3D-hybrid mathematical model for analysis of abnormal
neurological movements for the purposes of diagnosis and
treatment of limb tremor
1
Mykhaylo Bachynskyia, Mykhaylo Petryka, Vitaly Brevusa , Ivan Mudryka , and Bohdan
Glovaa
a
Ternopil Ivan Puluj National Technical University, 56 Ruska str., Ternopil 46001, Ukraine
Abstract
The suggested hybrid neural biosystem model provides an explanation for the condition and
behavior of limb tremors by utilizing the propagation of wave signals. Specifically, it focuses
on the segmental depiction of 3D trajectories of atypical neurological movements in the
examined part of the body, while considering the matrix of cognitive influences from groups
of neuroobjects in the central nervous system. Through the application of a hybrid integral
transformation, which incorporates Fourier, Bessel, and Hilbert transforms, we have achieved
a high-speed analytical solution to the model. This solution is presented in the form of a vector
function that characterizes the 3D elements of trajectories during each segment of movement.
Additionally, we introduce a methodology for calculating the hybrid spectral motion function,
a system of orthogonal basic functions, and spectral values. These components form the
foundation of the proposed hybrid transformation, offering integral vector solutions for the
model. These solutions describe the elements of abnormal neurological movements trajectories
and the distribution of absorbed components, taking into account feedback effects at both
macro and micro levels.
Keywords
Tremor, abnormal movements, mathematical model, computer modeling, 3D-hybrid models,
hybrid integral transforms, cognitive feedback signals
1. Introduction
The development of cutting-edge scientific models was driven by the collaborative efforts with
French research institutions, including the University of Pierre and Marie Curie Sorbonne Paris 6, the
Institute of Brain and Spinal Cord, and the Higher School of Industrial Physics and Chemistry in Paris.
In this research, the authors have introduced innovative hybrid models to analyze the propagation of
wave signals, aiming to understand the state and behavior of abnormal neurological movements (ANM)
in specific body parts of a subject, referred to as T-objects. These movements are influenced by a
particular group of neural nodes known as cerebral cortex (CC) neuro-objects.
These models are built on the foundation of integrated transforms and spectral analysis techniques
tailored for various types of media. The research employs parallelization and component-wise
assessment of interactions, resulting in explicit expressions for gradients of incoherent functionals. This
approach facilitates the implementation of gradient methods for the identification of internal and
external parameters.
The proposed hybrid model for neuro-biosystems provides a comprehensive description of the state
and behavior of T-objects, focusing on the segmental depiction of 3D-trajectories associated with
1
ITTAP’2023: 3nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24,
2023, Ternopil (Ukraine), Opole (Poland)
EMAIL: m.bachynskyi@gmail.com (A. 1); mykhaylo_petryk@tu.edu.te.ua (A. 2); v_brevus@tntu.edu.ua (A. 3); i1mudryk@ukr.net (A. 4);
bogdanglova2014@gmail.com (A. 5)
ORCID: 0000-0001-6612-7213 (A. 1); 0000-0003-4139-7633 (A. 2); 0000-0002-7055-9905 (A. 3); 0000-0002-4305-1911 (A. 4); (A. 5).
©️ 2023 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
abnormal neurological movements in a specific part (limb) of the T-object's body, taking neuro-objects
into account. High-speed analytical solutions for the model, which describe trajectories for each
neuromovement segment in vector form, are obtained using hybrid integral Fourier transformations. A
novel method for calculating the hybrid spectral function of movement, a system of orthogonal basic
functions, and spectral values form the basis of the proposed hybrid transformation, offering an
integrated vector solution for the model.
2. A comprehensive approach and analytical instruments for diagnosing
neurological conditions in T-objects utilizing the hybrid ANM model
These studies primarily emphasize the investigation of parameters pertaining to normal
physiological conditions and behaviors, particularly the typical wave-like movements observed in
specific body regions. To analyze these parameters, conventional digital processing techniques relying
on integral Fourier transforms were employed. In prior works [2-3], cognitive feedback connections
were roughly approximated through the utilization of methods and software technologies related to
neural networks.
The methodology is founded on a hybrid model of the neuro-system, incorporating СС nodes and
the tremor-object. This model was developed based on the theory of wave signal propagation and serves
to delineate the states and behaviors of T-objects. It facilitates a segment-by-segment description of the
3D elements within the movements trajectories of the examined T-object, particularly focusing on the
limbs of the hand. The model takes into consideration the matrix of cognitive influences emanating
from groups of СС neuro-nodes on motion segments. These elements encompass the components of
the hybrid spectral function across all signal segments. To dissect intricate ANM movements into more
straightforward elements, the number of partitions can be flexibly selected, depending on the
complexity of the ANM patterns. The mathematical model is designed to yield quantitative parameters
related to tremors. An integral aspect of this method for analyzing the ANM data of a T-object is the
remarkable capability to obtain a frequency response. This is achieved by applying a hybrid integral
Fourier transform and employing digital signal processing techniques on hybrid spectral functions and
spectral values [7, 8].
An electronic pen is employed for the recognition of handwritten content, including numbers, text,
and template drawings, as well as for capturing and digitizing arbitrary movements of the hand. We
have introduced a graphic digital pen device equipped with a built-in 3D microaccelerometer for
conducting diagnostic assessments. The microcontroller is responsible for reading and processing data
obtained from a three-axis acceleration sensor (microaccelerometer). In accordance with the proposed
formulas, the system calculates the instantaneous coordinates of the accelerometer's position in space
[8, 9]. Concurrently, data regarding the electronic pen's motion across the graphics tablet surface is
collected.
Upon detecting zero pressure from the pen on the tablet's sensitive surface, signifying the pen's
detachment from the surface, crucial information about the pen's movements is extracted from the
microaccelerometer displays. This facilitates the determination of the instantaneous coordinates of the
Micro-Electro-Mechanical Systems (MEMS) accelerometer's position in space, thereby ensuring the
comprehensive acquisition of data pertaining to the trajectory of the ANM for the T-object and
enhancing the data's reliability.
Figure 1: A 3D model of the ANM in the T-object, derived from data acquired via a microaccelerometer
The digitized pen position data is transmitted to the PC. This enhancement significantly bolsters
the system's reliability in identifying the ANM of the T-object by integrating the tablet's sensitive
element with an electronic pen and the embedded MEMS accelerometer.
The data pertaining to pen movements are utilized to generate a 3D model of the T-object's ANM,
displayed in a graphical interface. This interface allows for the deconstruction of intricate 3D
movements into three possible projections, enabling subsequent analysis of each projection. This
analytical capability supports the selection of the most decisive parameters for ANM identification
and comprehensive assessment [10].
3. The formulation and approach for the direct solution of inhomogeneous
boundary value problems in the context of ANM analysis, considering the
impact of cognitive feedback influences
As a result of this phenomenon, the patterns observed in electroencephalography (EEG) signals from
neural nodes governing oscillatory neurological movements exhibit correlations. In essence, these
correlations play a pivotal role in shaping the dynamics of the ANM for each segment of the track
(referred to as the j-th segment), where n1 represents the number of division points for the abnormal
neurological movements trajectories track (as depicted in Fig. 1). The division can be automatically
configured in a flexible manner, accommodating any finite number of segments, each of which may
possess varying lengths contingent upon the intricacy of the traffic areas and the selection of suitable
basis functions. These basis functions are crucial for constructing acceptable dependencies for their
approximation [13, 14]. Various criteria can be employed to determine the lengths of the partition
elements, such as the amplitude characteristics of individual trends within the oscillating ANM
trajectory [6, 7].
Mathematical Formulation of the Problem:
In accordance with the specified physical principles within the realm of neurological analysis, we
can express the direct inhomogeneous initial-boundary value problem for ascertaining the parameters
associated with the ANM of a T-object through a system of equations [6, 7].
2 u j k ( t , xk ) 2 u jk
t 2
=b 2
jk
xk 2
( )
+ S *jk ( t , xk ) , xk l jk −1 , l jk , j = 1, n1 + 1, k = 1..3 (1)
with initial conditions (homogeneous):
u jk
u jk ( t , xk ) = 0, =0 , jk = 1, n1 + 1, k = 1..3 , (2)
t =0 t t =0
Additionally, it involves homogeneous boundary conditions and a set of interface conditions:
u1 ( t , xk ) x =0 = 0, un ( t , xk ) x =l = 0, (3)
xk z
u jk ( t , xk ) − u jk ( t , xk ) = 0, k = 1..3 b2 uk ( t , xk ) − b2 u j +1 ( t , xk ) = 0, jk = 1, n1 , k = 1..3
z =l jk
jk
z jk +1
z k z =l jk
(4)
(l , l ) ; l = 0, l l , k = 1..3 .
n1 +1
in the multicomponent region Dn1 = ( t , xk ) : t (0;T), xk I n1 =
+
jk −1 jk 0 n1 +1
jk =1
Here (1.1) is a system of wave equations describing the ANM trajectories of tremor on each j-th
segment of the trajectory jk = 1, n1 + 1, k = 1..3 depending on the resulting action of the set of signals
S *j ( t , z ) , arriving from EEG-sensors for a certain set of KGM neural nodes that control the behavior of the
studied T-object, b jk , jk = 1, n1 + 1, k = 1..3 - components of the phase velocity of propagation of the ANM
waves, which are the amplitude characteristics of the wave tremor motion;
n2
S *jk ( , ) = jk i Si ( , ) , jk i , jk = 1, n1, i = 1, n2 , k = 1..3. - an adaptive matrix determines
i =1
the connections and feedback-effects of specific KGM neuronodules on individual small segments of the
ANR-track. The matrix element j i is a weighting coefficient (from 0 to 1), which determines the integral
k
influence of the i-th neuronode S i on the j k -th segment of motion (determined by machine learning methods
based on data mining [13]. The interface conditions (1.3), (1.4) ensure the continuity and integrity of the
solution of the problem for the entire multicomponent domain of its definition.
Development of an analytical solution for the boundary value problem related to ANM.
To establish an analytical solution for the direct inhomogeneous problem, denoted as (1) - (4), we
employ the Hybrid Integral Fourier Transform (HIFT) as previously defined in [12]. This
transformation relies on hybrid integral operators presented in matrix form.
- of direct action:
l1 l2 ln1 ln1+1
Fn1 ... = ...V1 ( xk , m ) 1dz ...V2 ( xk , m ) 2 dx... ...Vn1 ( xk , m ) n1 dz ...Vn1 +1 ( xk , m ) n1 +1dx , (5)
l0 l1 ln1−1 ln1
- of reverse action:
...V1 ( xk , m ) V ( xk , m )( 2 −1
)
m =1
−1
Fn1 ... = m =1
(
...V2 ( xk , m ) V ( xk , m )
2 −1
.
) (6)
..............................................
( )
−1
...V ( x , ) V ( x , ) 2
n1 +1
m =1 k m k m
Here V l ( xk , m ) - is the vector of the hybrid spectral function defined as follows:
,l =1, nk +1
n1
i +1 m 02 ( m)111 m xk − 01 ( m)121 m xk
b i +1 b1 b1
i =1
V 1 ( xk , m )
...
...
n1 m 2 . (7)
11 m 21 m
V l ( xk , m ) = i +1 l −1 ( m)l xk − l −1 ( m)l
1
xk , l = 2,n
i =l b i +1 bk bk 1
...
V n +1 ( z , ) .. .
1 m
m m
n1 ( m )n1 +1
2 11
xk − n1 ( m )n1 +1
1 21
xk
b n1 +1 b n1 +1
m m=0 - the set of spectral values of the GIPF, which are the roots of the transcendental equation
n2 ( )n11+1
− ( ) = 0.
bn +1 1
1 21
l l (8)
1 bn +1 1
n +1 n1 n1 +1 n +1
1 1
It has been determined that the array of spectral values forms a monotonically ascending sequence
that extends to positive infinity + ∞. Leveraging this observation, a recursive approach is introduced
for computing the constituents of the hybrid spectral function of ANM. This methodology focuses on
the identification of a set of orthogonal basic functions and serves as the foundation for the proposed
hybrid transformation. The process ultimately results in an exhaustive vector solution pertinent to the
theoretical mode:
ij ( ) = 2j −1 ( ) ij−1 k
l jk − 1jk −1 ( ) 2jki l jk ,
l jk , l jk
b jk b jk +1 b jk b jk +1
k k
i1 i 2
ijk lk , lk = jk l jk jk l jk − ij2k l jk ij1k l jk , i, = 1,2, jk = 1, n1 , k = 1..3
b k b k +1 b jk b jk +1 b jk b jk +1
21
11j l j = cos l j , j l j = sin l j
bs bs bs bs
k k k k k k
12jk l jk = − s sin l jk , 22jk l jk = s cos l jk , s l , l + 1
bs bs bs bs bs bs
k1 ( ) = − 11k l0k , 02k ( ) = − 021k l0k .
b1 b1
1
j = , j = 1, n1 , k = 1..3 .
k
b 2jk
Following this, we represent the system of equations (7) and the conditions (8) for the boundary
value problem (1) - (4) using matrix notation:
2 2
2
2 1(
2 − b1 u t , xk )
t xk S (t, x ) u1 ( t , xk ) u1 ( t , xk )
2 1 k
− b22 u2 ( t , xk )
2
S 2 ( t , xk ) u 2 ( t , xk ) u 2 ( t , x k )
(9)
t 2
xk 2
= , ............ = 0, t ............ = 0
............
.................................................. S
n1 +1 ( t , xk )
un1 +1 ( t , xk ) un1 +1 ( t , xk )
2
2 − bn2 +1 2 un +1 ( t , xk )
2 t =0 t =0
t 1
xk 1
Applying to problem (9) the direct-action HIPF integral operator Fn1 (5), where 𝐹𝑛1 [𝐿𝑛1 [( xk )]] =
n1 +1 2
2
−𝛽𝑚 𝑢𝑚 Ln ... = b 2j ( xk − l j −1 ) ( l j − xk ) d - hybrid Fourier differential operator, - is the
2
1
j =1 dxk
Heaviside step unit function, we obtain the Cauchy problem:
d2 2 d
2 + mk umk ( t ) = S mk ( t ) ; umk ( t ) um ( t ) = 0 .
= 0,
t =0 t =0
dt dt k
whose solution is the function [12, 13]:
t
sin mk ( t − )
umk ( t ) = S m k ( ) d (10)
0
m k
Applying to (10) the inverse integral GIPF operator Fn−1 1 (6), after transformations, we obtain a
unique solution to the homogeneous boundary value problem of ANM (1.1) - (1.4):
n1 +1 t mk
u j ( t , xk ) = mk k ( t − , xk , ) S k* ( , ) k d d , j = 1, n1 + 1, k = 1..3 . (11)
l =1 0 mk −1
Here, the impact matrix is the response of the ANM system to the influence of the k-th segment of
*
the resulting action of signals S k a certain set of CC-neural nodes on the j-segment of the ANM track:
( t , xk , ) =
(
sin mk t V j xk , mk Vl , mk ) ( ) ; j = 1, n + 1, k = 1..3 . (12)
m V ( z, )
jk 2 1
mk =1 k
mk
4. Identification of AMM amplitude components. Inverse inhomogeneous boundary value
problem taking into account the cognitive feedback-influences of the neuro-nodes of the CC
Choice of residual functional. It is assumed that the amplitude components of the phase velocity
of propagation of the ANM wave bk , k = 1, n1 + 1 boundary value problem (1) - (4) are unknown
functions of time. However, on the surfaces of the regions k k , k = 1, n1 + 1 , of an
inhomogeneous medium, traces of solutions (trajectories of the ANR)
ulk ( t , x k ) = U lk ( t , xk ) (13)
k k
Thus, we have obtained problem (1) - (4), (13), which consists in finding the functions
bk , l = 1, n1 + 1, k = 1..3 D , where D = ( t , z ) : C kT , 0, l = 1, n1 + 1, k = 1..3 .
kT
( )
The residual functional, which determines the deviation of the desired decoupling from the traces
of the decoupling, obtained empirically on surfaces k , can be written as follows:
1 n+1
T
J ( bk ) = usk ( , z, bk ) − U k
2
d (14)
2 0 k =1 L2 ( k ) k
where L ( ) = d k − squared norm. In this case L ( ) = ( t , z )
2 2
.
2 k 2 k z = k
k
The challenge of functionally identifying the amplitude parameters of ANM. The issue, as
described in (1) - (4), requires a solution that involves implementing a procedure for the functional
identification of the amplitude components of the phase velocity of ANM propagation.
bk2 , k = 1, n1 + 1 as a function of time and conditions, known decoupling traces for each sufficiently
thin k-th segment, k = 1, n1 + 1 , is transformed into a direct boundary value problem (15) - (17) as a
system of homogeneous initial boundary value problems for successive thin segments of the ANR:
2 2
( )
2
u t , x = b u + Sl*k ( t , xk ) (15)
t 2 lk
xk
lk lk 2 lk
with initial conditions:
ulk
u l k ( t , xk ) = 0, = 0, l = 1, n1 + 1, k = 1..3 (16)
t =0 t t =0
Boundary conditions on each of the thin segments of the ANM on Z coordinate:
ul −1 ( t , xk ) x = L = U Ll , u l k ( t , xk ) = U l , l = 1, n1 + 1, k = 1..3 (17)
k lk −1 k −1 xk = Llk k
Choice of residual functional. It is assumed that the components of the phase velocity of
propagation of the ANM wave b , k = 1, n1 + 1 of the boundary value problem (15) - (17) are unknown
functions of time. With known values of the pen position uk (t , z ) at observation points on segments
of the ANM k k , k = 1, n1 + 1 :
u lk ( t , xk ) = U Ll ( t , xk ) (18)
lk k lk
the initial-boundary value problem (15) - (17) can be considered for each point z for each thin k1-th
segment of the ANM trace and will consist in finding the functions bk D , where
D = (t, z ) :
k T
1
( )
C k1T , 0, k = 1, n1 + 1 .
The residual functional of the deviation of the solution from its traces on k1 k1 , and can be
obtained as follows
T
( ) (
= ulk t , xk , blk − U lk dt
1
)
2
J lk blk k (19)
2 0
Approach for addressing the direct boundary value identification problem. The process of
constructing and providing mathematical support for the problem's solution is achieved through the
application of the finite integral Fourier transform [12, 13]. By utilizing integral operators [10], we
apply them to the problem outlined in (15) - (17)::
F ulk ( t , xk ) = k ulk ( t , xk )Vm ( m , xk ) dxk U lk m (t ) ,
Ll
lk −1
Vm ( m , xk )
F −1 U lk m ( t ) = U lk m ( t ) ulk ( t , xk ) , (20)
Vm ( m , xk )
2
m =0
2 U Ll
F 2 uk ( t , xk ) = − m2U lk m (t ) + mU Ll 1 − ( −1) = − m2U lk m (t ) + mU Ll − m ( −1) U Ll ,
m k
m
z k −1
U Ll
k −1
k −1 k
( )
Vm ( m , xk ) = sin m xlk − Llk −1 , m = m , Vm 2 = Vm ( m , xk ) dxk = l ,
lk
2
h lk −1
2
the Cauchy problem is obtained:
d2 U ll *
2 km (
U t , xk ) = −bk mU km (t ) + bk mU ll 1 − ( −1) + Skm ( t )
2 2 2 m k
(21)
dt k −1
U ll
k −1
ulk m
u l k m ( t , xk ) = 0, = 0, lk = 1, n1 + 1, k = 1..3 (22)
t =0 t t =0
The unique solution to the Cauchy problem (1.21), (1.22) has the form:
sin blk m ( t − )
( )
t
U lk m ( t ) = S * ( ) + b 2 U − ( −1) m U d (23)
0
blk m lk m lk m Llk −1 Llk
Passing to the originals in (23), we obtain a unique solution to the original boundary value
problem (15) - (17) in the classical form.
t Llk
( t − , xk , )Sl* ( , ) d d + ( (t − , x , L )U )d .
t
u l k ( t , xk ) = 1
lk k
21
lk ( t − , x , L )U
k lk −1 llk −1 − 22
lk k lk Llk
(24)
0 Llk −1 0
Here, the components of the influence vectors have the form:
2 sin bk m ( t − )
1
k ( t − , xk , ) = b
h m=0
sin m ( − lk −1 ) sin m ( xk − lk −1 )
k m
2bk
k
21
( t , xk , lk −1 ) = sin bk mt sin m ( xk − lk −1 )
h m=0
(25)
(t, x , L ) = h sin (b t ) ( −1) sin ( x − L ) .
22
2blk m
k k xk lk m m k lk −1
m=0
The solution (24) to the problem described in (15) - (17) undergoes a sequence of transformations,
leading it to a format that is both convenient and efficient for numerical iterative computations and for
application in parameter identification procedures. By integrating, substituting explicit expressions for
influence functions, and conducting several transformations, the equations (25) are simplified into
straightforward algebraic expressions that are highly suitable for the identification process. This
eliminates the necessity for iterations in this phase of the regularization identification process, markedly
enhancing the overall efficiency. Thus, after integration, we arrive at the following result:
t Llk
2 * 1 − cos blk mt 1 ( )
S
* 1
( t − , xk , )d d = S lk ( −1) − 1 sin m xk − Llk −1
m
( ) ( )
h m=à m( )
lk k 2
0 Llk −1 b lk m
2 1 − cos ( bk mt )
t
U ll
k −1 k
21
( t , xk , lk −1 )d = Ull k −1
h m=0
sin m ( xk − lk −1 ) (26)
0 m
2 m 1 − cos ( bk mt )
t
U ll
k
k
22
( t , xk , lk −1 )d = U l lk
h m=0
( −1)
m
sin m ( xk − lk −1 )
0
After substituting expressions (26) into (24), we finally obtain:
2 1 − cos ( bk mt ) U ll
uk ( t , z ) =
h m=0 m
sin m ( x k − l k −1 ) S * 1
k ( bk m ) 2
( −1)
m
− 1 + U llk −1 (
1 − ( −1)
m
) k
U ll
(27)
k −1
Formulas for the gradient components. We derive analytical expressions for the gradient
components of the residual functional:
T lk 2
J b = k (t , xk ) uk (t , xk )dxk dt . (28)
k
0 lk −1 z 2
In the context of the functional identification problem, we derive the subsequent formulas for the
gradient components of the residual functional:
lk
2
J b ( t ) = k (t , xk ) 2 uk (t , xk )dxk , (29)
k
lk −1 xk
2 1 − сh ( bk m (T − t ) )
k ( t , xk ) = (
sin m k sin m ( xk − lk −1 ) U k − uknk , l = 1, n1 + 1, k − 1..3)
h m=0 ( bk m )
2
2 1 − cos ( bk mt ) U ll
u k ( t , xk ) =
h m=0 m
sin m ( xk − lk −1 ) S k*
1
( bk m ) 2 (
m
(
−1) − 1 + U ll 1 − ( −1)
k −1
)m k
U ll
k −1
U ll
2
z 2 k (
u t , xk ) = −
2
h m=0
m (1 − cos ( bk mt ) ) sin m ( xk − lk −1 ) S k*
1
( bk m ) 2 ( ( )
−1) − 1 + U ll 1 − ( −1)
m
k −1
m k
U ll
k −1
Regularization expressions for the n+1 -th step of defining the identifying functional
dependency. Using the method of minimum errors to determine the dependence of the identification of
the amplitude components of the phase velocity of propagation of the ANM-wave 𝑏̃ 𝑛+1 on time for
lk
each lk - th element of the ANM lk = 1, n1 + 1, k = 1..3 , we obtain
( )
2
ulnk t , lk , blnk − U lk
blnk +1 ( t ) = blnk ( t ) − J bnl ( t ) 2
, t ( 0, T ) , lk = 1, n1 , k = 1..3 (30)
J (t )
k n
blk
lk
A valuable and efficient approach to scrutinize the acquired outcomes is through cyclic
computations, wherein the analyzed data sets are progressively reduced in proportion. In essence,
estimates are derived and compared at each iteration of the analyzed data constraints. These results,
illustrated in the form of frequency and amplitude characteristics, serve as the fundamental components
for assessing a patient's condition via computerized diagnostic methods. Integral aspects of this
development encompass algorithms for obtaining simulated system parameters, the capability to
visually represent the obtained outcomes, and the necessity for dynamically adjusting system
parameters.
These factors collectively enhance the presentation of results, providing greater clarity, and
promoting the focused utilization of the technology. A successful feature of this advancement is its
deployment as an autonomous module, functioning as a library that permits the continuous
enhancement of methods and the sustainability of research relevance.
5. Displaying the digital analysis of the patient's movement trajectory
As illustrated in Fig. 2, these movements exhibit significant heterogeneity, featuring numerous segments
with pronounced high-amplitude and high-frequency abnormal movements. To enhance the visualization of
the trajectory graph for the ANM of the T-object, as depicted in Fig. 1, it is presented in a temporal-spatial
format. In this format, the sections of the oscillating abnormal movement trajectories become clearly
discernible, revealing their dependence on time and their remarkable variation within short time intervals
(Fig. 2).
To conduct a more detailed examination of these ANM movement segments, they can be subdivided
according Tremor-model
to specific time intervals under study. This allows for the investigation of their real amplitude and
frequency characteristics concerning the integral time distributions of cognitive signals from the CC nodes.
0
brusque 1 l
in
tbrsq1
out
l tbrsq1
in
tbrsq 2 brusque 2
out
tbrsq 2
t
Figure 2: Temporal-spatial representation of the ANM in the T-object, highlighting specific segments
characterized
МихайлоbyПЕТРИК
intense vibrational
(ТНТУ) abnormal movements
кафедра програмноїthat vary with
інженерії 121time within
– Software short intervals.
Engineering 4/365
6. Conclusions
We have developed a hybrid model of a neuro-bio-system that elucidates the state and behavior of
the 3D elements within the trajectories of abnormal movements in T-objects. This model takes into
consideration the matrix of cognitive influences originating from groups of neuro-nodes in the cerebral
cortex. Utilizing the techniques of hybrid integral Fourier transforms, we propose high-performance
algorithms for the identification of parameters in the studied feedback systems. These algorithms enable
component-wise estimation of mutual influences by explicit expressions for the gradients of the residual
functional, facilitating parallel computations on multi-core computers.
In contrast to the conventional classical approach, our proposed hybrid model prioritizes a deep
decomposition of the system while preserving its integrity and essential connections. This approach
allows for a more comprehensive description of the complex underlying mechanisms, especially those
involving numerous internal connections and cognitive feedback influences. It enhances data
completeness, which was previously overlooked during conventional statistical processing.
The software implementation in this manner enhances adaptability and ease of integration into
diverse systems for research purposes. Mathematical methods, specifically their calculation algorithms,
have been translated into a set of classes with associated methods that emulate their functionality.
Software modules, classes, and their interactions have been consolidated into a unified library module,
fostering the versatile use of the input data analysis method across various practical applications and
programs.By incorporating the 3D microaccelerometer module within the digital pen of a graphics
tablet, we maintain the existing high measurement accuracy while additionally gaining the capability
to monitor the separation of the pen from the surface along the Xz-axes.
7. References
[1] A.N. Khimich, M.R. Petryk, D.N. Mykhalyk, I.V. Boyko, A.V. Popov, V.A. Sydoruk, Methods
for mathematical modeling and identification of complex processes and systems based on
visoproductive computing (neuro- and nanoporous cyber-physical systems with feedback,
models with sparse structure data, parallel computing). Monograph, Kiev: National Academy
of Sciences of Ukraine. Glushkov Institute of Cybernetics. 2019. - 176 p. ISBN: 978-966-02-
9188-1.
[2] Petryk, M., Pastukh, O., Bachynskiy, M., Mudryk, I., Stefanyshyn, V. Processing of Cerebral
Cortex Neurosignals from EEG Sensors and Recognizing Specific Types of Mechanical
Movements Elements of Pacient Limbs under the Cognitive Feedback Influenses. CEUR
Workshop Proceedingsthis link is disabled, 2023, 3468, pp. 61–70
[3] R. Bhidayasiri, Z. Mari, Digital phenotyping in Parkinson's disease: Empowering neurologists
for measurement-based care. Parkinsonism Relat Disord. 2020 Nov; 80:35-40. DOI:
10.1016/j.parkreldis.2020.08.038.
[4] V. Rajaraman, D. Jack, S.V. Adamovich, W. Hening, J. Sage, H. Poizner, A novel quantitative
method for 3D measurement of Parkinsonian tremor. Clinical neurophysiology, 11(2), 187-369
(2000)
[5] D. Haubenberger, D. Kalowitz, F.B. Nahab, C. Toro, D. Ippolito, D.A. Luckenbaugh, L.
Wittevrongel, M. Hallett, Validation of Digital Spiral Analysis as Outcome Parameter for
Clinical Trials in Essential Tremor. Movement Disorders 26 (11), 2073-2080, (2011)
[6] E.D. Louis, A. Gillman, S. Böschung, C.W. Hess, Q. Yu, & S.L. Pullman, High width
variability during spiral drawing: Further evidence of cerebellar dysfunction in essential tremor.
Cerebellum, 11, 872-879 (2012).
[7] P. Viviani, P.R. Burkhard, S.C. Chiuvé, C.C. dell’Acqua, P. Vindras, Velocity control in
Parkinson’s disease: a quantitative analysis of isochrony in scribbling movements. Exp Brain
2009; 194:259–83.
[8] G. LO, A.R. Suresh, L. Stocco, S. González-Valenzuela, and V. C. Leung, “A wireless sensor
system for motion analysis of Parkinson's disease patients,” (PERCOM Workshops), 2011
IEEE International Conference on. IEEE, pp. 372-375, 2011
[9] Bhidayasiri R., Mari Z. Digital phenotyping in Parkinson's disease: Empowering neurologists
for measurement-based care. Parkinsonism Relat Disord. 2020 Nov;80:35-40. DOI:
10.1016/j.parkreldis.2020.08.038.
[10] Grimaldi G., Manto M. Neurological tremor: sensors, signal processing & emerging
applications. Sensors. 2010;10(2):1399–1422. DOI: 10.3390/s100201399.
[11] Mykhaylo Petryk, Tomasz Gancarczyk, Oleksander Khimich (2021) Methods mathematical
modeling and identification of complex processes and systems on the basis of high-
performance calculations (neuro- and nanoporous feedback cyber systems, models with sparse
structure data, parallel computations), Akademia Techniczno-Humanistyczna w Bielsku-
Białej, Bielsko-Biała, p.195.
[12] Mudryk I., Petryk M.. High-performance modeling, identification and analysis of
heterogeneous abnormal neurological movement’s parameters based on cognitive neuro
feedback-influences. Innovative Solutions in Modern Science. 2021. New York. TK Meganom
LLC. V1(45). P. 235-249 doi: 10.26886/2414-634X.1(45)2021.16.
[13] Glova B., Mudryk I. Application of Deep Learning in Neuromarketing Studies of the Effects
of Unconscious Reactions on Consumer Behavior. 2020 IEEE Third International Conference
on Data Stream Mining & Processing (DSMP): Conference, Lviv, 21-25 August 2020. P. 337–
340. DOI: 10.1109/DSMP47368.2020.9204192