Dispersion Derivative HRF

Evaluate the analytic derivative of the caller double-gamma HRF with respect to its common scale parameter; this differs from time-shift differentiation and is not an orthogonalized design column.

Description

Changing the common scale of a double-gamma HRF changes both timing and width.

Changing the common scale of a double-gamma HRF changes both timing and width. The scale derivative is a parameter derivative rather than a derivative with respect to time.

The tool returns scaleDerivative (scale derivative). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Dispersion Derivative HRF

Use Dispersion Derivative HRF when you already have timesSeconds, peakShape, undershootShape, scaleSeconds, undershootRatio and want scaleDerivative. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Dispersion Derivative HRF and understand its results

Enter timesSeconds, peakShape, undershootShape, scaleSeconds, undershootRatio using the field table below. Match the units, ordering and data boundaries described in the calculation.
Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
Read scaleDerivative. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.

Input field	Meaning	Value type and bounds	Requirement
timesSeconds	Times seconds	list of number	Required
peakShape	Peak shape	integer	Required
undershootShape	Undershoot shape	integer	Required
scaleSeconds	Scale seconds	number	Required
undershootRatio	Undershoot ratio	number	Required

Validate nonnegative ordered times and the double-gamma shapes, scale and undershoot ratio. Subtract the scaled undershoot derivative from the peak derivative.

∂gammaDensity(t,k,s)/∂s = gammaDensity(t,k,s)×(t/s²−k/s)

The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.

json

{
  "input": {
    "timesSeconds": [
      0,
      1,
      2
    ],
    "peakShape": 2,
    "undershootShape": 3,
    "scaleSeconds": 1,
    "undershootRatio": 6
  },
  "output": {
    "scaleDerivative": [
      0,
      -0.30656620097620196,
      0.0451117610788709
    ]
  }
}

Limitations and assumptions

The derivative is analytical for this shared-scale model and is not normalized or orthogonalized to a canonical basis.

Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.

Alternative or Complementary methods

Temporal Derivative HRF: Differentiate a supplied HRF with centered interior differences and one-sided endpoints; this is the time derivative, not an automatically orthogonalized GLM basis. HRF Peak: Evaluate a supplied double-gamma HRF and identify its unique sampled peak; integer gamma shapes and caller scale/undershoot ratio define the model, not a universal response.

Methodological background: fMRI model specification: haemodynamic response functions and derivatives.^[1]

References

fMRI model specification: haemodynamic response functions and derivatives — SPM developers, University College London

Similar or alternative tools

Temporal Derivative HRF
Differentiate a supplied HRF with centered interior differences and one-sided endpoints; this is the time derivative, not an automatically orthogonalized GLM basis.
HRF Peak
Evaluate a supplied double-gamma HRF and identify its unique sampled peak; integer gamma shapes and caller scale/undershoot ratio define the model, not a universal response.