*SF* are caused by sporadic and spontaneous (i.e. unrelated to experimentally presented events) activity of the sudomotor nerve (

Boucsein, 1992). Spontaneous firing occurs in short bursts with a duration of around 500–1000 ms, separated by longer intervals, and is followed by opening of sweat glands (

Macefield and Wallin, 1996; Nishiyama et al., 2001). The number of sweat glands recruited is linearly related to the amplitude of a firing burst (

Nishiyama et al., 2001). Consequently, it is plausible to assume that the amplitude of an

*SF* is linearly related to the amplitude of the firing burst. Further, it appears from previous research that both the number and the amplitude of bursts reflect sympathetic arousal.

It is biophysically plausible that the measured signal has some relationship with sudomotor nerve [

*SN*] firing and reasonable to assume that this relation is constant (that is, time-invariant), and that two subsequent responses will build up in a linear fashion. Under these LTI assumptions, and in the absence of noise, it is easy to see that the time-integral (or area under the curve) of an

*SC* time series is simply the

*SCL*, plus the number of responses

*n*, scaled by their amplitude

*a*, and multiplied by a constant

*c*. This constant is the time-integral of a single response to an input of unit amplitude (i.e., the response function;

*RF*).

where we describe

*SN* as a Dirac delta function, and where

*ā* is the mean amplitude of spontaneous fluctuations occurring at times

*T*_{i}:

*i* 1,…,

*n*.

^{2} The

*SCL*-corrected time-integral, or area under curve [

*AUC*], which is simple to compute, should therefore reflect the number and amplitude of sudomotor bursts and the status of the sympathetic nervous system. In reality, LTI assumptions are unlikely to be met completely. However, we can posit that

where

*e* denotes some error that absorbs random fluctuations and any violations of time-invariance and linearity assumptions. This is the model that we seek to validate in this paper. Note that a similar measure has been used in applied psychophysiology to quantify arousal during anaesthesia (

Ledowski et al., 2007), but has not been formally derived or validated. Here, we provide a measure for the integrated time series, corrected for

*SCL* by subtracting the lowest signal value, which we refer to as the area under the curve.

An alternative measure that has been previously used is the spectral power of the signal (

Shimomura et al., 2008). If we regard the skin conductance time series

*SC* as a convolution of a time series of sudomotor firing bursts

*SN* with a time-invariant response function

then, according to the convolution theorem, we can write the Fourier transform [

*FT*] of the skin conductance time series as a product of the

*FT* of nerve firing and response function. This is the same as Eq.

(1) but now we are treating the sudomotor input as a continuous times series (as opposed to a series of discrete events). The overall spectral power of the (

*SCL*-corrected) skin conductance time series will vary with the amplitude of sudomotor firing, while the frequency of sudomotor bursts will influence low frequencies of the spectral power (because the inter-burst interval determines the lower frequencies): an increase in the (low) frequency of bursts will shift the frequency of spectral power in lower ranges. More formally, for a rectangular (sudomotor pulse) wave of duration

*d*, occurring every 1/

*n* seconds (i.e., a burst frequency of

*n*), the Fourier coefficients are:

This simply says that the change in the Fourier coefficients with burst frequency is greatest at low frequencies (low

*i*) because cos(

*iπnd*) decreases with increasing

*i*, given that

*nd* < 1. This is why it has been proposed previously to quantify sympathetic arousal by integrating the spectral power of the SC signal over low frequencies (

Shimomura et al., 2008). However, Eq.

(4) describes the power spectrum of the (unknown)

*SN* and does not directly apply to the

*SC* power spectrum. In fact, Eq.

(3) means that the burst frequency will have its greatest impact on spectral power of the

*SC* when it matches the peak frequencies of the response function. Therefore, the

*SC* power spectrum captures the frequency overlap between the response function and sudomotor firing, but not the sudomotor firing itself.

If the spectral power of the response function is known, it is possible to recover the firing frequency, or even the time series of sudomotor firing using

However, noise and response variability render Eq.

(5) useless for practical purposes (see

Alexander et al., 2005 for a similar deconvolution approach in the time domain that does not account for noise). Although classical methods are available for deconvolution with known noise spectra (e.g., Wiener deconvolution and related approaches), we pursue the time domain formulation in Eq.

(2), because its application does not rely on knowing the noise spectrum.