• Functional fMRI maps localized hemodynamic blood-flow fluctuations.
• The primary dependent metric of interest is the BOLD signal profile.
• Metabolic energy demands shift directly during active sensory stimulation.
• We model BOLD response variations programmatically using linear arrays.
• Core model predictors include absolute visual stimulus and user attention.

• This slide utilizes standard \(\LaTeX\) mathematical syntax array notations:

\[Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \epsilon_i\]

• \(Y_i\) corresponds directly to the target fMRI BOLD response values.
• \(X_{1i}\) tracks the percentage stimulus intensity contrast parameters.
• \(X_{2i}\) maps the specific measured behavioral participant attention metrics.
• \(\epsilon_i\) effectively accounts for stochastic background scanner noise elements.

• Parameter optimization calculations leverage standard matrix algebraic equations:

  \[\min \sum_{i=1}^{n} \epsilon_i^2 = \min \sum_{i=1}^{n} (Y_i - \hat{Y}_i)^2\]

  • Neural noise residuals \(\epsilon\) assume a perfectly normal distribution variance.

• High scanner baseline signal drift parameters must be tightly controlled.

ggplot(neuro_data, aes(x = Stimulus, y = BOLD)) +
  geom_point(color = "#8C1D40", alpha = 0.7, size = 3) +
  geom_smooth(method = "lm", color = "black") +
  theme_minimal() +
  labs(x = "Stimulus Intensity (% Contrast)", y = "BOLD Signal Change (%)")

`geom_smooth()` using formula = 'y ~ x'

ggplot(neuro_data, aes(x = BOLD)) +
  geom_histogram(bins = 15, fill = "#8C1D40", color = "white") +
  theme_minimal() +
  labs(x = "BOLD Signal Change (%)", y = "Voxel Count")

Hemodynamic 3D Linear Modeling Surface map:

# 1. Fit the linear regression model
fit <- lm(BOLD ~ Stimulus + Attention, data = neuro_data)

# 2. Extract sequence grids using bracket syntax
axis_x <- seq(min(neuro_data[["Stimulus"]]), max(neuro_data[["Stimulus"]]), length.out = 20)
axis_y <- seq(min(neuro_data[["Attention"]]), max(neuro_data[["Attention"]]), length.out = 20)

# 3. Create the mathematical surface matrix
surface_matrix <- t(outer(axis_x, axis_y, function(x, y) {
  predict(fit, newdata = data.frame(Stimulus = x, Attention = y))
}))

# 4. Build and print the interactive plot directly
plot_ly(neuro_data, x = ~Stimulus, y = ~Attention, z = ~BOLD, height = 350, width = 800) %>%
  add_markers(marker = list(color = '#8C1D40', size = 4, opacity = 0.6)) %>%
  add_surface(x = ~axis_x, y = ~axis_y, z = ~surface_matrix, opacity = 0.5)

• Sensory input scales directly alongside measured BOLD variations.
• Attentional focus parameters alter baseline neural response curves.
• Multi-predictor regressions chart complex interactive localized spaces.
• Modern R modeling engines unlock automated reproducible brain workflows.