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.

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 nt_s), which could be 5-HT₄, D_1,5, 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 nt_h) are assumed for 5-HT_1A, 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.

To explain our hypothesis better, we briefly revisit the first two models from our previous work 43.

The terms L_a, L_p, R_a and R_p 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; α_i(i = 1,2,3,4) denotes the activation rates and γ_i(i = 1,2,3,4) the natural degradation rates; n_j(j = 2,3) are the Hill coefficients; and h is the threshold value of the neuron. The corresponding model may be defined by:

d d t ( L p ) = α 1 ( S t ) − γ 1 ( L p ) d d t ( L a ) = α 2 h n 2 + ( S t ) n 2 − γ 2 ( L a ) d d t ( R p ) = α 3 h n 3 + ( S t ) n 3 − γ 3 ( R p ) d d t ( R a ) = α 4 ( S t ) − γ 4 ( R a ) d d t ( S t ) = f ( S t )       { 1 } 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} MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzcqGGOaakcqWGtbWucqWG0baDcqGGPaqkcqGH9aqpcqaHXoqydaWgaaWcbaGaeGynaudabeaakiabcIcaOiabdofatjabdsha0jabcMcaPmaacmqabaGaeGymaeJaeyOeI0YaaSaaaeaacqGGOaakcqWGtbWucqWG0baDcqGGPaqkaeaacqWGlbWsaaaacaGL7bGaayzFaaaaaa@4408@, where (K) is the carrying capacity for stress and α₅ is the intrinsic growth rate of stress. Hence system {1} becomes

d d t ( L p ) = α 1 ( S t ) − γ 1 ( L p ) d d t ( L a ) = α 2 h n 2 + ( S t ) n 2 − γ 2 ( L a ) d d t ( R p ) = α 3 h n 3 + ( S t ) n 3 − γ 3 ( R p ) d d t ( R a ) = α 4 ( 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=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@5C1B@

Where x₅(τ₀) > 0 is the initial stress when τ = τ₀.

The time dependent solutions of L_p and R_a 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 = e − δ 4 τ ∫ { β 4 x 5 ( τ 0 ) ℜ e δ 4 τ x 5 ( τ 0 ) + [ ℜ − x 5 ( τ 0 ) ] e − β 5 τ } d τ + C R a e − δ 4 τ       { 8 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@7C3A@

Also, the time dependent solutions of L_a and R_p in dimensionless form are given by

x 2 = e − δ 2 τ ∫ β 2 [ x 5 ( τ 0 ) + { ℜ − x 5 ( τ ) } e − β 5 τ ] n 2 { x 5 ( τ 0 ) ℜ } n 2 + [ x 5 ( τ 0 ) + { ℜ − x 5 ( τ 0 ) } e − β 5 τ ] n 2 d τ + C L a e − δ 2 τ       { 9 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@9C9A@

x 3 = e − δ 3 τ ∫ β 3 [ x 5 ( τ 0 ) + { ℜ − x 5 ( τ ) } e − β 5 τ ] n 3 { x 5 ( τ 0 ) ℜ } n 3 + [ x 5 ( τ 0 ) + { ℜ − x 5 ( τ 0 ) } e − β 5 τ ] n 3 d τ + C R p e − δ 3 τ       { 10 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaWgaaWcbaGaeG4mamdabeaakiabg2da9iabdwgaLnaaCaaaleqabaGaeyOeI0IaeqiTdq2aaSbaaWqaaiabiodaZaqabaWccqaHepaDaaGcdaWdbaqaamaalaaabaGaeqOSdi2aaSbaaSqaaiabiodaZaqabaGcdaWadiqaaiabdIha4naaBaaaleaacqaI1aqnaeqaaOGaeiikaGIaeqiXdq3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkcqGHRaWkdaGadeqaaiabgYriclabgkHiTiabdIha4naaBaaaleaacqaI1aqnaeqaaOGaeiikaGIaeqiXdqNaeiykaKcacaGL7bGaayzFaaGaemyzau2aaWbaaSqabeaacqGHsislcqaHYoGydaWgaaadbaGaeGynaudabeaaliabes8a0baaaOGaay5waiaaw2faamaaCaaaleqabaGaemOBa42aaSbaaWqaaiabiodaZaqabaaaaaGcbaWaaiWabeaacqWG4baEdaWgaaWcbaGaeGynaudabeaakiabcIcaOiabes8a0naaBaaaleaacqaIWaamaeqaaOGaeiykaKIaeyihHimacaGL7bGaayzFaaWaaWbaaSqabeaacqWGUbGBdaWgaaadbaGaeG4mamdabeaaaaGccqGHRaWkdaWadiqaaiabdIha4naaBaaaleaacqaI1aqnaeqaaOGaeiikaGIaeqiXdq3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkcqGHRaWkdaGadeqaaiabgYriclabgkHiTiabdIha4naaBaaaleaacqaI1aqnaeqaaOGaeiikaGIaeqiXdq3aaSbaaSqaaiabicdaWaqabaGccqGGPaqkaiaawUhacaGL9baacqWGLbqzdaahaaWcbeqaaiabgkHiTiabek7aInaaBaaameaacqaI1aqnaeqaaSGaeqiXdqhaaaGccaGLBbGaayzxaaWaaWbaaSqabeaacqWGUbGBdaWgaaadbaGaeG4mamdabeaaaaaaaaWcbeqab0Gaey4kIipakiabdsgaKjabes8a0jabgUcaRiabdoeadnaaCaaaleqabaGaemOuai1aaSbaaWqaaiabdchaWbqabaaaaOGaemyzau2aaWbaaSqabeaacqGHsislcqaH0oazdaWgaaadbaGaeG4mamdabeaaliabes8a0baakiaaxMaacaWLjaWaaiWabeaacqaIXaqmcqaIWaamaiaawUhacaGL9baaaaa@9DB0@

Where CLp,CLa,CRa MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqdaahaaWcbeqaaiabdYeamnaaBaaameaacqWGWbaCaeqaaaaakiabcYcaSiabdoeadnaaCaaaleqabaGaemitaW0aaSbaaWqaaiabdggaHbqabaaaaOGaeiilaWIaem4qam0aaWbaaSqabeaacqWGsbGudaWgaaadbaGaemyyaegabeaaaaaaaa@3A29@ and CRp MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGdbWqdaahaaWcbeqaaiabdkfasnaaBaaameaacqWGWbaCaeqaaaaaaaa@30AB@ are the constants of integration, which can be obtained from the initial condition τ = τ₀.

A detailed numerical solution is shown graphically in Figures 4 and 5 and the values of the parameters are given Table 2. The MATHCAD 13 computer software was used to obtain these numerical solutions.

To solve system {3} we used the Romberg method of Integration with TOL (tolerance) to the order of 10 ^-3.

The computer-simulated outcomes of model-1 are depicted in Figures 4 and 5. The R_a and L_p growth curves show similar outcomes. The L_a and R_p growth curves are also analogous.

The outcomes of this model show that L_p concentration heads towards a saturation point (carrying capacity), whereas L_a concentration gradually diminishes. This indicates that stress alone can lead the brain to a catastrophic state in which depression may become uncontrollable. An unpredictable event may arise beyond this catastrophic point (maximum sustainable carrying capacity). It also shows the imbalance and dynamically opposite characteristics implicit in the lateral hemispheric division of the brain. However, model-1 does not consider the effects of exercise and stress together; that is incorporated in model-2.

As a non-pharmacological intervention, we have introduced 'exercise' into the stress dynamics. The schematic diagram shown in Figure 6 represents the functional characteristics of brain dynamics in presence of stress-induced exercise activities. In this particular schema we assume that both stress and exercise are acting simultaneously where the stress activity (not counting "moderate" exercise itself as a stressor, whereas "heavy" exercise may qualify as a stressor) develops independently from various sources and/or systems over which the individual has no control.

A person who is not under the influence of stress can do exercise. On the other hand one can do the exercise when one knows that one is under influence of stress. We call this situation 'stress-induced exercise activity'. In the present study, our approach is based on the latter scenario.

In this scenario, the effects of exercise positively activate the left-anterior and right-posterior of the brain but they negatively activate (feedback mechanism) the left-posterior and right anterior of the brain. As such, the exercise effect conteracts the stress effect on the brain.

Based on the above schematic diagram we have developed the following mathematical model.

Model-2 (Figure 6) may be defined as:

d d t ( L p ) = α 1 ( S t ) h n 1 + ( E x ) n 1 − γ 1 ( L p ) d d t ( L a ) = α 2 ( E x ) h n 2 + ( S t ) n 2 − γ 2 ( L a ) d d t ( R p ) = α 3 ( E x ) h n 3 + ( S t ) n 3 − γ 3 ( R p ) d d t ( R a ) = α 4 ( S t ) h n 4 + ( E x ) n 4 − γ 4 ( R a ) d d t ( S t ) = α 5 ( S t ) − γ 5 ( S t ) ( E x ) d d t ( E x ) = α 6 ( S t ) ( E x ) − γ 6 ( E x )       { 11 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@FD66@

Where (Ex) denotes the exercise activity and n₁, n₄ are Hill coefficients; α₆ is the exercise generation due to stress, γ₅ is the degradation of stress due to exercise and γ₆ is the degradation of exercise effects.

The non-trivial steady state of the above system is as follows:

L p 0 = ( γ 1 α 1 ) [ ( S t ) h n 1 + ( E x ) n 1 ] > 0 , L a 0 = ( γ 2 α 2 ) [ ( E x ) h n 2 + ( S t ) n 2 ] > 0 R p 0 = ( γ 3 α 3 ) [ ( E x ) h n 3 + ( S t ) n 3 ] > 0 , R a 0 = ( γ 4 α 4 ) [ ( S t ) h n 4 + ( E x ) n 4 ] > 0 S t 0 = γ 6 α 6 > 0 , E x 0 = α 5 γ 5 > 0       { 12 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@CDF8@

The dimensionless form of Eq. {11} is:

d x 1 d τ = ξ 1 x 5 1 + x 6 n 1 − ζ 1 x 1 d x 2 d τ = ξ 2 x 6 1 + x 5 n 2 − ζ 2 x 2 d x 3 d τ = ξ 3 x 6 1 + x 5 n 3 − ζ 3 x 3 d x 4 d τ = ξ 4 x 5 1 + x 6 n 4 − ζ 4 x 4 d x 5 d τ = ξ 5 x 5 − ζ 5 x 5 x 6 d x 6 d τ = ξ 6 x 5 x 6 − ζ 6 x 6       { 13 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@CCCA@

Where

Let (x10,x20,x30,x40,x50,x60 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaeGymaedabaGaeGimaadaaOGaeiilaWIaemiEaG3aa0baaSqaaiabikdaYaqaaiabicdaWaaakiabcYcaSiabdIha4naaDaaaleaacqaIZaWmaeaacqaIWaamaaGccqGGSaalcqWG4baEdaqhaaWcbaGaeGinaqdabaGaeGimaadaaOGaeiilaWIaemiEaG3aa0baaSqaaiabiwda1aqaaiabicdaWaaakiabcYcaSiabdIha4naaDaaaleaacqaI2aGnaeaacqaIWaamaaaaaa@4674@) be the dimensionless steady state values; then for u_i = x_i - xi0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG4baEdaqhaaWcbaGaemyAaKgabaGaeGimaadaaaaa@309B@ (i = 1,.......,6) the linearization version of the above system is:

d u 1 d τ = − ζ 1 u 1 + ( ζ 1 x 5 0 ) u 5 − ζ 1 2 n 1 ( x 6 0 ) n 1 − 1 ( x 1 0 ) 2 ξ 1 x 5 0 u 6 d u 2 d τ = − ζ 2 u 2 − ζ 2 2 n 2 ( x 5 0 ) n 2 − 1 ( x 2 0 ) 2 ξ 2 x 6 0 u 5 + ( ζ 2 x 6 0 ) u 6 d u 3 d τ = − ζ 3 u 3 − ζ 3 2 n 3 ( x 5 0 ) n 3 − 1 ( x 3 0 ) 2 ξ 3 x 6 0 u 5 + ( ζ 3 x 6 0 ) u 6 d u 4 d τ = − ζ 4 u 4 + ( ζ 4 x 5 0 ) u 5 − ζ 4 2 n 4 ( x 6 0 ) n 4 − 1 ( x 4 0 ) 2 ξ 4 x 5 0 u 6 d u 5 d τ = − ζ 5 x 5 0 u 6 d u 6 d τ = − ξ 6 x 6 0 u 5       { 15 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@429E@

The characteristic equation of the above system is given by:

( ζ 1 x 1 0 + λ ) ( ζ 2 x 2 0 + λ ) ( ζ 3 x 3 0 + λ ) ( ζ 4 x 4 0 + λ ) ( λ 2 + ξ 6 ζ 5 x 5 0 x 6 0 ) = 0       { 16 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@70F6@

The possible roots of the characteristic equation are:

λ 1 = − ζ 1 x 1 0 , λ 2 = − ζ 2 x 2 0 , λ 3 = − ζ 3 x 3 0 , λ 4 = − ζ 4 x 4 0 , λ 5 = i ξ 6 ζ 5 x 5 0 x 6 0 , λ 6 = − i ξ 6 ζ 5 x 5 0 x 6 0       { 17 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8977@

It is evident from the linear stability analysis that un-damped oscillations exist there and that the steady state is a centre. This implies that system with stress-induced exercise is structurally stable.

Numerical solutions of system {15} are shown in figure 8 and figure 9; the values of the parameters are given Table 2. MATHCAD 13 computer software was used to obtain these numerical solutions.

To solve system {15} we used the fourth-order Runge-Kutta fixed-step method for solving systems of differential equations.

The oscillatory behavior in response to concomitant stress and exercise (in any one quadrant, here LA) is depicted in Figure 7. The oscillatory nature of the behavior of two antero-posterior quadrants on the same side (left), L_p and L_a, is given in Figure 8. Similarly, the oscillatory nature of the behavior of two antero-posterior quadrants on the same side (right), R_p and R_a, is presented in Figure 9.

The outcomes of this model (where stress and exercise act together) show that there is a perfect harmony between the lateral hemispheric divisions of brain by exhibiting limit cycle solutions for (L_p, L_a) and (R_p, R_a). In this situation neither L_p nor L_a (and neither R_a nor R_p) can reach the maximum sustainable stage. Therefore, depression may not be manifested.

To extrapolate the behavior of this model biologically, it is highly possible that regular exercise in a moderate dose can counteract the lateralized effects of chronic stressors, over a substantial period of time, without allowing the harmful effects of stressors to take the upper hand.

This is the most important part of this paper. We wanted to see whether our assumptions could lead to valid behavioral outcomes in real life.

Here we consider the activities/concentrations of the neurotransmitter receptor subtypes within L_p, L_a, R_p and R_a. As shown in Figures 3 and 6, Stress increases nt_s activity/concentration in the R_a and L_p areas of the brain, and/or reduces nt_h activity/concentration in the L_a and R_p areas, leading to a sad state. Exercise has the converse effects and elicits a happy mood.

We denote the activities/concentrations of these neurotransmitter receptor subtypes in the L_p and R_a regions by C and those in the R_p and L_a regions by G.

The time-dependent changes in activity/concentration may be modeled by the following equations.

d C d T = α 11 C ( Ω − G ) − β 11 C d G d T = α 22 G ( C − Ω ) − β 22 G       { 18 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5D9A@

Where Ω is the neurotransmitter receptor subtype threshold level; α₁₁, α₂₂ and β₁₁, β₂₂ are the growth and decay parameters of neurotransmitter receptors sub-types.

If stress increases, eventually nt_s activity/concentration in L_p and R_a regions increases at the same time nt_h as activity/concentration in L_a and R_p regions decreases. Therefore, dCdT MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdoeadbqaaiabdsgaKjabdsfaubaaaaa@319E@ ≥ 0 and dGdT MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdEeahbqaaiabdsgaKjabdsfaubaaaaa@31A6@ ≤ 0.

Vice versa, if exercise increases, eventually nt_h activity/concentration in L_a and R_p regions increases at the same time nt_s as activity/concentration in L_p and R_a region decreases. Therefore, dCdT MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdoeadbqaaiabdsgaKjabdsfaubaaaaa@319E@ ≤ 0 and dGdT MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKjabdEeahbqaaiabdsgaKjabdsfaubaaaaa@31A6@ ≥ 0. Here, the threshold level Ω plays the pivotal role.

Numerical solutions of system {18} are shown in figure 10 and the parameter values are chosen arbitrarily (since experimental data are not available). MATHCAD 13 computer software was used to obtain these numerical solutions. Figure 10 compares their relative behaviors over time.

The models described in this article show one of the likely mechanisms for the action, on the brain, of concomitant stress and exercise. They mathematically prove our assumptions that 'Stress' nonlinearly increases the activity/concentration of particular subtypes of receptors (designated nt_s) for each of the known (and unknown) neurotransmitters in the right anterior (RA or R_a) and left posterior (LP or L_p) regions of the brain, and/or nonlinearly decreases the activity/concentration of another set of receptor subtypes (designated nt_h) for each of these neurotransmitters in the left anterior (LA or L_a) and right posterior (RP or R_p) activity areas. Exercise elicits the opposite (nonlinear) effects.

In other words, the behavior of the models can be experimentally verified by suitable designs, as outlined in the section on implications of the hypothesis.