A method for solving nonlinear cardiac beat signals based on discrete implicit mapping

Abstract

The application discloses a solving method of a nonlinear heart beat signal based on a discrete implicit mapping, and the method comprises the following steps: defining an implicit mapping system of a nonlinear heart beat dynamic model through an implicit discrete method, and solving a semi-analytic solution of the nonlinear heart beat signal by defining a discrete vector function and a Jacobian matrix of the implicit mapping system; after obtaining the semi-analytic solution of the heart beat signal, the stability of the signal is judged and the bifurcation is defined, the nonlinear characteristics of the nonlinear heart beat signal are calculated and solved, and the user can comprehensively understand the nonlinear mechanism of the nonlinear heart beat signal; compared with a traditional numerical solution method, the application can solve an unstable solution of the nonlinear heart beat signal, so that the dynamic characteristics of the heart beat model can be better analyzed; in addition to the advantages of high precision, fast convergence and strong robustness, the application can also realize the prediction and simulation of a periodic beat signal and a quasi-periodic beat signal in a nonlinear heart beat system.

Description

Technical Field

[0001] This invention relates to the field of pulsation signal processing technology in the human heart, and particularly to a method for solving nonlinear cardiac pulsation signals based on discrete implicit mapping, which is used to realize semi-analytical numerical simulation and nonlinear characteristic analysis of the nonlinear action potentials of the sinoatrial node and atrioventricular node during heart pulsation. Background Technology

[0002] Cardiovascular disease is one of the major threats to human life. The cardiac pulsation signal is a comprehensive reflection of cardiac activity on the body surface. It is valuable not only for diagnosing certain cardiovascular diseases such as chronic ischemic heart disease, acute coronary syndrome, myocarditis, pericarditis, pulmonary embolism, and arrhythmias, but also for diagnosing hereditary ion channel diseases, cardiac structural abnormalities, and electrolyte imbalances. Solving the semi-analytical solution of the human cardiac pulsation signal is crucial for understanding the various components of the electrocardiogram (ECG) signal, cardiovascular lesions, and the detection and diagnosis of cardiovascular diseases. It also has significant research value from the perspective of life information science. Existing ECG signal simulation technologies, such as the dynamic ECG signal simulation technology disclosed in patent application CN202010566297.3, can output ECG signals as low as 1uV-10uV according to their true voltage values, thereby ensuring the accuracy of dynamic ECG performance testing. The generator includes a touchscreen display, a data update interface, a flash memory, a processor, a digital-to-analog converter (DAC), a digital comparator, a signal amplitude adjustment circuit, and a 3.5mm coaxial analog signal output interface. The processor is connected to the touchscreen display for user operation; it is also connected to the flash memory to store ECG signals; the DAC converts the ECG signals from digital to analog; and the digital comparator determines whether the output ECG signal is below a set threshold and outputs the comparison result to the signal amplitude adjustment circuit. The DAC is electrically connected to the 3.5mm coaxial analog signal output interface via the signal amplitude adjustment circuit. However, its drawback is that the proposed analog technology cannot systematically solve for the semi-analytical solution of the heartbeat signal, and it does not systematically analyze and solve for the stability and instability characteristics of the heartbeat signal. Summary of the Invention

[0003] To address the problems of the prior art, the present invention aims to provide a solution method for nonlinear cardiac pulsation signals based on discrete implicit mapping. This method designs a discrete cardiac pulsation dynamics model, defines it as an implicit mapping system using implicit discrete theory, and then uses the Gaussian method to solve the nonlinear semi-analytical solution of the cardiac pulsation system. The method of the present invention can more accurately obtain complex nonlinear pulsation signals in the cardiac pulsation system, and also has advantages such as high accuracy, fast convergence, and strong robustness.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for solving nonlinear cardiac pulsation signals based on discrete implicit mapping includes the following steps:

[0006] (1) Using the implicit midpoint method to solve the nth-order nonlinear dynamic model of cardiac pulsation. Discretize, in t∈[t k-1 ,t k Within the time interval, the expression is obtained:

[0007]

[0008] in, For the cardiac pulsation dynamics model t k Vector field at time , For the process of heartbeat in t k The potential vector or electric potential vector of the pulsating electrical signal at a given moment. h = t k -t k-1 Represents discrete time intervals. These are parameters of the heartbeat model, and at the same time

[0009] Let N be the number of discrete points within a period. Let m be the number of beats in a heartbeat signal. The following condition must be met for a heartbeat signal to produce m times the period of a beat: Conditions (m = 1, 2, 3, ...);

[0010] (2) Establish a mapping system;

[0011] Based on the discretized cardiac pulsation dynamics model in step (1), a general implicit mapping is defined, expressed as:

[0012]

[0013] in This represents an implicit mapping. According to the above general implicit mapping, the heartbeat signal from the initial time t0 to a certain time t... k Or at a certain time t, which is m times the period mN The implicit mapping of the pulsation signal is:

[0014]

[0015] (3) To solve the implicit mapping in step (2), define a discrete vector g. k The function transforms the expression in step (2) as follows:

[0016]

[0017] in, The discretized cardiac pulsation dynamics model in step (1) is then transformed as follows:

[0018]

[0019] (4) Define and solve the Jacobian matrix, and solve the vector field of the discrete cardiac pulsation dynamics model in step (3). The implicitly mapped discrete Jacobian matrix is ​​as follows:

[0020]

[0021] Substituting the initial conditions into the Jacobian matrix Dg, and combining steps (3) and (4), we can solve for... Iterate through the loop until the accuracy requirement is met. The solution for the nonlinear cardiac pulsation signal can then be obtained.

[0022] The stability analysis of the solution for the nonlinear cardiac pulsation signal is as follows:

[0023] definition There is a tiny perturbation at that point. Then, after this disturbance is propagated in the general implicit mapping in step (2), it becomes:

[0024]

[0025] definition Then the initial disturbance After the implicit mapping evolution, at t mN The perturbation at any given moment evolves into:

[0026]

[0027] Here This is the final stability solution matrix for the discrete mapping of the m-period nonlinear cardiac pulsation signal, obtained by applying the eigenvalues ​​λ of DP. i (i = 1, 2, 3, 4, ...) Determine the stability and bifurcation of the heartbeat signal.

[0028] The stability is determined if the modulus of all eigenvalues ​​is less than 1, i.e., |λ| i If |<1, the heartbeat signal is stable; otherwise, the heartbeat signal is unstable.

[0029] The determination criteria for the bifurcation are as follows:

[0030] (1) If DP has one and only one characteristic root equal to 1, and the modulus of the remaining characteristic roots is less than 1, then the heart pulsation signal undergoes a node-saddle bifurcation.

[0031] (2) If DP has one and only one eigenvalue equal to -1, and the modulus of the remaining eigenvalues ​​is less than 1, then the heart pulsation signal undergoes a period-doubling bifurcation.

[0032] (3) If DP has only one pair of complex eigenvalues ​​with a modulus of 1, and the modulus of the remaining eigenvalues ​​is less than 1, then the heart pulsation signal undergoes a Neimark bifurcation.

[0033] Compared with the prior art, the present invention has the following advantages and innovations:

[0034] (1) Since the implicit discretization algorithm is used in step (1) of this invention, this method has more stable convergence for the discretization of the nonlinear dynamic model of heartbeat, and also has a faster convergence effect.

[0035] (2) Since the number of discrete points of the nonlinear semi-analytical solution in step (1) of the present invention is N, this number of discrete points is flexible and has the characteristics of equal step size discretization and variable step size setting. It can be flexibly set according to different heart pulsation models to improve the solution accuracy.

[0036] (3) Due to the discrete vector g defined in step (3) of this invention k The parameters can be flexibly set according to different solution accuracies, time, and computational energy consumption, using a high-order discrete vector g. k It can provide computational accuracy by using a low-order discrete vector g. k It can provide computing speed.

[0037] (4) Since the Jacobian matrix Dg defined in step (4) of this invention forms a sparse matrix through design and layout, and the internal elements can be designed to have the same or similar patterns to improve the calculation speed.

[0038] (5) Because the stability solution matrix of this invention is obtained by multiplying multiple matrices, the stability dimension of a high-dimensional nonlinear system can be transformed into a low-dimensional matrix eigenvalue problem, simplifying the complex problem and making it convenient and simple to solve while also achieving higher accuracy. Furthermore, the obtained stability solution matrix not only provides the stability of the nonlinear semi-analytical solution of the heartbeat model but also gives the bifurcation judgment criterion for the semi-analytical solution, thus systematically completing the nonlinear analysis.

[0039] In summary, this invention employs a discrete implicit mapping system to solve complex nonlinear cardiac pulsation systems. By discretizing through implicit mapping, the nonlinear differential equations are transformed into a discrete implicit mapping system for solution, reducing the difficulty and improving the accuracy of the solution. Furthermore, the designed solution steps and methods can solve complex high-dimensional cardiac pulsation models with high accuracy, overcoming problems such as the difficulty in deriving complex terms in the "harmonic balance method" and the inability of the "Runge-Kutta method" to solve unstable semi-analytical solutions. This provides a new method for solving semi-analytical solutions of nonlinear dynamic systems. Attached Figure Description

[0040] Figure 1 These are the time-domain diagram and phase diagram of the periodic discharge of the nonlinear cardiac pulsation signal of the present invention, wherein, Figure 1 Figure (a) in the figure represents the time domain diagram of a stable cardiac pulsation signal with period m = 2 and frequency f1 = 0.8. Figure 1 Figure (b) in the diagram represents the phase diagram of a stable cardiac pulsation signal with a period of m = 2 and a frequency of f1 = 0.8. Figure 1 Figure (c) shows the time-domain plot of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.745. Figure 1 Figure (d) in the figure represents the phase diagram of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.745. Figure 1 Figure (e) in the figure represents the time domain plot of a stable cardiac pulsation signal with period m = 3 and frequency f1 = 1.2. Figure 1 The (f) diagram in the figure represents the phase diagram of a stable cardiac pulsation signal with period m = 3 and frequency f1 = 1.2; Figure 1 The (g) plot in the diagram represents the time-domain plot of an unstable cardiac pulsation signal with a period of m = 2 and a frequency of f1 = 1.05. Figure 1 The (h) diagram in the figure represents the phase diagram of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.05.

[0041] Figure 2 This is a bifurcation diagram of the periodic discharge of the nonlinear cardiac pulsation signal of the present invention. In the diagram, the solid black line represents the stable solution, and the dashed red line represents the unstable solution; SN represents the nodal sella bifurcation, NB represents the Neimark bifurcation, and PD represents the period-doubling bifurcation; wherein... Figure 2 Figure (a) in the figure shows the bifurcation diagram of the period 1 discharge of the nonlinear cardiac pulsation signal. The bifurcation in the figure is the Neimark bifurcation. Figure 2 Figure (b) in the figure shows the bifurcation diagram of the period 2 discharge of the nonlinear cardiac pulsation signal. The bifurcation in the figure is a period-doubling bifurcation. Figure 2 Figure (c) in the figure shows the bifurcation diagram of the periodic 3-wave discharge of the nonlinear cardiac pulsation signal. The bifurcation in the figure is the period-doubling bifurcation and the nodal saddle bifurcation. Figure 2Figure (d) in the figure represents an atrial magnification of the bifurcation diagram of the period 2 discharge of the nonlinear cardiac pulsation signal, which better illustrates the period doubling and node sella bifurcation of the diagram.

[0042] Figure 3 This is a comparison between the present invention and the fourth-order Runge-Kutta method. In the figure, the solid black line represents a stable solution, and the dashed red line represents an unstable solution; among them, Figure 3 Figure (a) in the figure shows the calculation results of the fourth-order Runge-Kutta method; Figure 3 Figure (b) in the figure is a semi-analytical solution of the nonlinear cardiac pulsation signal based on discrete implicit mapping in this invention. Detailed Implementation

[0043] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. These embodiments are only for explaining the present invention and do not constitute a limitation on the scope of protection of the present invention.

[0044] A semi-analytical solution method for nonlinear cardiac pulsation signals based on discrete implicit mapping is proposed. The cardiac pulsation model employs an improved coupled van der Bohr equation. A specific embodiment includes the following steps:

[0045] (1) A discrete, improved coupled van der Bohr cardiac pulsation dynamics system is established based on the implicit midpoint method, defining the time interval as t∈[t k-1 ,t k When ], then there is

[0046] x 1,k =x 1,k-1 +hx 2,kc

[0047]

[0048] x 3,k =x 3,k-1 +hx 4,kc

[0049]

[0050] Where f1 is the control frequency of the heart pulsation model, and has

[0051]

[0052]

[0053] Here and

[0054]

[0055] in, For the cardiac pulsation dynamics model t kThe vector field at time, and the specific model of heartbeat are as follows: As shown in the formula. For the process of heartbeat in t k The potential vector or electric potential vector of the pulsating electrical signal at time t, h = t k -t k-1 Represents discrete time intervals. These are the parameters of the heartbeat model, here. at the same time The specific expression is as follows:

[0056]

[0057] Define N as the number of discrete points within a period. Define the cardiac pulsation dynamics model as requiring m times the periodic pulsation to satisfy the following conditions: That is (x) 1,mN ,x 2,mN ,x 3,mN ,x 4,mN ) T =(x 1,0 ,x 2,0 ,x 3,0 ,x 4,0 ) T condition.

[0058] (2) Establish a mapping system

[0059] Based on the discretized cardiac pulsation dynamics model in step (1), a general implicit mapping is defined, expressed as:

[0060]

[0061] in This represents an implicit mapping. Based on the above general implicit mapping, the cardiac firing signal from the initial time t0 to a certain time t... k Or at a certain time t, which is m times the period mN The implicit mapping of the pulsatile discharge signal is:

[0062]

[0063] (3) To solve the implicit mapping in step (2), define a discrete vector g. k The function transforms the expression in step (2) as follows:

[0064] g x1,k =x 1,k -x 1,k-1 -hx 2,kc

[0065]

[0066] g x3,k =x 3,k -x 3,k-1 -hx 4,kc

[0067]

[0068] in, but

[0069]

[0070] k = 1, 2, 3, ..., mN;

[0071] The discretized cardiac pulsation dynamics model in step (1) is then transformed as follows:

[0072]

[0073] (4) Define and solve the Jacobian matrix, and solve the vector field of the discrete cardiac pulsation dynamics model in step (3). The implicitly mapped discrete Jacobian matrix is ​​as follows:

[0074]

[0075] Substituting the initial conditions into the Jacobian matrix Dg, and combining steps (3) and (4), we can solve for... Iterate through the loop until the accuracy requirement is met. (ζ is a pre-set precision, for example, ζ = 16), and the solution for the nonlinear cardiac pulsation signal can be obtained.

[0076] in

[0077]

[0078] The expressions for each element are as follows:

[0079]

[0080]

[0081]

[0082]

[0083]

[0084]

[0085]

[0086]

[0087] Taking k = -1.45, ω1 = -0.2, ω2 = 1.9, b1 = 1.0, b2 = 0.666, f1 = 1.0, c1 = 0.1, c2 = 0.48 as an example, the action potentials of the sinoatrial node (x2) and atrioventricular node (x4) of the periodic motion discharge of the nonlinear cardiac pulsation signal with m = 1 are obtained. The results are as follows: Figure 1 As shown. The solution to the nonlinear cardiac pulsation system obtained by the discrete implicit mapping according to the definition is called the semi-analytical solution of this system. The results are shown in the appendix. Figure 1 .

[0088] Figure 1 Figure (a) in the figure shows the time-domain plot of a stable cardiac pulsation signal with period m = 2 and frequency f1 = 0.8. The stable periodic pulsation characteristics of the cardiac pulsation signal can be clearly seen from the time-domain plot. Figure 1 Figure (b) shows the phase diagram of a stable cardiac pulsation signal with period m = 2 and frequency f1 = 0.8. The periodicity of the cardiac pulsation signal is clearly visible. For stable cardiac pulsation signals, this method can achieve very good convergence results.

[0089] Figure 1 Figure (c) in the figure represents the time-domain plot of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.745. The blue line represents the results obtained by other numerical methods (such as the fourth-order Runge-Kutta method), and the red line represents the semi-analytical solution of the cardiac pulsation signal obtained by this method. For unstable cardiac pulsation periodic signals, this method can obviously obtain a convergent and accurate unstable analytical solution, while the calculation results of other methods (such as the fourth-order Runge-Kutta method) will diverge and ultimately fail to obtain a convergent and correct solution. Figure 1 Figure (d) in the diagram represents the phase diagram of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.745. It is clear that this method can obtain an unstable solution (represented by circles) for the cardiac pulsation signal, and the result is a convergent periodic signal result. Other methods (such as the fourth-order Runge-Kutta method) cannot obtain a convergent unstable periodic signal result, and the obtained solution is also divergent.

[0090] Figure 1 Figure (e) in the figure represents the time domain plot of a stable cardiac pulsation signal with period m = 3 and frequency f1 = 1.2. Figure 1 Figure (f) in the diagram represents the phase diagram of a stable cardiac pulsation signal with period m = 3 and frequency f1 = 1.2. The stable periodic pulsation characteristics of the cardiac pulsation signal are clearly visible in the time-domain plot, and the periodicity of the cardiac pulsation signal is clearly shown in the phase diagram. For stable cardiac pulsation signals, this method achieves very good convergence results.

[0091] Figure 1 Figure (g) in the figure represents the time-domain plot of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.05. The blue line represents the results obtained by other numerical methods (such as the fourth-order Runge-Kutta method), and the red line represents the semi-analytical solution of the cardiac pulsation signal obtained by this method. For unstable cardiac pulsation periodic signals, this method can obviously obtain a convergent and accurate unstable analytical solution, while the calculation results of other methods (such as the fourth-order Runge-Kutta method) will diverge and ultimately fail to obtain a convergent and correct solution. Figure 1 The (h) diagram in the figure represents the phase diagram of an unstable cardiac pulsation signal with period m = 2 and frequency f1 = 1.05. It is clear that this method can obtain the unstable solution (represented by circles) of the cardiac pulsation signal, and the result is a convergent periodic signal result. Other methods (such as the fourth-order Runge-Kutta method) cannot obtain a convergent unstable periodic signal result, and the obtained solution is also divergent.

[0092] After obtaining the nonlinear cardiac pulsation signal, a stability analysis of the results is required. The stability analysis of the results is as follows:

[0093] definition There is a tiny perturbation at that point. This perturbation, after being propagated in the general implicit mapping 2), becomes...

[0094]

[0095] Defined here: Then the initial disturbance After the implicit mapping evolution, at t mN The perturbation of time evolves into

[0096]

[0097] Here This is the final stability solution matrix for the discrete mapping of the m-period nonlinear cardiac pulsation signal, which can be obtained by applying the eigenvalues ​​λ of DP. i (i = 1, 2, 3, 4, ...) Determine the stability and bifurcation of the heartbeat signal.

[0098] Stability is determined if the modulus of all eigenvalues ​​is less than 1, |λ i If |<1, the heartbeat signal is stable; otherwise, the heartbeat signal is unstable.

[0099] Define the bifurcation determination criteria:

[0100] (1) If DP has one and only one characteristic root equal to 1, and the modulus of the remaining characteristic roots is less than 1, then the heart pulsation signal undergoes a node-saddle bifurcation.

[0101] (2) If DP has one and only one characteristic root equal to -1, and the modulus of the remaining characteristic roots is less than 1, then the heart pulsation signal undergoes a period-doubling bifurcation.

[0102] (3) If DP has only one pair of complex eigenvalues ​​with a modulus of 1, and the modulus of the remaining eigenvalues ​​is less than 1, then the heart pulsation signal undergoes Neimark bifurcation.

[0103] In this invention, the eigenvalues ​​of the stability determination matrix DP are obtained through programming as λ1 = -0.434811 + 0.186845i, λ2 = -0.434811 - 0.186845i, λ3 =

[0104] 0.220755, λ4 = 2.15729e-09. Therefore, it is determined that when f1 = 1.0, the cardiac beat signal with a period of 1 is stable and there is no bifurcation point. For different excitation frequencies, taking periodic motion with m = 1, 2, 3, and 4 cycles as examples, bifurcation diagrams are plotted as follows: Figure 2 As shown, the Neimark bifurcation, saddle node bifurcation, and period-doubling bifurcation data of the nonlinear cardiac pulsation signal are shown in Tables 1 and 2, respectively.

[0105] Table 1. Neimark bifurcation data of nonlinear cardiac pulsation signals.

[0106]

[0107] Table 2. Data on nonlinear cardiac pulsation signals, including node-saddle bifurcation and period-doubling bifurcation.

[0108]

[0109] Its advantages over other algorithms, such as the 4th-order Runge-Kutta method, are evident. Figure 3 .in, Figure 3 Figure (a) shows the calculation results of the fourth-order Runge-Kutta method. Figure 3 Figure (b) shows the semi-analytical solution of the nonlinear cardiac pulsation signal based on discrete implicit mapping. A comparison reveals that the semi-analytical solution of the nonlinear cardiac pulsation signal based on discrete implicit mapping can find the unstable solution of the cardiac pulsation system (represented by the red dashed line), while the fourth-order Runge-Kutta method unceremoniously obtains this unstable solution. Furthermore, the semi-analytical solution of the nonlinear cardiac pulsation signal based on discrete implicit mapping can accurately determine bifurcation and provide bifurcation determination criteria, while the fourth-order Runge-Kutta method's determination is inaccurate (bifurcation points are represented by vertical dashed lines).

[0110] The foregoing has shown and described the basic principles, general process, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of this invention is defined by the appended claims and their equivalents.

Claims

1. A semi-analytical solution method for cardiac pulsation signals based on discrete implicit mapping, characterized in that, Includes the following steps: (1) Using the implicit midpoint method to solve the nth-order nonlinear dynamic model of cardiac pulsation. Discretize, in t∈[t k-1 ,t k Within the time interval, the expression is obtained: in, For the cardiac pulsation dynamics model t k Vector field at time , For the process of heartbeat in t k The potential vector or electric potential vector of the pulsating electrical signal at a given moment. h = t k -t k-1 Represents discrete time intervals. These are parameters of the heartbeat model, and at the same time Let N be the number of discrete points within a period. Let m be the number of beats in a heartbeat signal. The following condition must be met for a heartbeat signal to produce m times the period of a beat: Conditions (m = 1, 2, 3, ...); (2) Establish a mapping system; Based on the discretized cardiac pulsation dynamics model in step (1), a general implicit mapping is defined, expressed as: in This represents an implicit mapping. According to the above general implicit mapping, the heartbeat signal from the initial time t0 to a certain time t... k Or at a certain time t, which is m times the period mN The implicit mapping of the pulsating signal is: (3) To solve the implicit mapping in step (2), define a discrete vector g. k The function transforms the expression in step (2) as follows: in, The discretized cardiac pulsation dynamics model in step (1) is then transformed as follows: (4) Define and solve the Jacobian matrix, and solve the vector field of the discrete cardiac pulsation dynamics model in step (3). The implicitly mapped discrete Jacobian matrix is ​​as follows: Substituting the initial conditions into the Jacobian matrix Dg, and combining steps (3) and (4), we can solve for... Iterate through the loop until the accuracy requirement is met. The solution for the nonlinear cardiac pulsation signal can then be obtained.

2. The method for solving a semi-analytical solution of a cardiac pulsation signal based on discrete implicit mapping according to claim 1, characterized in that, The stability analysis of the solution for the nonlinear cardiac pulsation signal is as follows: definition There is a tiny perturbation at that point. Then, after this disturbance is propagated in the general implicit mapping in step (2), it becomes: definition Then the initial disturbance After the implicit mapping evolution, at t mN The perturbation at any given moment evolves into: Here This is the final stability solution matrix for the discrete mapping of the m-times nonlinear cardiac pulsation signal, obtained by applying the eigenvalues ​​λ of DP. i (i = 1, 2, 3, 4, ...) Determine the stability and bifurcation of the heartbeat signal; The stability is determined if the modulus of all eigenvalues ​​is less than one, i.e., |λ| i If |<1, the heartbeat signal is stable; otherwise, the heartbeat signal is unstable. The determination criteria for the bifurcation are as follows: (1) If DP has one and only one characteristic root equal to 1, and the modulus of the remaining characteristic roots is less than 1, then the heart pulsation signal undergoes a node-saddle bifurcation. (2) If DP has one and only one characteristic root equal to -1, and the modulus of the remaining characteristic roots is less than 1, then the heart pulsation signal undergoes a period-doubling bifurcation. (3) If DP has only one pair of complex eigenvalues ​​with a modulus of 1, and the modulus of the remaining eigenvalues ​​is less than 1, then the heart pulsation signal undergoes Neimark bifurcation.

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