve SI a scaled version of the shuffle SI. Data in A-B show the mean across 100 cells from one example simulation. (C) Estimated SI (mean ± SEM) as a function of sample duration for the naïve calculation (blue), SR (red), SSR (magenta), and the true SI (cyan). (D) Correction instability (mean ± SEM), defined as the standard deviation of the estimated SI across sample durations, for the naïve calculation, SR, and SSR. SSR is more stable than the naïve calculation (matched-pairs two-sided t-test(8), t = 51.5, p = 2.2·10−11) and SR (matched-pairs two-sided t-test(8), t = 54.5, p = 1.4·10−11). Data in C-D were averaged across N = 9 simulations. Each simulation corresponds to behavioral data from a different mouse and consists of 100 simulated place cells. ***p < 0.001.
Fig 4.
Correcting the upward bias in the naïve SI using the BAE method.
(A) Demonstration of the asymptotic extrapolation (AE) method. Using an unbounded function of the form a+b/t+c/t2 (solid orange curve) to fit the naïve SI as a function of sample duration (blue) yields an inaccurate estimation (dashed orange curve) of the true SI (cyan). (B) Demonstration of the bounded asymptotic extrapolation (BAE) method. Using a bounded function of the form a+b/(1+ct) (solid green curve) to fit the naïve SI as a function of sample duration (blue) yields an accurate estimation (dashed green curve) of the true SI (cyan). The extrapolation of the fitted curve to an infinite sample duration is the estimated SI. Data in A-B show the mean across 100 cells from one example simulation. (C) Estimated SI (mean ± SEM) as a function of the sample duration for the naïve calculation (blue), AE (orange), BAE (green), and the true SI (cyan). (D) Absolute bias in the estimated SI (mean ± SEM) as a function of sample duration for the naïve calculation (blue), SR (red), SSR (magenta), AE (orange) and BAE (green). Data in C-D were averaged across N = 9 simulations. Each simulation corresponds to behavioral data from a different mouse and consists of 100 simulated place cells. ***p < 0.001.
Fig 5.
Validating the accuracy of the different bias-correction methods in estimating the true SI of individual simulated neurons.
(A) The naïve SI versus the true SI for simulated place cells with similar tuning properties to those observed in the hippocampal data. (B-C) The estimated SI of individual simulated neurons versus their true SI for the SR (B) and SSR (C) methods. (D) Distribution of the estimation errors of individual neurons for the naïve calculation (blue), SR (red), SSR (magenta), and the deviations between the naïve SI of each cell across different realizations (gray). (E-G) Same as in B-D but for AE (orange) and BAE (green). (H) Estimated SI of the same simulated neurons for the BAE method versus the SSR method. Inset, discrepancy between the estimated SI using SSR versus BAE. (I-J) Absolute estimation bias (mean ± SEM) as a function of the cells’ average firing rates (I) or number of active time bins (J) for SR (red), SSR (magenta), BAE (green) and AE (orange). Data were averaged across N = 9 simulations. (K-L) Differences in the absolute estimation bias between SR and SSR (K) and between AE and BAE (L) as a function of both the cells’ average firing rates and the sample duration. These results show that SSR and BAE yield a smaller bias than SR and AE, especially when the firing rates are low or the sample durations are short, indicating they are more robust. Data pooled from N = 9 simulations. Each simulation corresponds to behavioral data from a different mouse and consists of 100 simulated place cells.
Fig 6.
Validating the bias-correction methods on real data by tracking the same neurons across multiple sessions.
(A) Naïve SI as a function of the sampling duration for a 2-hour-long concatenated session, for real (blue) and shuffled (black) data. Inset, the same for simulated data, with the true SI shown in cyan. (B) Estimated SI as a function of sample duration from a 20-minute-long subsample for the naïve calculation (blue), shuffle (black), SR (red), SSR (magenta), and the full 2-hour-long concatenated session SI (cyan). (C) Estimated SI from a 20-minute-long subsample for the naïve calculation (blue), AE (orange), BAE (green), and the full 2-hour-long concatenated session (cyan). Data in A-C show the mean across the recorded cells from one example mouse. (D) Bias-corrected SI for individual neurons using SR (red) and SSR (magenta) on 20-minute subsamples versus the naïve SI of the full concatenated 2-hour-long session. (E) Bias-corrected SI for individual neurons using AE (orange) and BAE (green) on 20-minute subsamples versus the naïve SI of the full concatenated 2-hour-long session. (F) Estimated SI of the same neurons for the BAE method versus the SSR method. Inset, discrepancy between the estimated SI using SSR versus BAE. Data in D-F are for 947 place cells that were pooled from N = 9 mice (25–254 cells per mouse), tracked across 6 imaging sessions in the same environment and found to be active in at least 5 of those sessions.
Fig 7.
Bias-correction methods are applicable to quantifying information in recordings of neuronal activity from the visual cortex.
(A) Ca2+ imaging in the primary visual cortex (V1) of mice during the repeated presentation of drifting gratings. Data taken from the Allen Brain Observatory (see Methods) [42]. (B) Quantification of information carried about the direction of movement and temporal frequency of drifting gratings. Estimated SI as a function of the number of trials for the naïve calculation (blue), shuffle (black), SR (red), and SSR (magenta). (C) Estimated SI for the naïve calculation (blue), AE (orange), and BAE (green). Data in B-C show the mean across recorded cells from one example mouse. (D) Estimated SI (mean ± SEM) as a function of the number of trials obtained by the naïve calculation (blue), SR (red), SSR (magenta), AE (orange), and BAE (green). (E) The correction instability (mean ± SEM), defined as the standard deviation of the estimated SI across numbers of trials, for each method. SSR is more stable than SR (matched-pairs two-sided t-test(93), t = 33.8, p = 5.4·10−54) and BAE is more stable than AE (matched-pairs two-sided t-test(93), t = 25.3, p = 1.7·10−43). Data in D-E were averaged across N = 94 mice. (F) Estimated SI for the same individual neurons (10,225 cells pooled from 94 mice) for the BAE method versus the SSR method. Inset, discrepancy between the estimated SI using SSR versus BAE. ***p < 0.001.
Fig 8.
Bias-correction methods allow the estimation of spatial information content independently of the specific choice of spatial binning.
(A) Estimated SI (mean ± SEM) as a function of the spatial bin size for the naïve calculation (blue) and shuffle (black). (B) The naïve SI (mean ± SEM) as a function of the sample duration for different numbers of spatial bins. (C) Estimated SI (mean ± SEM) as a function of the number of spatial bins (log-scale) for the naïve calculation (blue), SSR (magenta), and BAE (green). While for the naïve calculation, the SI increases with the number of spatial bins, the SSR and BAE methods reach a stable estimation for a sufficient number of bins. (D) Estimated SI (mean) as a function of the bin size for the naïve calculation (blue), SSR (magenta), and BAE (green). The full-resolution SI (independent of bin size) was estimated using a linear extrapolation to bin size = 0 (indicated by the black arrow). Inset, the same analysis for different sample durations. Data were averaged across N = 9 mice. For each mouse, SI was averaged across the last four imaging sessions in each of the two environments when they were familiar.
Fig 9.
A bias-free estimation of spatial information uncovers hippocampal neuronal tuning properties that may be masked by the bias in the naïve estimation.
(A) The estimated SI (mean ± SEM) as a function of the within-field peak firing rates for the naïve calculation (blue), SSR (magenta), and BAE (green). SI was not found to increase with peak rate based on the naïve estimation (linear regression(10), R2 = 0.001, slope = -0.001 ± 0.015, p = 0.92), but did increase with peak rate based on SSR (linear regression(10), R2 = 0.95, slope = 0.07 ± 0.01, p = 1.8·10−6) and BAE (linear regression(10), R2 = 0.90, slope = 0.08 ± 0.02, p = 2.8·10−5). (B) Average number of track traversals (in each running direction; mean ± SEM) as a function of time in the experiment. Day one is the first exposure to the environment. (C) The estimated SI (mean ± SEM) as a function of time in the experiment for the naïve calculation (blue), SSR (magenta), and BAE (green). SI changed throughout learning for both the naïve estimation (repeated-measures ANOVA(7), F = 7.42, p = 2.7·10−6), and the bias-corrected estimations (repeated-measures ANOVA(7), F = 10.32, p = 3.1·10−8, and F = 10.10, p = 4.2·10−8 for SSR and BAE, respectively). Post hoc pairwise comparison tests revealed that the bias-corrected estimation, but not the naïve, significantly increased with learning. (D-F) Estimated SI (mean) as a function of the sample duration in both hippocampal subfields CA1 and CA3 (D), CA1 only (E), and CA3 only (F) during different learning sessions (colors) for the naïve calculation (dashed curves), and SSR (solid curves). Insets, the SI (mean ± SEM) as a function of time in the experiment was estimated from short subsamples of the data (3 minutes in D and 5 minutes in E-F) using the naïve calculation (left) and SSR (right). Note that even for short sample durations, the bias-corrected estimation clearly reveals an increase in information over time, while the naïve estimation fails to uncover this trend. Data were averaged across N = 9 mice. Data in B-F for each familiarity level (day in the experiment) were averaged across the two environments. For the analyses presented in panels C-F, only place cells with ≥ 10 active time bins in a given session were used. *p < 0.05; ***p < 0.001.