**3.5 Propagation of autowaves in a medium with local heterogeneity: simulation of cardiac pathology**

We assume that the region around the local inhomogeneity has a refractoriness time R1, different from the refractoriness time R2 of the medium. Therefore, the wave front reaches the section with refractoriness R1 and gets broken. Thus, arrhythmia occurs in oscillations between the surface of the cell as a whole and the

If there is a heterogeneity in the myocardium (muscle middle layer) of the heart, for example, a heart attack, which means that there is no blood supply in some area,

This experiment can be used as a model for processes occurring in the myocardium.

The physical mechanism of the autowave process in the near-electrode layer of a magnetic fluid is the following: as mentioned earlier, in an electric field, the charged particles of magnetite due to electro- and dipolar phoresis move to the nearest electrode and form a close-packed layer 0–170 nm thick around it (**Figure 11**).

A thin layer—the so-called structural-mechanical barrier—appears between the layer of particles and the electrode. It consists of a dielectric—a mixture of kerosene and oleic acid molecules (surfactant surrounding the magnetite particles). The thickness of this barrier is about 10 nm, but it prevents the particle discharge because of the contact with the electrode. With the growth of the near-electrode layer, the intensity increases in the barrier. At its critical value E � <sup>10</sup><sup>7</sup> V/m, the dielectric becomes conductive and magnetite particles get recharged and start

then a similar rhythm transformation may occur in the heart muscle.

*Oscillogram of relaxation oscillations of a current in a cell with local heterogeneity.*

**4. The physical model of the autowave process**

moving from the electrode—a single flat autowave passes.

area around the local heterogeneity.

*Propagation of autowaves in a medium with local heterogeneity.*

*DOI: http://dx.doi.org/10.5772/intechopen.85197*

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference*

**Figure 9.**

**Figure 10.**

**65**

Local heterogeneity was created in the near-electrode layer of the magnetic fluid cell, as shown in **Figure 9**. This figure also shows the propagation of the autowave, which shows that there is a region around the heterogeneity and the oscillations which occur with a phase different from the oscillations of the cell surface. In this case, the current waveform has the form shown in **Figure 10**.

**Figure 8.** *Experiment: the autowave diffraction around the obstacle. Frame width, 1.2 cm.*

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference DOI: http://dx.doi.org/10.5772/intechopen.85197*

**Figure 9.**

pacemaker is determined only by its own properties and cannot be adjusted by

Autowave is a special kind of wave that does not have standard wave properties. Autowaves do not interfere and do not transfer energy. For autowaves, the principle of superposition is not valid; they mutually destroy each other in a collision (a wave with a smaller period "destroys" a slower wave). The only common property of autowaves with conservative waves is diffraction or rounding of the obstacle. The bending of autowave near the obstacle in a cell with magnetic fluid is shown in

**3.5 Propagation of autowaves in a medium with local heterogeneity: simulation**

Local heterogeneity was created in the near-electrode layer of the magnetic fluid cell, as shown in **Figure 9**. This figure also shows the propagation of the autowave, which shows that there is a region around the heterogeneity and the oscillations which occur with a phase different from the oscillations of the cell surface. In this

case, the current waveform has the form shown in **Figure 10**.

*Experiment: the autowave diffraction around the obstacle. Frame width, 1.2 cm.*

external influence.

*Nanofluid Flow in Porous Media*

**Figure 7**

**Figure 8**.

**Figure 8.**

**64**

**of cardiac pathology**

**3.4 Autowave diffraction on an obstacle**

*Experiment: pacemakers with different periods. Frame width, 1.2 cm.*

*Propagation of autowaves in a medium with local heterogeneity.*

We assume that the region around the local inhomogeneity has a refractoriness time R1, different from the refractoriness time R2 of the medium. Therefore, the wave front reaches the section with refractoriness R1 and gets broken. Thus, arrhythmia occurs in oscillations between the surface of the cell as a whole and the area around the local heterogeneity.

If there is a heterogeneity in the myocardium (muscle middle layer) of the heart, for example, a heart attack, which means that there is no blood supply in some area, then a similar rhythm transformation may occur in the heart muscle.

This experiment can be used as a model for processes occurring in the myocardium.

#### **4. The physical model of the autowave process**

The physical mechanism of the autowave process in the near-electrode layer of a magnetic fluid is the following: as mentioned earlier, in an electric field, the charged particles of magnetite due to electro- and dipolar phoresis move to the nearest electrode and form a close-packed layer 0–170 nm thick around it (**Figure 11**).

A thin layer—the so-called structural-mechanical barrier—appears between the layer of particles and the electrode. It consists of a dielectric—a mixture of kerosene and oleic acid molecules (surfactant surrounding the magnetite particles). The thickness of this barrier is about 10 nm, but it prevents the particle discharge because of the contact with the electrode. With the growth of the near-electrode layer, the intensity increases in the barrier. At its critical value E � <sup>10</sup><sup>7</sup> V/m, the dielectric becomes conductive and magnetite particles get recharged and start moving from the electrode—a single flat autowave passes.

the electrode layer has already formed, (the voltage on the electrodes is less than the

When the voltage on the electrodes exceeds critical, the element gets excited and becomes active, and a discharge occurs in the elementary capacitor. This is the state of excitation or the so-called active transition, after which the elementary segment of the layer disappears, because the layer in this local region is destroyed. In this

Refractoriness is a state that comes after excitation when the medium accumu-

particles – structural-mechanical barrier – second electrode. The period of refractoriness corresponds to the charging of capacitors, one plate of which is an electrode

**5. Synchronization of the autowave process under the influence of a**

external influence in a thin near-electrode layer of a magnetic fluid.

important when designing computers with parallel architecture.

neighbors or, in a more complicated case, with several neighbors.

clusters that oscillated with the same frequency.

tion *| fe (t) - f0(t)- const| < 2π* is satisfied.

ment is the Belousov-Zhabotinsky chemical reaction. Petrov et al. [14]

In this section, we describe the synchronization of an autowave process by an

One of the main trends in the living world is the tendency to achieve a common rhythm in collective behavior—a tendency toward synchronization. An example of synchronization in nature is a colony of simultaneously flashing fireflies; these are simultaneously flapping birds flying in a flock. In technics, synchronization is

It was also noted that the most famous example of an active autowave environ-

experimented with the photosensitive form of such a reaction using periodic optical

Earlier we noted that the autowave process in the near-electrode layer of magnetic fluid is considered from the standpoint of oscillations of coupled oscillators. Each element of the layer (an ensemble of particles) interacts with its nearest

Each oscillator oscillates with its own frequency, but also we could see oscillator

We affected the self-oscillation medium (oscillators) with an external periodic force (pulsed electric field) and achieved synchronization. As the synchronization criterion, the conditions of frequency and/or phase adjustment are used. It is considered that the oscillator 1:1 is synchronized by an external periodic effect, if its own frequency f0 becomes equal to the frequency of the external signal *fe*. In addition to the criterion "frequency capture," another criterion is used —"phase capture." This criterion means that for any moment of time *t*, the condi-

We affected on the observed self-oscillations in a thin near-electrode layer of magnetic fluid by a periodic pulsed electric field and selected its frequency *fe* in such a way that it matched with the natural oscillation frequency of the layer elements. Thus, with a frequency capture of 1:1, we obtained a picture of autowave synchronization as shown in **Figure 12**. This picture allows us to determine the natural oscillation frequency *f0* of the elements of the near-electrode layer of magnetic fluid.

lates energy and cannot produce a new impulse. The medium must restore its properties with the energy coming from outside and prepare for the next impulse. The energy from the external source continues to flow, and the magnetite particles move to the nearest electrode, forming a system of successively connected capacitors: first electrode – structural-mechanical barrier – the layer of close-packed particles – magnetic fluid of low concentration – the layer of close-packed

critical one), but the autowave process has not yet begun.

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference*

case, a single wave front passes over the surface.

*DOI: http://dx.doi.org/10.5772/intechopen.85197*

and the other is near-electrode layer.

**periodic electric signal**

effects.

**67**

**Figure 11.**

*Schematic of a cell with a magnetic fluid with a formed near-electrode layer. (1) Magnetic fluid in the cell, (2) a layer of magnetite particles, (3) dielectric (structural-mechanical barrier), and (4) electrode (ITO).*

We assume that the medium is excitable and that it consists of a set of elementary areas of near-electrode layer. According to definition, each such site is an autonomous source of energy. Energy is accumulated due to the so-called polarization capacitance of near-electrode layer of particles [13]. The elements of an excitable medium interact with each other, transferring the energy of electrical and hydrodynamic interaction from one to another. This is how the excitation pulse transmits. This process is similar to the interaction of nonlinear coupled oscillators with a short coupling, where each oscillator is associated with only several adjacent ones. Thus, the physical meaning of the autowave process mechanism considered by us is a system of coupled nonlinear oscillators.

An excitable element has only one stable stationary state. An external impact that exceeds the threshold level can take an element out of a steady state and make it take action before it returns to that state. During the action, the active element can affect the elements associated with it and, in turn, remove them from the stationary state. As a result, an excitation wave propagates in such a medium. This is the most common type of autowaves in biological systems such as nerve tissue or heart muscle. In our medium, each section of the near-electrode layer affects the neighboring areas, transferring the charge and thereby removing them from the equilibrium state.

The element of an excitable medium can be in three states—rest, excitation, and refractoriness.

The local elements that make up the active medium under study (elements of the near-electrode layer) have "the only distinguished state of rest that is stable with respect to fairly weak external influences" [13]. This is a state of the medium, when

#### *Autowave Processes in Magnetic Fluid: Electrically Controlled Interference DOI: http://dx.doi.org/10.5772/intechopen.85197*

the electrode layer has already formed, (the voltage on the electrodes is less than the critical one), but the autowave process has not yet begun.

When the voltage on the electrodes exceeds critical, the element gets excited and becomes active, and a discharge occurs in the elementary capacitor. This is the state of excitation or the so-called active transition, after which the elementary segment of the layer disappears, because the layer in this local region is destroyed. In this case, a single wave front passes over the surface.

Refractoriness is a state that comes after excitation when the medium accumulates energy and cannot produce a new impulse. The medium must restore its properties with the energy coming from outside and prepare for the next impulse. The energy from the external source continues to flow, and the magnetite particles move to the nearest electrode, forming a system of successively connected capacitors: first electrode – structural-mechanical barrier – the layer of close-packed particles – magnetic fluid of low concentration – the layer of close-packed particles – structural-mechanical barrier – second electrode. The period of refractoriness corresponds to the charging of capacitors, one plate of which is an electrode and the other is near-electrode layer.

## **5. Synchronization of the autowave process under the influence of a periodic electric signal**

In this section, we describe the synchronization of an autowave process by an external influence in a thin near-electrode layer of a magnetic fluid.

One of the main trends in the living world is the tendency to achieve a common rhythm in collective behavior—a tendency toward synchronization. An example of synchronization in nature is a colony of simultaneously flashing fireflies; these are simultaneously flapping birds flying in a flock. In technics, synchronization is important when designing computers with parallel architecture.

It was also noted that the most famous example of an active autowave environment is the Belousov-Zhabotinsky chemical reaction. Petrov et al. [14] experimented with the photosensitive form of such a reaction using periodic optical effects.

Earlier we noted that the autowave process in the near-electrode layer of magnetic fluid is considered from the standpoint of oscillations of coupled oscillators. Each element of the layer (an ensemble of particles) interacts with its nearest neighbors or, in a more complicated case, with several neighbors.

Each oscillator oscillates with its own frequency, but also we could see oscillator clusters that oscillated with the same frequency.

We affected the self-oscillation medium (oscillators) with an external periodic force (pulsed electric field) and achieved synchronization. As the synchronization criterion, the conditions of frequency and/or phase adjustment are used. It is considered that the oscillator 1:1 is synchronized by an external periodic effect, if its own frequency f0 becomes equal to the frequency of the external signal *fe*.

In addition to the criterion "frequency capture," another criterion is used —"phase capture." This criterion means that for any moment of time *t*, the condition *| fe (t) - f0(t)- const| < 2π* is satisfied.

We affected on the observed self-oscillations in a thin near-electrode layer of magnetic fluid by a periodic pulsed electric field and selected its frequency *fe* in such a way that it matched with the natural oscillation frequency of the layer elements. Thus, with a frequency capture of 1:1, we obtained a picture of autowave synchronization as shown in **Figure 12**. This picture allows us to determine the natural oscillation frequency *f0* of the elements of the near-electrode layer of magnetic fluid.

We assume that the medium is excitable and that it consists of a set of elemen-

An excitable element has only one stable stationary state. An external impact that exceeds the threshold level can take an element out of a steady state and make it take action before it returns to that state. During the action, the active element can affect the elements associated with it and, in turn, remove them from the stationary state. As a result, an excitation wave propagates in such a medium. This is the most common type of autowaves in biological systems such as nerve tissue or heart muscle. In our medium, each section of the near-electrode layer affects the neighboring areas, transferring the charge and thereby removing them from the

The element of an excitable medium can be in three states—rest, excitation, and

The local elements that make up the active medium under study (elements of the near-electrode layer) have "the only distinguished state of rest that is stable with respect to fairly weak external influences" [13]. This is a state of the medium, when

tary areas of near-electrode layer. According to definition, each such site is an autonomous source of energy. Energy is accumulated due to the so-called polarization capacitance of near-electrode layer of particles [13]. The elements of an excitable medium interact with each other, transferring the energy of electrical and hydrodynamic interaction from one to another. This is how the excitation pulse transmits. This process is similar to the interaction of nonlinear coupled oscillators with a short coupling, where each oscillator is associated with only several adjacent ones. Thus, the physical meaning of the autowave process mechanism considered

*Schematic of a cell with a magnetic fluid with a formed near-electrode layer. (1) Magnetic fluid in the cell, (2) a layer of magnetite particles, (3) dielectric (structural-mechanical barrier), and (4) electrode (ITO).*

by us is a system of coupled nonlinear oscillators.

equilibrium state.

refractoriness.

**66**

**Figure 11.**

*Nanofluid Flow in Porous Media*

#### **Figure 12.**

*Synchronization of autowaves in a thin layer of concentrated magnetic fluid by an external pulsed electric field. The ratio of the natural oscillation frequency of the elementary section of the layer* f0 *and the frequency of the pulsed electric field* fe *is 1:1.*

**6. Mathematical model of the autowave process**

*Two clouds of synchronously oscillating points. The shift of phase between oscillations is π.*

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference*

*DOI: http://dx.doi.org/10.5772/intechopen.85197*

**Figure 14.**

**Figure 15.**

**69**

*Three uniformly oscillating clouds.*

in the conductivity of the structural-mechanical barrier.

*∂V ∂t* þ *∂*2 V

> *∂I ∂t*

To simulate spiral waves, pacemakers, and synchronization, we use the basic FitzHugh-Nagumo model (FHN). It describes an excitable medium and is named after Richard FitzHugh (1922–2007), who proposed the system in 1961, and Nagumo, who proposed a similar scheme the next year. It allows to adjust the characteristics of the autowave process widely and change the parameters of its behavior. Model solution can be obtained using the finite element method [16]. The model consists of two equations; the first equation describes a "fast" process—a change in the intensity in the near-electrode layer after the increase

The second equation describes a slow process—the change in charge of the near-electrode layer, when charged particles of magnetite accumulate in it.

*<sup>∂</sup>*t2 <sup>¼</sup> *<sup>D</sup>Δ<sup>V</sup>* � V3 <sup>þ</sup> <sup>V</sup> � <sup>I</sup>

where *V* is the function, depending on the field intensity in a thin near-electrode layer of a magnetic fluid (activator), *I* is the function associated with a change in

¼ *ε*ð Þ *V* þ *α* � *bV* (1)

**Figure 13.** *The formation of breaks and labyrinth structures with the ratio of frequencies:* fe 2f0*.*

If the ratio of the frequency of a pulsed electric field and the natural frequency are 2:1, then a few time after the beginning of the effect, we observed breaks and labyrinth structures, as shown in **Figure 13**, or two clouds, which oscillate with a phase shift of π, as shown in **Figure 14**.

Since the near-electrode layer of magnetic fluid is a heterogeneous medium, it can be assumed that the synchronization mechanism is similar to the formation of clusters in the intestine: oscillators with similar frequencies tend to group together.

With a ratio of 1:3, three uniformly oscillating clouds were observed (**Figure 15**).

The synchronization mechanism is the following: each point tends to synchronize with external force, as well as with neighboring points. If there were no interaction in the medium, then, with a frequency ratio of about 1:*m*, neighboring points would have an equal probability of the phase difference *2π/m i*, where i = 1, ..., *m*<sup>1</sup> . Because of the interaction, the points tend to have the same phases as the neighbors, and a compromise is achieved through the formation of clouds. Obviously, the phase difference between the clouds is *2π/m i* [15].

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference DOI: http://dx.doi.org/10.5772/intechopen.85197*

**Figure 14.**

*Two clouds of synchronously oscillating points. The shift of phase between oscillations is π.*

**Figure 15.** *Three uniformly oscillating clouds.*

#### **6. Mathematical model of the autowave process**

To simulate spiral waves, pacemakers, and synchronization, we use the basic FitzHugh-Nagumo model (FHN). It describes an excitable medium and is named after Richard FitzHugh (1922–2007), who proposed the system in 1961, and Nagumo, who proposed a similar scheme the next year. It allows to adjust the characteristics of the autowave process widely and change the parameters of its behavior. Model solution can be obtained using the finite element method [16].

The model consists of two equations; the first equation describes a "fast" process—a change in the intensity in the near-electrode layer after the increase in the conductivity of the structural-mechanical barrier.

The second equation describes a slow process—the change in charge of the near-electrode layer, when charged particles of magnetite accumulate in it.

$$\frac{\partial V}{\partial t} + \frac{\partial^2 V}{\partial t^2} = D\Delta V - \mathbf{V}^3 + \mathbf{V} - \mathbf{I}$$

$$\frac{\partial I}{\partial t} = \varepsilon (V + a - bV) \tag{1}$$

where *V* is the function, depending on the field intensity in a thin near-electrode layer of a magnetic fluid (activator), *I* is the function associated with a change in

If the ratio of the frequency of a pulsed electric field and the natural frequency are 2:1, then a few time after the beginning of the effect, we observed breaks and labyrinth structures, as shown in **Figure 13**, or two clouds, which oscillate with a

*The formation of breaks and labyrinth structures with the ratio of frequencies:* fe 2f0*.*

*Synchronization of autowaves in a thin layer of concentrated magnetic fluid by an external pulsed electric field. The ratio of the natural oscillation frequency of the elementary section of the layer* f0 *and the frequency of the*

Since the near-electrode layer of magnetic fluid is a heterogeneous medium, it can be assumed that the synchronization mechanism is similar to the formation of clusters in the intestine: oscillators with similar frequencies tend to group together. With a ratio of 1:3, three uniformly oscillating clouds were observed (**Figure 15**). The synchronization mechanism is the following: each point tends to synchro-

. Because of the interaction, the points tend to have the same phases as the

nize with external force, as well as with neighboring points. If there were no interaction in the medium, then, with a frequency ratio of about 1:*m*, neighboring points would have an equal probability of the phase difference *2π/m i*, where i = 1,

neighbors, and a compromise is achieved through the formation of clouds. Obvi-

ously, the phase difference between the clouds is *2π/m i* [15].

phase shift of π, as shown in **Figure 14**.

..., *m*<sup>1</sup>

**68**

**Figure 13.**

**Figure 12.**

*pulsed electric field* fe *is 1:1.*

*Nanofluid Flow in Porous Media*

charge (inhibitor), *D* is the diffusion coefficient of the activator, and ɛ is a small parameter; presumably this is the ratio of the time of single pulse passage and the time of the near-electrode layer formation.

Depending on the values of the parameters *α* and *β*, the medium element can be either in the oscillatory mode or in the excitable mode.

The proposed model of the autowave process was implemented by means of the COMSOL Multiphysics 5.2. The solutions that were obtained for spiral waves, pacemakers, and obstacle diffraction correspond to full-scale experiments and confirm the adequacy of the proposed model (1).

The simulation results and comparison with experimental data are shown in **Figures 16–18**.

We were able to show that the diffraction of autowaves (bending the obstacles) is a consequence of the existence of a region with nonlinear characteristics near the obstacles. One of the characteristics is the deceleration parameter *φ*.

The coefficient *φ* depends only on the distance to the nearest point of the obstacle and is calculated by the following formula:

<sup>φ</sup> <sup>¼</sup> <sup>d</sup> r <sup>4</sup>

*Autowave Processes in Magnetic Fluid: Electrically Controlled Interference*

and *H* is the Heaviside function.

*obstacle; (B) the result of a full-scale experiment.*

*DOI: http://dx.doi.org/10.5772/intechopen.85197*

studied in the electrically controlled medium.

eliminated. This is important for practical use.

**7. Conclusions**

**Figure 18.**

autowave modes.

scale experiment.

**71**

layer.

where *d* is the distance to the obstacle, *r* is the range of the deceleration function,

*Comparison of the autowave diffraction simulation results and experiment: (A) modeling the rounding of an*

This chapter describes the study of the autowave process in a layer of concentrated magnetic fluid that forms at the electrode in an electric field. The main modes of the autowave process are investigated: pacemakers and reverberators (spiral waves) and obstacle rounding (autowave diffraction). All visualizations are obtained by the author's method of electrically controlled interference in a thin

The uniqueness of the medium we study is that the autowave process is controlled by a weak electric field. It can be observed indefinitely, and this process is easily reproduced in the laboratory. Also synchronization by external influence was

We consider very important to investigate and control heterogeneities (obstacles) in the active medium. Heterogeneities are the main reason that causes the development of cardiac arrhythmias and heart attacks in the human heart. By controlling the movement of a wave around a heterogeneity, arrhythmia can be

A mathematical model of the autowave process, based on the FitzHugh-Nagumo basic model, is created for the near-electrode thin layer of a magnetic fluid. The model is implemented in the interactive environment for physical processes simulation—COMSOL Multiphysics 5.2. The coefficients in the equations of the modified FHN system are found; using these coefficients, we can simulate different

A criterion for the adequacy of the model is a visual correspondence to the full-

All the features of the formation and propagation of reverberators, pacemakers, and diffraction in the autowave medium observed in the natural experiment are

∗ H dð Þ <r (2)

#### **Figure 16.**

*Comparison of the results of pacemaker simulation and experiment. (A) The result of modeling a single pacemaker; (B) the result of a full-scale experiment.*

**Figure 17.**

*Comparison of the results of spiral wave simulation and experiment. (A) The result of modeling the development of a one-arm reverberator; (B) the result of a full-scale experiment.*

#### *Autowave Processes in Magnetic Fluid: Electrically Controlled Interference DOI: http://dx.doi.org/10.5772/intechopen.85197*

**Figure 18.**

charge (inhibitor), *D* is the diffusion coefficient of the activator, and ɛ is a small parameter; presumably this is the ratio of the time of single pulse passage and the

COMSOL Multiphysics 5.2. The solutions that were obtained for spiral waves, pacemakers, and obstacle diffraction correspond to full-scale experiments and con-

obstacles. One of the characteristics is the deceleration parameter *φ*.

The simulation results and comparison with experimental data are shown in

The coefficient *φ* depends only on the distance to the nearest point of the

*Comparison of the results of pacemaker simulation and experiment. (A) The result of modeling a single*

*Comparison of the results of spiral wave simulation and experiment. (A) The result of modeling the*

*development of a one-arm reverberator; (B) the result of a full-scale experiment.*

We were able to show that the diffraction of autowaves (bending the obstacles) is a consequence of the existence of a region with nonlinear characteristics near the

Depending on the values of the parameters *α* and *β*, the medium element can be

The proposed model of the autowave process was implemented by means of the

time of the near-electrode layer formation.

*Nanofluid Flow in Porous Media*

firm the adequacy of the proposed model (1).

obstacle and is calculated by the following formula:

*pacemaker; (B) the result of a full-scale experiment.*

**Figures 16–18**.

**Figure 16.**

**Figure 17.**

**70**

either in the oscillatory mode or in the excitable mode.

*Comparison of the autowave diffraction simulation results and experiment: (A) modeling the rounding of an obstacle; (B) the result of a full-scale experiment.*

$$\mathfrak{q} = \left(\frac{\mathbf{d}}{\mathbf{r}}\right)^4 \ast \mathbf{H}(\mathbf{d} < \mathbf{r}) \tag{2}$$

where *d* is the distance to the obstacle, *r* is the range of the deceleration function, and *H* is the Heaviside function.

#### **7. Conclusions**

This chapter describes the study of the autowave process in a layer of concentrated magnetic fluid that forms at the electrode in an electric field. The main modes of the autowave process are investigated: pacemakers and reverberators (spiral waves) and obstacle rounding (autowave diffraction). All visualizations are obtained by the author's method of electrically controlled interference in a thin layer.

The uniqueness of the medium we study is that the autowave process is controlled by a weak electric field. It can be observed indefinitely, and this process is easily reproduced in the laboratory. Also synchronization by external influence was studied in the electrically controlled medium.

We consider very important to investigate and control heterogeneities (obstacles) in the active medium. Heterogeneities are the main reason that causes the development of cardiac arrhythmias and heart attacks in the human heart. By controlling the movement of a wave around a heterogeneity, arrhythmia can be eliminated. This is important for practical use.

A mathematical model of the autowave process, based on the FitzHugh-Nagumo basic model, is created for the near-electrode thin layer of a magnetic fluid. The model is implemented in the interactive environment for physical processes simulation—COMSOL Multiphysics 5.2. The coefficients in the equations of the modified FHN system are found; using these coefficients, we can simulate different autowave modes.

A criterion for the adequacy of the model is a visual correspondence to the fullscale experiment.

All the features of the formation and propagation of reverberators, pacemakers, and diffraction in the autowave medium observed in the natural experiment are

repeated in computer simulation, which will allow us to find further methods for controlling the development of autowave process. This has practical application either in fundamental field or in practical purposes.

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