**2.3.1 The hidden-input metrics**

88 Neuroscience – Dealing with Frontiers

**S T I M U L U S**

m

**u(t)**

**y'''**

b

Fig. 2. State-diagram for the mathematical model of the skin conductance response (SCR):

**S C R**

**y**

**y''**

**y'**

ubiquitous regulatory motif performing sensitive robustness in biological systems (Zhang et al., 2007). The foremost activated positive feedback rapidly induces the "on" state transition of the signaling system, the delayed positive feedback robustly maintains this "on" state, while the negative feedback reinstates the system in the original "off" state, prevents excessive response due to multiple positive

feedback loops, and suppresses noise effects (Kim et al., 2006; Pfeuty & Kaneko, 2009)

y'''+a·y''+b·y'+c·y=m·u(t); y=SCR

c

a

Through the graphical solving of the hidden input problem, relying on the jump discontinuities of the third derivative of the SCR signal (see 2.1), we approximated the initial neural event in the SCR process with series of impulses and square pulses (see Figure 1). The method enables quantification of the four features of the hidden input. In other words, we can define four dimensions in the hidden-input metrics: 1) number of pulses in the hidden input, 2) the amplitude (height, strength) of the pulses, 3) duration of the pulses, and 4) timing of the pulses (e.g. inter-pulse interval). This forms the hidden-input metrics.

The count of pulses in a hidden-input in our sample varied from 1 to 6. The amplitude of the pulses was in the range 0.023-7.74. The duration of the pulses in the hidden-inputs varied in the interval 0.1-2.8 s. The inter-pulse interval in the hidden-inputs was in the range 0.1-3.5 s.

Exploration of the within-subject distribution of values of the described hidden-input measures revealed normal distribution of *pulse duration* and log-normal distribution of *pulse amplitude* <sup>3</sup> (Figure 3).

#### **2.3.2 System parameters of the SCR process**

The set of the parameters which refer to the regulatory or control system aspect of the SCR process is composed of: *input gain*, *fast positive feedback loop gain*, *negative feedback loop gain*, *slow positive feedback loop gain*, *static system gain*, and *fidelity period* (Branković, 2011).

*Input gain.* This parameter corresponds to the constant "m" in the differential equation of the system. It reflects the initial step in the process of the regulation of emotional arousal. We have found the range of values 0.0013 – 0.025 for this parameter.

*Fast positive feedback loop (FPFL) gain.* Of the three feedback loops in the model (Figure 2) the first is proportional to the second derivative of the SCR signal (y'') and it is positive. We denote the parameter that characterizes this feedback loop as fast positive feedback loop gain. It equals the constant "-a" in the differential equation of the system. The found range of values of this parameter in healthy subject is 2.61 – 2.83. The within-subject coefficient of variation of this *fast enhancement* is about 1 %.

*Negative feedback loop (NFL) gain.* The second feedback loop in our model is proportional to the rate of change of the emotional arousal (y') and it is negative. It reflects a feedback inhibition in the arousal process. The parameter that characterizes this feedback inhibition equals the constant "-b" in the differential equation of the system. The found values of this

 3 It is interesting that within-subject variations of the pulse amplitudes sometimes manifested quantal nature (i.e. different values relate as integer multiplications to each other) (see Figure 3).

Assessment of Brain Monoaminergic Signaling

the *system gain* in our sample are 1.13 – 48.57.

the SCR models is about 13 seconds.4

points to the reliability of measurement of these parameters.

other available method including invasive ones.

(Breska et al., 2011; Benedek & Kaernbach, 2010).

**3.1 Evidence for the brain mechanisms involvement in the SCR model** 

We have recently found (Branković, 2011) that different types of SCRs (orienting response, SCR to pleasant emotional stimuli, weak movement, and respiratory SCR) have distinct regulation (i.e. different system parameters). This finding raises the question of adequacy of the assumption that mathematical models of the SCR process describe the sudomotor innervation and sweat gland activity (Alexander et al., 2005; Bach et al., 2009, 2010a, 2010b, 2011; Benedek & Kaernbach, 2010). The doubt has been recently expressed by Bach and

4 The estimation of the duration of *fidelity period* of the SCR system in our sample corresponds well to the empirical data on breaking values of the interstimulus interval in the SCR measurement research

Through Mathematical Modeling of Skin Conductance Response 91

*Static system gain.* The value of low frequency or static system gain of linear time-invariant system can be obtained as a direct result of the MATLAB® function "dcgain". The values of

*Fidelity period.* Some but not all models of the SCRs to pleasant emotional stimuli show potential for resonance. That is why we can not use resonant frequency as a parameter to characterize the all SCR models. Instead, we use the time period of periodic stimulation when the amplitude of response drops 3 dB below (or to 70.7% of) its zero-frequency value (amplitude of response which appears responding to a single stimulus). Expressed in seconds, *fidelity period* has a psychological meaning and face validity. Namely, with a further shortening of the inter-stimulus period in comparison with the *fidelity period* the amplitude of the SCR is rapidly diminishing. In that way, *fidelity period* gives an indication of the noisefiltering characteristics of the system. The found value of the fidelity period in our sample of

Exploration of the within-subject distribution of values of the described system parameters revealed normal distribution of the three *feedback loops' gains* and *fidelity period*, and lognormal distribution of *input gain* and *overall static system gain* (Figure 4). The high intra-trial reliability of the estimations of the feedback loop gains (coefficients of variations: 1-5 %)

**3. The mathematical model of the SCR in the context of neurobiological data**  The stereotyped nature of the SCR waveform has recently inspired several dynamic modeling approaches to the process of SCR. The published models differ in the neurobiological interpretation of the system's parameters. While several research groups consider the peripheral sympathetic sudomotor nervous signal as the impulse input in their models of the SCR (Alexander et al., 2005; Bach et al., 2009, 2010a, 2010b, 2011; Benedek & Kaernbach, 2010), we assumed that the initial neural event and feedback regulatory mechanisms that we could comprise with our models took place in central brain structures (Branković, 2008). In the last case we would be able to derive much more information about the central neural processing from the SCR signal than we used to do, and, what is even more important, to infer about the brain mechanisms in a way that is not achievable by any

parameter are in the range from -2.44 to -2.68. The within-subject coefficient of variation of the *inhibition loop* in the SCR model is about 3 %.

Fig. 3. Within-subject distributions of the hidden-input measures of four different subjects

*Slow positive feedback loop (SPFL) gain.* The third feedback loop in the model is proportional to the actual level of the emotional arousal, actual value of the SCR signal (y) and it is positive. The parameter that characterizes this feedback enhancement equals the constant "-c" in the differential equation of the system. The found range of values of this feedback loop gain is 0.65 – 0.86. The within-subject coefficient of variation of the *slow enhancement* is about 5 %.

parameter are in the range from -2.44 to -2.68. The within-subject coefficient of variation of

Fig. 3. Within-subject distributions of the hidden-input measures of four different subjects

*Slow positive feedback loop (SPFL) gain.* The third feedback loop in the model is proportional to the actual level of the emotional arousal, actual value of the SCR signal (y) and it is positive. The parameter that characterizes this feedback enhancement equals the constant "-c" in the differential equation of the system. The found range of values of this feedback loop gain is 0.65 – 0.86. The within-subject coefficient of variation of the *slow enhancement* is about 5 %.

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

Pulse Duration [sec]

0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Pulse Duration)


<sup>0</sup> 0.05 0.1 <sup>0</sup>

Inter-Pulse Pause [sec]


log (Inter-Pulse Pause)



Pulse Duration/Pause


log (Pulse Duration/Pause)


log (Pulse Amplitude)


Pulse Duration/Pause

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Pulse Duration/Pause)


log (Pulse Amplitude)

0.5 1 1.5 2

0.5 1 1.5 2

0.5 1 1.5 2

> 0.5 1 1.5 2

> 0.5

0.5

1

1

Pulse Amplitude

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

Inter-Pulse Pause [sec]

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

log (Inter-Pulse Pause)


Pulse Amplitude

0.5 1 1.5 2

> 1 2 3

0.5 1 1.5 2

> 0.5 1 1.5 2

> 0.5

0.5

1

1

Pulse Count in Hidden Input

<sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>0</sup>

Pulse Duration [sec]

<sup>0</sup> 0.5 <sup>1</sup> <sup>0</sup>

log (Pulse Duration)


Pulse Count in Hidden Input

0.5 1 1.5 2

0.5 1 1.5 2

0.5 1 1.5 2

the *inhibition loop* in the SCR model is about 3 %.

Pulse Amplitude

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>0</sup>

Inter-Pulse Pause [sec]

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Inter-Pulse Pause)


Pulse Amplitude

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

Inter-Pulse Pause [sec]

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Inter-Pulse Pause)



Pulse Duration/Pause

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>0</sup>

log (Pulse Duration/Pause)


log (Pulse Amplitude)


Pulse Duration/Pause

<sup>0</sup> <sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>0</sup>

log (Pulse Duration/Pause)


log (Pulse Amplitude)

0.5 1 1.5 2

0.5

0.5

2 4 6

0.5 1 1.5 2

> 1 2 3

> 1

1

0.5

0.5

1 2 3

1 2 3

1 2 3

1

1

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>0</sup>

Pulse Duration [sec]

0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Pulse Duration)


Pulse Count in Hidden Input

<sup>2</sup> <sup>4</sup> <sup>6</sup> <sup>0</sup>

Pulse Duration [sec]

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>0</sup>

log (Pulse Duration)


Pulse Count in Hidden Input

5 10 15

> 5

2 4 6

10

*Static system gain.* The value of low frequency or static system gain of linear time-invariant system can be obtained as a direct result of the MATLAB® function "dcgain". The values of the *system gain* in our sample are 1.13 – 48.57.

*Fidelity period.* Some but not all models of the SCRs to pleasant emotional stimuli show potential for resonance. That is why we can not use resonant frequency as a parameter to characterize the all SCR models. Instead, we use the time period of periodic stimulation when the amplitude of response drops 3 dB below (or to 70.7% of) its zero-frequency value (amplitude of response which appears responding to a single stimulus). Expressed in seconds, *fidelity period* has a psychological meaning and face validity. Namely, with a further shortening of the inter-stimulus period in comparison with the *fidelity period* the amplitude of the SCR is rapidly diminishing. In that way, *fidelity period* gives an indication of the noisefiltering characteristics of the system. The found value of the fidelity period in our sample of the SCR models is about 13 seconds.4

Exploration of the within-subject distribution of values of the described system parameters revealed normal distribution of the three *feedback loops' gains* and *fidelity period*, and lognormal distribution of *input gain* and *overall static system gain* (Figure 4). The high intra-trial reliability of the estimations of the feedback loop gains (coefficients of variations: 1-5 %) points to the reliability of measurement of these parameters.
