**2.1 Solution for the "hidden input" problem in the SCR modeling**

The prerequisite for the system identification procedure which is performed using the *System Identification Toolbox* included in the mathematical software MATLAB® is that both output and input signal of the system are known. But, while the output signal of the SCR system is observable and measurable, the input signal is not directly measurable. Therefore, a separate scientific task is to find an appropriate mathematical representation of the driving input to the SCR system, i.e. to solve the "hidden input" problem.

A hint to visualize the initial forcing event in the SCR process came from mathematical dealing with differential equations with discontinuous forcing functions: the highest derivative of the solution appearing in the differential equation has jump discontinuities at

<sup>1</sup> There is an aspect of the present approach which we want to point out. Many findings and concepts developed in neuroscience during the last two decades have not yet been operationalized for clinical use. The method which is presented here could contribute to lessen this gap translating some concepts of modern neuroscience and making them visible in everyday clinical work. The following concepts are involved: tonic and phasic function of neurotransmitters, gain modulation of neural activity by monoaminergic inervation, feedback regulation in the arousal process, resonance and frequency characteristics of neural systems, central neural code, and signal-to-noise ratio of neural signals.

Assessment of Brain Monoaminergic Signaling

Likova, 2011).

of SCR input-output data.

interactions is shown in Figure 2.

the system:

Through Mathematical Modeling of Skin Conductance Response 87

The method showed simple to be performed and happened to be crucial for the feasibility and accuracy for the system identification approach to SCR. It enables easily obtaining almost perfect fit (>97%) between measured SCR and the corresponding simulated model's output. Recently we have automated the graphical procedure of estimating the neural input for the SCR system through a program application in the MATLAB® software environment (which will be available online at www.psychophysiological-computation.com and www.psychophysiological-computation.info). The tool could be regarded as a "temporal microscope" for the neural signals which had initiated the recorded SCRs (cf. Tyler &

**2.2 Positive and negative feedback loops drive the SCR to pleasant emotional stimuli**  System identification of the SCRs to pleasant emotional stimuli (textual fragments of short stories, see *Subjects and Methods* in Branković, 2011) has been performed using the *System Identification Toolbox* v. 6.0 included in the mathematical software MATLAB® v.7.0 (R14). We defined the model structure and set the fixed parameters and nominal (initial) values of the free parameters that enabled successful system identification of the SCRs to pleasant emotional stimuli for 27 healthy volunteers in the sample (see *Subjects and Methods* in Branković, 2011). The third-order state-space model appeared to be appropriate for two reasons. First, it yields accurate mathematical modeling of SCR providing high fit (>97%) between measured and simulated outputs, and second, it does it in a reliable, robust way, guarantying solution, convergence of the software algorithm dealing with the great variety

Through the system identification procedure we estimate the parameters of the state-space models, which correspond to the coefficients in the equivalent linear differential equation of

y''' + a·y'' + by' + cy = mu(t) That enables us to regard the process of emotional arousal as an interaction among the following signals: 1) neural input to the SCR system, i.e. u(t), 2) the third derivative of the SCR signal (y'''), 3) the second derivative of the SCR signal (y''), 4) the first derivative of the SCR signal (y'), and 4) the SCR signal itself (y=SCR). Graphical representation of the

Through modeling of over thousand SCRs to pleasant emotional stimuli we were consistently encountering the same resulting feedback structure: the fast positive feedback loop (occurring early in the feedback scheme), the negative (in the middle), and the slow positive feedback

2 Interlinked positive and negative feedback loops have been identified in many biological systems as the key regulatory scheme (Tsai et al., 2008; Brandman et al., 2005; Brandman & Meyer, 2008). The wide occurrence of dual-time (fast and slow) positive feedback loops combined with a negative feedback loop has motivated investigations of properties and functions of that regulatory motif during the last decade. Mathematical simulations revealed that such coupled feedback circuits enable systems in noisy environment to produce perfect responses with respect to response duration and amplitude. The dualtime switch, consisting of interconnected fast and slow positive feedback loops, has been suggested as a

loop (originating at the peripheral end of the regulatory chain in the SCR process)2.

the same points as the forcing function, but the solution itself and its lower derivatives are continuous even at those points (Boyce & DiPrima, 2001). Considering the SCR as an output (solution) of a serial integration (in both mathematical and neurophysiological (computation) meaning of the word) and having in mind this simple mathematical rule we assumed that a hidden neural input left its trace and had a fingerprint on the highest derivative of the SCR signal.

Indeed, an inspection of the SCR signal and its derivatives reveals that while the SCR signal and its first two derivates appear to be continuous, jump discontinuities features the third derivative of the SCR signal. This observation suggested that we could be able to read the traces of the discontinuous forcing function (the hidden input) in the third derivative of the SCR signal. The task is to determine an approximation of the unmeasured input signal looking at its fingerprints on the third derivative of the SCR (Figure 1). We approximated the hidden inputs with series of impulses and square pulses. The timing and duration of the pulses have been determined according to the jump discontinuities of the third derivative of the SCR signal. We also exploited the possibility to quantify the strength of the hidden input through measuring the magnitudes of these jump discontinuities. The height of the putative input pulses (the strength of the hidden neural input approximations) has been determined according to the height of the jump discontinuities of the third derivative of the SCR signal.

Fig. 1. Hidden-inputs to the SCR system determined observing the jump discontinuities in the third derivative of the SCR signal (SCR'''). Left column: the hidden-input consisting of two pulses of different strength and duration. Middle column: the input consisting of an initial pulse and two impulses of different strength. Right column: the input consisting of an initial pulse and three impulses.

5:26.5 5:27 5:27.5 5:28 5:28.5 5:29 5:29.5 5:30 5:30.5 5:31 5:31.5 5:32 5:32.5 5:33

7:58 7:58.5 7:59 7:59.5 8:00 8:00.5 8:01 8:01.5 8:02 8:02.5 8:03 8:03.5 8:04 8:04.5 8:05 8:05.5 8:06

5:07.5 5:08 5:08.5 5:09 5:09.5 5:10 5:10.5 5:11 5:11.5 5:12 5:12.5 5:13 5:13.5 5:14 5:14.5 5:15

the same points as the forcing function, but the solution itself and its lower derivatives are continuous even at those points (Boyce & DiPrima, 2001). Considering the SCR as an output (solution) of a serial integration (in both mathematical and neurophysiological (computation) meaning of the word) and having in mind this simple mathematical rule we assumed that a hidden neural input left its trace and had a fingerprint on the highest

Indeed, an inspection of the SCR signal and its derivatives reveals that while the SCR signal and its first two derivates appear to be continuous, jump discontinuities features the third derivative of the SCR signal. This observation suggested that we could be able to read the traces of the discontinuous forcing function (the hidden input) in the third derivative of the SCR signal. The task is to determine an approximation of the unmeasured input signal looking at its fingerprints on the third derivative of the SCR (Figure 1). We approximated the hidden inputs with series of impulses and square pulses. The timing and duration of the pulses have been determined according to the jump discontinuities of the third derivative of the SCR signal. We also exploited the possibility to quantify the strength of the hidden input through measuring the magnitudes of these jump discontinuities. The height of the putative input pulses (the strength of the hidden neural input approximations) has been determined according to the height of the jump discontinuities of the third derivative of the SCR signal.

ja-price

danica ciric2

10/6/2005 4:53:21.077 PM <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>0</sup>

7:58 7:58.5 7:59 7:59.5 8:00 8:00.5 8:01 8:01.5 8:02 8:02.5 8:03 8:03.5 8:04 8:04.5 8:05 8:05.5 8:06

y ''' ( V )

0.1 0.2 0.3 0.4 0.5 0.6 0.7

> -1.0 -0.5 0.0 0.5

y '' ( V )


y ' ( V )


S C R ( V ) -0.2 -0.0 0.2 0.4

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Fig. 1. Hidden-inputs to the SCR system determined observing the jump discontinuities in the third derivative of the SCR signal (SCR'''). Left column: the hidden-input consisting of two pulses of different strength and duration. Middle column: the input consisting of an initial pulse and two impulses of different strength. Right column: the input consisting of an

5:26.5 5:27 5:27.5 5:28 5:28.5 5:29 5:29.5 5:30 5:30.5 5:31 5:31.5 5:32 5:32.5 5:33

derivative of the SCR signal.

initial pulse and three impulses.

SCR (y)

SCR'''

HIDDEN INPUT (u)

y ''' ( ) -5 0 5

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

> y '' ( )


y ' ( )


S C R ()

8.4 8.6 8.8 9.0 9.2 ja-priceSel

1/17/2006 3:49:33.156 PM <sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>0</sup>

y ''' ( )

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

> -6 -4 -2 0 2 4 6

y '' ( )


y ' ( ) -0.1 -0.0 0.1 0.2 0.3 0.4

S C R ()

8.4 8.6 8.8 9.0 9.2

5:07.5 5:08 5:08.5 5:09 5:09.5 5:10 5:10.5 5:11 5:11.5 5:12 5:12.5 5:13 5:13.5 5:14 5:14.5 5:15

SCR''

SCR'

The method showed simple to be performed and happened to be crucial for the feasibility and accuracy for the system identification approach to SCR. It enables easily obtaining almost perfect fit (>97%) between measured SCR and the corresponding simulated model's output. Recently we have automated the graphical procedure of estimating the neural input for the SCR system through a program application in the MATLAB® software environment (which will be available online at www.psychophysiological-computation.com and www.psychophysiological-computation.info). The tool could be regarded as a "temporal microscope" for the neural signals which had initiated the recorded SCRs (cf. Tyler & Likova, 2011).

#### **2.2 Positive and negative feedback loops drive the SCR to pleasant emotional stimuli**

System identification of the SCRs to pleasant emotional stimuli (textual fragments of short stories, see *Subjects and Methods* in Branković, 2011) has been performed using the *System Identification Toolbox* v. 6.0 included in the mathematical software MATLAB® v.7.0 (R14). We defined the model structure and set the fixed parameters and nominal (initial) values of the free parameters that enabled successful system identification of the SCRs to pleasant emotional stimuli for 27 healthy volunteers in the sample (see *Subjects and Methods* in Branković, 2011). The third-order state-space model appeared to be appropriate for two reasons. First, it yields accurate mathematical modeling of SCR providing high fit (>97%) between measured and simulated outputs, and second, it does it in a reliable, robust way, guarantying solution, convergence of the software algorithm dealing with the great variety of SCR input-output data.

Through the system identification procedure we estimate the parameters of the state-space models, which correspond to the coefficients in the equivalent linear differential equation of the system:

$$\mathbf{y}^{\prime\prime\prime} + \mathbf{a} \cdot \mathbf{y}^{\prime\prime} + \mathbf{b} \cdot \mathbf{y}^{\prime} + \mathbf{c} \cdot \mathbf{y} = \mathbf{m} \cdot \mathbf{u}(\mathbf{t})$$

That enables us to regard the process of emotional arousal as an interaction among the following signals: 1) neural input to the SCR system, i.e. u(t), 2) the third derivative of the SCR signal (y'''), 3) the second derivative of the SCR signal (y''), 4) the first derivative of the SCR signal (y'), and 4) the SCR signal itself (y=SCR). Graphical representation of the interactions is shown in Figure 2.

Through modeling of over thousand SCRs to pleasant emotional stimuli we were consistently encountering the same resulting feedback structure: the fast positive feedback loop (occurring early in the feedback scheme), the negative (in the middle), and the slow positive feedback loop (originating at the peripheral end of the regulatory chain in the SCR process)2.

<sup>2</sup> Interlinked positive and negative feedback loops have been identified in many biological systems as the key regulatory scheme (Tsai et al., 2008; Brandman et al., 2005; Brandman & Meyer, 2008). The wide occurrence of dual-time (fast and slow) positive feedback loops combined with a negative feedback loop has motivated investigations of properties and functions of that regulatory motif during the last decade. Mathematical simulations revealed that such coupled feedback circuits enable systems in noisy environment to produce perfect responses with respect to response duration and amplitude. The dualtime switch, consisting of interconnected fast and slow positive feedback loops, has been suggested as a

Assessment of Brain Monoaminergic Signaling

**2.3 Emerging metrics of the SCR process** 

system itself (Branković, 2011).

**2.3.1 The hidden-input metrics** 

*amplitude* <sup>3</sup> (Figure 3).

**2.3.2 System parameters of the SCR process** 

variation of this *fast enhancement* is about 1 %.

Through Mathematical Modeling of Skin Conductance Response 89

Since our mathematical dealing with the SCR process encompasses two steps – revealing the hidden input and system identification of the SCR – the approach yields to two sets of parameters. One set includes parameters which characterize the hidden initial neural input to the SCR system. The other set consists of parameters which characterize the SCR control

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* 

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*,

*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

*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

*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).

*slow positive feedback loop gain*, *static system gain*, and *fidelity period* (Branković, 2011).

have found the range of values 0.0013 – 0.025 for this parameter.

Fig. 2. State-diagram for the mathematical model of the skin conductance response (SCR): y'''+a·y''+b·y'+c·y=m·u(t); y=SCR

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)
