A method for phase-frequency synchronization based on coupled neuron oscillator system

Abstract

This invention discloses a method for achieving phase-frequency synchronization based on a coupled neuron oscillator system. First, a cross-frequency nonlinear coupling model is constructed by establishing a two-connection structure based on exogenous input and local interaction in the Poincaré model. By introducing nonlinear coupling, phase-frequency synchronization of the coupled neuron oscillators is achieved. Spectral analysis is then used to determine the effect of the coupling strength between low-frequency and high-frequency neurons on phase-frequency synchronization. Theoretical analysis is then used to determine the influence of the low-frequency oscillation phase on the high-frequency oscillation frequency and its mechanism.

Description

Technical field

[0001] This invention relates to the field of neural network synchronization, specifically the study of cross-frequency coupling and phase-frequency synchronization of neuronal oscillator systems. This technology can be applied to fields such as neuroscience, electroencephalogram (EEG) analysis, and neural regulation. Background Art

[0002] The collective behavior of coupled neurons with different frequencies is closely related to brain function. In the cerebral cortex, neuronal oscillations of different frequencies coexist and interact, forming diverse brainwave patterns such as alpha waves (8-13 Hz), beta waves (13-30 Hz), gamma waves (30-80 Hz), and theta waves (4-7 Hz). Neurons in different brain regions generate neural oscillations of various frequencies, and their phase, amplitude, and frequency interact in complex ways. Understanding the various forms of synchronization formed by these complex interactions is crucial for understanding cognitive function and disease pathology. Cross-frequency coupling, in particular, refers to the mutual regulation and interaction between neural oscillations within different frequency ranges, often manifesting as phase-amplitude coupling, such as the coupling between theta-gamma and alpha-gamma waves, phase-phase coupling, and phase-frequency coupling. Cross-frequency coupling is not only associated with normal cognitive function but also closely linked to the mechanisms of various brain diseases. It plays a particularly important role in understanding the mechanisms of many brain disorders, including epilepsy, Parkinson's disease, and schizophrenia.

[0003] Common representations of cross-frequency coupling include phase-amplitude coupling and phase-frequency coupling. Phase-amplitude coupling (PAC) refers to the phenomenon that the amplitude (or power) of high-frequency neuronal oscillations is controlled by the phase of low-frequency neuronal oscillations [e.g. Figure 1(a)]. This phenomenon was first discovered in the hippocampus, where the phase of low-frequency theta waves can effectively regulate the amplitude of high-frequency gamma waves, forming a typical pattern of phase-amplitude coupling. Phase-amplitude coupling reveals the synergistic effect between oscillations of different frequencies in the brain and plays an important role in brain activities such as memory, attention, and learning. Phase-frequency coupling (PFC) refers to the phenomenon in which the oscillation frequency of high-frequency oscillating neurons in a coupled neuronal system is modulated by the phase of low-frequency oscillations. It plays a key role in the realization of brain functions such as cognition. There are significant differences in the manifestation and measurement methods of the two in neural signals. Phase-amplitude coupling is usually measured by extracting the amplitude of the high-frequency oscillation signal and examining its relationship with the phase of the low-frequency oscillation. Phase-frequency coupling focuses more on the phenomenon of low-frequency phase modulating the frequency of high-frequency oscillations, rather than amplitude modulation. This phase-frequency modulation phenomenon was observed in a densely firing pyramidal-interneuron gamma high-frequency oscillatory network in a noise-free environment, which includes continuous or intermittent cross-frequency coupling settings. In the minimal pyramidal-interneuron γ model modulated by low-frequency oscillations, when the low-frequency oscillatory input reaches its peak, it causes the high-frequency oscillations to accelerate, regardless of whether the input target is a pyramidal cell or an inhibitory cell. Although research on phase-frequency coupling is still in its infancy, it frequently occurs in strongly coupled neural networks, especially between low-frequency and high-frequency oscillations under strong coupling. In these networks, the phase of low-frequency oscillations may not only modulate the amplitude of high-frequency oscillations but also have a significant impact on their frequency. Such coupling effects play an important role in different patterns of neural activity, especially in brain functions that require precise regulation of the interaction between oscillations of different frequencies.

[0004] In brain signals recorded by various neuroimaging techniques, such as electroencephalography (EEG) and intracranial recordings, researchers have discovered that phase-frequency coupling is widespread across multiple neural networks in the brain. Phase-frequency coupling not only reveals the interactions between oscillations of different frequencies in the brain but also provides important clues for a deeper understanding of the coordination of neural activity and the mechanisms of information transmission. Therefore, a deeper understanding of the phase-frequency coupling mechanisms between different brain regions in the human brain and their application in brain function is of great significance.

[0005] By constructing a cross-frequency coupling model based on the two main connection structures of the Poincaré model, namely exogenous input and local interaction, the phase-frequency coupling in the brain neural network was analyzed. Under the action of exogenous input, the introduction of frequency regulation can effectively stimulate the coupling between the phase of low-frequency neurons and the frequency of high-frequency oscillations, and this phenomenon is more significant under strong coupling conditions. In addition, under the local interaction structure, the interaction between neurons gives the phase-frequency coupling a new form of expression. The strengthening of local connections not only contributes to the generation of the phase-frequency coupling phenomenon, but also significantly enhances the influence of the phase of low-frequency oscillations on the frequency of high-frequency oscillations. These results provide important theoretical support for understanding the mechanism of cross-frequency coupling in brain neural networks and provide a new perspective for exploring the dynamic characteristics of brain function. Summary of the Invention

[0006] 1. Purpose of the Invention

[0007] By establishing a cross-frequency coupling model based on the exogenous input and local interaction structures of the Poincaré model, phase-frequency synchronization of coupled neuronal oscillators was achieved. Spectral analysis was used to determine the effect of the coupling strength of neurons at different frequencies on phase-frequency synchronization. Theoretical analysis was used to determine the influence of the phase of low-frequency oscillations on the frequency of high-frequency oscillations and the underlying mechanism.

[0008] 2. Technical solution

[0009] The present invention is achieved through the following technical solutions.

[0010] A method for realizing phase-frequency synchronization based on a coupled neuron oscillator system comprises the following steps:

[0011] (S01): Construct a cross-frequency coupled Poincaré model with exogenous input connections, introducing nonlinear coupling to simulate the dynamical synchronization of neuronal oscillators in the brain driven by exogenous signals. Determine the phase-frequency synchronization of high-frequency oscillating neurons driven by low-frequency oscillating neurons, and the phase-frequency synchronization of low-frequency oscillating neurons driven by high-frequency oscillating neurons.

[0012] (S02): Use the fourth-order Runge-Kutta method to numerically solve the model constructed in (S01), and determine the rules of how the frequency of high-frequency oscillating neurons is regulated by the phase of low-frequency oscillating neurons, and the rules of how the frequency of low-frequency oscillating neurons is regulated by the phase of high-frequency oscillating neurons.

[0013] (S03): Construct a cross-frequency nonlinear coupled Poincaré model with local interaction connection mode to simulate the synchronous dynamic behavior of high-frequency oscillating neuronal oscillators and low-frequency oscillating neuronal oscillators in the brain under interaction.

[0014] (S04): The model constructed in (S03) is numerically solved using the fourth-order Runge-Kutta method, and the synchronization between the phase of the low-frequency oscillator of the neuron and the frequency of the high-frequency oscillator of the neuron is determined under the interaction of different coupling strengths between the high- and low-frequency oscillators.

[0015] (S05): Based on the exogenous input model, theoretical analysis gives the relationship between the frequency of high-frequency oscillation neurons and the phase of the driving signal, and further determines the functional relationship between the frequency variation range of high-frequency oscillations and the intensity of the driving effect.

[0016] The specific implementation steps are as follows:

[0017] Step 1: Construct a cross-frequency nonlinear coupled Poincaré model with exogenous input connection mode.

[0018]

[0019] or

[0020]

[0021] Among them, x f ,y f is the variable of high-frequency oscillating neurons, x s ,y s is the variable of low-frequency oscillating neurons, ω f,s and a f =a s =a are the natural frequencies and natural amplitudes of high-frequency and low-frequency oscillating neurons respectively, γ = 1 represents the relaxation parameter, is the amplitude of the low- and high-frequency oscillating neurons. The interaction between high-frequency and low-frequency oscillating neurons is realized through multiplicative driving. In formula (1), the high-frequency oscillating neurons are only affected by the low-frequency oscillating neurons and will not reversely affect the low-frequency oscillating neurons, where ε s Indicates the driving coupling strength of the low-frequency oscillating neuron on the high-frequency oscillating neuron oscillator. In formula (2), the low-frequency oscillating neuron is only affected by the high-frequency oscillating neuron and will not reversely affect the high-frequency oscillating neuron, where ε f It represents the driving coupling strength of high-frequency oscillating neurons on low-frequency oscillating neuron oscillators.

[0022] Step 2: The cross-frequency nonlinear coupling Poincaré model (1) constructed in the first step is numerically solved using the fourth-order Runge-Kutta method to study how the frequency of high-frequency oscillating neurons is regulated by the phase of low-frequency oscillating neurons. The cross-frequency nonlinear coupling Poincaré model (2) constructed in the first step is numerically solved to study how the frequency of high-frequency oscillating neurons affects the phase of low-frequency oscillating neurons.

[0023] Step 3: Construct the local interaction nonlinear coupling Poincaré model as follows:

[0024]

[0025] Among them, the subscripts f and s represent high-frequency and low-frequency neurons respectively, and ε f ,ε s are the coupling strengths of high frequency to low frequency and low frequency to high frequency respectively. f ,y f is the variable of high-frequency oscillating neurons, x s ,y s is the variable of low-frequency oscillating neurons, ω fs and a f =a s =a are the natural frequencies and natural amplitudes of high-frequency and low-frequency oscillating neurons respectively, γ = 1 represents the relaxation parameter, is the amplitude of low- and high-frequency oscillating neurons.

[0026] Step 4: The cross-frequency nonlinear coupling Poincaré model (3) constructed in (S03) is numerically solved using the fourth-order Runge-Kutta method. The instantaneous frequency change amplitude Δf is calculated by Fourier transforming the time series to quantify the frequency change of neuronal oscillations. The regulatory effect of the phase change of low-frequency oscillating neurons on the frequency of high-frequency oscillations is further analyzed.

[0027] Step 5: Based on the exogenous input model (1) constructed in the first step, convert the formula (1) into polar coordinate form through x = ρcosθ, y = ρsinθ as shown in formula (4), and conduct a theoretical analysis of the relationship between the frequency change amplitude of high-frequency oscillating neurons and the driving force intensity.

[0028]

[0029] Among them, ε s is the driving force of low-frequency oscillating neurons on high-frequency neurons, ρ f ,ρ s are the amplitudes of high-frequency and low-frequency oscillations respectively, and a is the initial amplitude parameter of high-frequency and low-frequency neurons. We can get θ s =ω s t, so Among them, the instantaneous frequency of high-frequency oscillation is Right now,

[0030]

[0031] By |cos(ω st)|≤1, the frequency variation amplitude of the high-frequency oscillation neuron is,

[0032]

[0033] The amplitude of the low-frequency oscillating neuron ρ s Can be ordered Solving for ρ s =a, and substituting it into equation (6), we can get the frequency variation amplitude of high-frequency oscillation neurons as,

[0034]

[0035] Therefore, the instantaneous frequency of the high-frequency oscillating neuron is modulated by the low-frequency oscillating neuron, and its variation amplitude Δf is only affected by the inherent amplitude a of the low-frequency neuron and the coupling effect ε s impact.

[0036] 3.Beneficial effects:

[0037] The present invention can effectively simulate the cross-frequency coupling between brain neurons through a simple mathematical model, and produce phase-frequency synchronization between low-frequency oscillation neurons and high-frequency oscillation neurons. The regulation mechanism of low-frequency oscillation phase on high-frequency oscillation frequency is revealed under two different connection modes of exogenous input and local interaction, which is of great significance for understanding the functional coordination between different areas of the brain. By introducing polar coordinates into the model, it can be theoretically analyzed that the frequency of high-frequency oscillation neurons is regulated by the phase of low-frequency oscillation, and its frequency change amplitude Δf is related to the coupling strength ε of low-frequency oscillation neurons. s and the functional relationship between the inherent amplitude a, satisfying Δf=ε s a / π, we can get Δf as ε s and increases with the increase of a. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] Figure 1 Phase-frequency coupling between low-frequency and high-frequency signals when low-frequency signals are used as exogenous input signals; (a) Time series of low-frequency neuronal oscillations; (b) ε s = 2 when the time series of high-frequency neural oscillations driven; (c) ε s = 4; (d) Spectrum of high-frequency neuron oscillations; (e) The amplitude of high-frequency neuron frequency changes with the driving coupling strength ε s The changing relationship.

[0039] Figure 2 Phase-frequency coupling between low-frequency and high-frequency signals when high-frequency signals are used as exogenous input signals; (a) Oscillation time series of high-frequency neurons; (b) ε s =0,ε f= 2; (c) ε s =0,ε f =4; (d)-(f) are the frequency spectra of the neuron time series corresponding to (a)-(c).

[0040] Figure 3 ε f and ε s The amplitude of the instantaneous frequency change of neurons in parameter space, (a) The amplitude of the instantaneous frequency change of high-frequency oscillating neurons with coupling strength ε f and ε s (b) The amplitude of the instantaneous frequency change of the low-frequency oscillating neuron changes with the coupling strength ε f and ε s changes.

[0041] Figure 4 Phase-frequency synchronization under local interaction. (ac) Time series of low-frequency oscillations; (df) Time series of high-frequency oscillations; (gi) Spectrum diagram; where (a, d, g) ε f =2,ε s =2;(b,e,f)ε f =8,ε s =4;(c,f,i)ε f =10,ε s =10.

[0042] Figure 5 Functional relationship curve of the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO). (a) When ε f and ε s The functional relationship between the instantaneous frequency of the fast oscillation (FO) and the phase of the slow oscillation (SO) when both are zero. (b) When ε f = 0 and ε s = 2, the system has only a unidirectional influence of the low-frequency oscillation phase on the high-frequency oscillation frequency. (c) When ε f =ε s = 2, the functional relationship between the instantaneous frequency of the fast oscillation (FO) and the phase of the slow oscillation (SO). (d) When ε f =2,ε s = 8, the functional relationship between the instantaneous frequency of the fast oscillation (FO) and the phase of the slow oscillation (SO). The high frequency has a more obvious adverse effect on the phase of the low frequency oscillation, further causing the change of the oscillation amplitude.

[0043] Figure 6 (a) When a=4,6, Δf increases with the coupling strength ε s(b)ε s =2,3, the relationship between Δf and the natural amplitude a, where the solid line is the theoretical result and the dotted line is the numerical calculation result. DETAILED DESCRIPTION

[0044] In order to make the purpose, features and advantages of the present invention more clearly understood, specific embodiments are introduced below and further detailed description is given as follows:

[0045] (1) Phase-frequency coupling with exogenous input

[0046] Based on formula (3), take parameters a=6, ω s =3rad / s,ω f =30rad / s,ε f =0,ε s = 2, formula (3) is the exogenous signal input, at this time the low-frequency signal ( Figure 1 (a) drives the high-frequency signal. Under the action of the low-frequency oscillation neurons, the amplitude of the high-frequency oscillation is not affected by the low-frequency oscillation and maintains the initial amplitude value a=6 ( Figure 1 (b)), and its oscillation frequency changes periodically due to the phase change of the low-frequency oscillation, and the change period corresponds to the phase change period of the low-frequency oscillation. That is, the frequency of the high-frequency oscillation changes accordingly with the phase change of the low-frequency oscillation, resulting in phase-frequency coupling. As the coupling strength increases, as shown in ε s =4( Figure 1 As shown in (c), within a phase change cycle of a low-frequency oscillation (0 to 2π), the frequency fluctuation range of the high-frequency oscillation increases, and its time series shows a greater distinction between dense and sparse areas. This indicates that as the coupling strength increases, the regulatory effect of the low-frequency oscillation on the high-frequency oscillation frequency becomes more significant, and the frequency regulation range becomes wider. s =2,4, the spectrum shows that ε s = 2, under the drive of low-frequency oscillation neurons, the main frequency component of high-frequency oscillation neurons widens from the original 4.78Hz to 2.8~6.7Hz. Figure 1 (d), when the coupling strength ε s =4, the frequency range of the spectrum is significantly widened to 0.95~8.6Hz. Figure 1As shown in (d), the frequency broadening effect is more obvious, which indicates that the frequency regulation effect of the low-frequency oscillation phase change on the high-frequency oscillation becomes more significant. In order to determine the influence of the coupling strength of the low-frequency oscillating neurons on the phase-frequency coupling, the frequency change amplitude of the high-frequency component is calculated when the coupling strength ε = 0 to 4, so as to further quantify the influence of the low-frequency oscillation phase component on the frequency regulation of the high-frequency oscillation. The Hilbert transform is performed on the high-frequency oscillating neuron signal under different coupling strengths, and its instantaneous frequency is extracted to calculate the change amplitude of its instantaneous frequency (the difference between the maximum and minimum values ​​of the instantaneous frequency). Figure 1 (e) The results show that as the coupling strength increases, the change amplitude of the instantaneous frequency increases linearly. For example, when the coupling strength increases from 1 to 4, the frequency table change amplitude increases from 3.8Hz to 7.6Hz. s = 0, the high-frequency oscillation neuron shows a single-frequency oscillation, and the amplitude of its frequency change is 0. s = 2, the instantaneous frequency variation of high-frequency oscillation is about 3.8Hz, and ε s =4, the instantaneous frequency variation of the high-frequency oscillation is about 7.6Hz.

[0047] Based on formula (3), low-frequency oscillating neurons are more sensitive to high-frequency oscillating neurons ( Figure 2 (a)) under the driving action (ε s =0,ε f >0), the high-frequency oscillation time series will be superimposed on the low-frequency oscillation time series. Coupling strength ε f = 2, the superimposed high-frequency oscillation amplitude is small, such as Figure 2 (b) shows that when ε s =4, the amplitude of high-frequency oscillation increases, such as Figure 2 (c) shown. Figure 2 (d) shows the spectrum of a high-frequency oscillating neuron, with a dominant frequency of 4.78 Hz. Figure 2 (e) and (f) give ε respectively. s = 2,4. The dashed line represents the spectrum of the high-frequency driving signal. The dominant frequency of the low-frequency neuron is 0.478 Hz, and new frequencies of 4.3 Hz and 5.178 Hz are also generated. As the coupling strength increases, the dominant frequency peak of the driven low-frequency neuron decreases, while the newly generated secondary peaks increase. The low-frequency neuron is modulated by the high-frequency oscillating neuron.

[0048] (2) Phase-frequency coupling during local interaction

[0049] Based on formula (3), the local interaction between high-frequency and low-frequency neuron oscillators (ε s >0,ε f>0), and calculate the change amplitude of the instantaneous frequency of the high-frequency oscillating neurons (low-frequency oscillating neurons) with the two local coupling strengths ε f and ε s The change relationship diagram of Figure 3 (a)( Figure 3 (b) As shown in Figure 2, the driving coupling strength ε of the low-frequency oscillating neurons to the high-frequency oscillating neurons s As the oscillation frequency of high-frequency neurons increases, the amplitude of the oscillation frequency of high-frequency neurons gradually increases. The driving coupling strength ε of high-frequency neurons to low-frequency neurons f The increase of ε has little effect on the oscillation frequency of high-frequency neurons. s When it increases to a certain value, the modulation effect of low-frequency oscillation on high-frequency oscillation becomes dominant. s As ε increases, the instantaneous frequency variation also increases. f and ε s When both increase to a certain critical value, the amplitude of the instantaneous frequency of high frequency and low frequency begins to decrease gradually, showing the characteristics of the system tending to be stable ( Figure 3 (a) (b) (upper right corner). This phenomenon indicates that under strong coupling, the interaction within the network produces an intrinsic constraint mechanism that stabilizes the changes in high-frequency oscillation frequency. Figure 3 (b) further illustrates how the instantaneous frequency amplitude of low-frequency oscillating neurons is affected by ε f and ε s Different from high frequency oscillation, under this condition ε f It is the main controlling factor, which dominates the change trend of the instantaneous frequency amplitude of low-frequency oscillation. s value, when ε f When it increases to a certain value, the frequency variation of the low-frequency oscillation suddenly increases, and this variation is much greater than the frequency variation of the low-frequency oscillation. Figure 4 (a) The result is that when ε f and ε s At the same time, after increasing to a certain threshold, the instantaneous frequency amplitude of the low-frequency oscillation also begins to gradually decrease ( Figure 3 The upper right corner of (b) also shows the stability of the system. This phenomenon shows that when the coupling strength exceeds a certain critical point, whether it is high-frequency or low-frequency oscillation, the network tends to synchronize and stabilize, thereby reducing frequency fluctuations.

[0050] Phase-frequency coupling occurs when the interaction between two neuronal oscillators is small. The frequency of the high-frequency oscillation is not completely synchronized with the period of the low-frequency oscillation, but is adjusted at certain specific phase positions, which is manifested as a certain coupling relationship between the frequency fluctuation and the phase of the low-frequency oscillation. f =2,εs = 2, the two neuronal oscillations interact with each other. The phase of the low-frequency oscillation affects the frequency fluctuations of the high-frequency oscillation, and the frequency fluctuations of the high-frequency oscillation also have a feedback effect on the oscillations of the low-frequency neuron. It can be seen that within the cycle of the low-frequency oscillation, the low-frequency phase acts on the frequency component of the high-frequency oscillation, and the frequency fluctuations of the high-frequency oscillation are fed back to the low-frequency oscillation through phase regulation. Figure 4 (a, d, g) show the effect of this mutual coupling, indicating that the interaction between high-frequency and low-frequency oscillations is not just a one-way process, but a complex two-way regulation process. As the coupling strength increases, the interaction between high-frequency and low-frequency oscillations in the system gradually strengthens, showing more significant coupling characteristics. This enhanced interaction promotes the occurrence of partial phase synchronization phenomena, such as Figure 4 As shown in (b, e, h), it can be clearly seen from the spectrum analysis that the different frequency components of high-frequency and low-frequency oscillations gradually show a corresponding relationship with each other, indicating that the coupling between the two is not independent or unidirectional, but there is a complex bidirectional regulation mechanism. As the coupling strength increases further, the interaction between high-frequency oscillations and low-frequency oscillations in the system continues to strengthen, eventually leading to the occurrence of phase locking phenomenon. The results are shown in Figure 4 (c, f, i) are shown. When the coupling strength reaches a certain threshold, the phases of the high-frequency and low-frequency oscillations gradually become closely synchronized, forming a stable phase-locked state. This state indicates that the coupling between the two oscillations has reached a new equilibrium, and the dynamic behavior has transitioned from complex bidirectional modulation to a stable synchronization mode. This phenomenon can be further verified through spectrum analysis. When the coupling strength exceeds the threshold, the frequency components of the high-frequency and low-frequency oscillations exhibit clear synchronization characteristics, and the originally dispersed frequency components in the spectrum gradually converge and align. This frequency alignment behavior clearly reveals how the coupling mechanism between the high-frequency and low-frequency oscillations promotes the synchronization of their frequencies and phases during the phase-locking process.

[0051] (3) Response curve of high-frequency oscillation frequency to low-frequency oscillation phase

[0052] The intrinsic mechanism and characteristics of the phase of low-frequency oscillation affecting the frequency of high-frequency oscillation can be revealed by constructing the response curve of high-frequency oscillation. Figure 5 Shows the different ε f and ε s The corresponding relationship between the frequency and phase of two oscillating neurons under the following conditions.

[0053] 1) No interaction: When ε f and ε s When both are zero, there is no interaction between the two oscillators, and there is no obvious coupling effect between the phase of the low-frequency oscillation and the frequency of the high-frequency oscillation. Figure 5(a) shows that the frequency of the high-frequency oscillation remains constant and is not affected by the phase change of the low-frequency oscillation, so there is no phase-frequency coupling.

[0054] 2) Only the low-frequency oscillation phase affects the high-frequency oscillation frequency: When ε f = 0 and ε s = 2, the system only has a one-way influence of the low-frequency oscillation phase on the high-frequency oscillation frequency. At this time, the phase of the low-frequency oscillation regulates the frequency of the high-frequency oscillation, causing the instantaneous frequency of the high-frequency oscillation to fluctuate periodically as the phase of the low-frequency oscillation increases (such as Figure 5 (b) shows that the coupled system has a phase-frequency coupled synchronization state, that is, the low-frequency phase has a significant modulation effect on the high-frequency frequency.

[0055] 3) Bidirectional interaction: When ε f >0 and ε s > 0, the interaction between the phase of low-frequency oscillations and the frequency of high-frequency oscillations becomes more complicated. f =ε s =2, such as Figure 5 As shown in (c), the instantaneous frequency of the high-frequency oscillation is modulated by the phase of the low-frequency oscillation as a whole, and also undergoes disturbances when the high-frequency frequency has a certain counter-effect on the phase of the low-frequency oscillation. That is, the frequency change of the high-frequency oscillation is not only consistent with the low-frequency phase, but also accompanied by some small disturbances; when ε f =2,ε s = 8, the reaction of high frequency to low frequency oscillation phase becomes more obvious, further resulting in changes in oscillation amplitude, such as Figure 5 (d) shown.

Claims

1. A method for realizing phase-frequency synchronization based on a coupled neuron oscillator system, comprising the following steps: (S01): Construct a cross-frequency nonlinear coupled Poincaré model with exogenous input connection mode; or in, x f ,y f is the variable of high-frequency oscillating neurons, x s ,y s is the variable of low-frequency oscillating neurons, ω f,s and a f =a s =a are the natural frequencies and natural amplitudes of high-frequency and low-frequency oscillating neurons respectively, γ = 1 represents the relaxation parameter, is the amplitude of low- and high-frequency oscillating neurons. The interaction between high-frequency and low-frequency oscillating neurons is realized through multiplicative driving. In formula (1), high-frequency oscillating neurons are only affected by low-frequency oscillating neurons and will not reversely affect low-frequency oscillating neurons, where ε s represents the driving coupling strength of the low-frequency oscillating neuron on the high-frequency oscillating neuron oscillator. In formula (2), the low-frequency oscillating neuron is only affected by the high-frequency oscillating neuron and will not reversely affect the high-frequency oscillating neuron, where ε f It represents the driving coupling strength of high-frequency oscillating neurons on low-frequency oscillating neurons; (S02) The cross-frequency nonlinear coupling Poincaré model (1) constructed in (S01) is numerically solved by the fourth-order Runge-Kutta method to study how the frequency of high-frequency oscillating neurons is regulated by the phase of low-frequency oscillating neurons. The cross-frequency nonlinear coupling Poincaré model (2) constructed in (S01) is numerically solved to study how the frequency of high-frequency oscillating neurons affects the phase of low-frequency oscillating neurons. (S03): Construct the local interaction nonlinear coupling Poincaré model as follows: Among them, the subscripts f and s represent high-frequency and low-frequency neurons respectively, and ε f ,ε s are the coupling strengths of high frequency to low frequency and low frequency to high frequency, respectively, x f ,y f is the variable of high-frequency oscillating neurons, x s ,y s is the variable of low-frequency oscillating neurons, ω fs and a f =a s =a are the natural frequencies and natural amplitudes of high-frequency and low-frequency oscillating neurons respectively, γ = 1 represents the relaxation parameter, is the amplitude of low- and high-frequency oscillating neurons; (S04): The cross-frequency nonlinear coupling Poincaré model (3) constructed in (S03) is numerically solved by using the fourth-order Runge-Kutta method. The instantaneous frequency change amplitude Δf is calculated by Fourier transforming the time series to quantify the frequency change of neuronal oscillations, and the regulatory effect of the phase change of low-frequency oscillating neurons on the frequency of high-frequency oscillations is further analyzed. (S05): Based on the exogenous input model (1) constructed in (S01), (1) is converted into polar coordinate form by x = ρcosθ, y = ρsinθ (see formula (4),) and the relationship between the frequency variation amplitude and driving force intensity of high-frequency oscillating neurons is theoretically analyzed; Among them, ε s is the driving force of low-frequency oscillating neurons on high-frequency neurons, ρ f ,ρ s are the amplitudes of high-frequency and low-frequency oscillations, respectively, and a is the initial amplitude parameter of high-frequency and low-frequency neurons; We can get θ s =ω s t, so Among them, the instantaneous frequency of high-frequency oscillation is Right now, By |cos(ω s t)|≤1, the frequency variation amplitude of the high-frequency oscillation neuron is, The amplitude of the low-frequency oscillating neuron ρ s Can be ordered Solving for ρ s =a, and substituting it into equation (6), we can get the frequency variation amplitude of high-frequency oscillation neurons as, Therefore, the instantaneous frequency of the high-frequency oscillating neuron is modulated by the low-frequency oscillating neuron, and its variation amplitude Δf is only affected by the inherent amplitude a of the low-frequency neuron and the coupling effect ε s impact.

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