# Hemodynamic Response Functions

#### Bradley R. Buchsbaum

#### 2026-02-09

## Introduction to Hemodynamic Response Functions (HRFs)

A hemodynamic response function (HRF) models the temporal evolution of the fMRI BOLD (Blood-Oxygen-Level-Dependent) signal in response to a brief neural event. Typically, the BOLD signal peaks 4-6 seconds after the event onset and then returns to baseline, often with a slight undershoot.

`fmrihrf` provides tools to define, manipulate, and visualize various HRFs commonly used in fMRI analysis.

## Pre-defined HRF Objects

`fmrihrf` includes several pre-defined HRF objects, which are essentially functions with specific attributes defining their type, number of basis functions (`nbasis`), and effective duration (`span`).

Let’s look at two common examples: the SPM canonical HRF (`HRF_SPMG1`) and a Gaussian HRF (`HRF_GAUSSIAN`).

    # SPM canonical HRF (based on difference of two gamma functions)
    print(HRF_SPMG1)
    #> -- HRF: SPMG1 ---------------------------------------------
    #>    Basis functions: 1
    #>    Span: 24 s
    #>    Parameters: P1 = 5, P2 = 15, A1 = 0.0833

    # Gaussian HRF
    print(HRF_GAUSSIAN)
    #> -- HRF: gaussian ------------------------------------------
    #>    Basis functions: 1
    #>    Span: 24 s
    #>    Parameters: mean = 6, sd = 2

These objects are functions themselves, so you can evaluate them at specific time points. The `plot_hrfs()` function provides a convenient way to compare multiple HRFs:

    time_points <- seq(0, 25, by = 0.1)

    # Compare HRFs using plot_hrfs() - normalize = TRUE scales to peak at 1.0
    plot_hrfs(HRF_SPMG1, HRF_GAUSSIAN,
              labels = c("SPM Canonical", "Gaussian"),
              normalize = TRUE,
              title = "Comparison of SPM Canonical and Gaussian HRFs",
              subtitle = "HRFs normalized to peak at 1.0 for shape comparison")

Note that the `span` attribute (e.g., 24 seconds) indicates the approximate time window over which the HRF is non-zero.

## Modifying HRF Parameters with `gen_hrf`

The `gen_hrf` function is a flexible way to create new HRF functions, often by modifying the parameters of existing ones.

For example, the `hrf_gaussian` function takes `mean` and `sd` arguments. We can use `gen_hrf` to create Gaussian HRFs with different peak times (`mean`) and widths (`sd`).

    # Create Gaussian HRFs with different parameters using gen_hrf
    # Note: hrf_gaussian is the underlying function, not the HRF object HRF_GAUSSIAN
    hrf_gauss_7_3 <- gen_hrf(hrf_gaussian, mean = 7, sd = 3, name = "Gaussian (Mean=7, SD=3)")
    hrf_gauss_5_2 <- gen_hrf(hrf_gaussian, mean = 5, sd = 2, name = "Gaussian (Mean=5, SD=2)")
    hrf_gauss_4_1 <- gen_hrf(hrf_gaussian, mean = 4, sd = 1, name = "Gaussian (Mean=4, SD=1)")

`gen_hrf` can also directly incorporate lags and durations (see later sections).

## Modeling Event Duration with `block_hrf`

fMRI events often have a duration (e.g., a stimulus presented for several seconds). The `block_hrf` function (or `gen_hrf` with a `width` argument) modifies an HRF to model the response to a sustained event of a specific `width` (duration). Internally, it convolves the original HRF with a boxcar function of the specified width.

The `precision` argument controls the sampling resolution used for this convolution.

    # Create blocked HRFs using the SPM canonical HRF with different durations
    hrf_spm_w1 <- block_hrf(HRF_SPMG1, width = 1)
    hrf_spm_w2 <- block_hrf(HRF_SPMG1, width = 2)
    hrf_spm_w4 <- block_hrf(HRF_SPMG1, width = 4)

### Normalization

By default, longer durations lead to higher peak responses (assuming summation, see next section). Setting `normalize=TRUE` in `block_hrf` (or `gen_hrf`) rescales the response so the peak amplitude is approximately 1, regardless of duration.

    # Create normalized blocked HRFs
    hrf_spm_w1_norm <- block_hrf(HRF_SPMG1, width = 1, normalize = TRUE)
    hrf_spm_w2_norm <- block_hrf(HRF_SPMG1, width = 2, normalize = TRUE)
    hrf_spm_w4_norm <- block_hrf(HRF_SPMG1, width = 4, normalize = TRUE)

### Modeling Saturation with `summate`

The `summate` argument in `block_hrf` controls whether the response accumulates over the duration (`summate=TRUE`, default) or saturates (`summate=FALSE`). Saturation implies that the response reaches a plateau and doesn’t increase further with longer stimulation.

    # Create non-summating blocked HRFs
    hrf_spm_w2_nosum <- block_hrf(HRF_SPMG1, width = 2, summate = FALSE)
    hrf_spm_w4_nosum <- block_hrf(HRF_SPMG1, width = 4, summate = FALSE)
    hrf_spm_w8_nosum <- block_hrf(HRF_SPMG1, width = 8, summate = FALSE)

We can combine `summate=FALSE` and `normalize=TRUE`:

    # Create normalized, non-summating blocked HRFs
    hrf_spm_w2_nosum_norm <- block_hrf(HRF_SPMG1, width = 2, summate = FALSE, normalize = TRUE)
    hrf_spm_w4_nosum_norm <- block_hrf(HRF_SPMG1, width = 4, summate = FALSE, normalize = TRUE)
    hrf_spm_w8_nosum_norm <- block_hrf(HRF_SPMG1, width = 8, summate = FALSE, normalize = TRUE)

## Modeling Temporal Shifts with `lag_hrf`

Sometimes, the hemodynamic response might be delayed or advanced relative to the event onset. The `lag_hrf` function (or `gen_hrf_lagged`) shifts an existing HRF in time by a specified `lag` (in seconds). A positive lag delays the response, while a negative lag advances it.

    # Create lagged versions of the Gaussian HRF
    hrf_gauss_lag_neg2 <- lag_hrf(HRF_GAUSSIAN, lag = -2)
    hrf_gauss_lag_0 <- HRF_GAUSSIAN # Original (lag=0)
    hrf_gauss_lag_pos3 <- lag_hrf(HRF_GAUSSIAN, lag = 3)

## Combining Lag and Duration

We can combine `lag_hrf` and `block_hrf` using the pipe operator (`%>%`) from `dplyr` (or `magrittr`).

    # Create HRFs that are both lagged and blocked
    hrf_lb_1 <- HRF_GAUSSIAN %>% lag_hrf(1) %>% block_hrf(width = 1, normalize = TRUE)
    hrf_lb_3 <- HRF_GAUSSIAN %>% lag_hrf(3) %>% block_hrf(width = 3, normalize = TRUE)
    hrf_lb_5 <- HRF_GAUSSIAN %>% lag_hrf(5) %>% block_hrf(width = 5, normalize = TRUE)

Alternatively, `gen_hrf` can apply lag and width directly:

    # Using gen_hrf directly
    hrf_lb_gen_3 <- gen_hrf(hrf_gaussian, lag = 3, width = 3, normalize = TRUE)
    resp_lb_gen_3 <- hrf_lb_gen_3(time_points)

    # Compare (should be very similar to hrf_lb_3 from piped version)
    # plot(time_points, resp_lb_3, type = 'l', col = 2, lwd = 2, main = "Piped vs gen_hrf")
    # lines(time_points, resp_lb_gen_3, col = 1, lty = 2, lwd = 2)
    # legend("topright", legend = c("Piped", "gen_hrf"), col = c(2, 1), lty = c(1, 2), lwd = 2)

## Multivariate HRFs: Basis Sets

Instead of assuming a fixed HRF shape, we can model the response using a linear combination of multiple basis functions. This allows for more flexibility in capturing variations in HRF shape across brain regions or individuals. The resulting HRF function returns a matrix where each column corresponds to a basis function.

### SPM Basis Sets

`fmrihrf` provides pre-defined HRF objects for the SPM canonical HRF plus its temporal derivative (`HRF_SPMG2`), and additionally its dispersion derivative (`HRF_SPMG3`).

    # SPM + Temporal Derivative (2 basis functions)
    print(HRF_SPMG2)
    #> -- HRF: SPMG2 ---------------------------------------------
    #>    Basis functions: 2
    #>    Span: 24 s

    # SPM + Temporal + Dispersion Derivatives (3 basis functions)
    print(HRF_SPMG3)
    #> -- HRF: SPMG3 ---------------------------------------------
    #>    Basis functions: 3
    #>    Span: 24 s

### B-Spline Basis Set

The `hrf_bspline` function generates a B-spline basis set. We typically use it within `gen_hrf` to create an HRF object. Key parameters are `N` (number of basis functions) and `degree`.

    # B-spline basis with N=5 basis functions, degree=3 (cubic)
    hrf_bs_5_3 <- gen_hrf(hrf_bspline, N = 5, degree = 3, name = "B-spline (N=5, deg=3)")
    print(hrf_bs_5_3)
    #> -- HRF: B-spline (N=5, deg=3) -----------------------------
    #>    Basis functions: 5
    #>    Span: 24 s

    # B-spline basis with N=10 basis functions, degree=1 (linear -> tent functions)
    hrf_bs_10_1 <- gen_hrf(hrf_bspline, N = 10, degree = 1, name = "Tent Set (N=10)")
    print(hrf_bs_10_1)
    #> -- HRF: Tent Set (N=10) -----------------------------------
    #>    Basis functions: 10
    #>    Span: 24 s

### Sine Basis Set

The `hrf_sine` function creates a basis set using sine waves of different frequencies.

    hrf_sin_5 <- gen_hrf(hrf_sine, N = 5, name = "Sine Basis (N=5)")
    print(hrf_sin_5)
    #> -- HRF: Sine Basis (N=5) ----------------------------------
    #>    Basis functions: 5
    #>    Span: 24 s

### Half-Cosine Basis Set (FLOBS-like)

The `hrf_half_cosine` function implements the basis set described by Woolrich et al. (2004), often used in FSL’s FLOBS (FMRIB’s Linear Optimal Basis Sets). It uses four half-cosine functions to model initial dip, rise, fall/undershoot, and recovery.

## Other HRF Shapes

### Gamma HRF

The `hrf_gamma` function uses the gamma probability density function.

````

    hrf_gam <- gen_hrf(hrf_gamma, shape = 6, rate = 1, name = "Gamma (shape=6, rate=1)")
    print(hrf_gam)
    #> -- HRF: Gamma (shape=6, rate=1) ---------------------------
    #>    Basis functions: 1
    #>    Span: 24 s

### Mexican Hat Wavelet HRF

The `hrf_mexhat` function uses the Mexican hat wavelet (second derivative of a Gaussian).

    hrf_mh <- gen_hrf(hrf_mexhat, mean = 6, sd = 1.5, name = "Mexican Hat (mean=6, sd=1.5)")
    print(hrf_mh)
    #> -- HRF: Mexican Hat (mean=6, sd=1.5) ----------------------
    #>    Basis functions: 1
    #>    Span: 24 s

### Inverse Logit Difference HRF

The `hrf_inv_logit` function creates an HRF shape by subtracting two inverse logit (sigmoid) functions, allowing control over rise and fall times.

    hrf_il <- gen_hrf(hrf_inv_logit, mu1 = 5, s1 = 1, mu2 = 15, s2 = 1.5, name = "Inv. Logit Diff.")
    print(hrf_il)
    #> -- HRF: Inv. Logit Diff. ----------------------------------
    #>    Basis functions: 1
    #>    Span: 24 s

## Boxcar and Weighted HRFs (No Hemodynamic Delay)

Traditional HRFs model the hemodynamic delay—the sluggish blood flow response that peaks several seconds after neural activity. However, sometimes you want to extract signal from specific time windows without assuming any hemodynamic transformation. This is useful for: - Extracting raw signal averages from specific post-stimulus windows - Trial-wise analyses where you want the mean (or weighted mean) of the BOLD signal - Comparing signal in different temporal windows directly - Creating custom temporal weighting schemes

### Simple Boxcar HRF (`hrf_boxcar`)

The `hrf_boxcar` function creates a simple step function that is constant within a time window and zero outside. Unlike traditional HRFs, there is no built-in hemodynamic delay—the HRF starts at time 0 (event onset) and extends for the specified `width`.

    # Create a boxcar of width 5 seconds (from 0 to 5 seconds)
    hrf_box <- hrf_boxcar(width = 5)
    print(hrf_box)
    #> -- HRF: boxcar[5] -----------------------------------------
    #>    Basis functions: 1
    #>    Span: 5 s

To create a boxcar that starts at a later time point—useful for capturing signal in a specific post-stimulus window—use `lag_hrf()`:

    # Boxcar from 4-8 seconds post-stimulus (capturing the expected BOLD peak)
    # Use lag_hrf() to delay a 4-second boxcar by 4 seconds
    hrf_delayed <- hrf_boxcar(width = 4) %>% lag_hrf(lag = 4)

### Normalized Boxcar: Estimating Mean Signal

When `normalize = TRUE`, the boxcar is scaled so its integral equals 1. This has an important interpretation: when used in a GLM, the regression coefficient β directly estimates the mean signal within the time window.

    # Normalized boxcar - integral = 1
    # A 4-second boxcar lagged by 4 seconds (captures 4-8s window)
    hrf_norm <- hrf_boxcar(width = 4, normalize = TRUE) %>% lag_hrf(lag = 4)

    # Check: amplitude should be 1/4 = 0.25
    t_fine <- seq(0, 12, by = 0.01)
    resp_norm <- evaluate(hrf_norm, t_fine)
    cat("Amplitude of normalized boxcar:", max(resp_norm), "\n")
    #> Amplitude of normalized boxcar: 0.25
    cat("Expected (1/width):", 1/4, "\n")
    #> Expected (1/width): 0.25

    # Verify integral ≈ 1
    integral <- sum(resp_norm) * 0.01
    cat("Integral of normalized boxcar:", round(integral, 3), "\n")
    #> Integral of normalized boxcar: 1

Interpretation: If you fit a GLM with this normalized boxcar HRF, the estimated β coefficient represents the average BOLD signal from 4-8 seconds post-stimulus.

### Weighted HRF (`hrf_weighted`)

The `hrf_weighted` function provides more flexibility by allowing you to specify different weights at different time points. You can either: - Use `width` + `weights`: evenly space the weights across the specified width - Use `times` + `weights`: explicitly specify the time points for each weight

This creates either a step function (`method = "constant"`) or a smoothly interpolated function (`method = "linear"`).

#### Using `width` for Evenly Spaced Weights

    # 6 weights evenly spaced over 10 seconds (at 0, 2, 4, 6, 8, 10)
    hrf_wt_width <- hrf_weighted(
      weights = c(0.1, 0.3, 1.0, 1.0, 0.3, 0.1),
      width = 10,
      method = "constant"
    )

#### Using Explicit `times` for Custom Spacing

    # Weighted step function with explicit time points
    hrf_wt <- hrf_weighted(
      weights = c(0.1, 0.3, 1.0, 1.0, 0.3, 0.1),
      times = c(2, 4, 6, 8, 10, 12),
      method = "constant"
    )

#### Smooth Weights (Linear Interpolation)

    # Smooth weights using linear interpolation
    hrf_smooth <- hrf_weighted(
      weights = c(0, 0.3, 1.0, 1.0, 0.3, 0),
      times = c(2, 4, 6, 8, 10, 12),
      method = "linear"
    )

#### Sub-second Precision

The `hrf_weighted` function supports sub-second time intervals, which is useful for fine-grained temporal weighting:

    # Sub-second intervals: create a Gaussian-shaped weight function
    times_fine <- seq(4, 10, by = 0.25)
    weights_gaussian <- dnorm(times_fine, mean = 7, sd = 1)

    hrf_gauss_wt <- hrf_weighted(weights_gaussian, times = times_fine, method = "linear")

### Normalized Weighted HRF: Estimating Weighted Mean

When `normalize = TRUE`, the weights are scaled to sum (constant) or integrate (linear) to 1. The regression coefficient then estimates a weighted mean of the signal:

    # Normalized weights - creates weighted average interpretation
    hrf_wt_norm <- hrf_weighted(
      weights = c(1, 2, 2, 1),  # Will be normalized
      times = c(4, 6, 8, 10),
      method = "constant",
      normalize = TRUE
    )

    # The coefficient β will estimate: (1*Y[4-6] + 2*Y[6-8] + 2*Y[8-10] + 1*Y[10+]) / 6
    # where Y[a-b] is the signal in that interval

    t_check <- seq(0, 12, by = 0.01)
    resp_wt_norm <- evaluate(hrf_wt_norm, t_check)

    # Verify: integral should be approximately 1
    integral_wt <- sum(resp_wt_norm) * 0.01
    cat("Integral of normalized weighted HRF:", round(integral_wt, 3), "\n")
    #> Integral of normalized weighted HRF: 2.002

### Practical Example: Comparing Early vs. Late Response Windows

A common analysis compares BOLD signal in early vs. late portions of a trial. Here’s how to set up HRFs for this:

    # Early window: 2-6 seconds (4-second boxcar lagged by 2 seconds)
    hrf_early <- hrf_boxcar(width = 4, normalize = TRUE) %>% lag_hrf(lag = 2)

    # Late window: 8-12 seconds (4-second boxcar lagged by 8 seconds)
    hrf_late <- hrf_boxcar(width = 4, normalize = TRUE) %>% lag_hrf(lag = 8)

Using these HRFs in separate regressors allows you to estimate and compare the mean BOLD signal in each window.

### Using Boxcar/Weighted HRFs with Regressors

These HRFs integrate seamlessly with the `regressor()` function:

    # Create a regressor with boxcar HRF (4-second window starting 4s after onset)
    reg_boxcar <- regressor(
      onsets = c(0, 20, 40),
      hrf = hrf_boxcar(width = 4, normalize = TRUE) %>% lag_hrf(lag = 4)
    )

    # Compare with traditional SPM HRF
    reg_spm <- regressor(onsets = c(0, 20, 40), hrf = HRF_SPMG1)

## Creating Custom Basis Sets with `gen_hrf_set`

The `gen_hrf_set` function allows you to combine any set of HRF functions into a single multivariate HRF object (a basis set).

For example, we can create a basis set from a series of lagged Gaussian HRFs:

    # Create a list of lagged Gaussian HRFs
    lag_times <- seq(0, 10, by = 2)
    list_of_hrfs <- lapply(lag_times, function(lag) {
      lag_hrf(HRF_GAUSSIAN, lag = lag)
    })

    # Combine them into a single HRF basis set object
    hrf_custom_set <- do.call(gen_hrf_set, list_of_hrfs)
    print(hrf_custom_set) # Note: name is default 'hrf_set', nbasis is 6
    #> -- HRF: hrf_set -------------------------------------------
    #>    Basis functions: 6
    #>    Span: 34 s

## Creating Empirical HRFs

### From a Single Measured Response (`gen_empirical_hrf`)

If you have a measured or estimated hemodynamic response profile (e.g., from deconvolution), you can turn it into an HRF function using `gen_empirical_hrf`. It uses linear interpolation between the provided points.

    # Simulate an average measured response profile
    sim_times <- 0:24
    set.seed(42) # For reproducibility
    sim_profile <- rowMeans(replicate(20, {
      h <- HRF_SPMG1 %>% lag_hrf(lag = runif(n = 1, min = -1, max = 1)) %>%
                        block_hrf(width = runif(n = 1, min = 0, max = 2))
      h(sim_times)
    }))

    # Normalize profile to max = 1 for better visualization
    sim_profile_norm <- sim_profile / max(sim_profile)

    # Create the empirical HRF function from the normalized profile
    emp_hrf <- gen_empirical_hrf(sim_times, sim_profile_norm)
    print(emp_hrf)
    #> -- HRF: empirical_hrf -------------------------------------
    #>    Basis functions: 1
    #>    Span: 24 s

### Empirical Basis Set via PCA

You can create an empirical basis set by applying dimensionality reduction (like PCA) to a collection of observed or simulated HRFs.

    # 1. Simulate a matrix of diverse HRFs
    set.seed(123) # for reproducibility
    n_sim <- 50
    sim_mat <- replicate(n_sim, {
      hrf_func <- HRF_SPMG1 %>%
                  lag_hrf(lag = runif(1, -2, 2)) %>%
                  block_hrf(width = runif(1, 0, 3))
      hrf_func(sim_times)
    })

    # 2. Perform PCA on the transpose (each column = one HRF, each row = one time point)
    pca_res <- prcomp(t(sim_mat), center = TRUE, scale. = FALSE)
    n_components <- 3

    # Print variance explained by top components
    variance_explained <- summary(pca_res)$importance[2, 1:n_components]
    cat("Variance explained by top", n_components, "components:",
        paste0(round(variance_explained * 100, 1), "%"), "\n")
    #> Variance explained by top 3 components: 67.1% 29.9% 2.7%

    # Extract the top principal components
    pc_vectors <- pca_res$rotation[, 1:n_components]

    # 3. Convert principal components into HRF functions
    list_pc_hrfs <- list()

    for (i in 1:n_components) {
      pc_vec <- pc_vectors[, i]
      pc_vec_zeroed <- pc_vec - pc_vec[1]
      max_abs <- max(abs(pc_vec_zeroed))
      pc_vec_norm <- pc_vec_zeroed / max_abs
      list_pc_hrfs[[i]] <- gen_empirical_hrf(sim_times, pc_vec_norm)
    }

    # 4. Combine PC HRFs into a basis set using gen_hrf_set
    emp_pca_basis <- do.call(gen_hrf_set, list_pc_hrfs)
    print(emp_pca_basis)
    #> -- HRF: hrf_set -------------------------------------------
    #>    Basis functions: 3
    #>    Span: 24 s

This empirical basis set can then be used in regression models just like any other pre-defined or custom basis set.