Quantitative evaluations of eigenvector properties in the cOFC and HPC populations. A, Time series of vector size estimated in the cOFC population for P and M of rewards. Vector sizes are estimated in the PC1-PC2 plane (top) and PC2-PC3 plane (bottom), respectively. a.u., Arbitrary unit. The solid-colored lines indicate interpolated lines using a cubic spline function to provide a resolution of 0.005 s. B, Time series of vector size estimated in the HPC population for the best and worst items. C, Boxplots of vector size estimated in the cOFC population for probability and magnitude of rewards. D, Boxplots of vector size in the HPC population for the best and worst items and locations. E, F, Boxplots of vector angle estimated in the cOFC (E) and HPC (F) populations. G, H, Boxplots of vector deviance from the mean estimated in the cOFC (G) and HPC (H) populations. In C-H, data after 0.1 s are used. *p , 0.05, ***p , 0.001.

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... quantitatively examine and compare these neural population structures using geometry, we compared the properties of the eigenvectors by estimating the vector size, angle, and deviance in each neural population (Fig. 8). We analyzed rank-ordered HPC data shown in Figure 7 as well as cOFC data shown in Figure 5. For the rank-ordered data, we evaluated the best and worst conditions as the typically used conditions in conventional rate-coding ...

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... the vector size evaluation provided clear time-dependent structures in both cOFC and HPC populations for probability and magnitude (Fig. 8A) and for the best and worst items (Fig. 8B). Such time-dependent changes were not clearly observed in the eigenvectors for the best and worst locations (Fig. 8B, right and second to right columns), because location information already has been provided to the monkeys before the samples appeared. The vector sizes during the period ...

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... the vector size evaluation provided clear time-dependent structures in both cOFC and HPC populations for probability and magnitude (Fig. 8A) and for the best and worst items (Fig. 8B). Such time-dependent changes were not clearly observed in the eigenvectors for the best and worst locations (Fig. 8B, right and second to right columns), because location information already has been provided to the monkeys before the samples appeared. The vector sizes during the period 0.1-0.6 s after the onset of the lottery cue were ...

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... the vector size evaluation provided clear time-dependent structures in both cOFC and HPC populations for probability and magnitude (Fig. 8A) and for the best and worst items (Fig. 8B). Such time-dependent changes were not clearly observed in the eigenvectors for the best and worst locations (Fig. 8B, right and second to right columns), because location information already has been provided to the monkeys before the samples appeared. The vector sizes during the period 0.1-0.6 s after the onset of the lottery cue were not significantly different between the two continuous parameters, probability and magnitude of rewards ( Fig. 8C; ...

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... locations (Fig. 8B, right and second to right columns), because location information already has been provided to the monkeys before the samples appeared. The vector sizes during the period 0.1-0.6 s after the onset of the lottery cue were not significantly different between the two continuous parameters, probability and magnitude of rewards ( Fig. 8C; Wilcoxon rank-sum test; PC1 to PC2: n ¼ 52, df ¼ 51, W ¼ 330, p ¼ 0.892; PC2 to PC3; n ¼ 52, df ¼ 51, W ¼ 341, p ¼ 0.964). In contrast, the vector sizes during the period 0.1-0.6 s after the onset of the sample significantly differed between the best and worst items ( Fig. 8D; Wilcoxon signed-rank test; PC1 to PC2, item: n ¼ 52, df ¼ ...

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... the two continuous parameters, probability and magnitude of rewards ( Fig. 8C; Wilcoxon rank-sum test; PC1 to PC2: n ¼ 52, df ¼ 51, W ¼ 330, p ¼ 0.892; PC2 to PC3; n ¼ 52, df ¼ 51, W ¼ 341, p ¼ 0.964). In contrast, the vector sizes during the period 0.1-0.6 s after the onset of the sample significantly differed between the best and worst items ( Fig. 8D; Wilcoxon signed-rank test; PC1 to PC2, item: n ¼ 52, df ¼ 51, W ¼ 502, p ¼ 0.002; PC2 to PC3, item: n ¼ 52, df ¼ 51, W ¼ 588, p , 0.001; PC1 to PC2, location: n ¼ 52, df ¼ 51, W ¼ 600, p , 0.001; PC2 to PC3, location: n ¼ 52, df ¼ 51, W ¼ 542, p , 0.001). This is because the regression coefficients for the best item were considerably ...

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... analyses of vector angles showed that all eigenvectors were stable in both populations in the top two dimensions (Fig. 8E,F, top; Wilcoxon rank-sum test; cOFC, PC1 Figure 7. Effects of preference ordering on the HPC categorical data. A, Three examples of HPC neurons for preference ordering. The activities were ordered by their preference to the items and locations (right, shown best to worst), while their activity have a preference to item or location during the ...

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... PC2: n ¼ 52, df ¼ 51, W ¼ 62, p , 0.001; HPC, PC1 to PC2, item: n ¼ 52, df ¼ 51, W ¼ 520, p , 0.001; HPC, PC1 to PC2, location: n ¼ 52, df ¼ 51, W ¼ 0, p , 0.001). The angles in the PC2-PC3 plane were not stable (Fig. 8E,F, bottom; Wilcoxon rank-sum test; cOFC, PC2 to ...

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... neural populations showed some vector deviance (,0.1), with some statistical differences ( Fig. 8G,H; Wilcoxon rank-sum test; cOFC, PC1 to PC2: n ¼ 52, df ¼ 51, W ¼ 361, p ¼ 0.683; cOFC, PC2 to PC3: n ¼ 52, df ¼ 51, W ¼ 300, p ¼ 0.496; HPC, PC1 to PC2, item: n ¼ 52, df ¼ 51, W ¼ 459, p ¼ 0.027; HPC, PC2 to PC3, ...

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... our state-space analysis in the regression subspace described neural modulation dynamics in the cOFC and HPC during two different cognitive tasks composed of continuous and categorical parameters. These dynamic structures, evaluated qualitatively (Figs. 5-7) and quantitatively ( Fig. 8), reflected the neural modulation properties described by the conventional rate-coding analyses (Fig. 3). The straight dynamics observed in both cOFC and HPC populations were captured by a combination of changes in vector size and stable vector angle across time, which cannot be captured by the conventional rate-coding ...

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... the state-space analysis in the regression subspace for the categorical task parameters; and (2) we found that the HPC neural population exhibited straight dynamics. Fair comparisons of neural modulation dynamics of the two parameters (Figs. 5-8) indicated that straight dynamics observed at the lower dimension exhibited a gradual development (Fig. 8A,B) and stable composition of the neural modulation structures at different angles (Fig. 8E,F, top). Thus, we conclude that the neural population structures obtained from different brain regions using different behavioral tasks were similarly stable in terms of geometric ...

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... and (2) we found that the HPC neural population exhibited straight dynamics. Fair comparisons of neural modulation dynamics of the two parameters (Figs. 5-8) indicated that straight dynamics observed at the lower dimension exhibited a gradual development (Fig. 8A,B) and stable composition of the neural modulation structures at different angles (Fig. 8E,F, top). Thus, we conclude that the neural population structures obtained from different brain regions using different behavioral tasks were similarly stable in terms of geometric ...

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... (Fig. 7C, Extended Data Fig. 7-1B). In the rate-coding analysis, the time course of neural modulation by item and location was observed in the percentages of modulated neurons (Fig. 3I) and in the magnitudes of regression coefficients (Fig. 3J). In the dynamic analysis, these characteristics were observed on the time course of the vector size (Fig. 8A,B), while the neural modulation structures were evaluated in terms of their similarity across time. Thus, the classical rate-coding analysis was well incorporated into the dynamic analyses, which specifically captured their neural population ...

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... components for categorical variables (Kobak et al., 2016). Our method first projects the neural population dynamics into a regression subspace that removes the activity change other than the neural modulations by task parameters and demonstrates the modulation dynamics. dPCA figures are represented in the study by Kobak et al. (2016, their Fig. 8). PSTH indicates peristimulus time histogram. the population level. In this step, both continuous and categorical task parameters were reliably used within the framework of the general linear model. However, it was reliably performed with one limitation; the conditions in any parameter would be orthogonalized as the experimental design ...

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... across time in the multidimensional neural subspace. If neural modulations are completely stable across time (i.e., all vectors are the same across time), then all three parameters become very stable (i.e., congruence in mathematics: constant vector angle, constant vector size, and very small deviance). Thus, our analysis of vector properties (Fig. 8) can quantitatively evaluate trajectory geometry in neural ...

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... our previous study, fluctuating neural population signals were observed in the dorsal striatum (DS) and medial OFC (mOFC) because of signal instability or weakness ( Yamada et al., 2021, their Fig. 5A,B; Imaizumi et al., 2022). As the signal carried by the mOFC population was weak ( Yamada et al., 2021, their Fig. 8, bottom row), eigenvector fluctuation in the mOFC population reflected the weak signal modulations by the probability and magnitude of rewards. In this case, moment-by-moment vector fluctuations were observed because there was no clear neural modulation structure in the mOFC population. Conversely, the fluctuating DS signal seemed to reflect the ...

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... seemed to reflect the functional role of the DS neural population in detecting and integrating the probability and magnitude of rewards related to the control of some actions ( Balleine et al., 2007;Enomoto et al., 2020;Inokawa et al., 2020). In the DS population, structural changes in eigenvectors occurred over time ( Yamada et al., 2021, their Fig. 8). Future studies are required to explore neural modulation geometry to elucidate how neural circuitry operates and computes ( Ebitz and Hayden, 2021;Humphries, ...