# An Adaptive Temporal-Causal Network Model for Stress Extinction Using Fluoxetine

S. Sahand Mohammadi Ziembari

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S. Sahand Mohammadi Ziembari. An Adaptive Temporal-Causal Network Model for Stress Extinction Using Fluoxetine. 15th IFIP International Conference on Artificial Intelligence Applications and Innovations (AIAI), May 2019, Hersonissos, Greece. pp.107-119, 10.1007/978-3-030-19823-7_8 . hal-02331346

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# An adaptive Temporal-Causal Network Model for Stress Extinction Using Fluoxetine

S. Sahand Mohammadi Ziarabi

Social AI Group, Vrije Universiteit Amsterdam
sahandmohammadiziabari@gmail.com

**Abstract.** In this paper, an adaptive temporal causal network model based on drug therapy named fluoxetine to decrease the stress level of post-traumatic stress disorder is presented. The stress extinction is activated by a cognitive drug therapy (here fluoxetine) that uses continuous usage of medicine. The aim of this therapy is to reduce the connectivity between some components inside the brain which are responsible for causing stress. This computational model aspires to realistically demonstrate the activation of different portions of brain when the therapy is applied. The cognitive model starts with a situation of strong and continuous stress in an individual and after using fluoxetine the stress level begins to decrease over time. As a result, the patient will have a reduced stress level compared to not using drug.

**Keywords:** temporal-causal network model, cognitive, extreme emotion, drug-therapy, fluoxetine.

## 1 Introduction

Stress is a vital response to physical and emotional threats with strong roots in human evolution. Stress is important to protect humans from dangerous conditions, where in early history it could have life-or-death consequences. A particular situation might trigger a fight or flight reaction, which could result in unnecessarily avoiding certain (social) circumstances. As it has been described in [1] depression is one of the most grueling psychiatric sicknesses and may decrease life-time expectancy with up to 20%. Recent literature [8] shows that fluoxetine suppresses or decreases synaptic changes associated with stress. It has been also mentioned that fluoxetine relatively suppresses the impact of stress on the infusion of synaptic plasticity in the medial prefrontal cortex which is responsible for receiving direct fibers from the hippocampus. There are some previous temporal causal network-oriented modeling literatures for decreasing stress have been proposed [20-30].

The paper is organized as follows. In Section 2 the underlying neurological principles concerning the parts of the brain involved in stress and in the suppression of stress are addressed. In Section 3 the integrative temporal-causal network model is introduced. In Section 4 the results of the simulation model are discussed, in section 5 the
mathematical analysis of the model is presented and eventually in the last section a
conclusion is presented.

## 2 Underlying Neurological Principles

In many recent research literature [12, 13] it has been proven that fluoxetine decreases or suppresses changes in synapses caused by stress. In many researches [55-57] it has been proven that repeated stressful conditions and experiences bring a remarkable effect on neural plasticity in many brain components, especially in limbic structures like hippocampal changes, prefrontal cortex (PFC), and Amygdala. As it has been clearly mentioned in [13]:

‘Acute stress inhibits long-term potentiation (LTP) at synapses from the hippocampus to prefrontal cortex in the rat, a model of the dysfunction in the anterior cingulate/orbitofrontal cortices which has been observed in human depression.

In major depressive disorder, decreased blood flow and metabolism have been regularly described in multiple areas of the prefrontal cortex (PFC) with occasional changes in the hippocampal region. Conversely, a beneficial response to antidepressants has been associated with reduced blood flow in the hippocampus and a return to baseline metabolism level or increase in blood flow in the anterior cingulate cortex.

Plasticity at hippocampal to PFC synapses can be regulated up and down, as assessed by long-term potentiation (LTP) and long-term depression (LTD), depending on specific patterns of after-activation and this circuit contributes to working memory processes.

Antidepressant effects may be obtained by several mechanisms, such as inhibition of serotonin uptake, for fluoxetine.’ [13]

Also, previous studies [15-16] revealed that chronic stress changes dendritic morphology not just in the hippocampus, but also in the mPFC (medial Prefrontal Cortex).

‘Depression is said to be caused by chronically low levels of serotonergic transmission. SSRIs interfere with the activity of the serotonin transporter (5-HTT), a reuptake molecule that removes serotonin from the synapses. The putative low levels of synaptic serotonin in the depressed patient are elevated, and depression is relieved.

These manipulations of serotonin levels have little effect on mood except in individuals who are depressed or recently recovered from depression.’ [11, pp.1]

‘Studies of neurotransmitter release with microdialysis have demonstrated that acute olanzapine significantly increases both dopamine and norepinephrine levels in rat prefrontal cortex, nucleus accumbens, and striatum, and the combination of olanzapine plus fluoxetine produces a greater increase in levels of dopamine and norepinephrine in the rat prefrontal cortex than fluoxetine alone.’ [2], pp.776.

The functionality of chronic stress on brain parts particularly on the Hippocampus is mentioned in [3, 4].
‘The volume of the hippocampus is decreased in patients with depression or posttraumatic stress disorder.’ [3], pp. 975

‘The reduction in hippocampal volume is inversely proportional to the amount of time a patient is medicated with an antidepressant, and reduced hippocampal volume is partially reversed after antidepressant treatment.’ [4], pp.577

‘In the striatum, there was a tendency for an increase in the number of BrdU-positive cells that is similar in magnitude to that in hippocampus. This effect is consistent with highly significant and robust induction of cell proliferation reported in a recent study, and the greater increase could be due to the higher dose of olanzapine used (10mg/kg) relative to the current study (2 mg/kg). In the current study, we found that the combination of olanzapine plus fluoxetine did not produce a greater increase in the number of BrdU-positive cells than either drug alone. This suggests that fluoxetine alone would be sufficient to produce a maximum response and therefore could not account for the augmentation that has been observed clinically; however, the clinical approach has been to add olanzapine after a patient has failed to respond to an SSRI like fluoxetine’ [4], pp.577.

Many researches illustrate that the hippocampus and other sections in the medial temporal lobe are interfered with detection of novelty [16, 17]. In other research [18] it has been proved that using of antidepressant drugs (Ads) will enhance the levels of extracellular epinephrine and serotonin.

‘The ability to detect unusual events occurring in the environment is essential for survival. Several studies have pointed to the hippocampus as a key brain structure in novelty detection, a claim substantial by its wide access to sensory information through the entorhinal cortex and also distinct aspects of its intrinsic circuity.’ [15], pp.18286

In [10] it has been shown that small amounts of fluoxetine might block stress-facilitated hippocampal LTD and eventually helps in memory retrieval impairment.

‘Chronic fluoxetine treatment reinstates ocular dominance plasticity in the primary visual cortex of adult rats, a form of developmentally regulated plasticity that is significantly reduced in the mature brain, and enhances long-term potentiation (LTP) in the dentate gyrus of adult mice. Results show that chronic fluoxetine treatment suppresses LTP in the primary auditory cortex and hippocampus of adult rats.

It has been well documented that exposure to acute stress impairs LTP and facilitate LTD in rats, as well as to produce learning and memory impairment in rats and monkeys. A single systematic injection of fluoxetine is able to reverse the impairment in LTP at synapses from the hippocampus to prefrontal cortex in the rats, caused by stress on elevated platform.’ [10], pp.1

In [10, pp.7] the influences of antidepressant agents such as fluoxetine is clearly mentioned:
‘The chronic effects of antidepressant agents including fluoxetine, are involved in the regulation of intracellular transduction pathways, implicating changes in the cyclic adenosine monophosphate (cAMP) second messenger system, cAMP response element binding protein (CREB) and brain-derived neurotrophic factor (BDNF) in antidepressant action.’

In [14] explicitly the effect of chronic stress on medial Prefrontal Cortex and Amygdala has been mentioned:

‘Chronic stress significantly suppressed cytogenesis in the mPFC and neurogenesis in the dentate gyrus, but had minor effect in nonlimbic structures. Fluoxetine treatment counteracted the inhibitory effect of stress. Hemispheric comparison revealed that the rate of cytogenesis was significantly higher in the left mPFC of control animals, whereas stress inverted this asymmetry, yielding a significantly higher incidence of newborn cells in the right mPFC. Fluoxetine treatment abolished hemispheric asymmetry in both control and stressed animals.

Structural alterations including suppressed dentate neurogenesis may contribute to the pathogenesis of depression.

Antidepressant treatment with fluoxetine or electroconvulsive seizure modulates cell proliferation not only in the dentate gyrus, but also in the medial PFC (mPFC) in adult rats. (pp.1490)'

## 3 The Temporal-Causal Network Model

First the Network-Oriented Modelling approach used to model the integrative overall process is briefly explained. As discussed in detail in [17, Ch 2, 18, 19] this approach is based on temporal-causal network models which can be represented at two levels: by a conceptual representation and by a numerical representation. A conceptual representation of a temporal-causal network model in the first place involves representing in a declarative manner states and connections between them that represent (causal) impacts of states on each other, as assumed to hold for the application domain addressed. The states are assumed to have (activation) levels that vary over time. In reality, not all causal relations are equally strong, so some notion of *strength of a connection* is used. Furthermore, when more than one causal relation affects a state, some way to *aggregate multiple causal impacts* on a state is used. Moreover, a notion of *speed of change* of a state is used for timing of the processes. These three notions form the defining part of a conceptual representation of a temporal-causal network model:

* **Strength of a connection** $\omega_{X,Y}$. Each connection from a state $X$ to a state $Y$ has a *connection weight value* $\omega_{X,Y}$ representing the strength of the connection, often between 0 and 1, but sometimes also below 0 (negative effect) or above 1.

* **Combining multiple impacts on a state** $c_Y(..)$. For each state (a reference to) a *combination function* $c_Y(..)$ is chosen to combine the causal impacts of other states on state $Y$.

* **Speed of change of a state** $\eta_Y$. For each state $Y$ a *speed factor* $\eta_Y$ is used to represent how fast a state is changing upon causal impact.
Combination functions can have different forms, as there are many different approaches possible to address the issue of combining multiple impacts. Therefore, the Network-Oriented Modelling approach based on temporal-causal networks incorporates for each state, as a kind of label or parameter, a way to specify how multiple causal impacts on this state are aggregated by some combination function. For this aggregation a number of standard combination functions are available as options and a number of desirable properties of such combination functions have been identified.

Fig. 1 represents the conceptual representation of the temporal-causal network mode. The components of the conceptual representation shown in Fig. 1 are explained here. The state $w_{sc}$ shows the world state of the contextual stimulus $c$. The states $s_{sc}$ and $s_{ee}$ are the sensor state for the context $c$ and sensor state of the body state $ee$ for the extreme emotion. The states $s_{rc}$ and $s_{ee}$ are the sensory representation of the contextual stimulus $c$ and the extreme emotion, respectively. The state $s_{ee}$ is a stimulus influencing the activation level of the preparation state. Furthermore, $p_{ee}$ is the preparation state of an extreme emotional response to the sensory representation $s_{sc}$ of the context $c$, and $f_{ee}$ shows the feeling state associated to this extreme emotion. The state $e_{ee}$ indicates the execution of the body state for the extreme emotion. All these relate to the affective processes. The (cognitive) goal state shows the goal for absorbing fluoxetine in the body. The (cognitive) state $p_{pil}$ is the preparation state of taking a pill (here fluoxetine). The state $e_{pil}$ is the execution state of taking pill (fluoxetine). The other states relate to biological brain parts (Norepinephrine, Hippocampus, Thalamus, Serotonin, Prefrontal Cortex, Amygdala, Lateral Cerebellum, Striatum) which are involved in the stress condition, and in the influence of the fluoxetine applied.

<caption>**Table 1** Explanation of the states in the model</caption>
<table><tr><td>X<sub>1</sub></td><td>w<sub>ee</sub></td><td>World (body) state of extreme emotion ee</td></tr><tr><td>X<sub>2</sub></td><td>ss<sub>ee</sub></td><td>Sensor state of extreme emotion ee</td></tr><tr><td>X<sub>3</sub></td><td>w<sub>c</sub></td><td>World state for context c</td></tr><tr><td>X<sub>4</sub></td><td>ss<sub>c</sub></td><td>Sensor state for context c</td></tr><tr><td>X<sub>5</sub></td><td>sr<sub>ee</sub></td><td>Sensory representation state of extreme emotion ee</td></tr><tr><td>X<sub>6</sub></td><td>sr<sub>c</sub></td><td>Sensory representation state of context c</td></tr><tr><td>X<sub>7</sub></td><td>f<sub>ee</sub></td><td>Feeling state for extreme emotion ee</td></tr><tr><td>X<sub>8</sub></td><td>ps<sub>ee</sub></td><td>Preparation state for extreme emotion ee</td></tr><tr><td>X<sub>9</sub></td><td>es<sub>ee</sub></td><td>Execution state (bodily expression) of extreme emotion ee</td></tr><tr><td>X<sub>10</sub></td><td>goal</td><td>Goal of using fluoxetine</td></tr><tr><td>X<sub>11</sub></td><td>ps<sub>pil</sub></td><td>Preparation state of using pill</td></tr><tr><td>X<sub>12</sub></td><td>es<sub>pil</sub></td><td>Execution of using pill</td></tr><tr><td>X<sub>13</sub></td><td>Norepinephrine</td><td>Brain part</td></tr><tr><td>X<sub>14</sub></td><td>Hippocampus</td><td>Brain part</td></tr><tr><td>X<sub>15</sub></td><td>Thalamus</td><td>Brain part</td></tr><tr><td>X<sub>16</sub></td><td>Serotonin</td><td>Brain part</td></tr><tr><td>X<sub>17</sub></td><td>Prefrontal Cortex</td><td>Brain part</td></tr><tr><td>X<sub>18</sub></td><td>Amygdala</td><td>Brain part</td></tr><tr><td>X<sub>19</sub></td><td>Lateral Cerebellum</td><td>Brain part</td></tr><tr><td>X<sub>20</sub></td><td>Striatum</td><td>Brain part</td></tr><tr><td>X<sub>21</sub></td><td>ps<sub>act</sub></td><td>Preparation of action inside the brain</td></tr></table>

The connection weights $\omega_i$ in Fig.1 are as follows. The sensor states $ss_{ee}$, $ss_{cc}$ have two incoming connections from $w_{s_{ee}}$ and $w_{s_{c}}$ (weights $\omega_1, \omega_2$). The world state of extreme emotion $w_{s_{ee}}$ has one arriving connection from $es_{ee}$, $\omega_{11}$ as a body-loop with weight. The sensory representation state of an extreme emotion $sr_{s_{ee}}$ has an incoming connection weights $\omega_8$ from state preparation state of an extreme emotion $ps_{s_{ee}}$. The feeling state $f_{s_{ee}}$ has one outgoing connection weight $\omega_5$ from $sr_{s_{ee}}$. The preparation state of an extreme emotion $ps_{ee}$ has two incoming connection weights $\omega_{36}, \omega_{37}$ from states Striatum and $es_{act}$, respectively. The preparation state of an extreme emotion $ps_{ee}$ has three outgoing connection weights, $es_{ee}$, Thalamus, and the connection weight between states Hippocampus and Prefrontal Cortex, ($\omega_{10}, \omega_{15}, \omega_{20}$) respectively.

The goal has one arriving connection weight from the sensory representation $sr_{s_{ee}}$ ($\omega_{27}$) and preparation state $ps_{pil}$ an entering connection from the goal with weight $\omega_{28}$. The execution of taking the drug (here fluoxetine) is named $es_{pil}$, and has an entering connection weight $\omega_{29}$ from preparation state of taking $ps_{pil}$. The state Thalamus has three entering connection weights $\omega_{12}, \omega_{14}$ and $\omega_{22}$ from preparation state of extreme emotion $ps_{ee}$, Hippocampus and Amygdala, respectively. The Norepinephrine of brain has an arriving connection weight $\omega_{16}$ from $es_{pil}$. The Hippocampus in brain has four incoming connection weights, $\omega_{17}, \omega_{24}, \omega_{15}$ and $\omega_{13}$ from Serotonin, Prefrontal Cortex, Norepinephrine and Thalamus. Note that the connection weight between states Prefrontal Cortex and Hippocampus is adaptive and using Hebbian learning means through time will be changed. The state Serotonin has an arriving connection from $es_{pil}$ ($\omega_{19}$). Prefrontal Cortex state has an incoming connection weight from amygdala with $\omega_{25}$ and three outgoing connection weights to Amygdala, Lateral Cerebellum and $ps_{act}$, $\omega_{26}, \omega_{30}$, and $\omega_{31}$. The state Lateral Cerebellum has two incoming connection weights from Prefrontal Cortex and Amygdala $\omega_{30}, \omega_{32}$, respectively and it has an outgoing connection weight to Striatum $\omega_{34}$. The state Striatum has two outgoing connection weights to $ps_{act}$,
ps<sub>ee</sub> named ω<sub>35</sub>, ω<sub>36</sub>. Finally, the state ps<sub>act</sub> has an outgoing connection weight to ps<sub>ee</sub> named ω<sub>37</sub>.

<figure><img src="image_2.png" /><figcaption>Figure 1. Conceptual representation of the temporal-causal network model</figcaption></figure>

This conceptual representation was transformed into a numerical representation as follows [17, Ch 2, 18, 19]:

* at each time point *t* each state *Y* in the model has a real number value in the interval [0, 1], denoted by *Y*(*t*)

* at each time point *t* each state *X* connected to state *Y* has an impact on *Y* defined as **impact**<sub>*X*,*Y*</sub>(*t*) = ω<sub>*X*,*Y*</sub> *X*(*t*) where ω<sub>*X*,*Y*</sub> is the weight of the connection from *X* to *Y*

* The aggregated impact of multiple states *X*<sub>*i*</sub> on *Y* at *t* is determined using a combination function **c**(*Y*):

$$
\begin{align*}
\text{aggimpact}_Y(t) &= c_Y(\text{impact}_{X_1,Y}(t), \dots, \text{impact}_{X_k,Y}(t)) \\
&= c_Y(\omega_{X_1,Y}X_1(t), \dots, \omega_{X_k,Y}X_k(t))
\end{align*}
$$

where *X*<sub>*i*</sub> are the states with connections to state *Y*

* The effect of **aggimpact**<sub>*Y*</sub>(*t*) on *Y* is exerted over time gradually, depending on speed factor η*Y*:
$$Y(t+\Delta t) = Y(t) + \eta_Y [\text{aggimpact}_Y(t) - Y(t)] \Delta t$$

$$\text{or} \quad \frac{dY(t)}{dt} = \eta_Y [\text{aggimpactY}(t) - Y(t)]$$

* Thus, the following *difference* and *differential equation* for Y are obtained:

$$Y(t+\Delta t) = Y(t) + \eta_Y [c_Y(\omega_{X1}, \omega_{X1}(t), \dots, \omega_{Xk}, \omega_{Xk}(t)) - Y(t)] \Delta t$$

$$\frac{dY(t)}{dt} = \eta_Y [c_Y(\omega_{X1}, \omega_{X1}(t), \dots, \omega_{Xk}, \omega_{Xk}(t)) - Y(t)]$$

For states the following combination functions $c_Y(...)$ were used, the identity function $\text{id}(.)$ for states with impact from only one other state, and for states with multiple impacts the scaled sum function $\text{ssum}_\lambda(...)$ with scaling factor $\lambda$, and the advanced logistic sum function $\text{alogistic}_{\sigma, \tau}(...)$ with steepness $\sigma$ and threshold $\tau$.

$$\text{id}(V) = V$$

$$\text{ssum}_\lambda(V_1, \dots, V_k) = (V_1 + \dots + V_k)/\lambda$$

$$\text{alogistic}_{\sigma, \tau}(V_1, \dots, V_k) = \left[ \left( \frac{1}{1+e^{-\sigma(V_1+\dots+V_k-\tau)}} \right) - \frac{1}{1+e^{\sigma\tau}} \right] (1+e^{-\sigma\tau})$$

## 4 Example Simulation

The simulation results of the cognitive temporal causal network model, which was constructed based on the neurological science which contains qualitative empirical information (such as fMRI) both for the mechanism by which the brain components work and for emerging result of the processes, has been shown in Figure 2.

Therefore, one can imply that the best option for declining the stress level has been chosen, given the usage of fluoxetine. The model used the Matlab codes which have been implemented in [22]. Using appropriate connections weights make the model numerical and adapted to qualitative empirical information. Table 2 illustrates the connection weights that has been used, where the values for are initial values as these weights are adapted over time. The time step was $\Delta t = 1$. The scaling factors $\lambda_i$ for the nodes with more than one arriving connection weights are mentioned in Table 2. At first, an external world state of an extreme emotion-stimuli context c (represented by $w_{sc}$) will influence the affective internal states of the individual by influencing the emotional response $e_{see}$ (via $s_{sc}$, $s_{rc}$, and $p_{see}$) conducted to manifest the extreme emotion by body state $w_{see}$. As a consequence, the stressed individual senses the extreme emotion (and at the same time all the biological brain components increased over time), so as a cognitive process, as a next step the goal becomes active to decrease this stress level by using fluoxetine at time around 300.

<table><caption>**Table 2** Connection weights and scaling factors for the example simulation</caption><thead><tr><th>Connection weight</th><th>ω<sub>1</sub></th><th>ω<sub>2</sub></th><th>ω<sub>3</sub></th><th>ω<sub>4</sub></th><th>ω<sub>5</sub></th><th>ω<sub>6</sub></th><th>ω<sub>7</sub></th><th>ω<sub>8</sub></th><th>ω<sub>10</sub></th><th>ω<sub>11</sub></th><th>ω<sub>12</sub></th><th>ω<sub>13</sub></th></tr></thead><tbody><tr><th>Value</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><th>Connection Weight</th><td>ω<sub>14</sub></td><td>ω<sub>15</sub></td><td>ω<sub>16</sub></td><td>ω<sub>17</sub></td><td>ω<sub>18</sub></td><td>ω<sub>19</sub></td><td>ω<sub>20</sub></td><td>ω<sub>21</sub></td><td>ω<sub>22</sub></td><td>ω<sub>23</sub></td><td>ω<sub>24</sub></td><td>ω<sub>25</sub></td></tr><tr><th>Value</th><td>1</td><td>1</td><td>1</td><td>1</td><td>-0.7</td><td>1</td><td>0.7</td><td>1</td><td>1</td><td>0.4</td><td>0.4</td><td>1</td></tr><tr><th>Connection Weight</th><td>ω<sub>26</sub></td><td>ω<sub>27</sub></td><td>ω<sub>28</sub></td><td>ω<sub>29</sub></td><td>ω<sub>30</sub></td><td>ω<sub>31</sub></td><td>ω<sub>32</sub></td><td>ω<sub>33</sub></td><td>ω<sub>34</sub></td><td>ω<sub>35</sub></td><td>ω<sub>35</sub></td><td>ω<sub>35</sub></td></tr><tr><th>Value</th><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>-0.9</td></tr><tr><th>Connection Weight</th><td>ω<sub>37</sub></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Value</th><td>-0.9</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><thead><tr><th>state</th><th>X<sub>5</sub></th><th>X<sub>8</sub></th><th>X<sub>14</sub></th><th>X<sub>15</sub></th><th>X<sub>17</sub></th><th>X<sub>18</sub></th><th>X<sub>19</sub></th><th>X<sub>20</sub></th><th>X<sub>21</sub></th></tr></thead><tbody><tr><th>λ<sub>i</sub></th><td>2</td><td>3</td><td>3.4</td><td>3</td><td>1.4</td><td>2</td><td>2</td><td>2</td><td>2</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Simulation results for temporal-causal network modeling of the therapy by fluoxetine</figcaption></figure>

As a biological process, the goal and in further steps, execution of taking drugs triggers the changes and suppression of execution of stress at the first state and this affects other brain components to be less active around time 300 and for stress level around 600. However, this effect is just temporary, and as the stressful context *c* still is present all the time, after a while the stress level goes up again, which in turn again leads to activation of the goal and performing another desire or prescription of eating fluoxetine, and so on and on repeatedly until the person or the doctor decides to stop taking drugs. The fluctuation in Figure 2 shows how in real life the repeated usage of medicine (here fluoxetine) decreases the stress level over each intake. It is worth to tell that all of this fluctuation is produced internally by the model; the environment is constant, external input for the model is only the constant world state $w_s$. Therefore, based on the simulation results it is illustrated that the model for the drug therapy (fluoxetine) works as expected.

In Figure 3, the equilibrium situation where there is not any active goal (intake) has been shown. Based on this figure, when there is no intake (the goal is blocked in an artificial manner here), the stress level and activity of brain parts go up and stay high.
<figure><img src="image_4.png" /><figcaption>Figure 3. Simulation results for equilibrium state without eating drug</figcaption></figure>

The adaptivity connection (Hebbian learning) and suppression of connection be-
tween two brain parts; Hippocampus and Prefrontal Cortex is shown in Figure 4. As it
can be seen from Figure the adaptivity, learning to cope with stress and decreasing that
over time starts at time around 100 and continues until time 600 to stay constant.

<figure><img src="image_5.png" /><figcaption>Figure 4. Simulation results for adaptivity connection weight between Prefrontal Cortex and Hippocampus</figcaption></figure>

5 Mathematical Analysis

Emerging dynamic properties of dynamical models can be analyzed by simulation ex-
periments, but some types of properties can be found by calculations in a mathematical
manner using the WIMS Linear Solver<fnref n="1" />. For verification of the proposed temporal-causal network model, stationary points are investigated.

To analyze the model mathematically, the solutions of linear equations of each state of the model are achieved and by comparing the outcome with the simulation results of the model using Matlab [22] one can verify the model.

x1=x9
x2=x1
x3=1
x4=x3
2*x5=x2+x8
x6=x4
x7=x5
2*x8=x6+x7-0.9*x20-0.9*x21
x10=0.5
x12=x11
x11=x10
x13=x12
4*x14=x13 + x15+ x16+ 0.47*x17
3*x15=x14+x12+x19
x16 = x12
2*x17=x18+0.47*x14
2*x18=x17+x15
2*x19=x17+x18
2*x20=x18+x19
2*x21=x17+20

To compare mathematical results with simulation results, in particular the ones il-
lustrated in Fig. 1 and 2, the parameter values for X3 =1 and the value of the X10 (goal)
= 0.5 were used, due to the fact that the other states that are goal dependent are not able
to go up and would not reach equilibrium. Some of the comparisons among simulation
and mathematical analysis of states is depicted in Table 3.

<table><caption>**Table 3. Comparing Analysis and Simulation**</caption>
  <thead>
    <tr>
      <th>State</th>
      <th>wS<sub>c</sub><br>X<sub>3</sub></th>
      <th>ss<sub>c</sub><br>X<sub>4</sub></th>
      <th>srS<sub>c</sub><br>X<sub>6</sub></th>
      <th>goal<br>X<sub>10</sub></th>
      <th>ps<sub>pil</sub><br>X<sub>11</sub></th>
      <th>es<sub>pil</sub><br>X<sub>12</sub></th>
      <th>Norepine<br>X<sub>13</sub></th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>Simulation</th>
      <td>1,0000</td>
      <td>1.0000</td>
      <td>1.0000</td>
      <td>0.5000</td>
      <td>0.5000</td>
      <td>0.5000</td>
      <td>0.5000</td>
    </tr>
    <tr>
      <th>Analysis</th>
      <td>0.9999</td>
      <td>0.9999</td>
      <td>0.9999</td>
      <td>0.4900</td>
      <td>0.4900</td>
      <td>0.4900</td>
      <td>0.4900</td>
    </tr>
    <tr>
      <th>Deviation</th>
      <td>0.0001</td>
      <td>0.0001</td>
      <td>0.0001</td>
      <td>0.0100</td>
      <td>0.0100</td>
      <td>0.0100</td>
      <td>0.0100</td>
    </tr>
  </tbody>
</table>

<footnote n="1">¹ https://wims.unice.fr/wims/wims.cgi?session=K06C12840B.2&+lang=nl&+module=tool%2Flinear%2Flinsolver.en</footnote>
<table><thead><tr><th>State</th><th>Thal<br>X<sub>15</sub></th><th>PFC<br>X<sub>17</sub></th><th>Amyg<br>X<sub>18</sub></th><th>Late.Cer<br>X<sub>19</sub></th><th>Striatum<br>X<sub>20</sub></th></tr></thead><tbody><tr><td>Simulation</td><td>0.3158</td><td>0.2460</td><td>0.2482</td><td>0.2144</td><td>0.2312</td></tr><tr><td>Analysis</td><td>0.3225</td><td>0.1891</td><td>0.2538</td><td>0.2195</td><td>0.2367</td></tr><tr><td>Deviation</td><td>0.0067</td><td>0.0569</td><td>0.0056</td><td>0.0051</td><td>0.0055</td></tr></tbody></table>

# 6 Conclusion

In this paper a cognitive temporal causal network-oriented model of therapy by using drug (fluoxetine) for individuals under stress is introduced in which usage of medicine is used. The proposed model can be used to test different hypothesis and neurological principles about the impacts of the brain and the effects that different brain areas have the extinction of stress, but also on other processes.

Some simulations have been implemented, one of which was presented in the paper. This model can be used as the basis of a chatbot, a virtual agent model and to get insight in such processes and to bring up a certain cure or treatment of individuals to perform the therapies of extreme emotions for post-traumatic disorder individuals.

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