Information Theoretic Approaches to Deciphering the Neural Code with Functional Fluorescence Imaging

Abstract

Information theoretic metrics have proven useful in quantifying the relationship between behaviorally relevant parameters and neuronal activity with relatively few assumptions. However, these metrics are typically applied to action potential (AP) recordings and were not designed for the slow timescales and variable amplitudes typical of functional fluorescence recordings (e.g., calcium imaging). The lack of research guidelines on how to apply and interpret these metrics with fluorescence traces means the neuroscience community has yet to realize the power of information theoretic metrics. Here, we used computational methods to create mock AP traces with known amounts of information. From these, we generated fluorescence traces and examined the ability of different information metrics to recover the known information values. We provide guidelines for how to use information metrics when applying them to functional fluorescence and demonstrate their appropriate application to GCaMP6f population recordings from mouse hippocampal neurons imaged during virtual navigation.

Keywords: behavior; calcium imaging; in vivo; information theory; place cells; two-photon microscopy.

Figure 1.

Procedures for generating a library of 10,000 neurons with known amounts of information. A, Five splines with a gradient of ground truth information ( $I_{AP}^E$) representing the steps in generating a continuous rate map ( $\lambda \left(x\right)$) matching the desired target information, in this case, 2 bits/AP. Red Xs indicate control nodes that were moved to change the shape of the spline and minimize the squared error to the target information. B, Cross-section of the error surface around the solution point as a function of the position of node 3, and the trajectory taken by the solver to minimize the error and arrive at the target. C, Histograms of ground truth information resulting from repeating the procedure in A, B 10,000 times to target a range of ground truth information values in bits per second ( $I_S^E$). D, Splines representing $\lambda \left(x\right)$) and bits per AP ( $I_{AP}^E$ at the solution point for a low ( $I_{AP}^E=0.04$ bits/AP, left) and high ( $I_{AP}^E$, 2 bits/AP, right) information neuron. E, Steps to generate mock AP and functional fluorescence data. (1) An example real behavior trace from a mouse running on a linear track that was used to generate the simulated spiking. (2) The behavior in combination with the rate maps generated in A–D were used to generate an instantaneous firing rate trace. (3) The instantaneous rate was used to pseudorandomly generate APs, as shown in this mock raster. (4) The AP raster was convolved with the GCaMP6f kernel (red, inset), and noise was added to generate a mock $\frac{\Delta F}{F}$ trace. (5) Large numbers of these traces were generated and used to assess the effects of many simulation parameters on the estimators. F–L, Spiking and fluorescence activity patterns generated from the example simulated neurons shown in D and using a mean firing rate of 1 Hz. F, Behavioral trace in blue with AP raster shown in red. G, Lap-by-lap raster of the neurons’ firing versus mouse track position. H, Lap by lap binned, firing rates versus mouse track position for the neurons. I, AP raster (red) and mock calcium traces for the same behavioral period shown in F. J, Lap by lap mean binned fluorescence versus mouse position for the neurons. K, Binned average firing rate ( ${\lambda }_i$, black) and fluorescence intensity ( $f_i$, green) maps for the two neurons. These maps were used for information analyses.

Figure 2.

Quantification of the precision of the SMGM bits per second metric using APs or functional fluorescence recordings. A, Three representative mock neurons spanning the range of ground truth information values in bits per second ( $I_S^E$). From top to bottom for each, Mouse track position versus time, AP raster, fluorescence calcium trace (green), and firing rate map ( ${\lambda }_i$, black) and change in fluorescence map ( $f_i$, green). B–D, The ground truth bits per second values are well recovered when measured from AP traces. B, Information measured from AP data using the SMGM bits per second metric ( $\widehat{I_s^E}$) versus ground truth information ( $I_s^E$). Each dot is a single mock neuron, the gray dashed line is the unity line (perfect measurement), the pink line is the line of best fit. Red circles show the examples in A. C, Percentage error for the information measurements shown in B. D, Heat map of percentage error measurements shown in C. Black lines are 2 SDs, the white line is the mean. E–G, Effects of applying the SMGM bits per second metric to fluorescence traces. E, Information measured from mock GCaMP6f traces using the SMGM bits per second metric ( $\widehat{I_s^F}$) versus ground truth information ( $I_s^E$). F, Percentage error for the information measurements shown in E. G, Heat map of percentage error measurements shown in F. H, Representative mock kernels mimicking responses from different indicators. I–K, The effect of kernel height on estimating ground truth information ( $I_s^E$) using the SMGM bits per second metric ( $\widehat{I_s^F}$). Kernel height for the kernels shown in H are indicated by colored triangles. I, Percentage error as a function of kernel height. J, Heat map of percentage error measurements shown in I with mean (white) and 2 SDs (black). K, The average percentage error as a function of kernel height and ground truth information in SMGM bits per second ( $I_s^E$). L–N, The effect of kernel width on estimating ground truth information ( $I_s^E$) using the SMGM bits per second metric ( $\widehat{I_s^F}$). Kernel widths for the kernels shown in H are indicated by colored triangles. L, Percentage error as a function of kernel width. M, Heat map of percentage error measurements shown in L with mean (white) and 2 SDs (black). N, The average percentage error as a function of kernel width. Recording density affected the metrics (Extended Data Figure 2-1). Changing the kernel to common indicators yielded qualitatively similar, but quantitatively different results (Extended Data Figure 2-2).

Figure 3.

Quantification of the precision of the SMGM bits per AP metric using APs or functional fluorescence recordings. A, Three representative mock neurons spanning the range of ground truth information values in bits per AP ( $I_{AP}^E$). From top to bottom for each, Mouse track position versus time, AP raster, fluorescence calcium trace (green), and firing rate map ( ${\lambda }_i$, black) and change in fluorescence map ( $f_i$, green). B–D, The ground truth bits per AP values are well recovered when measured from AP traces. B, Information measured from AP data using the SMGM bits per AP metric ( $\widehat{I_{AP}^E}$) versus ground truth information ( $I_{AP}^E$). Each dot is a single mock neuron, the gray dashed line is the unity line (perfect measurement). Red circles show the examples in A. C, Percentage error for the information measurements shown in B. D, Heat map of percentage error measurements shown in C. Black lines are 2 SDs, the white line is the mean. E–G, Effects of applying the SMGM bits per AP metric to fluorescence traces. E, Information measured from mock GCaMP6f traces using the SMGM bits per AP metric ( $\widehat{I_{AP}^F}$) versus ground truth information ( $I_{AP}^E$). F, Percentage error for the information measurements shown in E. G, Heat map of percentage error measurements shown in F. H–J, The effect of kernel height on estimating ground truth information ( $I_{AP}^E$) using the SMGM bits per second metric ( $\widehat{I_{AP}^F}$). Kernel height for the kernels shown in Figure 2H are indicated by colored triangles. H, Percentage error as a function of kernel height. I, Heat map of percentage error measurements shown in H with mean (white) and 2 SDs (black). J, The average percentage error as a function of kernel height and ground truth information in bits per AP ( $I_{AP}^E$). K–M, The effect of kernel width on estimating ground truth information ( $I_{AP}^E$) using the SMGM bits per AP metric ( $\widehat{I_{AP}^F}$). Kernel widths for the kernels shown in Figure 2H are indicated by colored triangles. L, Percentage error as a function of kernel width. M, Heat map of percentage error measurements shown in L with mean (white) and 2 SDs (black). N, The average percentage error as a function of kernel width. Changing the kernel to common indicators yielded qualitatively similar, but quantitatively different results (Extended Data Figure 3-1). These errors could not be resolved by changing the bin width (Extended Data Figure 3-2). Addition of a nonlinearity further distorted the measured information (Extended Data Figure 3-3).

Figure 4.

Alternative techniques for measuring MI from functional fluorescence traces. A–E, top, Information measured from mock GCaMP6f traces versus ground truth information. The gray line is the unity line, the pink line is the best fit saturating exponential. Middle, Percentage error for the information measurements shown on top. Bottom, Heat map of percentage error measurements shown in middle. A, FOOPSI deconvolved traces using the SMGM bits per second metric ( $\widehat{{\text{I}}_{\text{s}}^{\text{d}}}$). B, FOOPSI deconvolved traces using the SMGM bits per AP metric ( $\widehat{{\text{I}}_{\text{AP}}^{\text{d}}}$). The regularization coefficient had little effect on these results (Extended Data Figure 4-1). C, The KSG measure applied to GCaMP6f traces. D, The binned estimator applied to GCaMP6f traces using uniform bins. E, The binned estimator applied to GCaMP6f traces using equal occupancy bins. The binned estimators were less distorted on the raw AP traces (Extended Data Figure 4-2). F, Table of summary statistics for each measure. P exponential is the p value from the χ² test used to determine whether a saturating exponential fit is better than a linear fit for the measured versus ground truth information plots. An analytic solution yielded qualitatively similar, but quantitatively disparate results (Extended Data Figure 4-3).

Figure 5.

Application of SMGM information metrics to functional fluorescence imaging data from hippocampus during spatial behavior. A, Example field of hippocampal pyramidal neurons expressing GCaMP6f and imaged during linear track navigation. Active cell ROIs shown in yellow; traces for green cells shown in B. B, Fluorescence DF/F traces (green) from two neurons in the field shown in A and the track position during the recording (blue). C, Distribution of information values using the fluorescence SMGM bits per second metric ( $\widehat{I_s^F}$, top) and the fluorescence SMGM bits per AP metric ( $\widehat{I_{AP}^F}$, bottom). The gray line indicates the recommended cutoff for reliability using GCaMP6f. D, Plot of $\widehat{I_s^F}$ versus $\widehat{I_{AP}^F}$ for each neuron. Place cells indicated in red and nonplace cells in blue. E, Example non-place cells spanning the information ranges shown in C. Spatial fluorescence map ( $f_i$) shown on left, and average change in fluorescence per track traversal on right. F, Same as E, but for place cells. G, Bayesian decoding of mouse’s track position using different subpopulations of neurons for one example session. From top to bottom, All active neurons, all nonplace cells, and place cells, the first through third quantiles of the SMGM bits per second formulation ( $\widehat{I_s^F}$), and the first through third quantiles of the SMGM bits per AP formation ( $\widehat{I_{AP}^F}$). The white dashed line indicates the ground truth position of the animal, the color map indicates the decoded position probability (peak-normalized posterior distribution). H, Decoding accuracy (absolute position decoding error in units of % of track) pooled over all sessions for each neuron group indicated in G. Black bars indicate significant differences by Holm–Bonferroni corrected rank-sum tests ( $\alpha =0.05$). Consistent results were obtained when measuring information from real spiking data and simulated florescence traces (Extended Data Fig. 5-1).