Keywords

1 Introduction

Post-Traumatic Stress Disorder (briefly PTSD) [15] is a severe psychiatric mental problem that might happen in an individual after serious trauma or extreme stress. PTSD can generate intense apprehension, impuissance in persons [1]. It has been known that electrical stimulation of the amygdala has a remarkable effect and changes on emotional, perceptual, and behavioral functioning.

What is deep brain stimulation is completely explained in [6, p. 406, 12]:

‘The implementation of DBS electrodes is a neurosurgical procedure, often performed under local anesthesia with patients fully awake to facilitate precise stimulation mapping. After placement of the electrodes in the desired target below the convexity of the brain, they are connected to a programmable pulse generator, similar to modern cardiac pacemakers, that is implanted under the skin below the collarbone.’

There are variety of indications for PTSD such as insomnia, hypervigilance, and stress [14]. Some brain components are involved in having reaction to the stress like cerebral Cortex, Hippocampus, Hypothalamus, Amygdala [12]. The Network-Oriented Modeling approach based on temporal-causal network models described in [16] is such a modeling approach, and has been used here.

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

2 Underlying Biological and Neurological Principles

As have been discussed in the literature a deep brain stimulation (DBS) has an important role in treatment with PTSD; e.g., [1]. There are many other therapies which have been explained in [18,19,20,21,22,23,24,25,26,27]. For instance, the cognitive models of using Ritalin and doing yoga to decrease the stress level presented in [24] and [25], respectively. The efficiency of deep brain stimulation and also the method is introduced precisely in [1, p. 1]:

‘Deep Brain Stimulation (DBS) has shown promise in refractory movement disorders, depression and obsessive-compulsive disorder, with deep brain targets chosen by integration of clinical and neuroimaging literature. The basolateral amygdala (BLn) is an optimal target for high-frequency DBS in PTSD based on neurocircuitry findings from a variety of perspective.’

‘An electroencephalographic [EEG] telemetry session will test safety of stimulation before randomization to staggered-onset, double-blind sham versus active stimulation for two months.’

‘Deep Brain Stimulation (DBS) refers to the process of delivering an electrical current to a precise location in the brain. DBS is now a common clinical practice. DBS is an invasive treatment, and the potential for benefits must clearly outweigh the risks.’

It notable that only stimulation of the BLn [11] has remarkable effect on curing PTSD as stated in [1, p. 7]:

‘DBS of gray matter, such as the BLn, would likely have effects opposite those seen from DBS of the white matters in the case of the amygdala.

  1. 1.

    It supports the model that amygdala overactivity is responsible for the symptoms of PTSD. DBS of the stria terminals physiologically equates to amygdala overactivity. In this patient with no psychiatric history, an increase in amygdala output activity has led to symptoms commonly seen in combat PTSD (for example, irritability, helplessness, depression and suicidality.)

  2. 2.

    The correction of the amygdala activity led to resolution of the symptoms. As opposed to DBS of the stria terminals, BLn DBS is expected to reduce amygdala.

  3. 3.

    The patient did not suffer from a seizure or deterioration in neuropsychological status during chronic DBS for nine months. This suggests an acceptable safety profile of DBS of the amygdala.’

‘Notably, only stimulation of the BLn, but not the central nucleus, the amygdala outflow tract, or neighboring regions affected by other contacts of the stimulating electrodes, led to benefit. Also important was the absence of significant side effects, including seizures.

While we thus agree that the BLn is not the only potentially valuable target for DBS in PTSD, we believe, based on the rationale described above, that it is the optimal target and it can be modulated safely with current technology.’

Targeting the exact part of the amygdala named Basolateral nucleus (BLn) is more complicated than it is assumed the exact region is defined in [12];

‘Targeting the BLn is complicated due to anatomical variations in this region. Using a stereotactic atlas, the inferior limit of the BLn is located 16 mm lateral to the AC, 4 mm posterior to the AC and 18 mm inferior to the AC-PC plane.’

Changing in functionality in noradrenergic neurons is believed to be involved in hyperarousal and reexperiencing symptoms of PTSD [2].

PTSD is also defined clearly in [1, p. 2];

‘PTSD was characterized by three clusters of psychiatric symptoms. The first, re-experiencing, involves the emotional and perceptual reliving of traumatic events either spontaneously or in response to ‘triggers’ that remind one of the events because they bear some similarity to the original circumstance.’

There are many therapies for PTSD such as Exposure therapy (PE) [3], Cognitive Processing Therapy [4], and Eye Movement Desensitization and reprocessing (EMDR) [5].

The connection between cerebellum and striatum, amygdala and striatum has been explained in [7, p. 489], [9].

‘Inputs from the cortex primarily project to the striatum.’ ‘The striatum glutamatergic inputs from several areas, including the cortex, hippocampus, amygdala, and thalamus.’

In [10] they claimed that direct stimulation of the vmPFC can prevent the responsiveness of the amygdala. In [10] it has been claimed that amygdalotomy may regulate and moderate stress level.

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 [16, Chap. 2] 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 (Fig. 1 and Table 1).

  • Strength of a connection ωX,Y. Each connection from a state X to a state Y has a connection weight value ωX,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 cY(..). For each state a combination function cY(..) is chosen to combine the causal impacts of other states on state Y.

  • Speed of change of a state ηY. For each state Y a speed factor ηY is used to represent how fast a state is changing upon causal impact.

Fig. 1.
figure 1

Conceptual representation of the integrative temporal-causal network model

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Table 1. Explanation of some states and their relation to neurological principles
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4 Example Simulation

An example simulation of this process is shown in Fig. 2. Table 2 shows the connection weights used, where the value for ω24 is initial value as these weights are adapted over time. The model implemented in the Matlab template provided in [22].

Fig. 2.
figure 2

Simulation results for Deep Brain Stimulation (DBS)

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Table 2. Connection weights and scaling factors for the example simulation
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Table 3 shows the states which use scale sum function as their combination function. The other states except X3 (world state of context) and X10 (goal) all contains identity function as they have only one incoming connection weight.

Table 3. Scaling factors for the example simulation of states with scale sum function
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As a biological process, the electrode triggers the activation of Basolateral nucleus in Amygdala at the first state and this impacts other brain parts such as the remarkable one mPFC, left and right lateral nucleus.

Figure 3 shows a simulation result of using Hebbian learning principle between states basolateral nucleus in amygdala and medial Prefrontal Cortex in one period of using deep brain stimulation.

Fig. 3.
figure 3

Simulation results for Deep Brain Stimulation (DBS), Adaptive connections

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5 Mathematical Verification

To verify the model that has been proposed mathematical analysis used. For some states in the model the mathematical analysis has been done. Table 4 shows this result. By using the WIMS Linear SolverFootnote 1, the following (unique) algebraic solution was obtained for the general case of these equations:

Table 4. Scaling factors for the example simulation of states with scale sum function
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$$ \begin{aligned} & {\text{X1}} = {\text{X9}} \\ & {\text{X2}} = {\text{X1}} \\ & {\text{X3}} = 1\\ & {\text{X4}} = {\text{X3}} \\ & 2\, *\,{\text{X5}} = 1\, *\,{\text{X2}} + 1\, *\,{\text{X8}} \\ & {\text{X6}} = {\text{X4}} \\ & {\text{X7}} = {\text{X5}} \\ & 2\, *\,{\text{X8}} = 1\, *\,{\text{X6}} + 1\, *\,{\text{X7}} - 0. 9 9\, *\,{\text{X16}} - 0. 5\, *\,{\text{X2}}0 \\ & {\text{X9}} = {\text{X8}} \\ & {\text{X1}}0 = 0. 2\\ & {\text{X11}} = {\text{X1}}0 \\ & {\text{X12}} = {\text{X11}} \\ & 2\, *\,{\text{X13}} = 1\, *\,{\text{X5}} + 1\, *\,{\text{X14}} \\ & 2\, *\,{\text{X14}} = {\text{X13}} + {\text{X15}} \\ & {\text{X15}} = 1\, *\,{\text{X14}} \\ & {\text{X16}} = 0. 1^{ * } {\text{X17}} \\ & {\text{X17}} = 0. 8\, *\,{\text{X14}} \\ & {\text{X18}} = 1\, *\,{\text{X17}} \\ & 2\, *\,{\text{X19}} = 0. 9 5\, *\,{\text{X15}} \\ & {\text{X2}}0 = {\text{X19}} \\ \end{aligned} $$

6 Conclusion

In this paper an integrative cognitive and affective model of therapy by using deep brain stimulation (DBS) for Post-Traumatic Stress Disorder (PTSD) is introduced in which electrode is used. This affects the preparation state of stress and enables to release the stressed person from a chronic stress. This model can be used as the basis of a virtual agent model 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 persons.