Background
Regular, "moderate", physical exercise is a non-pharmacological form of adjunctive treatment for depressive disorders. External stimuli such as various stressors or exercise influence the higher functions of the brain (cognition and affect). These effects often do not follow a linear course. Even though exercise itself can be seen as a stressor, in moderate doses it has been shown to reduce the effects of other stressors. To explain our hypothesis better, we need to elaborate on certain concepts – encompassing a wide range of biological and mathematical domains – of stress, depression, exercise, neurotransmitters along with their receptor subtypes, brain lateralization and nonlinear dynamics. All these concepts (and their interactions) are discussed broadly in the following paragraphs in this section. The hypothesis is based on the numerous published data obtained from experimental research, and on logical assumptions made where experimental data are not yet available. We have tried to thread together the gems (some key studies) of experimental evidence presented in Table 1123456789101112131415161718192021222324252627. The approach is more akin to systems biology (generalization) than to detailed characterization of any particular pathway of exercise and stress actions. The reader is encouraged to ponder over the items in Table 1 before going through the rest of this section for elucidation of the relevant concepts. A highly focused "linear" thought process may not be conducive to comprehending the underlying essential nonlinearities in our proposed model.
<p>Table 1</p>
Highlights of some relevant literature (abbreviations expanded in the text)
Areas, Author (Year)
Summary
Relevance
A. Origin of the idea
Sarbadhikari (1995a) [1]
Exercise reduces behavioral and EEG effects of stress
Mechanism to be determined
B. Stress and lateralization
Mandal et al. (1996), Atchely et al. (2003); Neveu and Merlot (2003); Yurgelun-Todd & Ross (2006) [2&6]
Definite lateralization effects observed for affect and stress
Stress acts in a lateralized fashion; lateralization of emotion in depression; lateralized effects of stress may act at cellular levels
C. Chaos and nonlinear dynamics in depression
Toro et al. (1999); Levine et al. (2000); Thomasson et al. (2000); Jeong (2002) [7–10]
Chaotic oscillations in the brain may account for many conditions including depression, where there is proven correlation between clinical and electrophysiological dimensions, and associations between clinical remission and bifurcation are present
Chaotic oscillations form one of the mechanisms for depression
D. Exercise, lateralization and nonlinear dynamics
Petruzzello et al. (2001); Kyriazis (2003) [11,12]
Exercise influences affective responsiveness by regional brain activation and also increases physiological complexity in the brain
Exercise acts in a lateralized fashion and increases complexity, unlike stress
E. Nonlinear dynamics linking various physiological and pathological processes
Sarbadhikari and Chakrabarty (2001); Glass (2001); Savi (2005) [13–15]
Nonlinear dynamics can be the underlying commonalty between depression, exercise and lateralization
Depression, exercise and lateralization may all be nonlinearly linked; Stress and Exercise may operate counteractively through the same systems
F. Neurotransmitter receptor subtypes have varied functions and distributions
Tecott (2000); Pediconi et al. (1993); Bortolozzi et al (2003); Xu et al. (2005); Fukumoto et al. (2005), et al [16–22]
Receptor subtypes for all neurotransmitters; asymmetric distribution of acetylcholine and monoamine receptors in mammalian brain
Same neurotransmitter may act in opposing ways by binding with different receptor subtypes; asymmetric distributions of various neurotransmitters are possible in the brain
G. Cellular level interactions involving BDNF and CREB
Cotman et al. (2002); Garoflos et al. (2005) [23, 24]
BDNF increases with Exercise and decreases with Stress; phosphorylation of the transcription factor CREB and increased BDNF expression are positively correlated
BDNF and CREB may be intermediaries for activating the various receptor subtypes
H. Integrating hypothesis
Shenal et al. (2003) [25]
LF, RF and RP interactions in the brain are responsible for the manifestation of stress effects
LA/RA/RP/LP quadratic interactions could give rise to cross-coupling of the systems
I. Detailed expositions
Sarbadhikari (2005a, b) [26, 27]
Depressive and dementive disorders can be caused by nonlinear disturbances in lateralization
Stress and Exercise may operate counteractively through the same systems
Broadly: "Stress" refers to the mental or physical condition resulting from various disturbing physical, emotional, or chemical factors ("stressors"), which can be environmental or anthropogenic, and lead to a behavior or outcome that is commonly labeled "depressive". The effects of the stressors on the body constitute the "stress response", which may be measured by behavioral, biochemical, and genetic modifications. "Anxiety" may be defined as the emotional discomfort associated with "stress". "Depression" denotes a spectrum of disorders affecting many aspects of human physiology, and can be precipitated by various psychological (e.g., mental trauma), biophysical (e.g., loss of organ or function and genetic predisposition) and social (e.g., loss of job) stressors. However, under-diagnosis in general medical practice is quite common 1.
Depression (including its various subtypes) is a common global disorder. Apart from newer pharmacotherapeutic management, some non-pharmacological interventions also play a significant part in its alleviation 1. Regular, "moderate" physical exercise forms a pillar of such treatment. Our hypothesis concerns general mechanisms that give rise to the effects of exercise along with stress.
Cerebral hemispheric lateralization alludes to the localization of brain function on either the right or left sides of the brain, and is an important factor in the progress of depression 2. Incidentally, this lateralization is not confined to only the cerebral cortices, but also to the subcortical structures. A recent paper 3 indicates that mood state may be differentiated by lateralization of brain activation in fronto-limbic regions. The interpretation of fMRI (functional magnetic resonance imaging) studies in bipolar disorder is limited by the choice of regions of interest, medication effects, comorbidity, and task performance. These studies suggest that there is a complex alteration in regions important for neural networks underlying cognition and emotional processing in bipolar disorder. However, measuring changes in specific brain regions does not identify how these neural networks are affected. New techniques for analyzing fMRI data are needed in order to resolve some of these issues and identify how changes in neural networks relate to cognitive and emotional processing in bipolar disorder.
The relationship between exercise and stress is not a simple one. As succinctly pointed out by Mastorakos and Pavlatou 4: "Exercise represents a physical stress that challenges homeostasis. In response to this stressor, the autonomic nervous system and hypothalamus-pituitary-adrenal axis are known to react and participate in the maintenance of homeostasis and the development of physical fitness. This includes elevation of cortisol and catecholamines in plasma. However, physical conditioning is associated with a reduction in pituitary-adrenal activation in response to exercise." In our present model, we shall start at the point at which chronic moderate exercise has already led to the "baseline adaptive changes" and behaves in a different way from any other stressor. In future modifications, changes in the model's threshold for exhibiting this particular (bimodal) behavior can also be incorporated. This bimodal or hormetic response is characterized by low dose stimulation, high dose inhibition, resulting in either a J-shaped or an inverted U-shaped (nonlinear) dose response. A chemical pollutant or toxin or radiation showing hormesis therefore has the opposite effect in small doses to that in large doses. Therefore, we can assume regular moderate exercise as the mild, repeated "stressful" stimulation (which is good for health). While excessive and prolonged stress (as in heavy exercise) can lead to depression, mild and irregular (non-linearly applied, hormetic) stress can actually improve depression. Radak et al. 28 extend the hormesis theory to include reactive oxygen species (ROS). They further suggest that the beneficial effects of regular exercise are partly based on the ROS-generating capacity of exercise, which is in the stimulation range of ROS production. Therefore, they suggest that exercise-induced ROS production plays a role in the induction of antioxidants, DNA repair and protein degrading enzymes, resulting in decreases in the incidence of oxidative stress-related diseases.
External stimuli such as various stressors or exercise influence the higher brain functions, i.e., cognition and affect. These effects often do not follow a linear course. In nonlinear dynamics the rate of change of any variable cannot be written as a linear function of the other variables. Therefore, it may be better suited to modeling many phenomena, and putative global pathways, in the brain, that are attributable to such influences 7812131415.
Neurotransmitters convey the information to be passed and processed through some 1014 to 1016 interconnections linking approximately 1010 to 1011 neurons in the human brain. Each of the many neurotransmitters (including as yet unidentified ones) acts through a receptor, which in general will have numerous subtypes 16. The same neurotransmitter acting through two different receptor subtypes may have opposing actions. Most psychotropic drugs exert their therapeutic effects through various neurotransmitters, mainly through specific receptor subtypes. Some neurotransmitter receptor subtype interactions are depicted in Figure 1. It may be noted that 5-HT2 class receptors couple to Gq/G11 and do not primarily signal through cAMP pathways. Similarly, 5-HT3 receptors are ligand-coupled ion channels and do not primarily signal through cAMP as Figure 1 might seem to suggest. However, this only proves the existence of additional intracellular pathways such as the Gq/G11 coupled intracellular calcium/protein kinase C pathway, and also highlights the fact that signaling is much more complex than this model allows. Our oversimplification is essential for trying to grasp the overall complexity of all possible (known and as yet unknown) underlying mechanisms of the brain. The basic purpose of this figure is to show that (irrespective of the mechanisms of action) any neurotransmitter is capable of exerting opposing effects (e.g., increasing anxiety or 'anxiogenesis' and decreasing anxiety or 'anxiolysis') by acting through its diverse receptor subtypes.
<p>Figure 1</p>
Typical example of complementary action of some neurotransmitter receptor subtypes
Typical example of complementary action of some neurotransmitter receptor subtypes. Key: DA: Dopamine; NE: Norepinephrine; 5HT: 5-Hydroxytryptamine or Serotonin.
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Interestingly, there is a greater right-sided EEG abnormality in depression owing to impaired cerebral lateralization 2. Therapeutically, too, better antidepressant results are obtained with non-dominant unilateral electroconvulsive shock. It is generally believed that "affect" processing is a right hemisphere (RH) function. It is also believed that RH dysfunction is characteristic of depressive illness. Both these beliefs are oversimplified because the relationship between affect processing and affective illness, in terms of intra- and inter-hemispheric role-play, is not straightforward. There is exchange of information and action between the two hemispheres (inter-hemispheric, i.e., between left and right; intra-hemispheric i.e., between anterior and posterior; and also cross-hemispheric coupling i.e., similarities between the left anterior and right posterior quadrants). Very broadly, a sad mood is a function of positive coupling (stimulation) between the left posterior and right anterior areas and/or negative coupling (depression) between the left anterior and right posterior areas of the brain 2.
Brain functions are lateralized to the right or the left sides and there are observed differences in the expression of neurotransmitter receptor subtypes 16171819202122. Some of these results 21 are supported by a meta-analysis of various studies reported in the literature. Neuroanatomical asymmetries are known to be present in the human brain, and disturbed neurochemical asymmetries have also been reported in the brains of patients with schizophrenia 22. Not only neuroanatomical but also neurochemical evidence supports the loss or reversal of normal asymmetry of the temporal lobe in schizophrenia, which might be due to a disruption of the neurodevelopmental processes involved in hemispheric lateralization.
Neuropsychological research provides a useful framework for studying emotional problems such as depression and their correlates. Shenal et al. 25 review several prominent neuropsychological theories focusing on functional neuroanatomical systems of emotion and depression, including those that describe cerebral asymmetries in emotional processing. Following their review, they present a model comprising three neuroanatomical divisions (left frontal, right frontal and right posterior) and corresponding neuropsychological emotional sequelae within each quadrant. It is proposed that dysfunction in any of these quadrants could lead to symptomatology consistent with a diagnosis of depression. Their model combines theories of arousal, lateralization and functional cerebral space and lends itself to scientific investigation. Shenal et al. 25 conclude: 'As the existing literature appears to be somewhat confusing and controversial, an increased precision for the diagnostic term "depression" may afford a better understanding of this emotional construct. Future research projects and innovative neuropsychological models may help to form a better understanding of depression.' Their proposed model 'combines theories of arousal, lateralization, and functional cerebral space to better understand these distinct clinical pictures, and it should be noted that these regions may be differentially activated following various therapies to depressive symptomatology.' However, their excellent neuropsychological model does not take into account the different neurotransmitter receptor subtype distribution and functions.
The theory of dynamical systems ("chaos theory") allows one to describe the change in a system's macroscopic behavior as a bifurcation in the underlying dynamics. From the example of depressive syndrome, a correspondence can be demonstrated between clinical and electro-physiological dimensions and the association between clinical remission and reorganization of brain dynamics (i.e., bifurcation). Thomasson et al. 9 discuss the relationship between mind and brain in respect of the question of normality versus pathology in psychiatry on the basis of their experimental study.
Neuropharmacological investigations aimed at understanding the electrophysiological correlates between drug effects and action potential trains have usually involved the analysis of firing rate and bursting activity. Di Mascio et al. 29 selectively altered the neural circuits that provide inputs to dopaminergic neurons in the ventral tegmental area and investigated the corresponding electrophysiological correlates by nonlinear dynamic analysis. The nonlinear prediction method combined with Gaussian-scaled surrogate data showed that the structure in the time-series corresponding to the electrical activity of these neurons, extracellularly recorded in vivo, was chaotic. A decrease in chaos of these dopaminergic neurons was found in a group of rats treated with 5,7-dihydroxytryptamine, a neurotoxin that selectively destroys serotonergic terminals. The chaos content of the ventral tegmental area dopaminergic neurons in the control group, and the decrease of chaos in the lesioned group, cannot be explained in terms of standard characteristics of neuronal activity (firing rate, bursting activity). Moreover, the control group showed a positive correlation between the density-power-spectrum of the interspike intervals (ISIs) and the chaos content measured by nonlinear prediction S score; this relationship was lost in the lesioned group. It was concluded that the impaired serotonergic tone induced by 5,7-dihydroxytryptamine reduces the chaotic behavior of the dopaminergic cell-firing pattern while retaining many standard ISI characteristics. However, some difficulties remain. There is a suspicion that the determinism in the EEG may be too high-dimensional to be detected with current methods. Previously 30, ISIs of dopamine neurons recorded in the substantia nigra were predicted partially on the basis of their immediate prior history. These data support the hypothesis that the sequence-dependent behavior of dopamine neurons arises in part from interactions with forebrain structures. ISI sequences recorded from unlesioned rats demonstrated maximum predictability when an average of 3.7 prior events were incorporated into the forecasting algorithm, implying a physiological process, the "depth" of history-dependence of which is approximately 600–800 ms.
It has been repeatedly confirmed that the brain acts nonlinearly, especially when complex interactions are required, as in cognition or affect processing. In a cognitive study 31, although the nonlinear measures ranged in the middle field compared to the number of significant contrasts, they were the only ones that were partially successful in discriminating among the mental tasks. In another cognitive study 32, initial increase in complexity of both episodic and semantic information was associated with right inferior frontal activation; further increase in complexity was associated with left dorsolateral activation. This implies that frontal activation during retrieval is a non-linear function of the complexity of the retrieved information.
A broader view of stress is that not only do dramatic stressful events exact a toll, but also the many events of daily life elevate the activities of physiological systems and cause some measure of wear and tear. This wear and tear has been termed "allostatic load" 33, and it reflects the impact not only of life experiences but also of genetic load (predisposition); individual habits reflecting items such as diet, exercise and substance abuse, and developmental experiences that set life-long patterns of behavior and physiological reactivity. Hormones and neurotransmitters associated with stress and allostatic load protect the body in the short term and promote adaptation, but in the long run allostatic load causes changes in the body that lead to disease. These have been observed particularly in the immune system and the brain.
Zheng et al. 34 state that exercise has beneficial effects on mental health in depressed sufferers; however, the mechanisms underlying these effects remained unresolved. These authors found that (1) exercise reversed the harmful effects of chronic unpredictable stress on mood and spatial performance in rats and (2) the behavioral changes induced by exercise and/or chronic unpredictable stress might be associated with hippocampal brain-derived neurotrophic factor (BDNF) levels. Also, the HPA (hypothalamus-pituitary-adrenal axis) system might play different roles in the two processes. BDNF is the most widely-distributed trophic factor in the brain and participates in neuronal growth, maintenance and use-dependent plasticity mechanisms such as long-term potentiation (LTP) and learning. Huang et al. 35 observed that compulsive treadmill exercise with pre-familiarization acutely up-regulates expression of the BDNF gene in rat hippocampus. Duman 36 states that stress and depression decrease neurotrophic factor expression and neurogenesis in the brain, and that antidepressant treatment blocks or reverses these effects. In contrast, exercise and enriched environment increase neurotrophic support and neurogenesis, which could contribute to blockading the effects of stress and aging and produce antidepressant effects. BDNF, in turn, exerts its effects through the formation/suppression of specific neurons, neurotransmitters, and receptor subtypes. Another study 37 corroborates the substantial data implicating common pathways involving neurotransmitter action through neurotrophic factors in the regulation of neural stem cells. This transmitter-mediated neurotrophic pathway could be altered by environmental factors including enriched environment, exercise, stress, and drug abuse. The most notable neurotransmitters in this context are serotonin (5-HT), glutamate and gamma-amino-butyric acid (GABA). There is ample evidence that enhancement of neurotrophic support and associated augmentation of synaptic plasticity and function may form the basis for antidepressant efficacy 38. Although depression is not a homogeneous disorder, some commonalty may be expected in the final common pathway for all forms of depression. Incidentally, exercise has various other effects (as mentioned in the limitations section), which are not discussed here. Also, exercise, as a stimulus, is dependent on its timing (what time of day it is performed), frequency (how many times a day, or a week) and content (aerobic, weight bearing and so on). The very fact that these parameters can be varied is a stimulus itself, and variations in them have physical influences on brain function, including upregulation of trophic factors such as GDNF (glial cell line-derived neurotrophic factor), FGF-2 (Fibroblast growth factor-2), or BDNF 39.
The beneficial role of exercise is evident in many neurodegenerative disorders 40. Despite the paucity of human research, basic animal models and clinical data overwhelmingly support the notion that exercise treatment is a major protective factor against neurodegeneration of various etiologies. The final common pathway of degradation is clearly related to oxidative stress, nitrosative stress, glucocorticoid dysregulation, inflammation and amyloid deposition. Exercise training may be a major protective factor but in the absence of clinical guidelines, its prescription and success with treatment adherence remain elusive. In the present model, Moderate Exercise: 3.0 – 6.0 METs (3.5 – 7.0 kcal/min) 41 is assumed for the purpose of modeling.
Freeman 42 believes that the search for simple rules is one good reason for using the tools of chaos theory to model neural functions. The present effort is to integrate these clues theoretically in order to gain a better overview of the interactions of stress and exercise inside the brain. The next section describes our preliminary hypothesis based on some experimental evidence.
To sum up, it is not known whether the complex dynamics are an essential feature or if they are secondary to internal feedback and environmental fluctuations 13. Because of the complexity of biological systems and the huge jumps in scale from a single ionic channel to the cell to the organ to the organism, all computer models will be gross approximations to the real system for the foreseeable future. There is a rich fMRI literature on affect, stress and depression and this, together with a wealth of preclinical data, will enable the very general model proposed in this paper to be refined in the future. At present, our concern is to determine whether a broadly testable nonlinear dynamic model can be elaborated and to outline the preliminary experiments required to validate it. Only after this task is completed will detailed refinement, producing a more practically helpful model, become appropriate. It may be noted that the basic purpose of the model is to provide direction for experimental research, since there is a paucity of real life data, which we feel to be essential for understanding the precise role of neurotransmitter receptor subtypes in different areas of the brain.
The Hypothesis
Introduction
The preliminary general model described here is based on the assumptions that (a) some neurotransmitter cascade (primarily nonlinear) affects the whole brain in a lateralized fashion, and (b) with more prolonged exercise, more favorable receptor subtypes are recruited for all the neurotransmitters involved.
From our previous studies 14344, we found that the deleterious behavioral effects of stress were less pronounced in the "exercised and stressed" animals, and the beneficial effects became more pronounced with time (more prolonged exercise), as indicated by the results of the behavioral tests.
Let us cite another example of (nonlinear) interactions among diverse neurotransmitters. Di Mascio et al. 29 showed that a 5-HT antagonist impairs serotoninergic tone, which in turn reduces the chaotic behavior of dopaminergic cell firing patterns in the brain. Another study by Toro et al. 7 included pharmacological modification of neurotransmitter pathways, electroconvulsive therapy (ECT), sleep deprivation, psychosurgery, electrical stimulation through cerebral electrodes, and repetitive transcranial magnetic stimulation (rTMS). Stemming from a pathophysiological model that portrays the brain as an open, complex and nonlinear system, a common mechanism of action has been attributed to all therapies. This report suggests that the isomorphism among therapies is related to their ability to help the CNS deactivate cortical-subcortical circuits that are dysfunctionally coupled. These circuits are self-organized among the neurons of their informational (rapid conduction) and modulating (slow conduction) subsystems. The following speculative overview is based on the aforementioned review and the detailed expositions by Sarbadhikari 2627. Disease specific genes (and ipso facto proteins) give rise to individual variations in different receptor subtype populations (endowment). This is the basis of pharmacogenomic (individualized) therapy in modern medicine. Each of the conditions mentioned here leads to a (primarily nonlinear) imbalance among the endowed receptor subtype populations (in specific areas of the brain) and tilts the final common pathway in favor of depression or elation. In the previous section, we mentioned some reports that support this view.
It may be surmised that some neurotransmitter cascade (nonlinear or a combination of linear and nonlinear) takes place in different areas of the whole brain, and, with more prolonged exercise, more favorable receptor subtypes are recruited. Stress leads to more left sided (RH or right hemisphere) psychomotor activity, which causes RH inhibition (negative valence), ultimately giving rise to sadness or more negative interpretation. Very broadly, a sad mood is a function of positive coupling (stimulation) between the left posterior and right anterior areas and/or negative coupling (depression) between the left anterior and right posterior areas of the brain. Figure 2 presents a schematic diagram of stress activity within the brain.
<p>Figure 2</p>
Schematic diagram of stress activity within the brain
Schematic diagram of stress activity within the brain.
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Moderate exercise, in contrast, causes more right-sided (psychomotor) activity leading to LH (left hemisphere) inhibition (positive valence), facilitating assertiveness or less negative interpretation. However, a happy mood is broadly a function of positive coupling (stimulation) between the right posterior and left anterior areas and/or negative coupling (depression) between the right anterior and left posterior areas of the brain 25. These couplings are at least partly caused by the activation of Brain Derived Neurotrophic Factor (BDNF) in exercise and the suppression of BDNF in stress 22. BDNF activation and phosphorylation of the cAMP response element binding (CREB) protein are also positively correlated 23. Further, the results of a study 45 are consistent with the hypothesis that decreased expression of BDNF and possibly other growth factors contributes to depression and that upregulation of BDNF plays a role in the actions of antidepressant treatment. Another study 46 suggests that in the frontal cortex and amygdala of mice, caffeic acid can attenuate the down-regulation of BDNF transcription that results from stressful conditions. Recently, investigators 47 have shown that imipramine (IMI) and metyrapone (MET) significantly elevate the BDNF mRNA level in the hippocampus and cerebral cortex. Joint administration of IMI and MET induces a more potent increase BDNF gene expression in both the examined brain regions compared to the treatment with either drug alone.
This article assumes a particular subtype of neurotransmitter receptor (designated nts), which could be 5-HT4, D1,5, β adrenoceptors or yet unidentified types. These are mostly responsible for the "anxiogenic" effects, leading to a "sad" mood. These are assumed to be more active/concentrated in the RA (right anterior) and LP (left posterior) quadrants of the brain. Another set of receptor subtypes (designated nth) are assumed for 5-HT1A, D2, NE or yet unidentified transporters. These are mostly responsible for the "anxiolytic" effects, giving rise to a "happy" mood, and are assumed to be more active/concentrated in the LA (left anterior) and RP (right posterior) quadrants of the brain. The predictions of this proposed model are indicated in Figure 3.
<p>Figure 3</p>
Some putative biochemical aspects of the hypothesis
Some putative biochemical aspects of the hypothesis.
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To explain our hypothesis better, we briefly revisit the first two models from our previous work 43.
Model-1: The effects of stress on the four different quadrants of the brain
The terms La, Lp, Ra and Rp represent the release of neurotransmitters from the axons of neurons in the four different quadrants of the brain (left anterior, left posterior, right anterior and right posterior) due to stress activity. The left-posterior and right-anterior areas of the brain are positively activated by stress whereas left-anterior and right-posterior quadrants are negatively activated by a feedback mechanism.
St denotes the stress activity; αi(i = 1,2,3,4) denotes the activation rates and γi(i = 1,2,3,4) the natural degradation rates; nj(j = 2,3) are the Hill coefficients; and h is the threshold value of the neuron. The corresponding model may be defined by:
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MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B164@
Irrespective of the source, the effects of stress are cumulative, but we assume that they cannot accumulate indefinitely – there must be a point of 'sustainability'. Here, we consider this stage as a suicidal point(K). Therefore, effects of stress can go up to a saturation stage (K) beyond which a suicidal tendency will develop. It may be noted that whether a person not doing exercise will actually commit suicide depends on the chaotic or unpredictable behavior of the system in the individual.
To the best of the authors' knowledge, there currently exists no mathematical model to explain stress dynamics clearly. As a first attempt we have considered the Volterra equation to represent stress dynamics. The justification for this selection is that there exists a saturation level in the Volterra equation. As such we can choose f(St)=α5(St){1−(St)K}
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(
S
t
)
−
γ
4
(
R
a
)
d
d
t
(
S
t
)
=
α
5
(
S
t
)
{
1
−
(
S
t
)
K
}
{
2
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BE1E@
The non-trivial steady state solution of the system {2} is given by
S
0
=
{
α
1
K
γ
1
,
α
2
γ
2
(
h
n
2
+
K
n
2
)
,
α
3
γ
3
(
h
n
3
+
K
n
3
)
,
α
4
K
γ
4
,
K
}
T
{
3
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6A8F@
The dimensionless form of {2} can be expressed as {4}:
d
d
t
x
1
=
β
1
x
5
−
δ
1
x
1
d
d
t
x
2
=
β
2
1
+
x
5
n
2
−
δ
2
x
2
d
d
t
x
3
=
β
3
1
+
x
5
n
3
−
δ
3
x
3
d
d
t
x
4
=
β
4
x
5
−
δ
4
x
4
d
d
t
x
5
=
β
5
x
5
(
1
−
x
5
ℜ
)
{
4
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9D4E@
Where
x
1
=
h
−
1
(
L
p
)
,
x
2
=
h
−
1
(
L
a
)
,
x
3
=
h
−
1
(
R
p
)
,
x
4
=
h
−
1
(
R
a
)
,
x
5
=
h
−
1
(
S
t
)
,
β
1
=
α
1
h
2
,
β
2
=
α
2
h
−
n
2
+
1
,
β
3
=
α
3
h
−
n
3
+
1
,
β
4
=
α
4
h
2
,
β
5
=
α
5
h
2
,
δ
1
=
γ
1
h
2
,
δ
2
=
γ
2
h
2
,
δ
3
=
γ
3
h
2
,
δ
4
=
γ
4
h
2
,
ℜ
=
K
h
−
1
,
τ
=
h
−
2
t
{
5
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@E152@
The time dependent general solution of stress in dimensionless form is given by
x
5
(
τ
)
=
x
5
(
τ
0
)
ℜ
x
5
(
τ
0
)
+
{
ℜ
−
x
5
(
τ
0
)
}
e
β
5
τ
{
6
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaWgaaWcbaGaeGynaudabeaakiabcIcaOiabes8a0jabcMcaPiabg2da9maalaaabaGaemiEaG3aaSbaaSqaaiabiwda1aqabaGccqGGOaakcqaHepaDdaWgaaWcbaGaeGimaadabeaakiabcMcaPiabgYricdqaaiabdIha4naaBaaaleaacqaI1aqnaeqaaOGaeiikaGIaeqiXdq3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkcqGHRaWkcqGG7bWEcqGHCeIWcqGHsislcqWG4baEdaWgaaWcbaGaeGynaudabeaakiabcIcaOiabes8a0naaBaaaleaacqaIWaamaeqaaOGaeiykaKIaeiyFa0Naemyzau2aaWbaaSqabeaacqaHYoGydaWgaaadbaGaeGynaudabeaaliabes8a0baaaaGccaWLjaGaaCzcamaacmqabaGaeGOnaydacaGL7bGaayzFaaaaaa@5C1B@
Where x5(τ0) > 0 is the initial stress when τ = τ0.
The time dependent solutions of Lp and Ra in dimensionless form are given by
x
1
=
e
−
δ
1
τ
∫
{
β
1
x
5
(
τ
0
)
ℜ
e
δ
1
τ
x
5
(
τ
0
)
+
[
ℜ
−
x
5
(
τ
0
)
e
−
β
5
τ
}
d
τ
+
C
L
p
e
−
δ
1
τ
{
7
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7AEA@
and
x
4
=