Graphic methods for the conventional rate-coding analysis and state-space analysis in the regression subspace. Conventional analysis (top and middle rows): in each single neuron, activity modulations by task variables are detected in the fixed time window (top row) using linear regression and ANOVA for continuous (left, Exp. 1) and categorical (right, Exp. 2) task parameters (Fig. 2, see for the task details), respectively. The same analyses were applied in a fine time resolution in Exp. 1 and Exp. 2 (middle row). The conventional analyses using a general linear model (linear regression and ANOVA) provide the extent of neural

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... we showed that the general linear model usually used in the rate-coding model detects neural modulations using the same continuous and categorical parameters as a standard analysis procedure. This conventional approach can provide temporal changes in the selected metrics (e.g., proportion and extent of neural modulations; Fig. 4, top row), but they cannot provide trajectory geometry at the lower dimension. We note that detailed results from these conventional analyses have been previously reported (Chen and Naya, 2020, their Figs. 2, 5;Yamada et al., 2021, their Fig. ...

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... we used a general linear model to project a time series of each neural activity into a regression subspace composed of task parameters as continuous and categorical, as shown in the regression equations in Figure 4 (middle row, step 1; for details, see Materials and Methods). This step captures the across-trial variance caused by task-related parameters moment-by-moment at a population level, which demonstrates the extent of neural modulations by the task parameters across time. ...

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... at a population level, which demonstrates the extent of neural modulations by the task parameters across time. This corresponds to the estimation of the regression coefficients shown in Figure 3, E and J (for all time bins and conditions), which constructs the regression matrices detected in each neural population with a fine time resolution (Fig. 4, middle row, step 1, X). Second, we applied PCA one time to the time series of neural activity in the regression subspace in each neural population (Fig. 4, bottom row, step 2). This step determines the main features of the neural modulation signal across time in the predominant dimensions as trajectory geometry. These two steps identify how neural ...

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... regression coefficients shown in Figure 3, E and J (for all time bins and conditions), which constructs the regression matrices detected in each neural population with a fine time resolution (Fig. 4, middle row, step 1, X). Second, we applied PCA one time to the time series of neural activity in the regression subspace in each neural population (Fig. 4, bottom row, step 2). This step determines the main features of the neural modulation signal across time in the predominant dimensions as trajectory geometry. These two steps identify how neural modulations by task parameters change as a time series of eigenvectors in the regression ...

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... evaluated the properties of the extracted time series of the eigenvectors in the lower-dimensional space: the first three principal components (PC1 to PC3) in each neural population, in terms of vector angle, size, and deviance (Extended Data Fig. 4-1). The angles and sizes provide trajectory geometry that describes how neural modulation evolves after the visual presentation. ...

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... and for the six items and four locations; in total, 30 eigenvectors for each. Extended Data Figure 4-1 represents detail of the vector analyses. ...

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... at the population level by removing the average activity in each moment (Fig. 9, bottom row). Our methods are similar to the dPCA, but not the same. For example, we did not include the interaction component since the state-space analysis assumes a linear system. PCArs only focuses on neural dynamics in the subspace and requires only two steps (Fig. 4, middle, bottom rows) and minimal assumptions (orthogonality in the task design). Our PCArs is beneficial in terms of being simple and intuitive as it is an extension of the conventional rate-coding approach. Thus, our state-space analysis provides researchers with a significant advantage to find modulation dynamics in ...

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... this study, we evaluated neural modulation dynamics in terms of vector size, angle, and deviance in the regression subspace (Extended Data Fig. 4-1) that composes possible trajectory geometries. The combination of vector size and angle describes our straight geometry (i.e., constant angle), whereas deviance also reflects vector stability over time. ...