The assumption made in this paper is that the skin conductance response s
) is the output of a finite Linear Time-Invariant filter:
• The linearity property satisfies the superposition principle that the response at a given time caused by two or more stimuli is the sum of the responses which would have been elicited by each stimulus individually.
• The time-invariant property means that the output does not explicitly depend on time, which corresponds here to say that the impulse response function is not a function of time except expressed by input.
• The finite property states that the response settles to zero after a finite interval.
A LTI filter is entirely characterised by its impulse response function h
) in such a way that the SCR is simply the convolution of the input signal u
) with the impulse response function that models the time delay of the skin conductance response and its specific shape
Note that the convolution is a linear operator.1
Deconvolution is the process of filtering the observed output signal s(t) to recover the input signal u(t) by removing the effect of the convolution filter h(t).This is an ill-posed problem even in the absence of observation noise, thus requiring some form of regularisation.
In the following we will assume that the impulse response function h
) is known (estimated from a principal component analysis on an independent dataset) and we will call it the canonical response function
, or CRF. We assume that it has a parametric form using a Gaussian smoothed probability density function of a Gamma distribution:
) is a centered Gaussian function with a standard deviation σ
of 0.5657 s.
) is a probability density function of a Gamma distribution:
for t > t0
and 0 otherwisewhere the parameters h
describe the shape and the inverse scale (rate) of the distribution, respectively, t0
is the delay of the response and Γ
is the Gamma function defined as:
In practice, the Gaussian function was truncated within a centered window of 4 s and the CRF was specified for the time interval of 0–120 s and was set to 0 everywhere else.
The parameters of this analytical form of the CRF are estimated using the dataset presented in experiment 1 of this paper using a least-square approach. The estimated parameters are:
To take into account that this canonical
response might not fit equally well the actual impulse response function for different conditions or individuals, it is possible to define a constrained basis set spanning the space of possible responses: this is called an informed basis set
. One way to construct this basis set is to perform a multivariate first-order Taylor expansion of the canonical response along parameters that are expected to vary between experimental conditions or subjects. Here we will consider time and dispersion parameters. Thus, for the construction of the temporal derivative
, the canonical function is differentiated with respect to time. The dispersion derivative
is obtained by computing the difference between the original CRF and the same one convolved with a wider Gaussian window (8 s duration and a standard deviation of 1.1314 s).
The informed basis set
capturing the CRF is therefore:
This basis set was orthogonalised using a serial Gram-Schmidt process as implemented in the SPM software (spm_orth.m). This does not change the space spanned by the basis set but will make sure that the effect explained by the canonical response is not partialed out by the other regressors of the basis set.
The impulse response function for a particular dataset can now be appropriately modelled by a linear combination of the functions comprised in the informed basis set.
The skin conductance response s
) with M
experimental conditions can then be expressed as
is a time-dependent input parameter of interest (e.g. an event onset function) with unit amplitude for condition m
, defined as a sum of Dirac delta functions centered on the onsets
of the events of condition m
is a vector containing the weights of the linear combination.
Note that this framework allows to model parametric modulations such as time (adaptation effect) or any other factors and any non-linear effect of those using polynomial expansion. For example, if
is such a parametric modulator, defined for all events, then another input parameter can be defined by
to model quadratic interactions between the variate p
and the trials.
All of this can be rephrased in terms of a general linear model (GLM): let Y
observed datapoints of a skin conductance time-series and X
matrix containing L
regressors, called the design matrix
, constructed as explained above (convolution of all the input functions with the informed basis set), then the GLM can be expressed as
is defined as before (a L
× 1 vector of weights or effect sizes) and
is the error term that is assumed to be i.i.d. (independent and identically distributed) Gaussian noise.
Under these assumptions, the maximum likelihood estimators are obtained through the Gauss-Markov theorem and are equal to the ordinary least square estimates (OLS):
Note that this is a linear operation: the estimates
are obtained through a linear combination of the data Y
If X is not of full rank, the model can still be inverted using the Moore-Penrose pseudoinverse.
Inverting a GLM can then be seen as a way to perform linear deconvolution. If the basis set is chosen to be a series of boxcar functions for each time bin, then no assumption is made on the shape of the impulse response function (apart from its length). If some constraints are added by using an informed basis set (restricting the domain of variations spanned the response to some plausible ones) then it still corresponds to some deconvolution that will likely provide some more robust estimates of the response. The extreme case is to only use the canonical response and in such a case, the response is assumed to be known and only the amplitude of the response remains to be estimated.
At last, once inverted, it is possible to perform some statistical inference on the estimated parameters to, for example, compare the effect sizes elicited by two conditions, using T
tests. This can be done on the group level using a hierarchical summary statistics approach which is equivalent to a random effects model given equal variance for different subjects and a balanced experimental design (Friston et al., 2008