An alternating current stimulation hodgkin-huxley cardiac purkinje fiber model design method

Abstract

The application discloses a design method of an alternating current stimulation Hodgkin-Huxley cardiac Purkinje fiber model, and the dynamic characteristics of the model under different alternating current parameters and different initial conditions are observed through a phase plane orbit, a waveform, a bifurcation diagram and a Lyapunov exponent spectrum, and multiple coexisting membrane action potential modes are revealed, corresponding to the same or different periodic coexisting attractors in a phase diagram. The model experiences period, quasi-period and forward or reverse period-doubling bifurcation with the change of an amplitude A or a frequency f, and it is proved that the cardiac Purkinje fiber model with alternating current injection has complex nonlinear characteristics. By changing a current I m , a sodium ion equilibrium potential E Na and a potassium ion equilibrium potential E K , heartbeat frequency rules are observed, and state conversion of the cardiac Purkinje fiber between normal, abnormal and sudden cardiac arrest is found. The application provides a reference for exploring the evolution rules of nonlinear dynamics in the HH CPF model, protecting life and heart health.

Description

Technical Field

[0001] This invention belongs to the fields of nonlinear dynamics and memristor neural networks, and involves the numerical simulation implementation of a fourth-order memristor circuit and numerical analysis using bifurcation theory. Background Technology

[0002] Purkinje fibers are dendritic networks of terminal fibers branching from the left and right bundle branches and their offshoots. They intertwine in a network beneath the endocardium, distributing throughout the subendocardium and ventricular myocardium. Their terminals form an inverted "Y" shape, connecting to ventricular myocardial cells. In 1962, Noble proposed a Purkinje fiber model based on the Hodgkin-Huxley neuron model. This model can be used to describe the action potentials and pacemaker potentials of cardiac Purkinje fibers. The total membrane current consists of sodium ion channel currents, potassium ion channel currents, and anion leakage currents, successfully reproducing most of the intrinsic properties and physiological environment of the real Purkinje fiber cell membrane. In 2019, Zhang Xiaohong et al. demonstrated that the sodium and potassium ion channels in the Hodgkin-Huxley cardiac Purkinje fiber model are memristors, and through DC external stimulation, they discovered that the model migrates and changes between three different state domains: the local active domain, the chaotic marginal domain, and the local passive domain, and demonstrated the emergence of three equilibrium points.

[0003] Cells in real-world environments may encounter various external factors and respond accordingly to external stimuli. To better simulate the real world and discover richer nonlinear characteristics, many studies on neurons and neural networks have introduced different external factors, including current, magnetic fields, electric fields, and noise. Among these, electrical stimulation has wide applications in clinical cardiac pacing, defibrillation, and cardiomyoplasty. Alternating current (AC) is frequently used in electrical stimulation research and can produce more complex nonlinear behaviors. For example, AC stimulation of the Hindmarsh-Rose neuron model, the Hodgkin-Huxley neuron model, and the Hopfield neural network all exhibit diverse firing patterns. Similarly, for the Hodgkin-Huxley cardiac Purkinje fiber model, AC stimulation exhibits richer and more unique dynamic phenomena compared to DC excitation. Therefore, further research on the dynamic evolution and heart rate variation patterns of the Hodgkin-Huxley cardiac Purkinje fiber model using bifurcation diagrams, Lyapunov exponent spectra, waveforms, and phase diagrams is of great significance for protecting human heart health. Summary of the Invention

[0004] The purpose of this invention is to propose a design method for an AC-stimulated Hodgkin-Huxley cardiac Purkinje fiber model. First, AC current is used to replace the total membrane current in the model equations to construct the AC-stimulated Hodgkin-Huxley cardiac Purkinje fiber model and its circuit structure. Then, the AC equilibrium point is calculated, and the influence of initial conditions and AC amplitude on the equilibrium point trajectory is discussed, along with the stability of the equilibrium point. The dynamic characteristics of the model under different AC parameters and initial conditions are observed through phase plane trajectory, waveform, bifurcation diagram, and Lyapunov exponent spectrum, revealing multiple coexisting membrane potential modes, coexisting asymmetric bifurcation, and local positive or negative period-doubling bifurcation. When the two-dimensional AC parameters change, the maximum Lyapunov exponent is calculated, giving the main parameter regions where periodic and quasi-periodic states coexist and the distribution characteristics of the two states. Finally, by combining changes in external current, sodium ion and potassium ion equilibrium voltage, the transition law of the cardiac Purkinje fiber between normal, abnormal, and sudden arrest states is sought.

[0005] This invention is achieved through the following technical solutions.

[0006] The present invention discloses a method for designing a Hodgkin-Huxley cardiac Purkinje fiber model based on communication stimulation, comprising the following steps:

[0007] (S01): Construct a structure with alternating current I AC =Asin(2πft) injected into the Hodgkin-Huxley heart Purkinje fiber model and circuit structure, where A is the amplitude of the alternating current and f is the frequency. The goal is to analyze the influence of applied current or disturbance on the dynamic evolution of the Hodgkin-Huxley heart Purkinje fiber model.

[0008] (S02): When the initial conditions (V0, m0, h0, n0) and the AC amplitude A parameter of the Hodgkin-Huxley cardiac Purkinje fiber model are varied, the trajectory of the system's equilibrium point evolution over time is analyzed. The distribution and jumping characteristics of unstable saddle points, unstable saddle focal points, stable nodes, and unstable non-hyperbolic equilibrium points with zero eigenvalues ​​are observed under the influence of initial conditions and AC amplitude. This allows for the determination of the injected AC current I. AC The Hodgkin-Huxley model of the Purkinje fibers in the heart exhibits complex nonlinearity.

[0009] (S03): Under the condition of fixed AC amplitude A, the Jacobian matrix eigenvalues ​​of the Hodgkin-Huxley cardiac Purkinje fiber model under different initial conditions are analyzed to determine the type and stability of the associated equilibrium point.

[0010] (S04): By adjusting the AC frequency f or amplitude parameter A, numerical analysis methods such as bifurcation diagrams, Lyapunov exponent spectra, and phase diagrams were used to discover the coexistence of attractors and the bifurcation dynamics of intertwining under different initial conditions of the Hodgkin-Huxley heart Purkinje fiber model.

[0011] (S05): Using the maximum Lyapunov exponent, the distribution of the periodic and quasi-periodic states of the model under the two-dimensional AC parameter variation of AC frequency f and amplitude parameter A is shown, and the coexistence of periodic and quasi-periodic attractors listed in (S04) is verified.

[0012] (S06): Adjusting the sodium ion balance voltage E in the Hodgkin-Huxley cardiac Purkinje fiber model Na Potassium ion balance voltage E K The study investigated the effect of an applied (S01) current on the heart rate of Purkinje fibers, seeking patterns in the normal, abnormal, and cessation of heart waveform frequencies. It concluded that when a suitable direct current stimulates an abnormal state caused by bradycardia, or when the frequency of an alternating current stimulation is f∈[1,1.66]Hz, the stimulated heart rate will be within the normal range; otherwise, the normal heart rate will be transformed into an abnormal state, or the dangerous state will become even more dangerous.

[0013] The specific steps are as follows:

[0014] Step 1: Construct a Hodgkin-Huxley cardiac Purkinje fiber model for communication stimulation.

[0015] 1) The original Hodgkin-Huxley cardiac Purkinje fiber model:

[0016]

[0017] Among them, I m I is the total membrane current. Cm I is the current flowing into the membrane capacitor. Na For sodium ion current, I K For potassium ion current, I An E represents the current of anions such as chloride ions. Na E K E An The equilibrium potential of sodium ions, potassium ions, and anions, g An For anionic conductivity, C m V is the membrane capacitance, V is the membrane potential, and t is time.

[0018] 2) Design a Hodgkin-Huxley heart Purkinje fiber model with AC current injection. Ignoring the influence of anions, the circuit equations are:

[0019]

[0020] in,

[0021]

[0022] Among them, I AC The introduced alternating current is represented by the formula I. AC =Asin(2πft); m, h, n are the terms in the equation related to the calculation I. Na and I K Related variables; I Na and I K The calculation method is the same as in formula (1). The circuit structure of the Hodgkin-Huxley cardiac Purkinje fiber model of AC stimulation is as follows: Figure 1 As shown.

[0023] Step 2: Equilibrium point distribution in the Hodgkin-Huxley cardiac Purkinje fiber model under communication stimulation.

[0024] Equilibrium point curves can reflect the dynamic characteristics of a model to a certain extent and measure possible dynamic phenomena, including unstable saddle points, unstable saddle focal points, stable nodes, and the appearance of unstable non-hyperbolic equilibrium points with zero eigenvalues. This invention fixes the AC frequency f = 7.93 Hz and selects different AC amplitudes A = 3 μA, 6 μA, 7 μA, 7.93137 μA, 8.18 μA, and 12 μA to obtain six sets of different initial conditions (V). 01 ,m 01 ,h 01 ,n 01 )=(-40,3,30,5),(V 02 ,m 02 ,h 02 ,n 02 )=(-55.2058,0.1710,0.1241,0.3706), (V 03 ,m 03 ,h 03 ,n 03 )=(-66,0.1,0.38,0.2),(V 04 ,m 04 ,h 04 ,n 04 )=(-77.7715,0.0536,0.7723,0.0969), (V 05 ,m 05 ,h 05 ,n 05 ) = (-80, -0.05, 0.8, 0.463) and (V 06 ,m06 ,h 06 ,n 06 Equilibrium point curves Vt and VI within a single AC cycle under (-100, 0, 0, 0) AC In fact, by fixing different frequencies f, we can find that the initial conditions (V0,m0,h0,n0) and AC amplitude A of the Hodgkin-Huxley cardiac Purkinje fiber model can exhibit different nonlinear complex phenomena in a certain region.

[0025] Step 3: The correlation between initial conditions, equilibrium point type, and equilibrium point stability in the Hodgkin-Huxley cardiac Purkinje fiber model.

[0026] In the equilibrium point curve diagram of step 2, a set of representative AC parameters A = 8.18 μA and f = 7.93 Hz are selected. The AC waveform is obtained within one AC cycle with a step size of 0.001*(1 / f). The V value curves of the equilibrium points under the same six different initial conditions as in step 2 are obtained. The Jacobian matrix eigenvalues ​​of the equilibrium points are calculated. The relationship between the initial conditions, amplitude, frequency and corresponding equilibrium point type and equilibrium point stability of the Hodgkin-Huxley cardiac Purkinje fiber model is obtained, indicating that there is a mutually restraining dynamic relationship between them.

[0027] Step 4: Attractor coexistence and bifurcation dynamics of the Hodgkin-Huxley cardiac Purkinje fiber model under communication stimulation.

[0028] The diversity of equilibrium point curves under different initial conditions suggests the possibility of attractor coexistence. This invention selects a commonly used initial condition (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463), and also selects another set of initial conditions with significantly different equilibrium point curves and the largest V values ​​(V0,m0,h0,n0) = (-40,3,30,5). By fixing the AC amplitude A = 7.88μA, 8.18μA, and 9.50μA, and varying the AC frequency f, or by fixing the AC frequency f = 7.93Hz, 10.50Hz, and 11.00Hz and varying the AC amplitude A, numerical analysis methods such as bifurcation diagrams, Lyapunov exponent spectra, phase diagrams, and waveforms are used to search for coexisting attractors and intertwined bifurcation phenomena.

[0029] Step 5: Verify the coexistence of periodic and quasi-periodic attractors in Step 4.

[0030] First, by observing the range of attractors exhibiting both periodic and quasi-periodic states in step 4, the maximum Lyapunov exponent is calculated within a concentrated region of amplitude A ∈ [7.00, 10.00] μA and frequency f ∈ [9.50, 12.50] Hz. Then, based on the correspondence between the Lyapunov exponent and attractors in step 4, different AC parameters are divided into periodic or quasi-periodic states using the maximum Lyapunov exponent, demonstrating the two-dimensional region where periodic and quasi-periodic states coexist. Finally, the AC parameters exhibiting periodic and quasi-periodic states in step 4 are compared with the state judgments in this step to verify the consistency of the results.

[0031] Step 6: Investigate the effects of sodium and potassium ion balance voltage, DC and AC stimulation on heart rate in the Hodgkin-Huxley cardiac Purkinje fiber model.

[0032] Sodium ion balance voltage E in a normal Hodgkin-Huxley cardiac Purkinje fiber model Na =40mV, potassium ion balance voltage E K = -100mV. When the equilibrium voltage of any ion is changed beyond a certain level, the model malfunctions. The membrane action potential is used to observe whether the model exceeds the normal heart rate range of 60-100 beats / minute as defined by the World Health Organization, and whether dangerous states such as tachycardia, bradycardia, or cardiac arrest occur. Simultaneously, AC and DC electrical stimulation are applied to the abnormal Hodgkin-Huxley cardiac Purkinje fiber model, and the efficiency of DC and AC in heart rate regulation is compared. Finally, the frequency of AC current f∈[1,1.66]Hz is sought to adjust bradycardia or tachycardia to a normal heart rate during the transition between abnormal and normal heart rate states, and the optimal frequency is explored when the sodium ion equilibrium voltage E... Na =40mV, potassium ion balance voltage E K When the injection frequency is -100mV and the heart rate is 71.5137 beats / min, the injection frequency is... A strong alternating current or a large direct current can cause abnormal heartbeat.

[0033] The key features of this invention are: a proposed Hodgkin-Huxley cardiac Purkinje fiber model based on AC stimulation. Under different AC parameters and initial conditions, the AC equilibrium point jumps between three equilibrium point types. The diversity of equilibrium point stability and evolutionary trends enables Purkinje fibers to generate complex dynamic behaviors. Through bifurcation analysis, coexisting asymmetric bifurcation and locally positive or negative period-doubling bifurcation are discovered. Further numerical analysis reveals the coexistence behavior of attractors generated by AC stimulation and quasi-periodic attractors that are more complex than limiting cycles. Furthermore, by varying the two-dimensional AC parameter fA, the intertwined state distribution reflects the changing trend of the model state related to the AC parameters, while also providing the main parameter ranges for the coexistence of periodic and quasi-periodic attractors. Finally, by adjusting the external current I in the model... m Communication I AC and parameter E K E Na The study investigated the transitions between normal heartbeats, abnormal heartbeats, and cardiac arrest, which will provide valuable insights for cardiac protection and disease prevention. Attached Figure Description

[0034] Figure 1 The circuit structure for the Hodgkin-Huxley cardiac Purkinje fiber model of electrical stimulation constructed in this invention.

[0035] Figure 2 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 3 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC curve.

[0036] The initial conditions for subgraphs (a1) and (b1) are: (V 01 ,m 01 ,h 01 ,n 01 ) = (-40, 3, 30, 5);

[0037] The initial conditions for subgraphs (a2) and (b2) are:

[0038] (V 02 ,m 02 ,h 02 ,n 02 = (-55.2058, 0.1710, 0.1241, 0.3706);

[0039] The initial conditions for subgraphs (a3) ​​and (b3) are: (V 03 ,m 03 ,h 03 ,n 03 = (-66, 0.1, 0.38, 0.2);

[0040] The initial conditions for subgraphs (a4) and (b4) are:

[0041] (V 04 ,m 04 ,h 04 ,n 04 = (-77.7715, 0.0536, 0.7723, 0.0969);

[0042] The initial conditions for subgraphs (a5) and (b5) are: (V 05 m 05 ,h 05 ,n 05 = (-80, -0.05, 0.8, 0.463);

[0043] The initial conditions for subgraphs (a6) and (b6) are: (V 06 ,m 06 ,h 06 ,n 06 ) = (-100,0,0,0).

[0044] Figure 3 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 6 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC The correspondence between the curve, initial conditions, and the graph. Figure 2 Consistent.

[0045] Figure 4 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 7 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC The correspondence between the curve, initial conditions, and the graph. Figure 2Consistent.

[0046] Figure 5 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 7.93137 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC The correspondence between the curve, initial conditions, and the graph. Figure 2 Consistent.

[0047] Figure 6 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 8.18 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC The correspondence between the curve, initial conditions, and the graph. Figure 2 Consistent.

[0048] Figure 7 For this invention, under six different initial conditions (AC frequency f = 7.93 Hz, amplitude A = 12 μA), the equilibrium points Vt and VI are calculated. AC The curve, and the evolution of all graphs is in I. AC = Within one period of Asin(2πft). Where: all (a) sequence plots are Vt curves, all (b) sequence plots are VI curves. AC The correspondence between the curve, initial conditions, and the graph. Figure 2 Consistent.

[0049] Figure 8 For the present invention, under six initial conditions, the waveform of an externally injected sinusoidal AC wave with amplitude A = 8.18 μA and frequency f = 7.93 Hz, the V value at the equilibrium point, the trajectory of the first two larger real parts of the eigenvalues, and the evolution of the imaginary part of the complex eigenvalues ​​with t are shown. The other two real eigenvalues ​​are not shown in the figure because they are always less than 0.

[0050] Figure 9The figures show the curves of the two larger Lyapunov indices LE1 and LE2 as a function of amplitude A at fixed frequencies f = 7.93 Hz, 10.50 Hz, and 11.00 Hz. The initial conditions used for comparison are (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463). For clarity, the curves of the two smaller Lyapunov indices LE3 and LE4 are omitted from all figures.

[0051] Figure 10 This invention illustrates the interleaved evolution of the nA bifurcation diagram under two different initial conditions. The AC frequencies f injected into the Hodgkin-Huxley heart Purkinje model were 7.93 Hz, 10.50 Hz, and 11.00 Hz, respectively. The two sets of initial conditions used for comparison were (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463). The part indicated by the arrow represents the main parameter range for the coexistence attractor.

[0052] Figure 11 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 5.52μA, 5.56μA, and 5.70μA and the frequency f = 7.93Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0053] Figure 12 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 7.38 μA, 7.78 μA, and 7.91 μA, and the frequency f = 7.93 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0054] Figure 13 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 7.94 μA, 8.10 μA, and 8.34 μA and the frequency f = 7.93 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0055] Figure 14This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 4.50 μA, 5.88 μA, and 7.00 μA and the frequency f = 10.50 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0056] Figure 15 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 8.77μA, 8.98μA, and 9.05μA and the frequency f = 10.50Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0057] Figure 16 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 9.65μA, 9.67μA, and 10.17μA, and the frequency f = 10.50Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0058] Figure 17 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 7.00 μA, 7.72 μA, and 8.15 μA and the frequency f = 11.00 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0059] Figure 18 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 9.23 μA, 9.82 μA, and 9.92 μA, and the frequency f = 11.00 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0060] Figure 19This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the amplitude A = 9.94 μA, 9.99 μA, and 10.03 μA and the frequency f = 11.00 Hz. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0061] Figure 20 The figures show the curves of the two larger Lyapunov indices LE1 and LE2 as a function of frequency f at fixed amplitudes A = 7.88 μA, 8.18 μA, and 9.50 μA. The initial conditions used for comparison are (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463). For clarity, the curves of the two smaller Lyapunov indices LE3 and LE4 are omitted from all figures.

[0062] Figure 21 This invention illustrates the interleaved evolution of the nf bifurcation diagram under two different initial conditions. The AC amplitudes A injected into the Hodgkin-Huxley heart Purkinje model were 7.88 μA, 8.18 μA, and 9.50 μA, respectively. The two sets of initial conditions used for comparison were (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463). The part indicated by the arrow represents the main parameter range for the coexistence attractor.

[0063] Figure 22 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 1.20Hz, 7.93Hz, and 7.98Hz, and the amplitude A = 7.88μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0064] Figure 23 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 8.16Hz, 8.18Hz, and 8.64Hz, and the amplitude A = 7.88μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0065] Figure 24This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 9.51Hz, 9.54Hz, and 10.86Hz, and the amplitude A = 7.88μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0066] Figure 25 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 7.91Hz, 7.98Hz, and 8.02Hz, and the amplitude A = 8.18μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0067] Figure 26 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 8.12Hz, 8.35Hz, and 9.80Hz and the amplitude A = 8.18μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0068] Figure 27 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 10.58Hz, 10.76Hz, and 10.86Hz, and the amplitude A = 8.18μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0069] Figure 28 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 9.80Hz, 10.00Hz, and 10.32Hz and the amplitude A = 9.50μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0070] Figure 29This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 10.33Hz, 10.40Hz, and 10.51Hz, and the amplitude A = 9.50μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0071] Figure 30 This invention provides the trajectories of n1-V1 and n2-V2 and the waveforms of V1-t and V2-t when the frequencies f = 10.60Hz, 11.35Hz, and 11.50Hz, and the amplitude A = 9.50μA. (V0,m0,h0,n0) = (-40,3,30,5) and (V0,m0,h0,n0) = (-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively.

[0072] Figure 31 To illustrate the state distribution of the model under different initial conditions as the two-dimensional combination fA changes, black and white represent periodic and quasi-periodic states, respectively. The initial conditions of (a) are (V0,m0,h0,n0)=(-40,3,30,5), and the initial conditions of (b) are (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463).

[0073] Figure 32 For the present invention, when the potassium ion equilibrium potential E K At -100mV, there are different sodium ion equilibrium potentials E Na The membrane action potential waveform, parameter E Na The correspondence between the curve and the graph is shown in the legend.

[0074] Figure 33 For the potassium ion equilibrium potential E of this invention K = -100mV and different sodium ion equilibrium potentials E Na The primitive membrane action potential of the Hodgkin-Huxley cardiac Purkinje model and its response to DC and AC Ip respectively AC The waveform after stimulation with Asin(2πft). Based on the heart rate f hb The numerical values ​​distinguish the waveform into three states: normal heartbeat, dangerous heartbeat, and heartbeat stoppage.

[0075] Figure 34 For the present invention, when the sodium ion equilibrium potential E Na At 40mV, there are different potassium ion equilibrium potentials E K The membrane action potential waveform, parameter E KThe correspondence between the curve and the graph is shown in the legend.

[0076] Figure 35 For the sodium ion equilibrium potential E of this invention Na =40mV and different potassium ion equilibrium potentials E K The primitive membrane action potential of the Hodgkin-Huxley cardiac Purkinje model and its response to DC and AC Ip respectively AC The waveform after stimulation with Asin(2πft). Based on the heart rate f hb The numerical values ​​distinguish the waveform into three states: normal heartbeat, dangerous heartbeat, and heartbeat stoppage. Detailed Implementation

[0077] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0078] Example 1: Equilibrium point evolution trajectory analysis of the Hodgkin-Huxley cardiac Purkinje fiber model under different initial conditions with AC stimulation of fixed frequency and varying amplitude.

[0079] Substituting the current calculation formula in formula (1) into the differential equation of formula (2) yields the complete AC stimulation Hodgkin-Huxley cardiac Purkinje fiber model. Setting the left side of formula (2) to 0, the equation used to calculate the model's equilibrium point E can be derived, namely formula (4).

[0080]

[0081]

[0082] For the equilibrium point curve, only the period of the frequency is changed. To visually compare the changes in V value and the stability evolution of the equilibrium point, the AC amplitude was set to A = 3μA, 6μA, 7μA, 7.93137μA, 8.18μA and 12μA, respectively, with a fixed frequency of f = 7.93Hz. The initial conditions used are as listed in formula (5). When A = 3μA and f = 7.93Hz, Matlab simulation can obtain the Vt curves and VI values ​​of the equilibrium point under six different initial conditions when the AC amplitude changes. AC Curves, such as Figure 2 As shown:

[0083] The initial condition is (V) 01 ,m 01 ,h 01 ,n 01 Vt curve when ) = (-40, 3, 30, 5) ------------- Figure 2 (a1);

[0084] The initial condition is (V)01 ,m 01 ,h 01 ,n 01 VI when ) = (-40, 3, 30, 5) AC curve----------- Figure 2 (b1);

[0085] The initial condition is (V) 02 ,m 02 ,h 02 ,n 02 The Vt curve when ) = (-55.2058, 0.1710, 0.1241, 0.3706) ------ Figure 2 (a2);

[0086] The initial condition is (V) 02 ,m 02 ,h 02 ,n 02 VI when ) = (-55.2058, 0.1710, 0.1241, 0.3706) AC curve---- Figure 2 (b2);

[0087] The initial condition is (V) 03 ,m 03 ,h 03 ,n 03 The Vt curve when ) = (-66, 0.1, 0.38, 0.2) ------- Figure 2 (a3);

[0088] The initial condition is (V) 03 ,m 03 ,h 03 ,n 03 VI when ) = (-66, 0.1, 0.38, 0.2) AC curve------ Figure 2 (b3);

[0089] The initial condition is (V) 04 ,m 04 ,h 04 ,n 04 The Vt curve when ) = (-77.7715, 0.0536, 0.7723, 0.0969) ----- Figure 2 (a4);

[0090] The initial condition is (V) 04 ,m 04 ,h 04 ,n 04VI when ) = (-77.7715, 0.0536, 0.7723, 0.0969) AC curve--- Figure 2 (b4);

[0091] The initial condition is (V) 05 ,m 05 ,h 05 ,n 05 The Vt curve when ) = (-80, -0.05, 0.8, 0.463) --- Figure 2 (a5);

[0092] The initial condition is (V) 05 ,m 05 ,h 05 ,n 05 VI when ) = (-80, -0.05, 0.8, 0.463) AC curve---- Figure 2 (b5);

[0093] The initial condition is (V) 06 ,m 06 ,h 06 ,n 06 Vt curve when ) = (-100, 0, 0, 0) ------------- Figure 2 (a6);

[0094] The initial condition is (V) 06 ,m 06 ,h 06 ,n 06 VI when ) = (-100,0,0,0) AC curve--------- Figure 2 (b6);

[0095] Similarly, when the AC frequency f = 7.93 Hz and the AC amplitude A = 6 μA, the Vt curves and VI curves stretched due to the increase in AC amplitude under the six initial conditions are as follows: AC Curve plotted in Figure 3 In the middle. When the AC parameters are set to f = 7.93 Hz and A = 7 μA, the Vt curves and VI curves under various initial conditions are as follows. AC Curves Figure 4 As shown. Keeping the AC frequency f = 7.93 Hz constant, and varying the AC amplitude A = 7.93137 μA, the Vt curves and VI curves under different initial conditions are shown. AC Curve combinations are drawn in Figure 5 In the middle. The Vt curves and VI curves corresponding to each initial condition. AC The curve can be obtained at f = 7.93 Hz and A = 8.18 μA. Figure 6 Similarly, the equilibrium curves Vt and VI under different initial conditions f = 7.93 Hz and A = 12 μA are shown. AC See Figure 7 . Figures 3-7 The relationships between each curve and the six sets of initial conditions are all... Figure 2 Consistent.

[0096] Comparing the Vt curves of the equilibrium points under different initial conditions and with different amplitudes, the occurrence of three equilibrium points and the phenomenon of equilibrium point jumps demonstrates the influence of initial conditions on the solution of equilibrium points. When the initial condition is (V 01 ,m 01 ,h 01 ,n 01 ), (V 03 ,m 03 ,h 03 ,n 03 ) and (V 06 ,m 06 ,h 06 ,n 06 When the amplitude A changes, the equilibrium point type remains stable. However, under the other four initial conditions, the solved equilibrium point always shifts between different equilibrium point types. As the amplitude A increases, VI... AC Alternating current I in the curve AC The numerical range of V increases, and a new equilibrium point is calculated based on the new current value, thus expanding the range of V values ​​at the equilibrium point. Reflected on the Vt curve, this appears as if the equilibrium point curve has been stretched along the V-axis.

[0097] Example 2: Stability type analysis of equilibrium points under six initial conditions.

[0098] 1) Jacobian matrix of equilibrium point E

[0099]

[0100] Different equilibrium points have different Jacobian matrices and their eigenvalues. After obtaining the equilibrium point E according to formula (4), the Jacobian matrix of the model at the equilibrium point E is given by formula (6). Thus, the eigenvalues ​​of the Jacobian matrix can be solved using Matlab software to determine the stability type of the equilibrium point.

[0101] 2) Stability analysis of equilibrium points under six initial conditions

[0102] When the amplitude A = 8.18 μA, the frequency f = 7.93 Hz, and the initial condition is given by formula (5), the AC waveform, the V value at the equilibrium point, the first two larger real parts of the Jacobian matrix eigenvalues, and the imaginary parts of the complex eigenvalues ​​are combined to depict the AC waveform. Figure 8 In (a)-(f):

[0103] ■(V 01 ,m 01 ,h 01 ,n 01 )=(-40,3,30,5)--------------------------- Figure 8 (a)

[0104] ■(V 02 ,m 02 ,h 02 ,n 02 )=(-55.2058,0.1710,0.1241,0.3706)-------- Figure 8 (b)

[0105] ■(V 03 ,m 03 ,h 03 ,n 03 )=(-66,0.1,0.38,0.2)--------------------- Figure 8 (c)

[0106] ■(V 04 ,m 04 ,h 04 ,n 04 )=(-77.7715,0.0536,0.7723,0.0969)-------- Figure 8 (d)

[0107] ■(V 05 ,m 05 ,h 05 ,n 05 )=(-80,-0.05,0.8,0.463)------------------ Figure 8 (e)

[0108] ■(V 06 ,m 06 ,h 06 ,n 06 )=(-100,0,0,0)--------------------------- Figure 8 (f)

[0109] To reveal the stability of equilibrium points under six initial conditions, representative moments in each subgraph are marked with dashed lines, taking into account the real and imaginary parts of the eigenvalues. The equilibrium point, the eigenvalue of the Jacobian matrix, and the corresponding stability type at each moment are summarized in Table 1. The type and distribution of equilibrium point stability differ under different initial conditions. As the initial condition V0 decreases, the range of unstable saddle points decreases, while the time range of stable nodes increases, such as... Figure 8 As shown in (d)-(f). However, for the single equilibrium point region, i.e., t < 65.0694 ms, if an equilibrium point exists, the stability is either an unstable saddle focus or an unstable saddle point. Compared with DC, the injection of AC causes the Hodgkin-Huxley cardiac Purkinje fiber model to change from a single stability type to a dynamic evolution among unstable saddle points, unstable saddle focuses, stable nodes, and equilibrium points with zero eigenvalues, exhibiting a variety of dynamic characteristics.

[0110] Table 1 shows the values ​​for amplitude A = 8.18 μA and frequency f = 7.93 Hz. Figure 8 The diagram shows the AC equilibrium point E, equilibrium point eigenvalues, and stability at different times under six sets of initial conditions.

[0111]

[0112]

[0113] Example 3: Numerical simulation and dynamic analysis of the Hodgkin-Huxley heart Purkinje model with fixed external AC frequency and varying amplitude.

[0114] When the AC frequency f is set to 7.93Hz, 10.50Hz and 11.00Hz respectively, the amplitude A changes from 0μA to 12μA with a step size of 0.01μA. The initial conditions used for comparison in numerical analysis such as Lyapunov exponent spectrum, bifurcation diagram, and phase diagram are (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463).

[0115] 1) To observe the overall dynamic characteristics of the model under a fixed frequency and varying amplitude, numerical calculations were performed using Matlab to obtain the first two larger Lyapunov exponent curves with amplitude A as the variable under different initial conditions, as shown below. Figure 9 As shown. The initial conditions for the left-hand plot are (V0,m0,h0,n0)=(-40,3,30,5), while the initial conditions for the right-hand plot are (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463). The correspondence between the plots and the frequency parameters is as follows:

[0116] ◆f=7.93Hz------------------------------------------ Figure 9 (a)

[0117] ◆f=10.50Hz----------------------------------------- Figure 9 (b)

[0118] ◆f=11.00Hz----------------------------------------- Figure 9 (c)

[0119] 2) Similarly, using Matlab numerical simulation, we can obtain... Figure 10 (a)-(c) show the nA bifurcation diagrams under two sets of initial conditions with three different AC frequencies fixed respectively. The correspondence between the initial conditions and the diagrams is the same as in 1).

[0120] ◆f=7.93Hz------------------------------------------ Figure 10 (a)

[0121] ◆f=10.50Hz----------------------------------------- Figure 10 (b)

[0122] ◆f=11.00Hz----------------------------------------- Figure 10 (c)

[0123] 3) Based on the Lyapunov exponent spectrum and the interleaved nA bifurcation diagram, various types of coexisting attractors are obtained, including the coexistence of attractors with different periods, the coexistence of periodic and quasi-periodic attractors, the coexistence of periodic attractors with different topologies, and the coexistence of quasi-periodic attractors with different topologies. When the fixed frequencies f = 7.93 Hz, 10.50 Hz, and 11.00 Hz, the phase diagrams and membrane potential waveforms of representative attractors projected onto the Vn plane under different initial conditions are shown below. Figure 11-19 As shown.

[0124] Figure 11 The external injection current parameters for the models in (a)-(c) are: f = 7.93 Hz, A = 5.52 μA, 5.56 μA, and 5.70 μA.

[0125] Figure 12The external current parameters of the models in (a)-(c) are: f = 7.93 Hz, A = 7.38 μA, 7.78 μA, 7.91 μA.

[0126] Figure 13 The external current parameters of the models in (a)-(c) are: f = 7.93 Hz, A = 7.94 μA, 8.10 μA, 8.34 μA.

[0127] Figure 14 The external injection current parameters for the models in (a)-(c) are: f = 10.50 Hz, A = 4.50 μA, 5.88 μA, and 7.00 μA.

[0128] Figure 15 The external injection current parameters for the models in (a)-(c) are: f = 10.50 Hz, A = 8.77 μA, 8.98 μA, and 9.05 μA.

[0129] Figure 16 The external injection current parameters for the models in (a)-(c) are: f = 10.50 Hz, A = 9.65 μA, 9.67 μA, 10.17 μA.

[0130] Figure 17 The external current parameters of the models in (a)-(c) are: f = 11.00 Hz, A = 7.00 μA, 7.72 μA, and 8.15 μA.

[0131] Figure 18 The external injection current parameters for the models in (a)-(c) are: f = 11.00 Hz, A = 9.23 μA, 9.82 μA, and 9.92 μA.

[0132] Figure 19 The external current parameters of the models in (a)-(c) are: f = 11.00 Hz, A = 9.94 μA, 9.99 μA, 10.03 μA.

[0133] (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively. For the chosen AC parameters, the Lyapunov exponents and attractor states under the two initial conditions are summarized in Table 2.

[0134] Table 2. Lyapunov exponents and attractor states under two different initial conditions with a fixed frequency f and varying amplitude A.

[0135]

[0136]

[0137] Example 4: Numerical simulation and dynamic analysis of the Hodgkin-Huxley heart Purkinje model with fixed amplitude and varying frequency of external AC pulse.

[0138] When the AC amplitude A is set to 7.88μA, 8.18μA and 9.50μA respectively, the frequency f changes from 0Hz to 20Hz with a step size of 0.01Hz. The initial conditions used for comparison in numerical analysis such as Lyapunov exponent spectrum, bifurcation diagram and phase diagram are (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463).

[0139] 1) To observe the dynamic characteristics of the model under a fixed amplitude and varying frequency, numerical calculations were performed using Matlab to obtain the first two larger Lyapunov exponent curves with frequency f as the variable under different initial conditions, as shown below. Figure 20 As shown. The initial conditions for the left-hand plot are (V0,m0,h0,n0)=(-40,3,30,5), while the initial conditions for the right-hand plot are (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463). The correspondence between the plots and the amplitude parameters is as follows:

[0140] A=7.88μA--------------------------------------------- Figure 20 (a)

[0141] A=8.18μA---------------------------------------- Figure 20 (b)

[0142] A=9.50μA--------------------------------------------- Figure 20 (c)

[0143] 2) Similarly, using Matlab numerical simulation, we can obtain... Figure 21 (a)-(c) show the nf bifurcation diagrams when three different AC amplitudes are fixed under two sets of initial conditions. The correspondence between the initial conditions and the diagrams is the same as in 1).

[0144] A=7.88μA--------------------------------------------- Figure 21 (a)

[0145] A=8.18μA---------------------------------------- Figure 21 (b)

[0146] A=9.50μA--------------------------------------------- Figure 21 (c)

[0147] 3) Based on the Lyapunov exponent spectrum and the interleaved nf bifurcation diagram, various types of coexisting attractors are obtained, including the coexistence of attractors with different periods, the coexistence of periodic and quasi-periodic attractors, the coexistence of periodic attractors with different topologies, and the coexistence of quasi-periodic attractors with different topologies. When the fixed frequencies A = 7.88 μA, 8.18 μA, and 9.50 μA are fixed, the phase diagrams and membrane potential waveforms of representative attractors projected onto the Vn plane under different initial conditions are shown below. Figure 22-30 As shown.

[0148] Figure 22 The external injection current parameters for the models in (a)-(c) are: A = 7.88 μA, f = 1.20 Hz, 7.93 Hz, and 7.98 Hz.

[0149] Figure 23 The external injection current parameters of the models in (a)-(c) are: A = 7.88 μA, f = 8.16 Hz, 8.18 Hz, and 8.64 Hz.

[0150] Figure 24 The external injection current parameters for the models in (a)-(c) are: A = 7.88 μA, f = 9.51 Hz, 9.54 Hz, and 10.86 Hz.

[0151] Figure 25 The external injection current parameters for the models in (a)-(c) are: A = 8.18 μA, f = 7.91 Hz, 7.98 Hz, and 8.02 Hz.

[0152] Figure 26 The external injection current parameters for the models in (a)-(c) are: A = 8.18 μA, f = 8.12 Hz, 8.35 Hz, and 9.80 Hz.

[0153] Figure 27 The external injection current parameters of the models in (a)-(c) are: A = 8.18 μA, f = 10.58 Hz, 10.76 Hz, 10.86 Hz.

[0154] Figure 28The external injection current parameters of the models in (a)-(c) are: A = 9.50 μA, f = 9.80 Hz, 10.00 Hz, 10.32 Hz.

[0155] Figure 29 The external injection current parameters of the models in (a)-(c) are: A = 9.50 μA, f = 10.33 Hz, 10.40 Hz, 10.51 Hz.

[0156] Figure 30 The external injection current parameters of the models in (a)-(c) are: A = 9.50 μA, f = 10.60 Hz, 11.35 Hz, 11.50 Hz.

[0157] (V0,m0,h0,n0)=(-40,3,30,5) and (V0,m0,h0,n0)=(-80,-0.05,0.8,0.463) are the initial conditions for solving n1, V1 and n2, V2, respectively. For the chosen AC parameters, the Lyapunov exponents and attractor states under the two initial conditions are summarized in Table 3.

[0158] Table 3 shows the Lyapunov exponents and attractor states when the amplitude A is fixed and the frequency f is varied under two different initial conditions.

[0159]

[0160]

[0161] Example 5: Calculate the maximum Lyapunov exponent and analyze the attractor state and distribution under two-dimensional AC parameter variations.

[0162] Both the amplitude A and frequency f affect the dynamic behavior of the model. When the amplitude A ∈ [7.00, 10.00] μA and the frequency f ∈ [9.50, 12.50] Hz vary simultaneously with a step size of 0.01, the state distribution distinguished by the maximum Lyapunov exponent is plotted on... Figure 31 In the diagram, black and white clearly represent the periodic and quasi-periodic states under varying AC parameters. The difference in the state domain under the two sets of initial conditions reflects the influence of the initial conditions on the model's stability and state, and a "hook-shaped" region where quasi-periodic and periodic attractors coexist is given. Because the positive maximum Lyapunov exponent is always near 0, and combined with... Figure 9 , Figure 20 By observing the variation patterns of the Lyapunov index spectrum, it can be determined that attractors with a positive maximum Lyapunov index value are quasi-periodic attractors, while those without are periodic attractors. Figure 11-19 and Figure 22-30 In China Figure 31 Periodic and quasi-periodic attractors and their AC parameters within the range of variation of variables f and A. Figure 31The parameters are listed in black and white in Table 4.

[0163] Table 4 Figure 11-19 and Figure 22-30 The states of periodic and quasi-periodic attractors whose exchange parameters belong to the range A∈[7.00,10.00]μA and f∈[9.50,12.50]Hz. Figure 31 The black and white correspondence in the text.

[0164]

[0165] Example 6: Adjusting the sodium ion equilibrium voltage E Na Potassium ion balance voltage E K To observe the pattern of waveform frequency changes, and to adjust the waveform's heartbeat frequency state by injecting external DC or AC.

[0166] 1) Maintain potassium ion equilibrium voltage E K =-100mV remains constant, let the sodium ion equilibrium voltage E Na Given membrane action potentials of 35mV, 36mV, 37mV, 40mV, 58.1mV, and 60mV respectively, calculate the membrane action potential within the range [0, 5000] ms and plot it on [the graph]. Figure 32 The heart rate f of each waveform. hb Calculated using formula (7).

[0167]

[0168] The World Health Organization defines a normal heart rate range as 60-100 beats / min. Based on the waveform in the graph, calculate the heart rate f. hb The curve can be divided into different states:

[0169] E Na =35mV----- Cardiac arrest due to inability to depolarize (0 beats / min)

[0170] E Na =36mV-----Brain rate (30.8642 beats / min)

[0171] E Na =37mV-----Brain rate (50.5476 beats / min)

[0172] E Na =40mV----- Normal heart rate (71.5137 beats / min)

[0173] E Na =58.1mV---Tachycardia, slow repolarization and electrical diastole disappeared (197.3684 beats / min)

[0174] E Na =60mV------Incomplete repolarization leading to cardiac arrest (0 beats / min)

[0175] 2) For different E in 1) Na The original waveforms were stimulated using both DC and AC methods, and plotted using different combinations of curves. Figure 33 In order to facilitate comparison of the transitions between different states, the transitions will be... Figure 32 and Figure 33 The heart rate and waveform status are listed in Table 5.

[0176] Table 5 Figure 33 Different types of external current stimulation and E Na Transition of waveform states in the Hodgkin-Huxley cardiac Purkinje fiber model under changing conditions

[0177]

[0178] Depend on Figure 33 As shown in Table 5, the injection of external current causes a reversal in the heartbeat state. As listed in items 2, 5, and 8 of Table 5, direct current stimulation can improve the heart rate from the dangerous or stopped state caused by bradycardia to a normal state, but it can also introduce the normal frequency into the dangerous or more dangerous state caused by tachycardia, as listed in items 11 and 14 of Table 5. Furthermore, the correspondence between heart rate and AC frequency shows that AC has a strong modulation effect on the waveform, thus allowing calculation of the AC frequency range corresponding to the normal heart rate range. The normal heart rate range is 60-100 beats / min, corresponding to a waveform period range of 600-1000 ms. Therefore, when the AC frequency f∈[1,1.66]Hz, AC stimulation keeps the heart rate constant within the normal range, and choosing an AC frequency of 1.2Hz can keep the heart rate consistently around 72 beats / min.

[0179] 3) Maintain sodium ion equilibrium voltage E Na =40mV remains constant, let the potassium ion equilibrium voltage E K Given values ​​of -105mV, -100.8mV, -100mV, -97mV, -90mV, and -85mV, calculate the membrane action potential within the range [0, 5000] ms and plot it on [the graph]. Figure 34 In the middle. Parameter E K Heart rate f hbThe correspondence between the curve states is shown below:

[0180] ·E K =-105mV----- Cardiac arrest due to inability to depolarize (0 beats / min)

[0181] ·E K =-100.8mV---Brain rate (27.1616 beats / min)

[0182] ·E K =-100mV----- Normal heart rate (71.5137 beats / min)

[0183] ·E K =-97mV------Tachycardia (129.0323 beats / min)

[0184] ·E K =-90mV------Tachycardia, slow repolarization and electrical diastole disappear (158.7302 beats / min)

[0185] ·E K = -85mV ------ Cardiac arrest due to incomplete repolarization (0 beats / min)

[0186] 4) For different E in 3) K The original waveforms were stimulated using both DC and AC methods, and plotted using different combinations of curves. Figure 35 Besides increasing heart rate, direct current (DC) has no other effective effects. Therefore, DC cannot restore a dangerous or cardiac arrest caused by tachycardia to normal, while the normal heart rate after AC stimulation suggests that AC is more effective in electrical stimulation. The current-induced transition of heart rate between normal and abnormal states in Purkinje fibers and the advantages of sinusoidal AC over DC may provide insights for the clinical treatment of heart disease.

Claims

1. A design method for a Hodgkin-Huxley cardiac Purkinje fiber model based on communication stimulation, characterized by: Includes the following steps: (S01): Construct a structure with alternating current I AC =Asin(2πft) injected Hodgkin-Huxley heart Purkinje fiber model and circuit structure, where A is the amplitude of alternating current and f is the frequency, to analyze the influence of applied current or disturbance on the dynamic evolution of Hodgkin-Huxley heart Purkinje fiber model. With an alternating current injected into the Hodgkin-Huxley heart Purkinje fiber model, and neglecting the influence of anions, the circuit equations are as follows: in, I AC =Asin(2πft) I Na =(400m 3 h+0.14)(VE Na ) I K =(1.2exp((-V-90) / 50)+0.015exp((V+90) / 60)+1.2n 4 )(VE K ) Among them, I AC Let m, h, and n be the introduced alternating currents, and m, h, and n be the variables involved in the equation and the calculation of I. Na and I K Related variables; I Na For sodium ion current, I K For potassium ion current, E Na E K C represents the equilibrium potential of sodium and potassium ions. m V is the membrane capacitance, V is the membrane potential, and t is time; (S02): When the initial conditions (V0,m0,h0,n0) and the AC amplitude A parameter of the Hodgkin-Huxley cardiac Purkinje fiber model are varied, the trajectory of the system equilibrium point evolution over time is analyzed. The distribution and jumping characteristics of unstable saddle points, unstable saddle points, stable nodes, and unstable non-hyperbolic equilibrium points with zero eigenvalues ​​under the influence of initial conditions and AC amplitude are observed to determine the injected AC current I. AC The Hodgkin-Huxley model of the Purkinje fibers in the heart exhibits complex nonlinearity. (S03): Under the condition of fixed AC amplitude A, the Jacobian matrix eigenvalues ​​of the Hodgkin-Huxley cardiac Purkinje fiber model under different initial conditions are analyzed for the complete cycle at the same frequency f to determine the associated equilibrium point type and equilibrium point stability. (S04): By adjusting the AC frequency f or amplitude parameter A, and through numerical analysis methods such as bifurcation diagrams, Lyapunov exponent spectra and phase diagrams, the coexistence of attractors and the bifurcation dynamics of intertwining under different initial conditions of the Hodgkin-Huxley cardiac Purkinje fiber model were discovered. (S05): Using the maximum Lyapunov exponent, the distribution of the periodic and quasi-periodic states of the model under the two-dimensional AC parameter variation of AC frequency f and amplitude parameter A is shown, and the coexistence of periodic and quasi-periodic attractors listed in (S04) is verified. (S06): Adjusting the sodium ion balance voltage E in the Hodgkin-Huxley cardiac Purkinje fiber model Na Potassium ion balance voltage E K The study investigated the effect of an external (S01) current on the heart rate of Purkinje fibers, seeking the patterns of normal, abnormal, and stopped heart waveform frequencies. The results showed that when a suitable direct current stimulates an abnormal state caused by bradycardia, or when the frequency of AC stimulation is f∈[1,1.66]Hz, the heart rate after stimulation will be within the normal range. Otherwise, the normal heart rate will be transformed into an abnormal state or a dangerous state or become more dangerous.

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