Control system toolbox

No statistical methods were used to predetermine sample sizes; our sample sizes were, however, similar to those reported in previous studies (for example, see refs. ^{13 (link),30 (link)}). Data were assumed to be normally distributed, but this was not formally tested. Visual stimuli were presented pseudorandomly in each block. Data collection and analysis were not performed blinded to the conditions of the experiments. No data were excluded, except for (1) licking frequencies and licking modulations that were considered to be outliers based on median absolute deviations (MADs; threshold, 3 MAD) in MkA, (2) trials in which LFP values were considered to be outliers (threshold, 3 MAD), and (3) LFP data with problems of collinearity, nonstationarity, or heteroscedasticity for Granger causality analysis (see below). All statistical procedures were assessed by two-tailed tests, unless otherwise stated, and performed using commercial software [MATLAB 2018a (ver. 9.4) and 2020b (ver. 9.9) with Statistics and Machine Learning toolbox (ver. 11.4), Signal Processing toolbox (ver. 8.1), Parallel Computing toolbox (ver. 6.13), and Control System toolbox (ver. 10.5); MathWorks Inc., MA, USA].

Although statistical methods were not used to predetermine sample sizes, our sample sizes were similar to those used in previous studies^{13 (link),14 (link)}. All well-isolated neurons were included in the neuronal recordings to prevent sampling bias. Blinding was not performed for investigators involved in data collection and analysis. No data were excluded unless otherwise stated. All statistical procedures were assessed by two-tailed tests and carried out using MATLAB Statistics and Machine Learning Toolbox, Signal Processing Toolbox, Parallel Computing Toolbox, Control System Toolbox, and Multivariate Granger Causality Toolbox (version 2018b and 2020b; MathWorks Inc.).

Linearized grey-box model’s complexity (see S3 Table) was simplified with reduced-order approximations using the Model Reducer app in the Control System Toolbox (Mathworks, Inc., USA). Simpler models can preserve model characteristics while discarding states that contribute relatively little to system dynamics (Balanced Truncation Model Reduction, ‘balred’ function). Simpler transfer function models can provide insights into the linear time-invariant model dynamics that were derived from the minimal realization transfer function for the four tDCS perturbation pathways. A generic reduced dimension grey-box linear model was derived (‘tfest’ function) using the nested linearized grey-box models for the initial parameterization then fitted to averaged fNIRS-tHb data–see Fig 7. We used Chi-Square Goodness-of-Fit for comparing the four hypotheses for the nested pathways (see Fig 6) where Chi-Square difference test for nested pathways [117 (link)] determined the tDCS perturbation pathway model of the least order.