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A quantum information theory is derived for multidimensional signals scaling. Dynamical data modeling methodology is described for decomposing a signal in a coupled structure of binding synergies, in scale-space. Mass conservation principle, along with a generalized uncertainty relation, and the scale-space wave propagation lead to a polynomial decomposition of information. Statistical map of data, through dynamical cascades, gives an effective way of coding and assessing its control structure. Using a multi-scale approach, the scale-space wave information propagation is utilized in computing stochastic resonance synergies (SRS), and a data ensemble is conceptualized within an atomic structure. In this paper, we show the analysis of multidimensional data scatter, exhibiting a point scaling property. We discuss applications in image processing, as well as, in neuroimaging. Functional neuro-cortical mapping by multidimensional scaling is explained for two behaviorally correlated auditory experiments, whose BOLD signals are recorded by fMRI. The point scaling property of the information flow between the signals recorded in those two experiments is analyzed in conjunction with the cortical feature detector findings and the auditory tonotopic map. The brain wave nucleons from an EEG scan, along with a distance measure of synchronicity of the brain wave patterns, are also explained.

Multidimensional scaling (MDS) refers to a wide variety of methodologies [

Complexity of computation of NP-hard problems has been addressed in the Cerny’s work [

Data clustering has been used in various disciplines, like engineering, life and social sciences. The Bayesian inference statistics has been applied in inferring the clustering parameters [

In life sciences, a synergy hypothesis refers to a natural computation of coordinative structures, and handling biological complexity [

In this work, we propose a unifying approach to the computing and signaling, that ranges across various disciplines, from physics to the behavioral analysis. This quantum information theory lays down, in our view, a new perspective to a networked system dynamics, as well as computation. Dynamic equilibriums, based on bipolar positive-negative synergies [

The computational method captures the quantum information theoretical model for the generation of the underlying data. The networked system dynamics and computation is carried out by the scale-space wave information propagation, accompanied by the inherent uncertainty relation in the information expression. Despite its non-linear and dynamical nature, it aims for the simplest description, coding and control of data.

A common problem that may occur with an optimization procedure seeking a global minimum is depicted in _{1} to β_{2}.

We address this issue by evaluating a generalized uncertainty relation in a data decomposition. A polynomial data decomposition procedure is derived by computing second order statistics for a system of coupled oscillators, in the hierarchy of scales. A scale equilibrium is evaluated, as shown in

corresponding to the coupled data clusters, F_{1} and F_{2}. As a result, a binary tree structure of coupled oscillators minimizes synergy exchange among the clusters of data points.

We define a cluster of data points by its vector representative y, on a set of data points W, the cluster window of computation. Let d(x, y) denotes a distortion measure of a data point x, induced by the vector representative y. The distortion energy, or variance V of a cluster is defined by:

It can be shown [

subject to:

and

is the Gibbs distribution:

where

is the partition function on r number of data points within a window W, and β is the Lagrange’s multiplier.

A nonlinear dynamics of information segmentation in this work is derived from the model of free energy, originally used in statistical physics to model different complex systems. Along with the variance function V, the free energy F describes the state of r data points cluster, for a given scale parameter β,

These functions can be conveniently written within the path integrals in scale β:

and

We have explored a covariant differentiability in the computational model for a system of coupled oscillators. Here, we show that a scale-space wave information propagation allows motion in the scale-space that conserves information. In this methodology, we refer to the energy and the mass, interchangeably, conveniently using the term information. We therefore, denote this a quantum information theory, and the methodology applied as the scale-space computing.

The constrained equation of motion for the coupled oscillators, exchanging synergies among data clusters, is given by:

This system of equations has the determinant of the map:

For_{1} and F_{2}, apply:

We refer it as the generalized uncertainty relation. It is used here to assess the condition of numerical stability for the equation of motion, in the coupled system.

This system can be analyzed by the series expansion of the system’s free energies:

Given the covariant differentiability, the methodology is derived for minimizing the synergy exchange in a coupled system of binary oscillators. The update formula for the scale parameter β is given by:

where

and,

where

The mass conservation principle:

is accompanied by the scale-space wave information propagation:

Dynamic equilibriums are computed that conserve information based on bipolar data distributions for two waves, F_{1} and F_{2}. At the scale equilibrium,

the conservation principle can be written as:

The Green’s function:

gives a model of spatial coherency of information for a cluster of data points, at the scale β. The information is distributed into the positive and negative values of the scale-space wave, according to:

In the neighborhood of a data point W_{P}, the rotor and divergence orthogonal operators combine to shape the positive and negative envelopes of the scale-space wave. When applied for a high-dimensional signal, the bipolar distribution (17) can be conveniently scaled down along the most singular vectors of the covariance matrices,

The main procedure, as written in

The generalized uncertainty relation is used to test for the condition of resonance, as shown in

the case of the resonating waves, the information is encapsulated within the nucleon. In the other case, at the saddle point in scale, as shown in

The system of coupled oscillators is brought to an equilibrium in scale by the procedure, as written in

In this section, we assess the key properties of our methodology by showing data distributions in different applications. The level for resonance, in Algorithm 1, is set such that in all the applications shown, data decomposition falls within a single nucleon information. The four color code (red, blue, green, yellow) is used to code the positive/negative envelopes of the two scale-space waves (red/yellow and blue/green) that are being brought into the resonance, within a nucleon.

An example of data scaling is shown in

In all of the plots of Figures 5(b)-(g), the “upper” Gaussians are shaped by the opposing envelopes (yellow and green) of the two scale-space waves, forming the nucleons. The other two Gaussians are mostly shaped by the single envelopes (red and blue) of the two scale-space waves.

Sequences of two images are used in assessing image motion computed by the stochastic resonance. We use the optical flow equation [

where _{P} is used in applying the local operators in (17), the divergence and rotor.

The “ball” image, I_{1} in _{2} in _{3} in

motion sequences, I_{1} → I_{2} in _{2} → I_{3} in _{1} → I_{3} in

The dynamical data modeling enables a synchronicity measure for the brain wave signals, as we show in the EEG scan. Multidimensional signal scaling in neuroimaging is shown useful also in functional mapping of the neuroanatomy, by the fMRI signal scan analysis. In both of these applications, the information is decomposed within nucleons; in the first by employing the temporal synergy exchange, while in the second, the spatial.

Two channels EEG recording, shown in

Experiments were performed in [

cortex, and the fMRI brain scans were obtained for two behaviorally correlated sound distance perception conditions. The BOLD signal level difference in the brain scans for these two conditions reveal an area in the human auditory cortex particularly active in processing the sound distance, independent of the sound intensity.

In this work, we compute a coupling structure of the information flow for the two stimulus conditions. The scale-space wave information propagation segments the brain activation regions that resonate the information by the spatial exchange of synergies. The term information flow better explains the results obtained, since neither temporal continuity in the fMRI sequences is assumed, nor the order of the images in a sequence.

In Figures 8(g)-(h), two waves are shown with the four segments, the red and green in the L, and the blue and yellow in the R. They are brought into resonatating the information flow between the correlated signal activations: L_{1} & R_{1} vs. L_{2} & R_{2}, shown in Figures 8(a)-(d). Due to a more coherent flow in the R stream, the blue/yellow wave is opposed to the more structured, the red/green in the L. This information decomposition is brought into the resonance at β = 0.010128.

Two distinct patterns can be observed in the red segment: 1) the stripe-like pattern, going more orthogonally

from the main activation area for the sound distance perception, as reported in [

The coupled structure of binding synergies functionally conceptualize the information content, in time, as discussed in the EEG example, or in space, as described with the spatial distribution of the fMRI signals. We denote it the genotype signal analysis that elucidates our understanding of the Space-Time relations, dynamics, and evolution. When describing a multidimensional data decomposition, scaled along one-dimensional path in space, a principle of “least action” is constructed out of minimizing the synergy exchange. Generally, chaotic cycles result out of the scale-space network statistics. Their control is a challenging research, in our view, which opens up a new perspective, from subatomic to diseases treatment and regenerative medicine. We envision it therefore as a unifying approach to signaling and computing, that ranges across various disciplines, from quantum physics to a behavioral analysis.

We have tested this algorithm in different applications. Two dimensional data scatter, in

The scale parameter at the equilibrium, in this experiment, follows a monotonic distance relation to the displacement parameters. We propose this monotonic relation also to be used in assessing the synchronicity level of the brain wave pattern, that is shown in Figures 7(c)-(d).

Higher amplitudes and temporal frequencies can be observed in red and blue envelopes of the scale-space waves, as opposed to the green and yellow envelopes. The highest frequencies in the signal, that correspond to the REM phases of the sleep, have effectively been shut down in the yellow components, to a zero level. When taken as a whole, this indicates that the scale-space decomposition exchange temporal synergies sensitive to the local content of temporal frequencies. Two resonating waves balance the information content in the brain waves with components distributed distinctively in the time-frequency bands.

The fMRI neuroimaging application is accompanied in this work by the results of two-dimensional image motion computation, in the examples shown in _{1} → I_{2}, the nucleon of the image motion decomposition, shown in

The point scaling property of the information flow is analyzed in the fMRI images. Two experiments were conducted [

We extend this analysis showing the information flow point scaling effect in Figures 8(g) and

The theory of stochastic resonance synergies is derived and a new data modeling methodology is described for multidimensional signals scaling. We have proposed it within a quantum information theory [

Coding and control structures in data have been explored by our methodology for the analysis purposes and data mining. Computing and control by parallel computer architectures have been also studied in [

This work was supported in part by the Slovak National Stipends (SAIA) Program. I am grateful to Norbert Kopco for providing the fMRI data and useful comments while writing this manuscript.