4. Discussion

Feeling of stress or proximity to death is a basic feature of the living organisms and this feature exists already at a single cell [1]. The discovery of homeostasis W. Cannon [3] assumed that homeostasis results from tendency of the organisms to decrease the stress and avoid death. This point is a biological basis for our theory.

It has been shown in Ref. [17] that if system evolution complies with the causality principle and a system state space displays trivial local topology, system dynamics inevitably satisfy generalized Lagrangian equations (2a) and (2b) with an additional "S-variable." Since the above conditions are quite general, we believe that they are applicable to the living organisms. In the chapter, we identified S-variable with a level of feeling of stress (called S-index). It should be emphasized that the feeling of stress or discomfort is not metaphor for biological systems, but real feature of the living organisms (see Sections 1 and [1]). Note that S-index is a

Fuzzy Logic and *S*‐Lagrangian Dynamics of Living Systems: Theory of Homeostasis http://dx.doi.org/10.5772/66473 161

Figure 2. "Death-pathway" of the system. (A) System was heavily injured. (B) System was strongly distressed (initial Sindex was high; graph B5 begins from 20), although initial injury was small.

phenomenological quantity and cannot be directly measured.17 It should be noted that such a kind of the phenomenological variables (which cannot be directly measured) is widely used in physics (e.g., "mechanical action" of the physical systems, order parameter in superfluid phase transition, etc.).

Supposing that dynamics of the stress is determined by competition between damage and the protection mechanisms, we have obtained an S-Lagrangian and dynamical equations of

finite-difference equations and it was widely discussed in the literature. A particular example of such a behavior was considered in Ref. [17] and a general explanation of this phenomenon

Figure 1. Homeostasis for different initial conditions. Here, x<sup>1</sup> and x<sup>2</sup> are C-variables and ξ<sup>1</sup> and ξ<sup>2</sup> are R-variables. (A) Light injury. (B) The system cannot return to the main ground state, but finds another comfortable state without damage and distress. (C) Homeostasis cannot fully compensate for injury and distress, but some discomforting stable state exists.

Feeling of stress or proximity to death is a basic feature of the living organisms and this feature exists already at a single cell [1]. The discovery of homeostasis W. Cannon [3] assumed that homeostasis results from tendency of the organisms to decrease the stress and avoid death.

It has been shown in Ref. [17] that if system evolution complies with the causality principle and a system state space displays trivial local topology, system dynamics inevitably satisfy generalized Lagrangian equations (2a) and (2b) with an additional "S-variable." Since the above conditions are quite general, we believe that they are applicable to the living organisms. In the chapter, we identified S-variable with a level of feeling of stress (called S-index). It should be emphasized that the feeling of stress or discomfort is not metaphor for biological systems, but real feature of the living organisms (see Sections 1 and [1]). Note that S-index is a

can be found in Ref. [19].

This point is a biological basis for our theory.

4. Discussion

160 Lagrangian Mechanics

<sup>17</sup>Note, however, that in medical practice level of stress often is subjectively defined by the patients.

Figure 3. Pseudo-random behavior of the system with latent time of "decision making". Circle designates an initial state and Star designates the finish state.

homeostasis given in Eqs. (11a), (11b) and (11c). Moreover, since other systems, such as social systems, may also possess distress or discomfort, they may also undergo homeostasis.

Solutions of the dynamical equations of homeostasis show that there are four types of system behavior. In the first, the system generates activity that quickly takes it to the main ground state with zero damage and stress (Figure 1A). In the second, the main ground state cannot be achieved; however, the system finds another ground state without damage and stress as well (Figure 1B). In the third, homeostasis cannot find the state with zero damage and stress and the system arrives at the damaged and distressed, but stable stationary states (Figure 1C). In the last type of behavior, the system cannot achieve any stable state, level of stress dramatically increases, system variables leave the life-compatible region, and the system moves toward death (Figure 2). It should be noted that there is a critical value of injure, which leads to fatal instability of a system by violation of the condition (21). Apparently, there is a critical value of the stress as well, so if S-index exceeds this value, an organism inevitably moves toward death. Note that near the injured stable states, where Iðxc;ScÞ > 0, the critical value of the stress may be lower than near uninjured states,<sup>18</sup> that is, injured organism is more sensitive to the stress than the healthy one.

All types of behavior are described by the same system of Eqs. (11b) and (11c) and S-Lagrangian, but differ by initial and/or environmental conditions (which are described by parameters of the Lagrangian). It was found that systems exhibiting homeostasis may have at least two types of variables. The first type is C-variables, which have stationary values in the stable states of the system. Injury disturbs these values and excites protection mechanisms. The other types

<sup>18</sup>For <sup>I</sup>ðx;S<sup>Þ</sup> and <sup>A</sup>ðx;S<sup>Þ</sup> from Eqs. (29a) and (29b), the critical value of S-index is obtained from <sup>∂</sup>Φ<sup>1</sup> <sup>∂</sup>S� <sup>J</sup><sup>−</sup> <sup>Φ</sup>2ðS�Þþð <sup>S</sup>� <sup>∂</sup>Φ<sup>2</sup> ∂S� ÞΓ ¼ 0.

of variables are R-variables, which can run in a stable state without disturbing system comfort. This finding agrees with the experimental data. Examples of the C-variables are ATP level, intracellular pH level, intracellular and blood concentration of Naþ, Kþ, Caþ, and intracellular levels of certain proteins (caspases, cytokines, and antioxidants), and so on. Examples of the R-variables are blood flow, α-rhythm of brain, heart contraction, brain pacemakers, and so on.

Interestingly, if a system has a "latent time" between consequent actions ("decision-making time"), it imitates random trial-and-error behavior. This corresponds to a real situation in a brain. Although the physical parameters of the brain are continuously changing, time intervals that are shorter than the nerve impulse duration (milliseconds) do not have physiological sense. Moreover, decisions in the brain take tens of milliseconds. Therefore, psychological time is more discrete. Consequently, the chaotic behavior of nerve processes inevitably arises in nerve tissue and can serve as the basis of free decision-making target. This creates an opportunity for trial-and-error behavior. A random search will be targeted if instability fluctuations increase with increasing deviation from the optimum. For example, on/off switching of voltage-dependent channels in neurons can occur more than 100 times/s, which is an adequate speed for searching for the homeostatic optimum. This mechanism ensures that obstacles can be overcome [20]. Therefore, chaotic behavior, illustrated in Figure 3, can play a crucial role in homeostasis. It should be emphasized that this chaotic behavior is not determined by some stochastic process, but rather is governed by deterministic equations.

The simulation results displayed satisfactory agreement between the biological properties of homeostasis and theory. Figure 1 demonstrated direct homeostasis for a weak injury and indirect homeostasis with the restructuring of some parameters for more severe damage. Damage aggravation caused the model to transfer to a working state, although the discomfort was not completely removed. Modeling was also amenable to the process of system destruction (Figure 2), with the behavior of the model depending not only on damage severity but also on "subjective" assessment (i.e., death threats).

The theory predicts that increasing of the stress itself (even without internal injure) leads to disturbing of the physiological parameters, that is, to physiological damage of the organism. This prediction is supported by the recent experimental data, which show that both in human and in animal models, the expression of many genes changed in response to early and to late stresses [21].
