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External low energy
electromagnetic fields affect
heart dynamics: surrogate for
system synchronization, chaos
control and cancer patient’s
health
Frederico P. Costa
1
*, Jack Tuszynski
2
, Antonio F. Iemma
3
,
Willian A. Trevizan
4
, Bertram Wiedenmann
5
and Eckehard Schöll
6
1
Oncology Department, Hospital Sírio Libanês, São Paulo, Brazil,
2
Dipartimento di Ingegneria Meccanica
e Aerospaziale, Politecnico di Torino, Turin, Italy,
3
Mathematical and Statistics, Autem Therapeutics,
Hanover, NH, United States,
4
Physics and Mathematical Modeling, Autem Therapeutics, Hanover, NH,
United States,
5
Department of Hepatology and Gastroenterology, Charité - Universitätsmedizin Berlin,
Berlin, Germany,
6
Institut fu€r Theoretische Physik, Technische Universität Berlin, Berlin, Germany
All cells in the human body, including cancer cells, possess specific electrical
properties crucial for their functions. These properties are notably different
between normal and cancerous cells. Cancer cells are characterized by
autonomous oscillations and damped electromagnetic field (EMF) activation.
Cancer reduces physiological variability, implying a systemic disconnection
that desynchronizes bodily systems and their inherent random processes. The
dynamics of heart rate, in this context, could reflect global physiological network
instability in the sense of entrainment. Using a medical device that employs an
active closed-loop system, such as administering specifically modulated EMF
frequencies at targeted intervals and at low energies, we can evaluate the periodic
oscillations of the heart. This procedure serves as a closed-loop control
mechanism leading to a temporary alteration in plasma membrane ionic flow
and the heart’s periodic oscillation dynamics. The understanding of this
phenomenon is supported by computer simulations of a mathematical model,
which are validated by experimental data. Heart dynamics can be quantified using
difference logistic equations, and it correlates with improved overall survival rates
in cancer patients.
KEYWORDS
non-thermal electromagnetic fields, radiofrequency, cancer treatment, cancer cells,
oscillations, resonance, synchronization, chaos control
Introduction
The human organism functions as an integrated network of organs and systems (Haken,
1977;Haken, 2006;Haken and Portugali, 2016). This hierarchical functional organization of
life processes is well described within the formalism of systems biology (Salvador, 2008).
Organs, cells, and biomolecules are interacting across different levels to create a dynamic
physiological network (Ivanov 2021;Haken, 1977). This system showcases collective
behavior emerging from nonlinear and adaptive interactions involving biophysical and
OPEN ACCESS
EDITED BY
Igor Franović,
University of Belgrade, Serbia
REVIEWED BY
Alexander E. Hramov,
Immanuel Kant Baltic Federal University, Russia
Vladimir Semenov,
Saratov State University, Russia
*CORRESPONDENCE
Frederico P. Costa,
frederico.costa@hsl.org.br
RECEIVED 08 November 2024
ACCEPTED 03 December 2024
PUBLISHED 03 January 2025
CITATION
Costa FP, Tuszynski J, Iemma AF, Trevizan WA,
Wiedenmann B and Schöll E (2025) External low
energy electromagnetic fields affect heart
dynamics: surrogate for system
synchronization, chaos control and cancer
patient’s health.
Front. Netw. Physiol. 4:1525135.
doi: 10.3389/fnetp.2024.1525135
COPYRIGHT
© 2025 Costa, Tuszynski, Iemma, Trevizan,
Wiedenmann and Schöll. This is an open-access
article distributed under the terms of the
Creative Commons Attribution License (CC BY).
The use, distribution or reproduction in other
forums is permitted, provided the original
author(s) and the copyright owner(s) are
credited and that the original publication in this
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academic practice. No use, distribution or
reproduction is permitted which does not
comply with these terms.
Frontiers in Network Physiology frontiersin.org01
TYPE Original Research
PUBLISHED 03 January 2025
DOI 10.3389/fnetp.2024.1525135
biochemical control, along with communication between cells and
organs (Schöll, 2022). The human body exhibits a variety of rhythms
and oscillations, such as circadian rhythms, cell cycles, and hormone
secretion. These oscillators interact within and across tissues and
cells, producing a spectrum of behaviors ranging from
synchronization to chaos, which are crucial for biological
adaptability and evolution. These dynamics, including
synchronization phenomena in networks of coupled nonlinear
oscillators, are key to understanding biological systems across
scales. Such patterns, ranging from cluster synchronization to
chimera states with both coherent and incoherent domains,
illustrate the universal role of synchronization in both natural
and technological contexts (Pikovsky et al., 2001;Rosenblum and
Pikovsky, 2004;Boccaletti et al., 2018;Heltberg et al., 2021;Schöll,
2021;Sawicki et al., 2022a;Sawicki et al., 2022b;Berner et al., 2022).
Thus, the frequency of oscillation (intrinsic frequency) is the crucial
component for understanding the patterns of synchronization,
resonance, and chaos. In complex biological systems, endogenous
and external electromagnetic fields (EMF) operate as the “fast”
primary messenger for physiological and pathological network
behavior and control (Costa et al., 2024). Electromagnetic
oscillations and synchronization of biomolecules triggered by
internal and external pulses are the physical basis of the cellular
electromagnetic field (Sun et al., 2022).
Cancer cell behavior has been proposed to be characterized by
damped EMF activities (Pokorný et al., 2020). Damped system
dynamics in biological systems are influenced by the inherent
randomness of biochemical reactions, which disrupt phase
coherence and reduce oscillation amplitude. In cancer patients,
this leads to an energy deficit in the system, further dampening
oscillations and increasing entropy, or disorder. Damping causes
reduction in physiological variability that indicates systemic
isolation by decoupling systems and their stochastic processes
(Pincus, 1994). Physiological system coupling is often measured
by variability, especially with heart rate variability serving as a
common indicator of physiological variability and overall system
stability. Heart rate variability measures the differences between
consecutive heartbeats, known as R–R intervals (RRI), over time
(Force et al., 1996;D’Angelo et al., 2023;Tiwari et al., 2021).
From the perspective of non-linear dynamics, the human
cardiovascular system has a self-oscillatory character at the micro
(cellular properties) and macro level. In particular, the heart rate can
be synchronized such that the effect of phase locking is observed
under the effect of weak external forcing (Anishchenko et al., 2000).
The normal sequence and synchronous contraction of heart
myocytes (e.g., heart dynamics) results from spontaneous and
coordinated rapid flow of ions through ion channels located in
the plasma membrane, producing a sequence of action potentials
(Grant, 2009). The heart dynamics can be represented by a Van der
Pol oscillator (VPO) that is extensively used to model the nonlinear
behavior of heartbeats. It is a self-sustained nonlinear dissipative
oscillator that exhibits chaotic switching between two types of
regular motion, namely, periodic and quasiperiodic oscillations in
the principal resonance region under exposure to EMF (Kadji et al.,
2007;Van der Pol and Van der Mark, 1927). The driven VPO serves
as a paradigmatic model for chaos in low-dimensional systems.
When subjected to external forcing (e.g., Lorentz force), VPO can
show not only limit cycles (asymptotically stable periodic orbits), but
more complex dynamical behavior like strange attractors. Strange
attractors, key to nonlinear dynamics and chaotic systems, represent
an asymptotic chaotic state with fractal, i.e., non-integer dimension.
This results in a system highly sensitive to small changes in initial
conditions, leading to practically unpredictable temporal behavior.
Cancer cells, in autoregulated damped systems, may be conceived as
strange attractors that could interfere directly with the heart
dynamics described by a VPO (Uthamacumaran, 2020). This
interference leads to phase transition scenarios that determine
the system’s initial dynamics prior to EMF exposure (Davies
et al., 2011).
A predetermined active closed-loop control system is a type
of feedback control system designed to automatically correct
any deviations which includes perturbations with feedback
dynamics (Gad-el-Hak, 2000). It is more systematically
designable, and adaptable to handle noise and uncertainties,
making it superior in flexibility and robustness compared to
constantly active open-loop control. Closed-loop control
uniquely allows for the examination and stabilization of
otherwise inaccessible unstable states, offering significant
practical and theoretical benefits (Macau and Grebogi, 2007;
Schöll et al., 2016;Schöll, 2024). Moreover, it supports the
recent advances in controlling low-dimensional chaos in
nonlinear systems and its extension to spatiotemporal
dynamics. In this study we use a medical device-based
methodology to explore the impact of external EMF upon
the heart dynamics in cancer patients to assess if heart
dynamics could serve as a surrogate for system
synchronization, chaos control and ultimate the patient’s
initial health status.
Methods
Study population
We conducted a retrospective analysis of 22 patients submitted to
identical exposure procedure at their first exposure to EMF. This
corresponded to a subset of 66 adult patients enrolled in the
feasibility trial reported by Capareli et al. (2023). Patients were aged
18 or older with advanced, unresectable, or metastatic hepatocellular
carcinoma (HCC), confirmed histologically or clinically per American
Association for the Study of LiverDiseasesguidelines. They were
classified as Child–Pugh A or B cirrhosis, Barcelona Clinic Liver
Cancer stage B or C, and an Eastern Cooperative Oncology Group
performance status of 0–2. There were no restrictions based on disease
progression, hematological or organ function, or previous or current
therapies. Hospital Sírio Libanês (São Paulo, Brazil) was the sole study
site for the feasibility study where all participants provided written
informed consent before enrollment. The trial adhered to the
Declaration of Helsinki, with protocol approval by the hospital’s
institutional review board, ethics committee, and the Comissão
Nacional de Ética em Pesquisa. Following full enrollment, the ethics
committee approved a compassionate access program under the same
protocol but with a revised consent form. The trial was registered at
ClinicalTrials.gov (NCT01686412) prior to starting enrollment. More
information about the study population can be found elsewhere
(Capareli et al., 2023).
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EMF exposure system
All study participants were exposed to low-energy radio
frequency EMF using the AutEMdev prototype (Autem
Therapeutics NH United States), a high-precision radio frequency
emitter device that controls systemic exposure of a dual signal at a
carrier frequency of 27.12 MHz and modulates amplitudes from
10 Hz to 20 kHz (Tuszynski and Costa, 2022). Hemodynamic
responses were monitored non-invasively and continuously using
a high precision beat-to-beat recording device (Task Force Monitor
or CNAP500; CNSystems, Graz, Austria), which was synchronized
with the AutEMdev in the millisecond (ms). Patients underwent
exposure to a pre-programmed fixed range and sequence of
modulation frequencies via the AutEMdev during the discovery
phase in part 2 as described by Capareli et al. (2023). Thus, each
patient underwent hemodynamic monitoring for 15 min
immediately prior to EMF exposure, which consisted of a 15-
minute session where they were exposed to a pre-programmed
fixed sequence of modulation frequencies delivered by the
AutEMdev. The carrier wave amplitude-modulated in sinusoidal
form with frequencies ranging from 10 to 100Hz, where each
frequency increment of 1Hz was exposed sequentially for 10 s
from the lowest to the highest frequency.
Processing hemodynamic data
Digital data were saved and supported real-time cloud
computing that represented a closed-loop control system. The
EMF generator emitted a fixed sequence of modulation
frequencies, each lasting 10 s over a 15-minute session,
simultaneously with hemodynamic monitoring that collected
metrics every 10 milliseconds. Every participant’s 15-minute
tachogram was automatically processed: the autoregressive
method was used for detecting outliers and the electro-pressure
vectogram gradient phase time plots were used to study 10-second
consecutive periodic orbits as described elsewhere (Capareli et al.,
2023;García et al., 2023). This analysis allows the identification of
instances of outlier heartbeats with significantly longer or shorter
durations than expected. Next, the device produces a patient-specific
series of modulation frequencies based on computing the feedback
responses collected during the exposure to the fixed series of
modulation frequencies. Finally, data recorded from 22 cancer
patients, selected from their initial exposure to EMF were used in
a retrospective correlation with survival. Full description of the
exposure procedures can be found elsewhere (Capareli et al., 2023).
Model of isolated heart
There is a significant link between the autonomic nervous
system and cardiovascular dynamics resulting from sympathetic
activity or vagal activity observed in patients with chronic disease,
including cancer. However, in order to study potential direct effects
of EMF (e.g., external force on the cardiac myocytes), we developed a
computational heart model (e.g., isolated heart) focused on the
action potential of the cardiomyocytes. The simulation of action
potential in cardiomyocytes considers that the membrane is excited
beyond a certain threshold, activating ion channels. This leads to ion
currents flowing in or out, changing the cardiomyocytes’potential
and triggering an action potential. After this, the membrane
potential resets to its resting state, ready for another excitatory
input. During an action potential, the cardiomyocyte enters a
refractory phase where it is temporarily unresponsive to external
disturbances. The rate at which these cells depolarize varies,
affecting how quickly they generate action potential. Thus, the
action potential as a function of the time, where the shape of the
action potential remains constant while the frequency may be
changed in a wide range.
Time-continuous computer modeling
A computational heart model simulated the EMF signal
coupling with the cardiomyocytes and their action potentials
following Hodgkin-Huxley, FitzHugh-Nagumo, and modified
VPO equations. The oscillations, variable excitability,
refractoriness, and asymmetric action potentials with distinct
depolarization and repolarization rates were validated to
reproduce experimental data (García et al., 2023;Berenfeld and
Abboud, 1996;Kumai, 2017). In the following we focus on the
FitzHugh-Nagumo model, which simplifies the complex dynamics
of ionic voltage-gated channels described by the Hodgkin-Huxley
equations, by means of an electronic analogue that captures two
basic processes (Figure 1). 1) A current F(V)V3
3−Vthat depends
nonlinearly on the membrane potential V(activator), and responds
in phase with it, representing the relatively fast opening of gated
channels that allow a current influx in the cell, and 2) a delayed and
persistent current W (inhibitor) that restores equilibrium in a longer
time-scale, representing the slow response of channels closing
(especially the potassium channels), which is modeled through an
effective in-parallel inductor L and resistance R. A battery Eadjusts
the value of the resting potential of the membrane.
ϵdV
dτIτ
()
−FV
()
−W
dW
dτV+E−RW (1)
where τis a rescaled time variable defined as τt/L. This definition
introduces the parameter ϵC/Lin the dynamical equation for the
FIGURE 1
Schematics of the electronic analogue for the FitzHugh-
Nagumo model.
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potential, which expresses the ratio between the capacitive and
inductive relaxation times, making the time scale separation
between the two gating processes explicit.
The FitzHugh-Nagumo model allows the input of an external
current into the system, which in our case is an ionic current in
phase to the local electric field delivered in the proximities of the
membrane, therefore having the form:
Iτ
()
Aτ
()
cos ωcτ
() (2)
where ωcis the carrier wave angular frequency, and A(τ)is the
modulated envelope that evolves at a much slower pace.
Results
Time-continuous computer modeling
In the absence of an external oscillating current I = 0 and with
suitable choice of parameters, the solution of Equation 1 for the
action potential evolves as an oscillating function V0(t)in a
characteristic time scale of the order of a second, reaching a limit
cycle, as shown in Figures 2A, B (in the simulations for this section
the parameters were set to ϵ0.1; E0.5; R0.5).
This trajectory is a combination of a slow and a fast dynamic
part, as seen in the time series of Figure 2A and in the phase portrait
of Figure 2B, which arises as a combination of a slow motion (order
of ϵ) close to the two branches of the WV−V3/3 nullcline with
negative slope, and a fast motion when the trajectory of
(V0(τ),W
0(τ)) detaches from the W−F(V)nullcline (Schöll
et al., 2009;Omelchenko et al., 2013). A refractory region then
emerges near the left branch of the nullcline, highlighted in
Figure 2B, during which the system is expected to be more
robust to external stimuli. Outside the slow refractory part of the
trajectory and outside the slow part of the excitation pulse (right
branch of the nullcline), the system becomes vulnerable to an
external EMF oscillating stimulus.
The robustness/vulnerability to external stimuli can be
inspected by analyzing the system’s Phase Response Curve
(PRC), which quantifies the phase shifts induced in the
trajectory due to very short perturbations (δ-function like
stimuli), and that can be computed by solving for the unstable
periodic limit cycle of the adjoint system (Schultheiss et al., 2012;
Winkler et al., 2022). The calculated Phase Response Curve in
Figure 2C shows that a weak stimulus will have little effect in the
action potential when applied in the refractory region or during
the slow part of the excitation pulse. When the trajectory
detaches from the slow regions near the two negative-slope
branches of the nullcline, a weak stimulus may either retard
or advance the action potential depending on whether the sign of
the Phase Response Curve is negative or positive, respectively.
Figures 3A–Cand3D-Fshowtheseeffects(delayedand
accelerated action potentials, respectively) when an external
current like the one in Equation 2 is applied for a short time
of one cycle of the envelope function, which takes the form of:
Aτ
()
A0
21+cos ωmodτ
()() (3)
When subject to an external current of the form of Equation 2,
the forced solutions (V(τ),W(τ)) evolve on a fast time scale
imposed by the carrier frequency and a slow time scale which
governs average deviations from the free system (Figures 3A–C
and 3D-F). The effective pull felt by the system in the slower time
scale in the presence of the EMF stimulus can be modeled by
substituting the external current by an effective current, obtained
by averaging out the fast time scale in the original dynamical
equations (derivation in the following section):
FIGURE 2
Evolution towards the limit cycle of an unforced solution of the FHN model with parameters ϵ0.1, E0.5andR0.5(A). Time series V0(τ)
representing the evolution of the action potential (B). Representation in the (V0(τ),W0(τ)) phase space highlighting the refractory region of the limit cycle
(C). Phase Response Curve (PRC) superimposed to the limit cycle solution of the model. Near the refractory region, short external stimuli have little to no
effect on the undriven trajectory. Before the ascending and descending parts of the action potential, an external stimulus can either advance or
retard the trajectory, respectively, depending on the sign of the PRC.
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Ieff τ
()
−KA2τ
()
Vτ
() (4)
Kbeing a coupling parameter. This introduces a closed-loop control
since the driving term depends upon the system variable V. The
effective driving current evolves at a time scale closer to that of the
system and is similar to a train of pulses bursting at the
envelope frequency.
It has been shown that a train of pulses at appropriate
frequencies and phases is able to induce time-changes in the
orbit of modified Van der Pol systems (Zebrowski et al., 2007). In
fact, Figures 4A–Cshow the impact of the effective current when
theenvelopeofEquation 3 is applied continuously. When the
frequency and starting point of the envelope train are chosen
such that the external effective current is nearly zero for most of
the positive regions of the Phase Response Curve, the action
potential is slowed down (Figure 4A). Note that the sign in
Equation 4 makes the current impulse positive whenever
V<0. Different choices in parametrization can also speed up
the action potential (when the effective current is suppressed in
thenegativeregionsofthePRC(Figure 4B), or do not affect
it (Figure 4C).
Therefore, the proposed EMF coupling can model changes in
heartbeat intervals observed experimentally (Figure 8), based on ion
flow dynamics. Furthermore, if a significant difference in the model
parameters is expected (e.g., normal vs. cancer cells bioelectric
properties), we can likely consider radical behavioral shifts in ion
flow dynamics across the membrane.
Derivation of the effective current
When the EMF signal is delivered, the forced solution of
Equation 1 can be expressed in terms of the deviations (V1,W
1)
from the undriven free system, that is, V(τ)V0(τ)+
V1(τ),W(τ)W0(τ)+W1(τ)with the perturbation terms
satisfying:
FIGURE 3
(A). Comparison between the limit cycle solution for the undriven system (unforced) and the delayed trajectory forced by a short external stimulus of
the form of Equations 2,3(B). Short current stimulus applied during a negative portion of the PRC. The para meters for the current are A05, ωmod 12.6
and ωc88.0(C). Representation of the trajectories in the phase space (D). Comparison between the limit cycle solution for the undriven system
(unforced) and the advanced trajectory forced by a short external stimulus of the form of Equations 2,3(E). Short current stimulus applied during
positive portion of the PRC. The parameters for the current are A05, ωmod 12.6 and ωc88.0(F). Representation of the trajectories in the phase space.
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ϵdV1
dτIτ
()
−GV
0,V
1
()
−W1
dW1
dτV1−RW1
GV
0,V
1
()
FV
0+V1
()
−FV
0
() (5)
As shown in Figures 3A, D, because of the difference between the
carrier wave frequency ωcand all the other time scales of the system,
the solutions of Equation 5 are expected to evolve at two different
time scales, which we will call τFτ(fast) and τS1
ωc
τ(slow).
Treating them as independent variables in a dual time approach, one
can write V1(τ)V1(0)(τF,τS)+ 1
ωcV1(1)(τF,τS)+... and
W1(τ)W1(0)(τF,τS)+ 1
ωcW1(1)(τF,τS)+...,which has a solution
in the lowest order in 1
ωcof the form:
V10
() τF,τS
()
A
ϵωc
sin ωcτF+δ
()
+aτS
()
W10
() τF,τS
()
bτS
()
e−RτF+1
RaτS
()
a0
()
−Rb 0
()
− A
ϵωc
sin δ(6)
Here, the initial time was chosen such that the perturbations were
absent (V10, W10), when the EMF current was delivered to the
system at an arbitrary phase δ.Equation 6 show that in the presence of
the external EMF current, the potential responds as a fast wave that
oscillates around a slower changing function a(τS),asinFigures 3A, D.
Moving up one order in the dual time approach, it can be
shown that:
1
ϵωc
dA
dτS
sin ωcτF+δ
()
+da
dτS
−
ωc
ϵGV
0,V
10
()
+W10
()
)
(7)
Averaging over the fast variable (one cycle of τF) and over all the
possible initial conditions (0≤δ<2π),Equation 7 yields (in the
original time variable τand for very short times):
ϵda
dττ→0
− 1
ϵωc
2
A20
()
V00
() (8)
Therefore, the fast-varying current exerts an effective pull on
the slow time evolution of the perturbation that is proportional
to the potential (and opposite in sign with it), and proportional
to the square of the envelope amplitude. Equation 8 motivates
the substitution of the exact current in the original system
(Equation 1) by its effective slower evolving
counterpart (Equation 4).
Time-discrete mathematical modeling
Considering a train of pulses at appropriate frequencies and
phases that interferes with the heart dynamics causing changes in
heart rate variability, by time-domain, frequency domains and
nonlinear analysis, we examine how the heart’s qualitative
behavior changes with variations in action potential trajectories.
A difference equation model discretizes events in time by
expressing each term as a function of its immediately preceding
term, enabling step-by-step computation of the sequence (Buchner
and Jan Żebrowski, 2001;Buchner and Żebrowski, 2000;May, 1976).
We use the logistic function for the interval x of two
successive spikes:
Fx
()
ax 1 –x
()
which gives the logistic difference equation, defined for 0 <x<4,
and has the form:
xt+1axt1-x
t
()
where non-trivial behavior occurs only if 1 <a<4 and in this case,
the fixed point is x*1−1
a. as illustrated in Figure 5.
FIGURE 4
Comparison between the limit cycle solution for the undriven system (unforced) and the trajectory forced by a continuous external stimulus
represented by an effective current (Equation 4), with envelope evolving as Equation 3. Parameters K 0.03 and A01. (A) ωmod 3.9 and starting point of
the envelope were chosen to suppress the current train in most of the positive regions of the PRC, leading to a delayed action potential. (B) ωmod 3.7 and
starting point of the envelope were chosen to suppress the current train in most of the negative regions of the PRC, leading to an advanced action
potential. (C) ωmod 9.0 and starting point of the envelope were chosen as to leave the action potential unaffected.
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Four strata were considered: If 0 <a<1 then there is only the
trivial solution x = 0, to which all orbits are attracted. If 1 <a<3 the
fixed-point x* is stable and attracts all orbits. If 3 <a<4 stability is
defined by the curve slope, S = 2 - a, at the fixed point and
bifurcations and chaos may occur. And if a>4 then the fixed
point tends to infinity (Supplementary Figure 1S, supplement
information). Constant awas calculated for each patient, as a
function of sample estimates of time to event in seconds,
meaning the time for low-frequency signal exposure where an
outlier event was observed: median, standard deviation,
coefficient of variation and skewness (Supplementary Table 1S,
supplement information). It was observed a strong positive
correlation between the averages of the strata of aand the
corresponding average survival of patients as illustrated in Figure 5.
FIGURE 5
(A). displays Difference Equation curves x
t+1
=ax
t
(1−x
t
) for seven cancer patients, each characterized by a specific growth rate constant a. The
intersection points of these curves with the line f(x) = x identify fixed points (orthogonal projections on the X(t) axis) and their slopes indicate the stability of
these points (dashed lines). Figures (B–D) illustrate strong correlation trends: a positive correlation between survival rates and the growth rate constant a
and fixed points, and a negative correlation between the slope and survival rates. Figures (E–G) further highlight these trends, showing strong
positive correlations between the fixed points and the constant a, and strong negative correlations between the slope, the constant a, and the fixed points.
See Supplementary Table 1S and Supplementary Figure 1S for more information. Blue bars corresponds to mean survival. Green bars corresponds to
mean constant a values.
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Experimental data validation
The RRI time series displayed a nonlinear, quasi-periodic
oscillation with an average RRI of 813.257 ±
143.593 milliseconds. We observed a shift in heart rate variability
when comparing conditions of non-exposure and exposure in the
same patient as illustrated in Figure 6, representing a change in
nonlinear dynamics of the heart induced by EMF modulation.
Further results are detailed in Capareli et al. (2023). Outlier
heartbeats were identified from each participant’s 15-minute
FIGURE 6
Comparison between non-exposure vs. exposure 15-minute tachograms recorded from a cancer patient. EMF exposure caused a delay in the heart
periodic oscillations. In the time-domain, EMF interference resulted in increase in RRI duration, increase in SDNNI, decrease in RMSSD. In the frequency-
domain, there is a shift for high VLF power, large increment in total power and LF/HF ratio. Graphic produced by Kubios. Note: RRI: R-R intervals; RMSSD:
Square root of the mean squared differences between successive RR intervals; SDNNI: Mean of the standard deviations of RR intervals in 5-min
segments; VLF: very low band peak frequencies; LF: low frequency band peak frequencies; HF: high frequency band peak frequencies.
TABLE 1 Description of outlier heart beats identified by automated process in the 15-minute tachograms recorded from each participant during their first
EMF exposure.
Patient’s initial Outlier heart beats Neighborhood beats Outcome
Single Burst Mean Std. Dev Mean Std dev
HMC 7 4 684.91 22.67 731.53 20.43 Accelerated
NJP 13 2 889.96 207.99 800.42 31.02 Decelerated
PJF 28 4 1152.69 72.77 783.56 32.43 Decelerated
LGM 18 8 673.83 61.98 601.22 146.07 Decelerated
LS 24 24 789.70 53.73 686.07 22.57 Decelerated
VML 24 4 988.12 53.73 945.52 41.31 Decelerated
APG 14 1 772.16 56.05 956.88 62.57 Accelerated
MA 27 18 854.37 61.20 850.62 11.64 Decelerated
JAS 19 5 601.79 34.60 1161.63 34.78 Accelerated
CBC 35 8 744.21 30.92 600.40 12.05 Decelerated
GF 35 2 948.23 80.56 743.88 19.00 Decelerated
JBC 27 9 982.50 81.33 1021.82 48.06 Accelerated
General 271 89 840.20 68.13 823.63 40.16 Decelerated
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tachogram and are summarized in Table 1. All participants exhibited
outlier heartbeats, either as isolated incidents or cluster of events
(burst), associated with specific frequencies of amplitude
modulation exposure. These outlier heartbeats were significantly
longer or shorter than neigborhood RRI, as shown in Figure 7.
Additionally, there were more instances of decelerated (longer)
outlier heartbeats than accelerated (shorter) ones. However, the
mean difference in RRI was in the order of +16.57 ±
27.97 milliseconds. A number of heart rate variability metrics for
time domain, frequency domain and nonlinear analysis were studied
FIGURE 7
(A) The boxplot displays outlier heartbeats categorized into decelerated and accelerated types, compared to the neighboring R-R interval values.
The neighboring beats consist of a total of 8–10 heartbeats, divided equally before and after the outlier heartbeat. The compar ison of two sequential 10-
second tachograms shows (B) progressive larger RRI and (C) trajectory effect in gradient phase plot vectograms, when no EMF effect is observed, or the
deceleration effect is introduced into the time series of heart beats.
TABLE 2 Sensitivity and specificity by ROC curve for prediction of survival longer than 360 days from initial EMF exposure procedure in 22 patients with
advanced HCC.
Time and nonlinear domains Cutoff Sensitivity Especificity
S_Entropy ≥1.2549 0.7000 0.6667
Higuchi ≥1.2549 0.7000 0.6667
MFDFA ≤1.2335 0.9000 0.4167
RMSSD ≤10.5739 0.3000 0.8333
Frequency domains Cutoff Sensitivity Especificity
VLF ≤227.0441 0.9000 0.5000
LF nu ≤63.4010 0.6000 0.5833
HF nu ≥30.4039 0.6000 0.5833
LF/HF ≤1.7323 0.6000 0.5833
Total power ≤213.6301 0.5000 0.6667
Constant a ≤1.1758 0.6667 0.9231
Note: S_Entropy: sample entropy; Higuchi, Higuchi fractal dimension; MFDFA, modified detrended fluctuation analysis, RMSSD, root mean square of successive differences between normal
heartbeats; VLF, very low frequency spectrum; LF, nu: indexed low frequency spectrum, HF, nu: indexed high frequency spectrum; LF/HF, low frequency high frequency ratio.
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Costa et al. 10.3389/fnetp.2024.1525135
as potential predictors for <360-day survival from the first exposure,
using data from patients with advanced hepatocellular carcinoma
restricted to the first exposure for each patient p=1,2,....22. However,
all of the heart rate variability metrics showed a low predictive
accuracy determined by Receiver-Operating Characteristic (ROC)
curve. On the other hand, when a difference logistic equation was
used, there was a significant improvement in accuracy (Table 2). A
ROC curve was constructed using the constant awith a cutoff value
set at 1.758. Patients with a value greater than 1.758 had a median
overall survival of 21.5 months (CI 95%: 16.5 <>26.4), compared to
7.9 months for those with a value of 1.758 or less (CI 95%: 3.9 <>
11.8) (Figure 8). This difference in survival rates was statistically
significant, as indicated by a Log Rank test (p <0.0001). The
constant ademonstrated very effective predictive power for
longer survival times, particularly those related to specificity
(0.9231, CI 95%: 0.7782 <>1.0000), positive predictive value
(0.8571), negative predictive value (0.8000) and likelihood ratio
for positive tests (8.667) (Figure 8). We observed significant
correlation between sample entropy and constant a(2-tailed
Person’s correlation, p = 0.035) (Supplementary Table 2S,
supplement information).
Discussion
This retrospective study with 22 cancer patients is the first to
demonstrate a closed-loop control method to exam the heart
dynamics under the exposure to an external low energy EMF
signal at appropriate frequencies and phases acting upon the
limit cycle attractor of the oscillating heart.
This noninvasive, safe and simple method provides for the first
time a measurement of the deterministic interference with the ionic
flow dynamics of the plasma membrane observed in excitable cells
(e.g., cardiomyocytes) with a large potential of application in
medicine (Mattsson et al., 2009). Moreover, the determination of
heart dynamics in this setting may serve as a novel strategy to access
not only cancer patients’health status but a novel cancer treatment
approach. Therefore, this novel technology could open a new
treatment area combining diagnostics with therapy using
systemic EMF signals (e.g., theranostics).
The heart generates by far the strongest EMF in the body,
surpassing the brain’s output in both electric and magnetic
strength. As the predominant source of EMF oscillations, the
heart produces nonlinear interactions affecting the entire
complex system’s behavior of a human being (McCraty, 2016).
The heart may act as a predominant node in the body’s
physiological EMF network (Schubert, 1993;Becker and Selden,
1985). The heart’s dominant EMF oscillations exert influence over
practically all EMF (also called bioelectromagnetic fields) produced
by the human body through its biological processes because of the
electrical activities of the cells, tissues, and organs (Young et al.,
2021;Abdul Kadir et al., 2018;Kaestner et al., 2018). By introducing
resilient limit cycle attractors at appropriate frequency and phase
determined by feedback, one can leverage the sensitivity of chaotic
systems to initial conditions and provide control of chaos through
feedback control. These interventions are aimed to modify the
system’s energy landscape, potentially stabilizing chaotic
fluctuations, and steering the system towards predictable and
regular behavior. This method uses minimal energy, and it is yet
crucial for enhancing control in systems where chaos is a disruptive
FIGURE 8
A ROC curve for 22 cancer patients is presented (left). The Kaplan-Meier curve showed significantly better overall survival for cancer patients with
constant a>1.1758.
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Costa et al. 10.3389/fnetp.2024.1525135
force, making it applicable to the entire physiological system (e.g.,
human body). This study presents a straightforward control strategy
for managing chaotic oscillations in systems such as cardiac rhythms
by stabilizing a previously unstable periodic behavior inherent in the
system’s natural dynamics. Our approach uses a series of pulse trains
that introduce delays in the heart dynamics, potentially leading to
the natural stabilization of the periodic orbit, reducing the system’s
chaotic fluctuations with minimal energy loss (Pyragas, 1992;
Pyragas and Tamaševičius, 1993;Pyragas, 2006;Schöll and
Schuster, 2008;Purewal et al., 2014).
Heart dynamics alterations induced by EMF occur significantly
more frequently in cancer patients than in healthy volunteers as
observed by Capareli et al. (p <0.0001) (Capareli et al., 2023). Low-
energy EMF appear insufficient to disrupt heart dynamics in healthy
volunteers, based on our experimental observation. This suggests
that internal oscillators in healthy physiological states, governed by
stable dynamics with feedback and flexibility, are robust and
adaptable to external interference. However, as in disease,
damped systems exhibit instabilities that can drive physiological
networks toward phase transitions, making them more susceptible
to smooth subtle external disturbances (Pokorný et al., 2020).
Furthermore, the initial conditions of a system significantly
influence its long-term behavior, particularly in damped systems.
This importance arises from the nonlinear dynamics of these
systems, where even minor differences in initial states can result
in dramatically divergent outcomes, which is referred to as
sensitivity to initial conditions. Consequently, by examining the
heart dynamics as response to the first exposure to EMF, we can gain
insights into the individual health status as reflected in the dynamics
of the system’s responses. Besides, big data containing frequency-
time information combined with clinical outcomes represent a
powerful and unique data source. Patients exhibiting different
dynamics such as bifurcation of periodic oscillations, odd-cycle
periodic windows or other dynamics, appear to experience better
survival outcomes. Due to the limitations of the existing
experiments, the long-term effects at various timescales, such as
on circadian rhythms, cannot be determined from the current
clinical data. Further research using a larger dataset in
prospective clinical trials is required to explore the effects of
EMF on survival outcomes and long-lasting effects.
As to the underlying molecular mechanism, the resting
membrane potential V
m
varies across and within cell types.
Mature ‘quiescent’cells, such as neurons and cardiomyocytes,
typically maintain a membrane potential V
m
of about -70 mV,
while cancer cells are significantly more depolarized, V
m
in resting
state ranging from −50 to −10 mV or even less (Di Gregorio et al.,
2022). Additionally, cancer cells demonstrate heightened electrical
activity, with evoked V
m
ranging from 100 to 400 μV in comparison
to excitable cells ranging in mV (Restaino et al., 2023). Unlike
normal cells, the membrane potential of cancer cells fluctuates more
frequently, with hyperpolarization events occurring 2 ± 0.2 times per
cell per 1,000 s and showing about a 3% variation in V
m
(Quicke
et al., 2022;Quicke et al., 2021). Analysis of cellular V
m
time series
indicates both synchronous and asynchronous intercellular
crosstalk between cancer cells, with temporal events ranging from
0.01 to 1 Hz (Quicke et al., 2022). Additionally, within the realm of
cancer, membrane depolarization may play a crucial role in the
development, proliferation and persistence of cancer stem cells,
contributing to continuous tumor growth (Yang and
Brackenbury, 2013). Given that the systemic electric field
generated by the heart is around 50 mV/m, and the external
EMF signal delivers an electric field energy output 100 times
higher, we hypothesize that a train of pulses at specific
frequencies and phases could modulate the membrane potential
through ionic flow dynamics of cancer cells, reverting it to a
hyperpolarized V
m
through the activity of voltage-gated channels
as observed in the heart (Tsong, 1988;Nawarathna et al., 2005). It is
of note here that 5V/m produced by the AutEMdev is below the
safety limits of EMF exposure in humans (Mattsson et al., 2009). In
fact, the EMF coupling term of Equation 4 favours damping of the
action potential trajectory, depending on the bioelectrical
parameters of the cell (hence the resting potential), which may
drive the action potential towards a polarized state. Furthermore,
translational studies support the concept that the effects of EMF
under the given conditions cause a calcium flux in cell culture and
xenograft models leading to an activation of calcium gated channels
(Jimenez et al., 2019;Sharma et al., 2019).
In conclusion, the presented data offer a new theranostic
concept by a minimally invasive and rapid method of controlling
ionic flow dynamics via the plasma membrane with a large potential
of application in medicine. By assessing heart dynamics, this method
potentially allows the diagnostic recognition of individual EMF in
cancer patients (e.g., patient-specific frequencies) thereby finding
new means for the translation of these diagnostic signals into
therapy and possibly prognostication by EMF.
Data availability statement
The original contributions presented in the study are included in
the article/Supplementary Material, further inquiries can be directed
to the corresponding author.
Ethics statement
The studies involving humans were approved by Ethical in
Research Review Board, Hospital Sírio Libanês, São Paulo Brazil.
The studies were conducted in accordance with the local legislation
and institutional requirements. The participants provided their
written informed consent to participate in this study.
Author contributions
FC: Conceptualization, Data curation, Formal Analysis, Funding
acquisition, Investigation, Methodology, Project administration,
Resources, Software, Supervision, Validation, Visualization,
Writing–original draft, Writing–review and editing. JT:
Conceptualization, Data curation, Formal Analysis, Methodology,
Supervision, Validation, Writing–original draft, Writing–review and
editing, Investigation, Resources, Visualization. AI: Data curation,
Formal Analysis, Investigation, Methodology, Supervision,
Validation, Writing–original draft, Writing–review and editing.
WT: Formal Analysis, Investigation, Methodology, Supervision,
Validation, Writing–original draft, Writing–review and editing.
Frontiers in Network Physiology frontiersin.org11
Costa et al. 10.3389/fnetp.2024.1525135
BW: Formal Analysis, Investigation, Methodology, Supervision,
Validation, Writing–original draft, Writing–review and editing,
Project administration. ES: Formal Analysis, Methodology,
Supervision, Validation, Writing–original draft, Writing–review
and editing, Conceptualization, Data curation.
Funding
The author(s) declare that no financial support was received for
the research, authorship, and/or publication of this article.
Conflict of interest
The authors (FPC, BW, JT, AI) disclose their employment and
financial relationships with Autem Therapeutics (AT) (Autem
Therapeutics LLC, Suite 208, 35 South Main Street, Hanover, NH
03755, USA). The authors receive honoraria and have stock from AT.
AT produced a medical device using a non-thermal radiofrequency
amplitude modulated electromagnetic fields and is conducting new
clinical trials to confirm its application as a systemic cancer treatment.
The remaining authors declare that the research was conducted
in the absence of any commercial or financial relationships that
could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board
member of Frontiers, at the time of submission. This had no
impact on the peer review process and the final decision.
Generative AI statement
The author(s) declare that Generative AI was used in the
creation of this manuscript. The author(s) verify and take full
responsibility for the use of generative AI in the preparation of
this manuscript. Generative AI was used The authors declare the use
of generative artificial intelligence (AI) and AI-assisted technologies
only to improve readability, grammar and spelling check in the
writing process. The authors carefully reviewed and edited the
results produced by AI and take full responsibility for the
content of the publication.
Publisher’s note
All claims expressed in this article are solely those of the authors
and do not necessarily represent those of their affiliated
organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or
claim that may be made by its manufacturer, is not guaranteed or
endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found online
at: https://www.frontiersin.org/articles/10.3389/fnetp.2024.1525135/
full#supplementary-material
References
Abdul Kadir, L., Stacey, M., and Barrett-Jolley, R. (2018). ’Emerging roles of the
membrane potential: action beyond the action potential. Front. Physiol. 9, 1661. doi:10.
3389/fphys.2018.01661
Anishchenko, V. S., Balanov, A. G., Janson, N. B., Igosheva, N. B., and Bordyugov, G.
V. (2000). Entrainment between heart rate and weak noninvasive forcing self-
organization. Int. J. Bifurcation Chaos 10, 2339–2348. doi:10.1142/s0218127400001468
Becker, R. O., and Selden, G. C. 1985. “The body electric: electromagnetism and the
foundation of life,”in William Morrow Publishers.
Berenfeld, O., and Abboud, S. (1996). Simulation of cardiac activity and the ECG
using a heart model with a reaction-diffusion action potential. Med. Eng. Phys. 18,
615–625. doi:10.1016/s1350-4533(96)00028-8
Berner, R., Sawicki, J., Thiele, M., Löser, T., and Schöll, E. (2022). Critical parameters
in dynamic network modeling of sepsis. Front. Netw. Physiology 2, 904480. doi:10.3389/
fnetp.2022.904480
Boccaletti, S., Pisarchik, A., Genio, C., and Amann, A. 2018. Synchronization: from
coupled systems to complex networks. Cambridge University Press.
Buchner, T., and Jan Żebrowski (2001). Logistic map with a delayed feedback: stability
of a discrete time-delay control of chaos. Phys. Rev. E, Stat. nonlinear, soft matter Phys.
63, 016210. Cambridge University Press. doi:10.1103/PhysRevE.63.016210
Buchner, T., and Żebrowski, J. J. (2000). Logistic map with a delayed feedback:
stability of a discrete time-delay control of chaos. Phys. Rev. E 63, 016210. doi:10.1103/
PhysRevE.63.016210
Capareli, F., Costa, F., Tuszynski, J. A., Sousa, M. C., Setogute, Y. C., Lima, P. D., et al.
(2023). ’Low-energy amplitude-modulated electromagnetic field exposure: feasibility
study in patients with hepatocellular carcinoma. Cancer Med. 12, 12402–12412. doi:10.
1002/cam4.5944
Costa, F., Wiedenmann, B., Schöll, E., and Tuszynski, J. A. (2024). Emerging cancer
therapies: targeting physiological networks and cellular bioelectrical differences with
non-thermal systemic electromagnetic fields in the human body –a comprehensive
review. Front. Netw. Physiology 4, 1483401. doi:10.3389/fnetp.2024.1483401
D’Angelo, J., Ritchie, S. D., Oddson, B., Gagnon, D. D., Mrozewski, T., Little, J., et al.
(2023). ’Using heart rate variability methods for health-related outcomes in outdoor
contexts: a scoping review of empirical studies. Int. J. Environ. Res. Public Health 20,
1330. doi:10.3390/ijerph20021330
Davies, P. C., Demetrius, L., and Tuszynski, J. A. (2011). Cancer as a dynamical phase
transition. Theor. Biol. Med. Model. 8, 30. doi:10.1186/1742-4682-8-30
Di Gregorio, E., Israel, S., Staelens, M., Tankel, G., Shankar, K., and Tuszynski, J. A.
(2022). The distinguishing electrical properties of cancer cells. Phys. Life Rev. 43,
139–188. doi:10.1016/j.plrev.2022.09.003
Force,E.S.C.T.,Bigger,J.T.,Camm,A.J.,Kleiger,R.E.,Malliani,A.,Moss,A.J.,etal.
(1996). Heart rate variability: standards of measurement, physiological interpretation, and
clinical use. Eur. Heart J. 17, 354–381. doi:10.1093/oxfordjournals.eurheartj.a014868
Gad-el-Hak, M. (2000). Flow control: passive, active, and reactive flow management.
Cambridge University Press.
García, P., Acosta, A., Gallo, R., and Peluffo-Ordóñez, D. (2023). Positive invariant
regions for a modified Van Der Pol equation modeling heart action. Appl. Math.
Comput. 442, 127732. doi:10.1016/j.amc.2022.127732
Grant, A. O. (2009). Cardiac ion channels. Circulation Arrhythmia Electrophysiol. 2,
185–194. doi:10.1161/CIRCEP.108.789081
Haken, H. (1977). Synergetics: an introduction: Nonequilibrium phase transitions
and self-organization in Physics, Chemistry and Biology. Springer-Verlag.
Haken, H. (2006). Information and self-organization: a macroscopic approach to
complex systems. Berlin, Germany: Springer.
Haken, H., and Portugali, J. 2016. Information and self-organization: a unifying
approach and applications applications. Entropy 18, 197, doi:10.3390/e18060197
Heltberg, M. L., Krishna, S., Kadanoff, L. P., and Jensen, M. H. (2021). A tale of two rhythms:
locked clocks and chaos in biology. Cell. Syst. 12, 291–303. doi:10.1016/j.cels.2021.03.003
Ivanov, P.Ch (2021). The new field of network physiology: building the human
physiolome. Front. Netw. Physiology 1, 711778. doi:10.3389/fnetp.2021.711778
Jimenez, H., Wang, M., Zimmerman, J. W., Pennison, M. J., Sharma, S., Surratt, T.,
et al. (2019). Tumour-specific amplitude-modulated radiofrequency electromagnetic
fields induce differentiation of hepatocellular carcinoma via targeting Ca(v)3.2 T-type
voltage-gated calcium channels and Ca(2+) influx. EBioMedicine 44, 209–224. doi:10.
1016/j.ebiom.2019.05.034
Kadji, H. G., Enjieu, H., Chabi Orou, J. B., Yamapi, R., and Woafo, P. (2007).
Nonlinear dynamics and strange attractors in the biological system. Chaos, Solit. and
Fractals 32, 862–882. doi:10.1016/j.chaos.2005.11.063
Frontiers in Network Physiology frontiersin.org12
Costa et al. 10.3389/fnetp.2024.1525135
Kaestner, L., Wang, X., Hertz, L., and Bernhardt, I. (2018). Voltage-activated ion
channels in non-excitable cells-A viewpoint regarding their physiological justification.
Front. Physiol. 9, 450. doi:10.3389/fphys.2018.00450
Kumai, T. (2017). Isn’t there an inductance factor in the plasma membrane of nerves?
Biophys. Physicobiol 14, 147–152. doi:10.2142/biophysico.14.0_147
Macau, E. E. N., and Grebogi, C. (2007). “Controlling chaos,”in Handbook of chaos
control. Editors E. Schöll and H. G. Schuster (Weinheim: Wiley VCH).
Mattsson, M.-O., Auvinen, A., Bridges, J., Norppa, H., Schüz, J., Juutilainen, J., et al.
2009. SCENIHR (Scientific Committee on Emerging and Newly Identified Health
Risks), Research needs and methodology to address the remaining knowledge gaps on
the potential health effects of EMF, 6 July, 2009.
May, R. M. (1976). Simple mathematical models with very complicated dynamics.
Nature 261, 459–467. doi:10.1038/261459a0
McCraty, R. 2016. Science of the heart, volume 2 exploring the role of the heart in
human performance an overview of research conducted by the HeartMath Institute.
Nawarathna, D., Miller, J. H., Jr., Claycomb, J. R., Cardenas, G., and Warmflash, D.
(2005). Harmonic response of cellular membrane pumps to low frequency electric fields.
Phys. Rev. Lett. 95, 158103. doi:10.1103/PhysRevLett.95.158103
Omelchenko, I., Omel’chenko, O., Hövel, P., and Schöll, E. (2013). When nonlocal
coupling between oscillators becomes stronger: patched synchrony or multi-chimera
states. Phys. Rev. Lett. 110, 224101. doi:10.1103/PhysRevLett.110.224101
Pikovsky, A., Rosenblum, M., and Kurths, J. (2001). Synchronization: a universal
concept in nonlinear sciences. Cambridge: Cambridge University Press.
Pincus, S. M. (1994). ’Greater signal regularity may indicate increased system
isolation. Math. Biosci. 122, 161–181. doi:10.1016/0025-5564(94)90056-6
Pokorný, J., Pokorný, J., Kobilková, J., Jandová, A., and Holaj, R. (2020). Cancer
development and damped electromagnetic activity. Appl. Sci. 10, 1826. doi:10.3390/
app10051826
Purewal,A.S.,Postlethwaite,C.M.,andKrauskopf,B.(2014).Effectofdelaymismatchin
Pyragas feedback control. Phys.Rev.E90, 052905. doi:10.1103/PhysRevE.90.052905
Pyragas, K. (1992). Continuous control of chaos by self-controlling feedback. Phys.
Lett. A 170, 421–428. doi:10.1016/0375-9601(92)90745-8
Pyragas, K. (2006). Delayed feedback control of chaos. Philos. Trans. A Math. Phys.
Eng. Sci. 364, 2309–2334. doi:10.1098/rsta.2006.1827
Pyragas, K., and Tamaševičius, A. (1993). Experimental control of chaos by delayed
self-controlling feedback. Phys. Lett. A 180, 99–102. doi:10.1016/0375-9601(93)90501-p
Quicke, P., Sun, Y., Arias-Garcia, M., Acker, C. D., Djamgoz, M. B. A., Bakal, C., et al.
(2021). Membrane voltage fluctuations in human breast cancer cells. bioRxiv 2021.
doi:10.1101/2021.12.20.473148
Quicke, P., Sun, Y., Arias-Garcia, M., Beykou, M., Acker, C. D., Djamgoz, M. B. A.,
et al. (2022). Voltage imaging reveals the dynamic electrical signatures of human breast
cancer cells. Commun. Biol. 5, 1178. doi:10.1038/s42003-022-04077-2
Restaino, A. C., Walz, A., Vermeer, S. J., Barr, J., Kovács, A., Fettig, R. R., et al. (2023).
Functional neuronal circuits promote disease progression in cancer. Sci. Adv. 9,
eade4443. doi:10.1126/sciadv.ade4443
Rosenblum, M. G., and Pikovsky, A. S. (2004). Controlling synchronization in an
ensemble of globally coupled oscillators. Phys. Rev. Lett. 92, 114102. doi:10.1103/
PhysRevLett.92.114102
Salvador, A. (2008). ’Uri alon, an introduction to systems biology: design principles of
biological circuits, 215. London: Chapman and Hall/CRC, 193–195. ISBN 1584886420,
GBP 30.99, 2007 (320 pp.)’,Mathematical Biosciences - MATH BIOSCI.
Sawicki, J., Berner, R., Löser, T., and Schöll, E. (2022a). Modeling tumor disease and
sepsis by networks of adaptively coupled phase oscillators. Front. Netw. Physiology 1,
730385. doi:10.3389/fnetp.2021.730385
Sawicki, J., Hartmann, L., Bader, R., and Schöll, E. (2022b). Modelling the perception
of music in brain network dynamics. Front. Netw. Physiol. 2, 910920. doi:10.3389/fnetp.
2022.910920
Schöll, E. (2021). Partial synchronization patterns in brain networks. Europhys. Lett.
136, 18001. doi:10.1209/0295-5075/ac3b97
Schöll, E. (2022). Editorial: network physiology, insights in dynamical systems: 2021.
Front. Netw. Physiology 2, 961339. doi:10.3389/fnetp.2022.961339
Schöll, E. (2024). “Delayed feedback control of synchronization patterns: Comment
on “Control of movement of underwater swimmers: Animals, simulated animates and
swimming robots”by S.Yu. Gordleeva et al,”Phys. Life Rev., 49, 112–114. doi:10.1016/j.
plrev.2024.03.010
Schöll, E., Hiller, G., Hövel, P., and Dahlem, M. A. (2009). ’Time-delayed feedback in
neurosystems. Philos. Trans. A Math. Phys. Eng. Sci. 367, 1079–1096. doi:10.1098/rsta.
2008.0258
Schöll, E., Klapp, S., and Hövel, P. (2016). Control of self-organizing nonlinear systems
(Berlin: Springer).
Schöll, E., and Schuster, H. G. (2008). Handbook of chaos control (Weinheim:
Wiley VCH).
Schubert, E. (1993). Spatial and temporal informations from the cardiac electric field.
Physiol. Res. 42, 61–67.
Schultheiss, N. W., Prinz, A. A., and Butera, R. J. (2012). Phase response curves in
neuroscience: theory, experiment, and analysis. New York: Springer.
Sharma, S., Wu, S. Y., Jimenez, H., Xing, F., Zhu, D., Liu, Y., et al. (2019). Ca2+ and
CACNA1H mediate targeted suppression of breast cancer brain metastasis by AM RF
EMF. EBioMedicine 44, 194–208. doi:10.1016/j.ebiom.2019.05.038
Sun, G., Li, J., Zhou, W., Hoyle, R. G., and Zhao, Y. (2022). Electromagnetic
interactions in regulations of cell behaviors and morphogenesis. Front. Cell. Dev.
Biol. 10, 1014030. doi:10.3389/fcell.2022.1014030
Tiwari, R., Kumar, R., Malik, S., Raj, T., and Kumar, P. (2021). Analysis of heart rate
variability and implication of different factors on heart rate variability. Curr. Cardiol.
Rev. 17, e160721189770. doi:10.2174/1573403X16999201231203854
Tsong, T. Y. (1988). Active cation pumping of Na+,K+-ATPase and sarcoplasmic
reticulum Ca2+-ATPase induced by an electric field. Methods Enzymol. 157, 240–251.
doi:10.1016/0076-6879(88)57080-5
Tuszynski, J. A., and Costa, F. (2022). ’Low-energy amplitude-modulated
radiofrequency electromagnetic fields as a systemic treatment for cancer: review and
proposed mechanisms of action. Front. Med. Technol. 4, 869155. doi:10.3389/fmedt.
2022.869155
Uthamacumaran, A. (2020). ’Cancer: a turbulence problem. Neoplasia 22, 759–769.
doi:10.1016/j.neo.2020.09.008
Van der Pol, B., and Van der Mark, J. (1927). ’Frequency demultiplication. Nature
120, 363–364. doi:10.1038/120363a0
Winkler, M., Dumont, G., Schöll, E., and Gutkin, B. (2022). Phase response
approaches to neural activity models with distributed delay. Biol. Cybern. 116,
191–203. doi:10.1007/s00422-021-00910-9
Yang, M., and Brackenbury, W. J. (2013). ’Membrane potential and cancer
progression. Front. Physiol. 4, 185. doi:10.3389/fphys.2013.00185
Young, A., Hunt, T., and Ericson, M. (2021). The slowest shared resonance: a review
of electromagnetic field oscillations between central and peripheral nervous systems.
Front. Hum. Neurosci. 15, 796455. doi:10.3389/fnhum.2021.796455
Zebrowski, J. J., Grudziński,K.,Buchner,T.,Kuklik,P.,Gac,J.,Gielerak,G.,etal.
(2007). Nonlinear oscillator model reproducing various phenomena in the
dynamics of the conduction system of the heart. Chaos 17, 015121. doi:10.
1063/1.2405128
Frontiers in Network Physiology frontiersin.org13
Costa et al. 10.3389/fnetp.2024.1525135
