Phase-frequency synchronization implementation method based on coupled neuron oscillator system

Abstract

The invention discloses a phase-frequency synchronization implementation method based on a coupled neuron oscillator system. The method comprises the following steps: firstly, establishing two connection structures of exogenous input and local interaction of a Poincare model to construct a cross-frequency nonlinear coupling model, and introducing nonlinear coupling to realize phase-frequency synchronization of coupled neuron oscillators; and determining the influence of the coupling effect strength between the low-frequency neurons and the high-frequency neurons on phase-frequency synchronization by adopting a spectrum analysis method. And determining the influence of the low-frequency oscillation phase on the high-frequency oscillation frequency and the mechanism thereof based on theoretical analysis.

Description

Technical Field

[0001] The present invention relates to the field of neural network synchronization, in particular to the research on cross-frequency coupling and phase-frequency synchronization behavior of neuron oscillator systems. This technology can be applied to fields such as neuroscience, electroencephalogram analysis, and neuromodulation. Background Art

[0002] The collective behavior of coupled neurons with different frequencies is closely related to the functional realization of the brain. In the cerebral cortex, neuronal oscillations of different frequencies coexist and interact with each other, forming different forms of brain waves such as alpha waves (8 - 13 Hz), beta waves (13 - 30 Hz), gamma waves (30 - 80 Hz), theta waves (4 - 7 Hz), etc. Neurons in different regions of the brain generate neural oscillations of multiple frequencies, and there are complex interactions among their phases, amplitudes, and frequencies. Understanding the various forms of synchronization formed under these complex interactions is crucial for understanding the cognitive functions and disease pathological mechanisms of the brain. Among them, cross-frequency coupling refers to the phenomenon that there are mutual regulation and interaction between neural oscillations in different frequency ranges, often manifested as phase-amplitude coupling, such as the coupling between theta-gamma and alpha-gamma waves, phase-phase coupling, and phase-frequency coupling. The cross-frequency coupling phenomenon is not only related to normal cognitive functions but also closely related to the generation mechanisms of various brain diseases, especially playing an important role in understanding the generation mechanisms of many brain diseases such as epilepsy, Parkinson's disease, and schizophrenia.

[0003] Common manifestations of cross-frequency coupling include phase-amplitude coupling and phase-frequency coupling. Among them, 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 [such as Figure 1As shown in (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 cooperation between different frequency oscillations 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 a coupled neuron system where the oscillation frequency of high-frequency oscillating neurons is regulated by the phase of low-frequency oscillations, and it plays a key role in brain function realization such as cognitive processes. There are significant differences in their manifestations and measurement methods in neural signals. Phase-amplitude coupling is usually measured by extracting the amplitude of high-frequency oscillation signals and examining the relationship between it and the phase of low-frequency oscillations. Phase-frequency coupling focuses more on the phenomenon of low-frequency phase regulating the frequency of high-frequency oscillations rather than amplitude modulation. This phase-frequency modulation phenomenon was observed in a dense-spiking pyramidal-interneuron gamma high-frequency oscillation network in a noise-free environment, which covered continuous or intermittent cross-frequency coupling settings. In the minimal pyramidal-interneuron gamma model modulated by low-frequency oscillations, when the low-frequency oscillation input reaches its peak, it will cause the high-frequency oscillation to accelerate regardless of whether the input target is a pyramidal cell or an inhibitory cell. Although the research on phase-frequency coupling is still in its infancy, it frequently appears in strongly coupled neural networks, especially more significantly between low-frequency and high-frequency oscillations under strong coupling. In these networks, the phase of low-frequency oscillations may not only regulate the amplitude of high-frequency oscillations but also have a significant impact on their frequency. Such coupling effects play an important role in different neural activity patterns, especially in brain function processes that require precise regulation of the interaction between different frequency oscillations.

[0004] In brain signals recorded by various neuroimaging techniques such as electroencephalogram (EEG) and intracranial recordings, researchers have found that phase-frequency coupling widely exists in multiple neural networks of the brain. Phase-frequency coupling not only reveals the interaction between different frequency oscillations in the brain but also provides important clues for in-depth understanding of the mechanisms of neural activity coordination and information transmission. Therefore, it is of great significance to understand more deeply the application of the phase-frequency coupling mechanism generated between different brain regions of the human brain in brain function.

[0005] Construct a cross-frequency coupling model through two main connection structures of exogenous input and local interaction in the Poincaré model, and analyze the phase-frequency coupling in the brain neural network. Under the action of exogenous input, by introducing frequency regulation, the coupling between the phases of low-frequency neurons and the high-frequency oscillation frequency can be effectively excited, and this phenomenon is more significant under strong coupling conditions. In addition, under the local interaction structure, the interaction between neurons endows the phase-frequency coupling with new manifestations. The strengthening of local connections not only helps the generation of the phase-frequency coupling phenomenon, but also significantly enhances the influence of the low-frequency oscillation phase on the high-frequency oscillation frequency. These results provide important theoretical support for understanding the mechanism of cross-frequency coupling in the brain neural network and offer a new perspective for exploring the dynamic characteristics of brain functions. Summary of the Invention

[0006] 1. Object of the Invention

[0007] Construct a cross-frequency coupling model by establishing two connection structures of exogenous input and local interaction in the Poincaré model to achieve phase-frequency synchronization of coupled neuron oscillators. And determine the influence of the coupling strength of neurons with different frequencies on phase-frequency synchronization through spectral analysis, and determine the influence of the low-frequency oscillation phase on the high-frequency oscillation frequency and its mechanism based on theoretical analysis.

[0008] 2. Technical Solution

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

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

[0011] (S01): Construct a cross-frequency coupling Poincaré model with an exogenous input connection method, introduce a non-linear coupling effect, and simulate the synchronous dynamic behavior of neuron oscillators in the brain under the drive of exogenous signals. Respectively determine the phase-frequency synchronization of high-frequency oscillation neurons under the drive of low-frequency oscillation neurons, and the phase-frequency synchronization of low-frequency oscillation neurons under the drive of high-frequency oscillation neurons.

[0012] (S02): Numerically solve the model constructed in (S01) using the fourth-order Runge-Kutta method, and respectively determine the law of the regulation of the frequency of high-frequency oscillation neurons by the phase of low-frequency oscillation neurons, and the law of the regulation of the frequency of low-frequency oscillation neurons by the phase of high-frequency oscillation neurons.

[0013] (S03): Construct a cross-frequency non-linear coupling Poincaré model with a local interaction connection method, and simulate the synchronous dynamic behavior of high-frequency oscillation neuron oscillators and low-frequency oscillation neuron oscillators in the brain under their interaction.

[0014] (S04): Numerically solve the model constructed in (S03) using the fourth-order Runge-Kutta method, and respectively determine the synchronization between the phase of the low-frequency oscillating neuron oscillator and the frequency of the high-frequency oscillating neuron under the interaction of the coupling strengths between different high- and low-frequency oscillating neurons.

[0015] (S05): Based on the exogenous input model, theoretically analyze and give the relationship between the frequency of the high-frequency oscillating neuron and the phase of the driving signal, and further determine the functional relationship between the range of high-frequency oscillation frequency changes and the driving strength.

[0016] The specific implementation steps are as follows:

[0017] The first step: Construct a cross-frequency nonlinear coupling Poincaré model with an exogenous input connection method,

[0018]

[0019] Or

[0020]

[0021] where x f , y f are the variables of the high-frequency oscillating neuron, x s , y s are the variables of the low-frequency oscillating neuron, ω f,s and a f = a s = a are respectively the natural frequencies and natural amplitudes of the high- and low-frequency oscillating neurons, γ = 1 represents the relaxation parameter, is the amplitude of the low- and high-frequency oscillating neurons. The interaction between the high- and low-frequency oscillating neurons is achieved through multiplicative driving. In equation (1), the high-frequency oscillating neuron is only affected by the low-frequency oscillating neuron and does not affect the low-frequency oscillating neuron in reverse, where ε s represents the driving coupling strength of the low-frequency oscillating neuron on the oscillator of the high-frequency oscillating neuron. In equation (2), the low-frequency oscillating neuron is only affected by the high-frequency oscillating neuron and does not affect the high-frequency oscillating neuron in reverse, where ε f represents the driving coupling strength of the high-frequency oscillating neuron on the oscillator of the low-frequency oscillating neuron.

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

[0023] Step 3: Construct the local interaction non-linear coupling Poincaré model as follows:

[0024]

[0025] where 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 are the variables of high-frequency oscillatory neurons, and x s , y s are the variables of low-frequency oscillatory neurons. ω fs and a f = a s = a are the natural frequencies and natural amplitudes of high-frequency and low-frequency oscillatory neurons respectively. γ = 1 represents the relaxation parameter, and are the amplitudes of low-frequency and high-frequency oscillatory neurons.

[0026] Step 4: Numerically solve the cross-frequency non-linear coupling Poincaré model equation (3) constructed in (S03) by using the fourth-order Runge-Kutta method, and calculate the instantaneous frequency change amplitude Δf to quantify the frequency change of neuron oscillation by performing Fourier transform on the time series. Further analyze the regulation effect of the phase change of low-frequency oscillatory neurons on the high-frequency oscillation frequency.

[0027] Step 5: Based on the exogenous input model equation (1) constructed in Step 1, transform equation (1) into polar coordinate form as shown in equation (4) by x = ρcosθ, y = ρsinθ, and theoretically analyze the relationship between the frequency change amplitude of high-frequency oscillatory neurons and the driving strength.

[0028]

[0029] where ε s is the driving strength of low-frequency oscillatory neurons on high-frequency neurons, ρ f , ρ s are the amplitude quantities of high-frequency and low-frequency oscillations respectively, and a is the initial amplitude parameter of high-frequency and low-frequency neurons. From we can obtain θ s = ω s t, so where the instantaneous frequency of high-frequency oscillation is That is,[

[0030]

[0031] From |cos(ω s|t)| ≤ 1 gives the amplitude of the frequency variation of the high-frequency oscillating neuron as

[0032]

[0033] where the amplitude ρ of the low-frequency oscillating neuron s Let Solving for ρ s = a, substituting it into equation (6), the amplitude of the frequency variation of the high-frequency oscillating neuron can be obtained as

[0034]

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

[0036] 3. Advantageous effects:

[0037] The present invention can effectively simulate the cross-frequency coupling between brain neurons through a simple mathematical model, and generate a phase-frequency synchronization phenomenon between the low-frequency oscillating neuron and the high-frequency oscillating neuron. The regulation mechanism of the low-frequency oscillation phase on the 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 regions of the brain. By introducing polar coordinates into the model, it can be theoretically analyzed that the frequency of the high-frequency oscillating neuron is regulated by the low-frequency oscillation phase, and the functional relationship between its amplitude of frequency variation Δf and the coupling strength ε s of the low-frequency oscillating neuron and the inherent amplitude a satisfies Δf = ε s a / π, and it can be obtained that Δf increases with the increase of ε s as well as a. Description of the drawings

[0038] Figure 1 When the low-frequency signal is used as the exogenous input signal, the phase-frequency coupling between the low-frequency signal and the high-frequency signal; (a) Time series of the low-frequency neuron oscillation; (b) Time series of the driven high-frequency neural oscillation when ε s = 2; (c) Time series of the driven high-frequency neuron oscillation when ε s = 4; (d) Spectrum diagram of the high-frequency neuron oscillation; (e) Variation relationship of the amplitude of the high-frequency neuron frequency with the driving coupling strength ε s .

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

[0040] Figure 3 ε f and ε s Amplitude of the instantaneous frequency change of neurons in the parameter space. (a) Amplitude of the instantaneous frequency change of high-frequency oscillating neurons versus the coupling strength ε f and ε s variation. (b) Amplitude of the instantaneous frequency change of low-frequency oscillating neurons as the coupling strength ε f and ε s varies.

[0041] Figure 4 Phase-frequency synchronization under local interaction. (a-c) Time series of low-frequency oscillation; (d-f) Time series of high-frequency oscillation; (g-i) Spectrograms; 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 Function relationship curve between the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO). (a) When ε f and ε s are both zero, the function relationship between the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO). (b) When ε f = 0 and ε s = 2, the function relationship between the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO) when there is only a unidirectional influence of the low-frequency oscillation phase on the high-frequency oscillation frequency in the system. (c) When ε f = ε s = 2, the function relationship between the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO). (d) When ε f = 2, ε s = 8, the function relationship between the instantaneous frequency of fast oscillation (FO) and the phase of slow oscillation (SO). The reaction of the high-frequency frequency on the low-frequency oscillation phase is more obvious, further resulting in a change in the oscillation amplitude.

[0043] Figure 6 (a) When a = 4, 6, Δf varies with the coupling strength ε sVariation relationship. (b) ε s When ε = 2, 3, the variation relationship of Δf with the natural amplitude a, where the solid line is the theoretical result and the dotted line is the numerical calculation result. Specific implementation mode

[0044] To clearly understand the purpose, features and advantages of the present invention, the following introduces specific implementation examples and makes further detailed descriptions as follows:

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

[0046] Based on Equation (3), taking parameters a = 6, ω s = 3 rad / s, ω f = 30 rad / s, ε f = 0, ε s = 2, Equation (3) is for exogenous signal input. At this time, the low-frequency signal ( Figure 1 (as shown in (a)) drives the high-frequency signal. Under the action of the low-frequency oscillation neuron, the amplitude of the high-frequency oscillation does not change due to the low-frequency oscillation and remains the initial amplitude value a = 6 ( Figure 1 (as shown in (b))), while its oscillation frequency changes periodically with the phase change of the low-frequency oscillation, and the change period corresponds one-to-one with the phase change period of the low-frequency oscillation, that is, the frequency of the high-frequency oscillation changes correspondingly with the phase change of the low-frequency oscillation, resulting in phase-frequency coupling. As the coupling strength increases, such as ε s = 4 ( Figure 1 (as shown in (c))), within a phase change period (0 - 2π) of the low-frequency oscillation, the frequency fluctuation range of the high-frequency oscillation increases, and its time series shows a greater distinction between the dense area and the sparse area. This indicates that as the coupling strength increases, the regulation effect of the low-frequency oscillation on the high-frequency oscillation frequency becomes more significant, and the frequency regulation interval is wider. From the spectrograms of the high-frequency oscillation neuron when the coupling strength ε s = 2, 4, it can be seen that when ε s = 2, under the drive of the low-frequency oscillation neuron, the main frequency component of the high-frequency oscillation neuron broadens from the original 4.78 Hz to 2.8 - 6.7 Hz, as shown in Figure 1 (d). When the coupling strength ε s = 4, the frequency range of the spectrum significantly broadens to 0.95 - 8.6 Hz, as shown in Figure 1(d) shows that the frequency broadening effect is more obvious, indicating that the frequency regulation effect of the low-frequency oscillation phase change on the high-frequency oscillation becomes more significant. To determine the influence of the coupling strength of low-frequency oscillation neurons on phase-frequency coupling, the frequency change amplitudes of the high-frequency components are calculated respectively when the coupling strength ε = 0 - 4, so as to further quantify the influence of the low-frequency oscillation phase component on the high-frequency oscillation frequency regulation. Perform Hilbert transform on the high-frequency oscillation neuron signals under different coupling strengths, and extract their instantaneous frequencies to calculate the change amplitude of the instantaneous frequency (the difference between the maximum value and the minimum value of the instantaneous frequency) as shown in Figure 1 (e). The results show that with the increase of the coupling strength, the change amplitude of the instantaneous frequency shows a linear growth trend. For example, when the coupling strength increases from 1 to 4, the frequency change amplitude increases from 3.8 Hz to 7.6 Hz. When the coupling strength ε s = 0, the high-frequency oscillation neurons show single-frequency oscillation, and the change amplitude of its frequency is 0. When the coupling strength is ε s = 2, the change amplitude of the instantaneous frequency of the high-frequency oscillation is about 3.8 Hz, while when ε s = 4, the change amplitude of the instantaneous frequency of the high-frequency oscillation is about 7.6 Hz.

[0047] Based on equation (3), the low-frequency oscillation neurons are driven by the high-frequency oscillation neurons (as shown in Figure 2 (a)) (ε s = 0, ε f > 0), and the low-frequency oscillation time series will be superimposed with the high-frequency oscillation time series in time sequence. When the coupling strength ε f = 2, the superimposed high-frequency oscillation amplitude is small, as shown in Figure 2 (b), while when ε s = 4, the amplitude of the high-frequency oscillation increases, as shown in Figure 2 (c). Figure 2 (d) gives the spectrum of the high-frequency oscillation neurons, whose main frequency is 4.78 Hz. Figure 2 (e)(f) respectively give the spectra of the driven low-frequency neurons when ε s = 2, 4, where the dotted line is the spectrum of the high-frequency driving signal. The main frequency of the low-frequency neurons is 0.478 Hz, and new frequencies of 4.3 Hz and 5.178 Hz are also generated. With the increase of the coupling strength, the main frequency peak of the driven low-frequency neurons decreases, and the newly generated secondary peaks increase. The low-frequency neurons are modulated by the high-frequency oscillation neurons.

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

[0049] Based on equation (3), considering the local interaction between the high-frequency and low-frequency neuron oscillators (ε s > 0, ε f> 0), the variation amplitude of the instantaneous frequency of the high-frequency oscillating neurons (low-frequency oscillating neurons) is calculated separately with respect to the two local coupling strengths ε f and ε s to obtain the relationship diagrams as shown in Figure 3 (a) ( Figure 3 (b)). As the driving coupling strength ε s of the low-frequency oscillating neurons on the high-frequency oscillating neurons increases, the variation amplitude of the oscillation frequency of the high-frequency neurons gradually increases. However, the increase in the driving coupling strength ε f of the high-frequency neurons on the low-frequency neurons has a relatively small impact on the oscillation frequency of the high-frequency neurons. When ε s increases to a certain value, the modulation effect of the low-frequency oscillation on the high-frequency oscillation dominates. As ε s gradually increases, the variation amplitude of the instantaneous frequency also increases. When ε f and ε s both increase to a certain critical value, the variation amplitudes of the instantaneous frequencies of the high-frequency and low-frequency start to gradually decrease, showing the characteristic of the system tending to be stable ( Figure 3 the upper right part in (a)(b)). This phenomenon indicates that under strong coupling, the interactions within the network generate an internal constraint mechanism, making the variation of the high-frequency oscillation frequency tend to be stable. Figure 3 (b) further shows how the instantaneous frequency amplitude of the low-frequency oscillating neurons is modulated by ε f and ε s . Different from the high-frequency oscillation, under this condition, ε f acts as the main control factor, dominating the variation trend of the instantaneous frequency amplitude of the low-frequency oscillation. For a given value of ε s , when ε f increases to a certain value, the variation amplitude of the low-frequency oscillation frequency suddenly increases, and this variation amplitude is much larger than that of the low-frequency oscillation frequency. Similar to the result of Figure 4 (a), when ε f and ε s simultaneously increase to a certain threshold, the instantaneous frequency amplitude of the low-frequency oscillation also starts to gradually decrease ( Figure 3 the upper right part in (b)), also showing the stability of the system. This phenomenon indicates that whether it is high-frequency or low-frequency oscillation, when the coupling strength exceeds a certain critical point, the network tends to synchronize and reach a steady state, thus reducing frequency fluctuations.

[0050] The phase-frequency coupling phenomenon occurs when the interaction between two neuron oscillators is relatively small. The frequency of the high-frequency oscillation is not completely synchronized with the low-frequency oscillation period, but is regulated at certain specific phase positions, showing a certain coupling relationship between the frequency fluctuation and the phase of the low-frequency oscillation. When ε f = 2, εs When = 2, the effects of the two neuronal oscillations are mutual. The phase of the low-frequency oscillation affects the frequency fluctuations of the high-frequency oscillation, and at the same time, the frequency fluctuations of the high-frequency oscillation also have a feedback effect on the oscillation fluctuations of the low-frequency neurons. It can be seen that within the period of the low-frequency oscillation, the low-frequency phase acts on the frequency components of the high-frequency oscillation, and at the same time, 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 this mutual coupling effect, indicating that the interaction between the high-frequency and low-frequency oscillations is not only unidirectional, but a complex bidirectional regulation process. As the coupling strength continuously increases, the interaction between the high-frequency and low-frequency oscillations in the system gradually strengthens, showing more significant coupling characteristics. This enhanced interaction promotes the generation of partial phase synchronization phenomena, such as Figure 4 (b, e, h) shown. It can be clearly seen from the spectral analysis that a corresponding relationship gradually appears between the different frequency components of the high-frequency and low-frequency oscillations, indicating that the coupling between the two is not independent or unidirectional, but there is a complex bidirectional regulation mechanism. As the coupling strength further increases, the interaction between the high-frequency and low-frequency oscillations in the system continues to strengthen, and finally leads to the generation of the phase locking phenomenon, as shown in the results of Figure 4 (c, f, i). When the coupling strength reaches a certain threshold, the phases of the high-frequency and low-frequency oscillations gradually tend to be closely synchronized, forming a stable phase locking state. This state indicates that the coupling effect between the two oscillations reaches a new balance, and the dynamic behavior transitions from a complex bidirectional modulation to a stable synchronization mode. This phenomenon can be further verified through spectral analysis. When the coupling strength exceeds the threshold, the frequency components of the high-frequency and low-frequency oscillations show clear synchronization characteristics, and the originally scattered frequency components in the spectrum gradually concentrate and align. This frequency alignment behavior clearly reveals how the coupling mechanism between the high-frequency and low-frequency oscillations promotes their frequency and phase synchronization during the phase locking process.

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

[0052] The internal mechanism and characteristics of the influence of the low-frequency oscillation phase on the high-frequency oscillation frequency can be revealed by constructing the response curve of the high-frequency oscillation. Figure 5 Shows the corresponding relationship between the frequencies and phases of the two oscillating neurons under different ε f and ε s conditions. The specific situation is as follows:

[0053] 1) No interaction: When both ε f and ε s 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. Therefore, there is no phase-frequency coupling.

[0054] 2) Influence of only the low-frequency oscillation phase on the high-frequency oscillation frequency: When ε f = 0 and ε s = 2, there is only a unidirectional influence of the low-frequency oscillation phase on the high-frequency oscillation frequency in the system. At this time, the phase of the low-frequency oscillation regulates the frequency of the high-frequency oscillation, resulting in periodic fluctuations in the instantaneous frequency of the high-frequency oscillation as the phase of the low-frequency oscillation increases (as shown in Figure 5 (b)). At this time, the coupled system exhibits a phase-frequency coupling gait, 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 low-frequency oscillation phase and the high-frequency oscillation frequency becomes more complex. When ε f = ε s = 2, as shown in Figure 5 (c), the instantaneous frequency of the high-frequency oscillation is generally modulated by the low-frequency oscillation phase, and there are also perturbations when there is a certain reaction of the high-frequency frequency on the low-frequency oscillation phase. 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 perturbations; when ε f = 2, ε s = 8, the reaction of the high-frequency frequency on the low-frequency oscillation phase is more obvious, further resulting in a change in the oscillation amplitude, as shown in Figure 5 (d).

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 of exogenous input connection mode; or in, x f ,y f is the variable of the high-frequency oscillating neuron, 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), 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 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 by (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 by (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, and x f ,y f is the variable of the high-frequency oscillating neuron, 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 the fourth-order Runge-Kutta method, and 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 high-frequency oscillation frequency 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 as shown in (4), and the relationship between the frequency change 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 change amplitude of the high-frequency oscillating neuron 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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