An interdisciplinary approach to study individuality in biological and physical systems functioning

Signals of system functioning of different nature are presented in the parameter space (state-velocity-acceleration) as a trajectory of dynamic events. Such signals geometrization allows to reveal the hidden spatio-temporal correlation in dynamics of systems functioning. It is shown that the nature of relationship between the dynamic parameters of signal determines the natural cycle of sensor functioning. Its restructuring displays the inherited features of systems functioning in signature package. The universal differential-geometry parameters and new integrative indexes of system functioning are used to analyze the signatures of biological and physical signals.

of dynamic components of signals are found in various sensors 12,13,16 . Individuality in sensor functioning is evident in signal's signature configuration, i.e. after signal geometrization. The area spanned by signature is also informative. This area can be represented as a power of subset of possible system microstates 16 . Consequently, the signatures of signals from sensors and biosensors can be analyzed by complementary statistical and dynamical methods. This study is focuses on further development of interdisciplinary approach and search for universal tools. These are necessary for system analysis and control the complex systems functioning.

Sensor response transformation into a subset of microstates
Main ideas of the approach are considered by example of simple I(t) transient photoresponse (TPR) and complex V(t) cardiovascular signal geometrization in phase (time-state-velocity) space. Such signal geometrization is accompanied by I(t) and V(t) signals transformation into a sequence of dynamic states (phase space trajectory). This allows natural allocating of segments which are differing in linear density of states (Fig. 1). Interrelations between the dynamic states in these segments are most evident in projections of these trajectories on the phase plane (state-velocity). These projections I(t) − dI(t)/dt and V(t) − dV(t)/dt (Fig. 1) are individual phase portraits of bio-and semiconductor sensor and should be considered as a sort of signatures. As you can see, geometrization of different signals is accompanied by the natural decomposition on geometrically ordered segments of constant slope or curvature. Differential-geometric parameters of these segments are individual for each sensor. For an arbitrary signal x(t) they reflect the dynamic components. Partial contributions P are proportional to segment length and slope product or segment length and curvature product 17 . Values of partial contributions are sensitive to external and internal stimuli and fields. Sequence of partial contributions P 1 , P 2 , P 3 , … is the "marker identifier" of operation cycle. Dynamic balance is peculiar to TPR signatures with symmetric configurations. Area S enclosed by the I(t) − dI/dt TPR signature is very sensitive to the intensity and wavelength of radiation. Area of TPR signature can be statistically represented as a power of possible photoinduced quantum microstates subset W i 18 . From the microscopic point of view, the number of possible quantum microstates W can be presented as the statistical weight 19 . Consequently, the power of a microstates subset statistically characterizes the H entropy of TPR that is proportional to the natural logarithm of W, i.e . ∝ H W ln . This allows us to analyze the ordering of transient photoresponse from sensor. Therefore, the configuration and area of signature is naturally connected. This allows examining the functioning by the complementary dynamic and statistical methods. This establishes a natural connection between the macroscopic and microscopic characterization of the system. Thus, an integral index of dynamic balance B dyn is the area ratio of S sup superior and S inf inferior parts of signature, i.e. B dyn = S sup /S inf . TPR entropy increasing during sensor exploitation in extreme conditions is accompanied by violations of the B dyn dynamic balance and rearrangement of partial contributions of TPR components.
Phase portraits of human ECG are informative but their analysis is ambiguous 8 . Within the framework of approach the configuration of V(t) − dV(t)/dt signature for each wave of cardiac cycle can be analyzed dynamically and statistically. Thus, the V QRS (t) − dV QRS (t)/dt signature of QRS-complex includes both signatures of Pand T-wave of cardiac cycle, i.e. V P (t) − dV P (t)/dt and V T (t) − dV T (t)/dt (Fig. 1b). Therefore W QRS is a dynamic subset includes both P-and T-subsets of induced microstates. Operations between these subsets and their relationships provide new information which is usually hidden for the other approaches. However, for the system analysis of this information it is necessary to know the cycle of the myocardium, which is genetically inherited.

Sensor Response Transformation into a Dynamic Event Trajectory
TPR I(t) geometrization in (state-velocity-acceleration) space allows us to realize a natural transition from dynamic state sequence to dynamic event trajectory. It is possible to increase the number of tools for detection of induced individuality of photoresponse and its analysis 18 . Trajectory consists of curved sections which vary in linear density of dynamic events (Fig. 2). Spatio-temporal correlation between dynamic events is the most evident in the orthogonal projections of trajectory on three planes (state -velocity), (state -acceleration) and (velocity -acceleration). Trajectory projection onto (state -velocity) plane is the I(t) − dI(t)/dt 1 st order signature of TPR which is presented in Section 1. Trajectory projection onto (state -acceleration) plane is the I(t) − d 2 I(t)/dt 2 2 nd order signature of TPR.
Naturally, extremes on the I(t) − d 2 I(t)/dt 2 signature coincide with the extremes on dI(t)/dt − d 2 I(t)/dt 2 signatures. The I(t) − d 2 I(t)/dt 2 signature reflects the energy components of sensor photoresponse (see Fig. 2a). Individuality of antiphase components 1, 2 and 1' , 2' is shows up in the area ratio between the corresponding extremes, which is energy balance index for them. Spatio-temporal relationship between the dynamic events determines the configuration of dI/dt − d 2 I/dt 2 2 nd order signature. Signature configuration is located at 4 quadrants of (velocity -acceleration) plane and represents relationship between dI/dt and d 2 I/dt 2 dynamical variables in main phases of TRP (Fig. 2a, "+ + ", "+ − ", "− − " and "− + " quadrants). Technology inherited defect structure of a crystal determines the complex frequency spectra of resonant vibrations 20,21 . It also affects the configuration of the dI/dt − d 2 I/dt 2 signature. It should be noted that parameters of coupled resonant vibrations are sensitive to photoexcitation.
The areas covered by the dI/dt − d 2 I/dt 2 signature in each quadrant represent the power of basic phases of the bicycle 22 . Indeed, transition to a new dynamic variable Y = dI/dt allows us to convert the dI/dt − d 2 I/dt 2 2 nd order TPR signature into the Y(t) − dY/dt 1 st order signature. Configuration of the Y(t) − dY/dt signature occupies 4 quadrants and represents the basic phases of functioning cycle. Therefore, the dI/dt − d 2 I/dt 2 TPR signature is a natural geometrical model of the cycle control. The dimensionless indexes B ij of the power balance between the basic phases of TPR characterize the spatial-temporal coherence of dynamical processes. The B ij indexes are the ratios between areas covered by the dI/dt − d 2 I/dt 2 signature in each quadrant, e.g.
, etc. In a square matrix of the B ij indexes represents relations between the basic phases of sensor functioning ( Table 1). Individuality of sensor functioning is most evident in matrix ( Table 1).
The dV/dt − d 2 V/dt 2 signature configuration is a geometrical model of CVS functioning (Fig. 2b). It revealed the variety of geometric patterns in myocardium cycle. It is established the high sensitivity of B ij indexes of cardicycle to external and internal factors (magnetic storms, drugs, food, etc.). All of this opens up new opportunities for the system analysis of cardiac cycle in contrast to known methods. For instance, atlas creation for the typical geometrical models of CVS functioning simplifies not only their parametric identification and classification but also express diagnostics.

Integral Individuality of System Functioning
Identification of inherited functioning individuality is promoted by the fruitful idea of the ECG 23 (Fig. 3) and TPR 11 sequences presentation in a package form. The series of highly stable TPR cycles from sensor are shown in signature package as an attractor 23,24 . The characteristic features of the stable sensor functioning are: reversibility of cycles and their dynamic balance. These characteristic features are interrelated and combined with the minimum of covered area, i.e. entropy minimum of TPR. They correspond to thermodynamic criteria of cycle reversibility Δ H → 0, where H-entropy. Exploitation of these sensors in extreme conditions is accompanied by dynamical and energy imbalance of antiphase processes and can also lead to local distortion of the I(t) − dI/dt (Fig. 3a-c) and dI/dt − d 2 I/dt 2 (Fig. 4a-c) TPR signatures within the packages. However electroacoustic treatment at frequencies corresponding to a certain piezoelectric resonant vibrations improves the sensor's functional characteristics. Temperature treatment should be performed at T i temperature, where T i is a certain temperature at which TPR signature configuration becomes most symmetrical (Fig. 4d) 25 .
Package presentation of physiological signal is a quite effective to analyze integral individuality of system functioning. Analysis of more than 50 packages of the V(t) − dV/dt signature of QRS-complex shows diversity of nature of dynamic restructuring. The V(t) − dV/dt signature packages for three monitored people that have features of myocardium functionality altering 26 are showed in Fig. 3. The 1 st order signature restructuring over the package is qualitatively presented in: 1. Phase trajectories distribution character within the signature package (homogeneous, heterogeneous, steplike et al.).

Appearance of local variations Δ V(t) in signature over the cycle under influence of stress factor that is proportional to Kolmogorov entropy 27 .
Functional individuality is quantitatively manifested in entropy and entropy production time dependences, i.e. H(t) and dH(t)/dt The H(t) and dH(t)/dt time dependences converting into package of integro-differential H-signatures H(t) − dH(t)/dt leads to a typical chaos-gram 28 . However, in contrast to chaos-gram the package of H-signatures allow us to study individuality of human CVS. A duality of biological order which represents the relation between the structures and functioning processes 29 is most manifested in H-signature packages. It can be assumed that H-signature packages with a high enough resolution will provide us information about hidden relations between human body subsystems. Partly they appear in nature rearrangement of the dV/dt − d 2 V/dt 2 signatures of QRS-complex in the package (Fig. 4a-c).
CVS control system rearrangement can be analyzed dynamically (signature configuration change) and statistically (density of trajectories and area covered by each quadrant). It can be assumed that information about individual structure of interrelations is hidden in the character of evolution of dV/dt − d 2 V/dt 2 signature configuration (Fig. 4a-c) and area covered by dV/dt − d 2 V/dt 2 signature of Q, R and S waves. Individuality of adaptation processes is the most evident in evolution of dV/dt − d 2 V/dt 2 signature configuration. Package of the dV/dt − d 2 V/dt 2 signatures can be considered as subsets of operation cycles. Integrated individuality manifests in restructuring of operation cycles. Obviously, the systemically important function (inherited scenario of system functioning) is hidden in operation cycles and relationship between them. This is indicated by results of comparative analysis of ECG signature of 7 relatives. It turned out that configuration of the 1 st and 2 nd order signature for 1, 2 and 3 leads are similar only for father (70 years old), daughter (41) and grandson (8 years). These signatures are characterized by similar integrative indexes (B ij , H etc.) Package of these signatures also identified the same characteristic features of signature restructuring.
It was found that package of 1 st and 2 nd order signatures ECG contains information on the main indexes of CVS (level of functioning, functional reserve and degree of tension of regulatory mechanisms). Today, scientists use different methods of diagnosis to determine the main indexes of CVS. The versatility of parameters, indexes and criteria allows avoiding the ambiguity during analysis of complex study results in ergonomics, medicine, sports, etc. Natural geometrization of signals from different type of sensors simplifies analysis of both consistency and compatibility in manned subsystems. Natural cycles of biosystem operation for cybernetic systems are of particular interest for cyber physical system developers.

Conclusions
At different scale levels of signals the dynamic, energetic and cybernetic aspects of biological and physical systems functioning are hidden. They manifested after converting any cyclic signal X(t) and its derivatives in the trajectory of dynamic events. In fact, the trajectory is a geometric interpretation of variational Hamilton's principle of least action. As a result of such geometrization the natural decomposition of signal onto geometrically ordered sections that reflect its dynamic and energetic components is carried out. Universal differential-geometrical parameters of these sections (length, slope and curvature) are mapped to the physical values (state, velocity and acceleration). Therefore, the orthogonal projection of the trajectory are X(t) − dX/dt, X(t) − d 2 X/dt 2 and dX/dt − d 2 X/dt 2 signatures. Individual configuration of the signatures is naturally combined with a statistical regularity. Indeed, the spatial and temporal correlation of dynamic events are converted into geometrically ordered sections (dynamic and energy components of X(t) − dX/dt, X(t) − d 2 X/dt 2 signature configurations). The nature of relationship between dX/dt and d 2 X/dt 2 components determines individual cycle of system functioning that displayed by the dX/dt − d 2 X/dt 2 signature. The matrix of power balance indexes is proposed for analysis of the basic phases of functioning cycle.
Individuality is most evident in the character of rearrangement of system functioning cycle and manifested in a packet presentation of cyclic signal signatures. Both configuration and area changes of signatures are interrelated which allows to analyze functioning artifacts. Thus, various stress factors rise to local instability Δ X and imbalances of the main phases of functioning cycle. Presentation of area covered by the X(t) − dX/dt signature as a subset of microstates allows to provide entropy analysis of system functioning. Therefore, the character of signature configuration and area of changes in a natural way manifested the relation between dynamic and statistical regularities in system functioning. The results of integration of signatures in the package (H(t) dependence) and its derivative dH/dt are informative. Consistent signature configuration change and entropy H is peculiar to biological systems. For their studies the integro-differential H signature H(t) − dH/dt and their packages are proposed for the first time. They naturally represent biological order based on the inherited relation of structures and functions. In essence, the proposed approach is a kind of bridge between the macroscopic and microscopic description of systems functioning.
The approach has also proved effective for the signature analysis of other functional characteristics of sensors. Their signatures displayed the influence of "frozen" dynamics of defect structure onto individuality of functional characteristics of semiconductor sensors. For example, spectral signatures 30 and temperature 31,32 PR dependencies allowed to establish a complex energy spectrum of defects which had previously been identified by various methods. The proposed approach is also compatible with other signal processing methods. Thus, the transformation of wavelet spectrograms of photoresponse to wavelet signatures at different scales allowed to reveal usually hidden information by means of universal tools of approach 33 . Search studies have shown that higher-order signatures are promising for the analysis of multi-scale signals.
In general, our approach and universal tools for its implementation provide new opportunities for diagnostic systems, harmonization of human-computer interaction (human-computer interaction) and ensure intercomputer interaction (machine-machine interaction) in cyberphysical systems. Obviously, application of approach tools will simplify the control of consistency in subsystems of technical systems.