Nerve regulation and control intervention system based on time domain interference electrical stimulation

Abstract

The invention discloses a nerve regulation and control intervention system based on time domain interference electrical stimulation, and the system comprises a magnetic resonance image registration module which receives and registers an imported magnetic resonance image, and obtains a to-be-processed MRI image; the head model tissue segmentation module is used for performing head model tissue segmentation on the to-be-processed MRI image; the stimulation simulation module is used for performing stimulation simulation based on the tissue segmentation structure to obtain a front lead field matrix; the intervention coordinate selection module is used for determining a target brain region corresponding to a target spot brain region coordinate position needing stimulation intervention and putting the target brain region into a running sequence; the electrode position arrangement optimization module is used for verifying, optimizing and finely adjusting the electric field intensity of the target brain region under different electrode arrangement schemes to obtain an electrode configuration parameter result and displaying the electrode configuration parameter result; the parameter importing module is used for receiving the selected electrode configuration parameters and imported preset electrical stimulation parameters; and the electrical stimulation module is used for performing electrical stimulation on the target brain region of the target object. According to the application, the accuracy and the effectiveness of nerve regulation and control treatment can be realized.

Description

Technical Field

[0001] This application relates to the field of neural technology, and particularly to a neural regulation intervention system based on time-domain interference electrical stimulation. Background Art

[0002] In the fields of neuro-medicine and brain science research, precise neural regulation technology is crucial for treating neurological diseases and studying brain functions. Especially in the field of deep brain stimulation (DBS), the demand for non-invasive technologies is increasing to avoid the risks and side effects of traditional invasive surgeries. A typical application scenario is the treatment of Parkinson's disease, which causes movement control disorders due to the loss of dopamine neurotransmitters in the brain. By non-invasively stimulating specific regions deep in the brain, the symptoms of patients can be significantly improved.

[0003] In the prior art, as an emerging non-invasive neural regulation means, time-domain interference electrical stimulation (TI) technology generates a low-frequency envelope current in the target area by using two high-frequency currents with slightly different frequencies to achieve the stimulation of deep brain regions. The advantage of this method is that it can reduce the interference to surface tissues while precisely affecting deep brain tissues. In addition, finite element analysis (FEA) technology is used to simulate and predict the distribution of TI currents in the brain to optimize the stimulation parameters.

[0004] However, existing TI software usually has a complex interface and numerous operation steps. It not only requires users to have high professional knowledge but also greatly reduces the efficiency of practical applications. Secondly, the adjustment of current parameters is not precise enough. Especially when setting high-frequency currents and adjusting the frequency difference, it is very difficult to make a certain small adjustment to the parameters to slightly shift the target stimulation area, which limits the accuracy of treatment. Summary of the Invention

[0005] Embodiments of this application provide a neural regulation intervention system based on time-domain interference electrical stimulation. To have a basic understanding of some aspects of the disclosed embodiments, a simple summary is given below. This summary part is not a general review, nor is it to identify key / important constituent elements or delineate the protection scope of these embodiments. Its sole purpose is to present some concepts in a simple form as a prelude to the subsequent detailed description.

[0006] In a first aspect, embodiments of this application provide a neural regulation intervention system based on time-domain interference electrical stimulation, the system comprising: A magnetic resonance image registration module, configured to receive and register the magnetic resonance image of a target object imported into the client to obtain a to-be-processed MRI image; A head model tissue segmentation module, configured to perform head model tissue segmentation on the to-be-processed MRI image by using a Gaussian mixture model and a Markov random field model to obtain the tissue segmentation structure of the head model; A stimulation simulation module, configured to perform time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge based on the tissue segmentation structure of the head model, so as to obtain the forward lead field matrix of the head model; An intervention coordinate selection module, configured to determine the target brain region coordinate position that needs to be stimulated and intervened in response to a position selection instruction for the to-be-processed MRI image, and put the target brain region corresponding to the target brain region coordinate position into the running sequence; An electrode position arrangement optimization module, configured to verify and optimize the electric field intensity of the target brain region under different electrode arrangement schemes based on the forward lead field matrix, in combination with the convex optimization methods of the genetic algorithm and the particle swarm optimization algorithm, so as to determine the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs and display them, and in response to a fine-tuning instruction for the displayed information, fine-tune the displayed information to obtain and display the electrode configuration parameter result; A parameter import module, configured to receive the selected electrode configuration parameters and the imported preset electrical stimulation parameters as the configured stimulation parameters in response to a parameter selection instruction for the displayed electrode configuration parameter result; An electrical stimulation module, configured to output waveforms for the configured stimulation parameters through a communication protocol and a hardware device in response to an electrical stimulation instruction, so as to perform electrical stimulation on the target brain region of the target object.

[0007] Optionally, the system further includes: An initialization and user login module, configured to load and initialize computing resources and visualization components into the memory; receive a user login request, and perform verification based on the username and password carried in the login request; after successful verification, display the actual operation page of the software; A hardware device connection module, configured to automatically scan the port numbers of the currently connected serial port devices; establish a connection with them according to the scanned port numbers of the serial port devices; after the connection is successfully established, periodically send a heartbeat request based on the communication protocol to the serial port devices; when receiving a heartbeat packet returned by the hardware device, display a parameter configuration interface; A data storage and reuse module, configured to store the tissue segmentation structure of the head model and the configured stimulation parameters when the electrical stimulation is completed for reuse during the next electrical stimulation.

[0008] Optionally, use a Gaussian mixture model and a Markov random field model to perform head model tissue segmentation on the to-be-processed MRI image to obtain the tissue segmentation structure of the head model, including: Adopt a Gaussian mixture model to perform probability modeling on the gray value of each pixel in the to-be-processed MRI image to obtain the probability distribution of the pixel gray value; Initialize the parameter set of the Gaussian mixture model, where the parameter set includes mixing coefficients, means, and variances; According to the probability distribution of pixel gray values and combined with the EM algorithm, calculate the posterior probability that each pixel in the MRI image to be processed belongs to each Gaussian component; According to the posterior probability and the Markov random field model, generate a three-dimensional model after tissue segmentation and an MRI model with segmentation labels as the tissue segmentation structure of the head model.

[0009] Optionally, according to the posterior probability and the Markov random field model, generating a three-dimensional model after tissue segmentation and an MRI model with segmentation labels includes: Update the mixing coefficient, mean, and variance according to the posterior probability; Continue to execute the step of calculating the posterior probability that each pixel in the MRI image to be processed belongs to each Gaussian component according to the probability distribution of pixel gray values and combined with the EM algorithm. When the parameter update no longer changes, the parameter iteration ends. Based on the current posterior probability that each pixel belongs to each Gaussian component, assign each pixel to a tissue type to obtain the label category classification based on GMM; Use the Markov random field to smooth the label category classification based on GMM, and process it based on the data term and smooth term included in the energy function to obtain a three-dimensional model after tissue segmentation and an MRI model with segmentation labels; where the data term measures the matching degree between the pixel label and its gray value, and the smooth term encourages adjacent pixels to have the same label.

[0010] Optionally, based on the tissue segmentation structure of the head model, perform time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge to obtain the forward lead field matrix of the head model, including: Mesh the tissue segmentation structure of the head model to create a mesh model composed of nodes and elements; Construct a conductivity stiffness matrix according to the nodes and node distances in the mesh model; Use the conductivity stiffness matrix and the preset control equation for finite element simulation to calculate the steady-state current distribution of the meshed head model; Based on the steady-state current distribution of the head model, perform stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge to obtain the forward lead field matrix of the head model.

[0011] Optionally, the preset control equation for the steady-state current distribution is the Laplace equation or the Poisson equation; where, The Laplace equation or the Poisson equation is: , where, is the conductivity distribution, is the potential distribution, is the current source density, is a gradient operator, which is a vector differential operator used to represent the gradient of a scalar function or the divergence of a vector field in multivariable calculus.

[0012] Optionally, based on the steady-state current distribution of the head model, stimulation simulation is performed at each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge, and the forward lead field matrix of the head model is obtained, including: During the process of performing stimulation simulation at each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge, on each finite element cell, a shape function is defined according to the steady-state current distribution; The potential and test function in the stimulation simulation process are expanded using the shape function to obtain the algebraic equation at the element level; For the elements in the mesh model, the stiffness matrix is obtained; The stiffness matrices of all elements are assembled into a global stiffness matrix, and in the assembly process, the stiffness matrix is mapped to the corresponding position of the global matrix according to the global numbering of the nodes; According to the algebraic equation at the element level and the global stiffness matrix, the electric field and current density are calculated; Based on the electric field and current density, a forward lead field matrix is constructed, and the forward lead field matrix describes the relationship between the current source and the potential measurement.

[0013] Optionally, the algebraic equation at the element level includes:

[0014] Among them, is the stiffness matrix, is the potential vector of the element nodes, is the element load vector.

[0015] Optionally, the magnetic resonance image of the target object imported into the client is received and registered to obtain the MRI image to be processed, including: Receiving the magnetic resonance image of the target object imported into the client; Based on a preset spatial transformation matrix, format conversion and spatial pose adjustment are performed on the magnetic resonance image; Taking the adjusted magnetic resonance image as the MRI image to be processed.

[0016] Optionally, the probability distribution calculation formula of the pixel gray value is:

[0017] Among them, is the probability distribution of the gray value of the th pixel in the MRI image to be processed, is the The gray - scale value of a pixel, is the total number of all pixels, is the number of Gaussian components in the Gaussian mixture model, is the mixing coefficient of the -th Gaussian component, satisfying is the mean of the -th Gaussian component, is the variance of the is a Gaussian distribution with a mean of and a variance of .

[0018] In the embodiments of the present application, on the one hand, through a magnetic resonance image registration module, a head - model tissue segmentation module, a stimulation simulation module, etc., an automated and simplified operation process is realized. The user only needs to perform simple instruction input to complete the whole process from the import, registration, tissue segmentation of magnetic resonance images to stimulation simulation. This simplified operation method greatly reduces the dependence on the user's professional knowledge, enabling even non - professionals to quickly get started, thus significantly improving the efficiency of practical applications. The electrode position arrangement optimization module and the parameter import module further simplify the process of electrode configuration and parameter setting, making the entire treatment preparation process faster and more accurate. On the other hand, through the electrode position arrangement optimization module, combined with the convex optimization method of genetic algorithm and particle swarm optimization algorithm, the electric field intensity of the target brain region under different electrode arrangement schemes is verified and optimized. This optimization method based on the forward lead - field matrix can accurately calculate the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs, ensuring the precise adjustment of current parameters. The electrical stimulation module responds to the electrical stimulation instruction and outputs waveforms for the configured stimulation parameters through a communication protocol and hardware devices to achieve precise electrical stimulation of the target brain region. This precise adjustment of current parameters and electrical stimulation output significantly improves the precision of treatment, thereby enhancing the treatment effect and safety.

[0019] It should be understood that the above general description and the following detailed description are only exemplary and explanatory, and cannot limit the present application. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The drawings herein are incorporated into the specification and constitute a part of this specification, showing embodiments consistent with the present application and used together with the specification to explain the principles of the present application.

[0021] Figure 1 is a schematic structural diagram of a neuro - regulation intervention system based on time - domain interference electrical stimulation provided by an embodiment of the present application; Figure 2It is a schematic structural diagram of another neuroregulation intervention system based on time-domain interference electrical stimulation provided by an embodiment of the present application; Figure 3 It is a schematic flowchart of a neuroregulation intervention method based on time-domain interference electrical stimulation provided by an embodiment of the present application. Detailed implementation manners

[0022] The following description and drawings fully illustrate the specific implementation manners of the present application so that those skilled in the art can practice them.

[0023] It should be clear that the described embodiments are only a part of the embodiments of the present application, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present application without creative efforts belong to the scope of protection of the present application.

[0024] When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The implementation manners described in the following exemplary embodiments do not represent all implementation manners consistent with the present application. On the contrary, they are only examples of systems consistent with some aspects of the present application as detailed in the appended claims.

[0025] In the description of the present application, it should be understood that terms such as "first", "second", etc. are only used for descriptive purposes and cannot be construed as indicating or implying relative importance. For those of ordinary skill in the art, the specific meanings of the above terms in the present application can be understood according to specific circumstances. In addition, in the description of the present application, unless otherwise specified, "a plurality of" means two or more. "And / or" describes the association relationship of associated objects and indicates that three relationships can exist. For example, A and / or B can represent: A exists alone, A and B exist simultaneously, and B exists alone. The character " / " generally represents an "or" relationship between the associated objects before and after.

[0026] Please refer to Figure 1 , Figure 1 It is a schematic structural diagram of a neuroregulation intervention system based on time-domain interference electrical stimulation provided by an embodiment of the present application. The system includes: a magnetic resonance image registration module, a head model tissue segmentation module, a stimulation simulation module, an intervention coordinate selection module, an electrode position arrangement optimization module, a parameter import module, and an electrical stimulation module. The magnetic resonance image registration module, the head model tissue segmentation module, the stimulation simulation module, the intervention coordinate selection module, the electrode position arrangement optimization module, the parameter import module, and the electrical stimulation module are communicatively connected.

[0027] In some embodiments of the present application, a magnetic resonance image registration module is configured to receive and register magnetic resonance images of a target object imported into a client to obtain MRI images to be processed; a head model tissue segmentation module is configured to perform head model tissue segmentation on the MRI images to be processed using a Gaussian mixture model and a Markov random field model to obtain a tissue segmentation structure of the head model; a stimulation simulation module is configured to perform time-domain interference electrical stimulation simulation on set electrode positions in the form of dipole discharge based on the tissue segmentation structure of the head model to obtain a forward lead field matrix of the head model; an intervention coordinate selection module is configured to determine the coordinate positions of the target brain regions that need to be stimulated and intervened in response to a position selection instruction for the MRI images to be processed, and place the target brain regions corresponding to the coordinate positions of the target brain regions into a running sequence; an electrode position arrangement optimization module is configured to verify and optimize the electric field intensity of the target brain regions under different electrode arrangement schemes based on the forward lead field matrix in combination with convex optimization methods of a genetic algorithm and a particle swarm optimization algorithm to determine the optimal placement positions of the electrodes and the amplitude parameters of the electrode pairs and display them, and perform fine-tuning on the displayed information in response to a fine-tuning instruction for the displayed information to obtain and display electrode configuration parameter results; a parameter import module is configured to receive the selected electrode configuration parameters and imported preset electrical stimulation parameters as configured stimulation parameters in response to a parameter selection instruction for the displayed electrode configuration parameter results; an electrical stimulation module is configured to output waveforms for the configured stimulation parameters through a communication protocol and a hardware device in response to an electrical stimulation instruction to electrically stimulate the target brain regions of the target object.

[0028] Among them, the client refers to the user terminal that operates the neuromodulation device, and this user terminal is electrically connected to the neuromodulation device. The magnetic resonance image is an image of the internal structure of the head obtained by using magnetic fields and radio waves. Registration refers to adjusting the image to ensure that image data from different sources is consistent in spatial position. The Gaussian mixture model (GMM) is a probability model used to model data with a mixture of multiple Gaussian distributions. The Markov random field (MRF) is a statistical model used to describe the spatial relationship between pixels in an image. Head model tissue segmentation refers to the process of separating different tissues (such as brain tissue, cerebrospinal fluid, bone, etc.) in head MRI images. A dipole is the discharge of a single electrode unit current, and the forward lead field matrix is the electric field distribution obtained based on the dipole discharge at each electrode position based on the electrode positions in the 10-10 system through finite element analysis. Based on the linear characteristics of current propagation, we can use this data for multiple rounds of iteration when using particle swarm optimization and genetic algorithms to obtain the electrode distribution that maximizes the target brain region. The position selection instruction is the specific brain region specified by the user where stimulation intervention is required. The target brain region coordinate position is the coordinate of the specific brain region to be stimulated in three-dimensional space. The operation sequence is the order of operations that need to be executed during the treatment process. The genetic algorithm is a search algorithm that simulates the process of natural selection and is used to solve optimization problems. The particle swarm optimization algorithm is an optimization technique based on group cooperation that finds the optimal solution by simulating the social behavior of bird flocks or fish schools. Convex optimization is a mathematical optimization method used to find the global minimum of a given function. The parameter selection instruction is the selection made by the user based on the displayed electrode configuration parameter results. The preset electrical stimulation parameters are the electrical stimulation parameters default in the system, such as current intensity, frequency, etc. The electrical stimulation instruction is the command to start the electrical stimulation process. The communication protocol is the rules and standards for data transmission between devices. The waveform output is the electrical stimulation signal generated according to the configured stimulation parameters.

[0029] Specifically, the specific process of receiving and registering the magnetic resonance image of the target object imported into the client to obtain the MRI image to be processed includes: receiving the magnetic resonance image of the target object imported into the client; based on the preset spatial transformation matrix, performing format conversion and spatial pose adjustment on the magnetic resonance image; and using the adjusted magnetic resonance image as the MRI image to be processed.

[0030] In some embodiments of the present application, for example Figure 2 as shown, the system further includes: an initialization and user login module, a hardware device connection module, and a data storage and reuse module. Among them, the initialization and user login module, the hardware device connection module, and the data storage and reuse module are communicatively connected.

[0031] Specifically, the initialization and user login module is used to load and initialize computing resources and visualization components into memory; receive user login requests, and verify them based on the user name and password carried in the login request; after successful verification, display the actual operation page of the software; the hardware device connection module is used to automatically scan the port numbers of currently connected serial devices; establish a connection with them according to the scanned port numbers of the serial devices; after the connection is successfully established, periodically send heartbeat requests based on the communication protocol to the serial devices; when receiving the heartbeat packet returned by the hardware device, display the parameter configuration interface; the data storage and reuse module is used to store the tissue segmentation structure of the head model and the configured stimulation parameters when the electrical stimulation is completed, for reuse during the next electrical stimulation.

[0032] Among them, computing resources refer to the data processing capabilities required for software operation, such as CPUs, GPUs, etc. Visualization components refer to the visual elements used to display data in the software interface, such as charts, graphs, etc. Memory is the short-term storage space of a computer, used to temporarily store the data and programs being used. A login request is a request sent when a user attempts to enter the system, containing authentication information. The user name and password are the user's identity identifier and the corresponding password, used to verify the user's identity. Successful verification means that the system confirms that the user name and password provided by the user are correct. The actual operation page is the software interface where users can perform specific operations after logging in. Automatic scanning is a search process automatically performed by the software to discover connected hardware devices. A serial device is a hardware device connected through a serial communication interface. The port number is the number identifying the serial device, used to distinguish different communication interfaces. Establishing a connection is to establish a communication link between the software and the hardware device for data exchange. Successful connection establishment means that the communication link between the software and the hardware device has been successfully established. A heartbeat request is a signal sent periodically to detect and maintain the connection status between devices. The communication protocol is a standard that stipulates the data transmission format and rules. A heartbeat packet is the signal returned by the hardware device in response to the heartbeat request. The parameter configuration interface is the software interface where users can set the parameters of the hardware device. Reuse means using the previously saved data and settings again in subsequent operations to improve efficiency.

[0033] For example, in the initialization and user login module, after the software starts, in order to facilitate subsequent calculations and operations, it is necessary to load and initialize the relevant computing resources and visualization components, and pre-load some large files or models with relatively high rendering complexity into memory. Then, the user needs to log in to the system according to the registered user name and password. If the input user name and password match the system database successfully, they enter the actual operation page of the software system. The specific technologies used in this process are the logic of adding, deleting, modifying, and querying databases and creating databases and tables based on SQLite, the asynchronous loading of VTK components based on grid files, and the secondary development of program components based on the Qt framework, etc.

[0034] For example, in the hardware device connection module, since it is necessary to control the hardware device, the system needs to first construct the communication protocols for software and hardware and establish device connections based on the communication protocols. There are multiple connection methods available. The hardware device mainly uses the serial port for communication. Specifically, the software provides a serial port scanning program function that can scan the serial port devices currently connected to the computer of the current software system, and then establish a serial port connection object based on the port numbers of the serial port devices. After the connection is established, the software system will regularly send a heartbeat request based on the communication protocol to the serial port device. After the hardware device receives the request sent by the software system, it will also return a heartbeat response packet with a fixed rule through the serial port connection object. After the software receives the heartbeat packet returned by the hardware device, it proves that the communication construction is normal and subsequent parameter configuration operations can be performed. Conversely, if the communication connection is disconnected, the disconnection will be prompted on the software interface and the user will be guided to reconnect the device. The technologies specifically used in this process are: the serial port communication protocol based on hexadecimal, the setting of encoding and decoding methods, the connection detection of hardware devices based on the heartbeat packet mechanism, etc.

[0035] For example, in the magnetic resonance image registration module, in actual treatment, although the brain structure of the target object (such as a patient) has a statistical brain atlas and brain region division, in individualized treatment, due to the morphological and distribution differences between the brain tissue structure of the subject and the brain atlas, it often becomes a more effective strategy to use the individualized MRI images of the patient for guiding the stimulation intervention plan. The brain atlas mentioned above refers to a reference tool for systematically and standardly describing the structure and function of the brain. It divides the brain into different regions, structures or networks, and each region has a specific name, number and coordinate. This standardized division method helps researchers and clinicians accurately locate specific parts of the brain in neuroscience research, medical diagnosis and treatment, and promotes data sharing and result comparison between different research and medical institutions. In the software process, the user can import MRI medical images that have been taken from other MRI imaging devices, and then the software system will perform registration on the imported MRI images. Registration is to enable MRI images in different formats and spatial poses to be segmented and calculated in a standard coordinate space. The entire process involves the translation and rotation of coordinate positions based on the spatial transformation matrix, and the visualization and interaction based on MRI images.

[0036] In some embodiments of the present application, the process of using a Gaussian mixture model and a Markov random field model to perform head model tissue segmentation on the MRI image to be processed and obtain the tissue segmentation structure of the head model specifically includes: using a Gaussian mixture model to perform probability modeling on the gray value of each pixel in the MRI image to be processed to obtain the probability distribution of the pixel gray value; initializing the parameter set of the Gaussian mixture model, where the parameter set includes mixing coefficients, means, and variances; calculating the posterior probability that each pixel in the MRI image to be processed belongs to each Gaussian component according to the probability distribution of the pixel gray value in combination with the EM algorithm; generating a three-dimensional model after tissue segmentation and an MRI model with segmentation labels according to the posterior probability and the Markov random field model as the tissue segmentation structure of the head model.

[0037] Among them, probability modeling refers to using statistical methods to describe the probability distribution of data in order to capture the uncertainty and randomness in the data. In MRI image analysis, probability modeling is used to describe the distribution of pixel gray values, which helps to identify and distinguish different tissue types. The parameter set refers to a set of parameters used to define the characteristics of each Gaussian component in the Gaussian mixture model (GMM). This includes the mixing coefficient including , , , is the mixing coefficient of the th Gaussian component, is the mean of the th Gaussian component, is the The variances of the Gaussian components. These parameters together determine the shape and position of the Gaussian components, as well as their contributions to the overall distribution. The EM algorithm is an iterative statistical algorithm used to find the maximum likelihood or maximum a posteriori probability estimates of parameters in a statistical model. In a Gaussian mixture model, the EM algorithm is used to calculate the model parameters to maximize the likelihood of the observed data. The algorithm alternates between two steps: the E-step (expectation step), which calculates the posterior probability that each pixel belongs to each Gaussian component; and the M-step (maximization step), which updates the model parameters based on these posterior probabilities. The posterior probability is the probability that a certain hypothesis is true given the observed data and the model parameters. In the EM algorithm, the posterior probability is used to measure the relative likelihood that each pixel belongs to each Gaussian component. The Markov random field model is a statistical model used to describe the dependencies between pixels in spatial data. In image segmentation, the MRF is used to capture the spatial correlations between neighboring pixels, encouraging adjacent pixels to have the same label, thereby smoothing the segmentation result and reducing the influence of noise. A three-dimensional model refers to a mathematical model that represents an object or a scene in three-dimensional space. In medical image analysis, a three-dimensional model can be reconstructed from MRI image data for a more intuitive display and analysis of the patient's anatomical structure. A segmentation label refers to the label assigned to each pixel or voxel during the image segmentation process, indicating the tissue type it belongs to. These labels help identify and distinguish different structures in the image and are the basis for subsequent analysis and processing. A segmentation label refers to the label assigned to each pixel or voxel during the image segmentation process, indicating the tissue type it belongs to. These labels help identify and distinguish different structures in the image and are the basis for subsequent analysis and processing.

[0038] In some embodiments of the present application, the specific process of generating a three-dimensional model after tissue segmentation and an MRI model with segmentation labels based on the posterior probability and the Markov random field model includes: updating the mixing coefficients, means, and variances according to the posterior probability; continuing to execute the step of calculating the posterior probability that each pixel in the MRI image to be processed belongs to each Gaussian component by combining the EM algorithm based on the probability distribution of pixel gray values until the parameter update no longer changes and the parameter iteration ends, and based on the current posterior probability that each pixel belongs to each Gaussian component, assigning each pixel to a tissue type to obtain a label category classification based on GMM; using the Markov random field to smooth the label category classification based on GMM and processing it based on the data term and the smooth term included in the energy function to obtain a three-dimensional model after tissue segmentation and an MRI model with segmentation labels; wherein, the data term measures the matching degree between the pixel label and its gray value, and the smooth term encourages adjacent pixels to have the same label.

[0039] Specifically, the probability distribution calculation formula of pixel gray values is:

[0040] wherein, is the probability distribution of the gray value of the -th pixel in the MRI image to be processed, is the gray value of the -th pixel in the MRI image to be processed, is the total number of all pixels, is the number of Gaussian components in the Gaussian mixture model, is the -th mixing coefficient of the Gaussian component, satisfying , is the mean of the -th Gaussian component, is the variance of the -th Gaussian component, is the Gaussian distribution with mean and variance .

[0041] For example, in the head model tissue segmentation module, due to the difference in conductivity between head tissues, when deriving the electrode scheme, the head tissue model cannot be directly used for calculation and derivation. The tissue segmentation technology of the head model needs to be used to segment the MRI image of the subject into different tissues. For example, the head tissue is divided into scalp, eyeball, cancellous bone, cortical bone, blood vessel, cerebrospinal fluid, white matter, and gray matter. To determine the parameter set of GMM, the Expectation-Maximization (EM) algorithm is adopted. The EM algorithm first needs to go through initialization: Select the initial parameters , where there is an E step (Expectation): Calculate the posterior probability (responsibility) of each pixel belonging to the k-th category:

[0042] Then there is an M step (Maximization): Update the parameters:

[0043]

[0044]

[0045] Repeat the E-step and M-step iterations until the parameters converge, and you can get the label category classification based on the Gaussian mixture model assumption. However, there are still some boundary judgment problems, which require the introduction of the Markov field model (MRF) concept for smoothing. MRF is a probabilistic model for modeling spatial correlation. In image segmentation, MRF can capture the spatial dependency between neighboring pixels and encourage neighboring pixels to have the same label, thereby smoothing the segmentation results and reducing the impact of noise. Assume that there is a random field ,in Indicates The label (category) of pixels. According to the Markov property, .in is the neighborhood of pixel i. The joint probability distribution of MRF can be expressed as Gibbs distribution: ; Here, Z is the partition function that ensures that the probabilities sum to 1. is the energy function, defined as the sum of all group potential functions. The energy function is in the form of:

[0046] in, is a single-pixel potential, which is usually absorbed into the data term when combining GMM. is a two-pixel potential, representing the interaction between neighboring pixel labels.

[0047] In MRI image segmentation, GMM is used to model the relationship between pixel grayscale values and labels, while MRF is used to model the spatial correlation between labels. The combination of the two can be achieved by constructing the maximum a posteriori probability (MAP) estimation problem.

[0048] The goal is to find the label field The maximum a posteriori probability According to Bayes' theorem Among them, the likelihood term Modeled by GMM, it means that in a given label field Gray value observed The probability of . Prior ) is modeled by MRF, which represents the spatial smoothness of the label field.

[0049] Convert maximizing the posterior probability into minimizing the energy function: ; Total energy function It can be expressed as:

[0050] Data Item : Measure the matching degree between the label of pixel i and its grayscale value. Smooth term : Encourage adjacent pixels to have the same label. Commonly used smooth terms: β controls the smoothing intensity, is the Kronecker delta function, which is 1 when and 0 otherwise.

[0051] Therefore, the input of the head model tissue segmentation module process is the registered MRI image to be processed, and the output is the three-dimensional model after tissue segmentation and the MRI model with segmentation labels.

[0052] In some embodiments of the present application, based on the tissue segmentation structure of the head model, the specific process of performing time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge to obtain the forward lead field matrix of the head model includes: meshing the tissue segmentation structure of the head model to create a grid model composed of nodes and elements; constructing a conductivity stiffness matrix according to the nodes and node distances in the grid model; using the conductivity stiffness matrix and a preset control equation for finite element simulation to calculate the steady-state current distribution based on the meshed head model; based on the steady-state current distribution of the head model, performing stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge to obtain the forward lead field matrix of the head model.

[0053] Among them, meshing refers to the process of discretizing a complex geometric structure (such as a head model) into a mesh model composed of many small and simple geometric units (nodes and elements). Nodes are the basic points in the mesh model, representing the vertices or intersection points of the mesh. Elements are geometric shapes formed by connecting nodes, such as triangles, quadrilaterals, hexahedrons, etc., and they are the basic units in the meshed model. The conductivity stiffness matrix is a mathematical matrix that contains conductivity information and is used to describe the ability of a material to conduct electric current. In finite element simulation, this matrix is used to calculate the electric field and current distribution. Finite element simulation is a numerical analysis method that approximately solves complex physical problems by discretizing continuous physical problems into a finite number of small elements and applying physical laws to these elements. The preset control equations refer to the mathematical equations defined before finite element simulation, and these equations describe the basic laws of physical phenomena, such as the distribution laws of electric field and current. In this example, the control equations may be the equations describing the propagation of electric current in media with different conductivities. Steady-state current distribution refers to the distribution state of electric current in a conductor without time variation. This distribution can be obtained by solving the control equations and is one of the results of finite element simulation. Steady-state current distribution refers to the distribution state of electric current in a conductor without time variation. This distribution can be obtained by solving the control equations and is one of the results of finite element simulation. Dipole discharge is a way to simulate electrode discharge, where the electrode is regarded as a dipole, that is, a system composed of two point charges with opposite charges and equal magnitudes. This way is used to simulate the behavior of electrodes in brain stimulation. The 10-10 system is an international standard used to standardize the positions of electrodes in electroencephalogram (EEG) and related fields. This system defines the positions and naming rules of electrodes to ensure consistency and comparability between different studies and applications. The forward lead field matrix is a matrix that describes the relationship between electrode positions and potential measurements. It is based on the calculation results of steady-state current distribution and is used to guide electrode placement and setting of stimulation parameters.

[0054] Specifically, the preset control equation for steady-state current distribution is the Laplace equation or the Poisson equation; among them, the Laplace equation or the Poisson equation is: , where, is the conductivity distribution, is the potential distribution, is the current source density, is the gradient operator. The gradient operator is a vector differential operator used to represent the gradient of a scalar function or the divergence of a vector field in multivariable calculus.

[0055] Specifically, based on the steady-state current distribution of the head model, the specific process of performing stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge to obtain the forward lead field matrix of the head model includes: In the process of performing stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge, on each finite element unit, define the shape function according to the steady-state current distribution; Expand the potential and test function in the stimulation simulation process with the shape function to obtain the algebraic equation at the element level; For the elements in the mesh model, obtain the stiffness matrix; Assemble the stiffness matrices of all elements into a global stiffness matrix, and in the assembly process, map the stiffness matrix to the corresponding position of the global matrix according to the global numbering of the nodes; Calculate the electric field and current density according to the algebraic equation at the element level and the global stiffness matrix; Based on the electric field and current density, construct the forward lead field matrix, and the forward lead field matrix describes the relationship between the current source and the potential measurement.

[0056] Specifically, the algebraic equation at the element level includes:

[0057] Among them, is the stiffness matrix, is the potential vector of the element nodes, is the element load vector.

[0058] For example, in the stimulation simulation module, after obtaining the tissue segmentation structure of the head model, it is necessary to apply the physical properties based on the electromagnetic field to the tissue according to the segmentation result. The basic method is to mesh the tissue structure and construct the conductivity stiffness matrix based on the nodes and node distances after meshing. The software system will perform stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge to obtain the forward lead field matrix of the head model, and the forward lead field matrix of the head model is the basis for calculating the electrode position iteration.

[0059] Performing finite element simulation on the head model requires calculating the steady-state current distribution of the meshed head model. For the steady-state current distribution, the control equation is the Laplace equation or the Poisson equation: , among which, is the conductivity distribution, is the potential distribution, is the current source density. Convert the control equation into the weak form to adapt to the finite element method: , is the test function, is the computational domain. On each finite element unit, define the shape function , the potential and the test function Expanding with shape functions gives: ， , substituting the weak form gives the algebraic equations at the element level: . Where is the element stiffness matrix, is the element nodal potential vector, is the element load vector. Calculate the element stiffness matrix: For element e, the elements of the stiffness matrix are: , where is the conductivity tensor of the element, is the gradient of the shape function. Then the global stiffness matrix needs to be assembled, assembling the stiffness matrices of all elements into the global stiffness matrix , and in the assembly process, according to the global numbering of the nodes, the element matrices are mapped to the corresponding positions in the global matrix.

[0060] Then the linear equations can be solved: , is the potential vector of all nodes. Then the electric field and current density need to be calculated, and the electric field and current density can be calculated from the potential gradient: , V , Then the forward lead field matrix can be constructed, and the forward lead field matrix L describes the relationship between the current source and the potential measurement, , where is the potential vector, is the current source vector. For each independent current source (such as each electrode), the following process needs to be repeated: Apply a unit current source at the electrode position , and solve the corresponding potential distribution . Store the resulting potential as a column of the forward lead field matrix. Repeat the above process, traverse all current sources, and construct the complete forward lead field matrix. If there are nodes, electrodes (current sources), then the forward lead field matrix has a size of . This process can be carried out after the head model tissue segmentation or after the following selection of intervention targets. The calculation of the lead field matrix will greatly facilitate the operation of the subsequent convex optimization algorithm.

[0061] For example, in the intervention coordinate selection module, after the head model tissue segmentation process is completed, the user can enter the stage of selecting the intervention target after waiting for the operation to finish. The user operates the mouse to click on the MRI image component on the software system interface to determine the coordinate position of the target brain region for stimulation intervention, preparing for the subsequent iterative optimization calculation. After the user selects the three-dimensional coordinates, they can click the "Add Target Location" on the software system interface to put the selected target brain region into the subsequent running sequence.

[0062] For example, in the electrode position arrangement optimization module, after determining the target stimulation target position, the software system verifies and optimizes the electric field intensity of the target brain region under different electrode arrangement schemes based on the forward lead field matrix and the convex optimization methods based on genetic algorithms and particle swarm optimization algorithms, and obtains the optimized solution of electrode configuration for maximizing the stimulation effect of the target brain region. The output of this process is the position of the electrode placement and the amplitude parameters of the electrode pairs.

[0063] For example, in the parameter import module, after obtaining the electrode configuration parameter results of the previous steps, the user can click a button to import relevant calculation parameters, and parameters such as frequency, stimulation time, fade-in and fade-out time, pre-stimulation, and pseudo-stimulation need to be manually input by the user, and these parameters have certain default values. This not only saves the filling of the calculated click scheme configuration but also gives the user a certain degree of autonomy to flexibly change other parameters, which plays a certain role in conducting control experiments or stimulation configurations under different parameters.

[0064] For example, in the electrical stimulation module, the software triggers the relevant logic of the button through mouse click operations, and then outputs the waveform of the configured stimulation parameters through the communication protocol and the hardware device, thereby achieving the purpose of stimulation intervention treatment. In addition, when the hardware device outputs the waveform, it will also send the impedance of each stimulation channel of the device to the software according to the communication protocol and dynamically display and update it on the software.

[0065] For example, in the data storage and reuse module, for each treatment, the data based on MRI segmentation and navigation positioning is automatically stored in the local database, facilitating subsequent treatment reuse and related historical queries and documentation records.

[0066] In the embodiments of the present application, on the one hand, through a magnetic resonance image registration module, a head model tissue segmentation module, a stimulation simulation module, etc., an automated and simplified operation process is achieved. The user only needs to perform simple instruction input to complete the whole process from the import, registration, tissue segmentation of magnetic resonance images to stimulation simulation. This simplified operation method greatly reduces the dependence on the user's professional knowledge, enabling even non-professionals to quickly get started, thus significantly improving the efficiency of practical applications. The electrode position arrangement optimization module and the parameter import module further simplify the process of electrode configuration and parameter setting, making the entire treatment preparation process faster and more accurate. On the other hand, through the electrode position arrangement optimization module, combined with the convex optimization method of genetic algorithm and particle swarm optimization algorithm, the electric field intensity of the target brain region under different electrode arrangement schemes is verified and optimized. This optimization method based on the forward lead field matrix can accurately calculate the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs, ensuring precise adjustment of the current parameters. The electrical stimulation module responds to the electrical stimulation instruction and outputs waveforms for the configured stimulation parameters through a communication protocol and hardware device to achieve precise electrical stimulation of the target brain region. This precise adjustment of current parameters and electrical stimulation output significantly improves the accuracy of treatment, thereby improving the treatment effect and safety.

[0067] Please refer to Figure 3 , which is a schematic flow chart of a neuroregulation intervention method based on time-domain interference electrical stimulation provided by the embodiments of the present application. As Figure 3 shown, the detection method of the embodiments of the present application may include the following steps: S101, receiving and registering the magnetic resonance image of the target object imported by the client to obtain the MRI image to be processed; S102, using a Gaussian mixture model and a Markov random field model to perform head model tissue segmentation on the MRI image to be processed to obtain the tissue segmentation structure of the head model; S103, based on the tissue segmentation structure of the head model, performing time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge to obtain the forward lead field matrix of the head model; S104, in response to a position selection instruction for the MRI image to be processed, determining the coordinate position of the target brain region that needs to be stimulated and intervened, and putting the target brain region corresponding to the coordinate position of the target brain region into the running sequence; S105, based on the forward lead field matrix, combining the convex optimization method of genetic algorithm and particle swarm optimization algorithm to verify and optimize the electric field intensity of the target brain region under different electrode arrangement schemes to determine the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs and display them. In response to a fine-tuning instruction for the displayed information, fine-tuning the displayed information to obtain the electrode configuration parameter result and display it; S106. In response to a parameter selection instruction for the displayed electrode configuration parameter result, receive the selected electrode configuration parameter and the imported preset electrical stimulation parameter as the configured stimulation parameter; S107. In response to an electrical stimulation instruction, output a waveform for the configured stimulation parameter through a communication protocol and a hardware device to perform electrical stimulation on the target brain region of the target object.

[0068] In the embodiments of the present application, on the one hand, through a magnetic resonance image registration module, a head model tissue segmentation module, a stimulation simulation module, etc., an automated and simplified operation process is realized. The user only needs to perform simple instruction input to complete the whole process from the import, registration, tissue segmentation of magnetic resonance images to stimulation simulation. This simplified operation method greatly reduces the dependence on the user's professional knowledge, enabling even non-professionals to quickly get started, thus significantly improving the efficiency of practical applications. The electrode position arrangement optimization module and the parameter import module further simplify the process of electrode configuration and parameter setting, making the entire treatment preparation process faster and more accurate. On the other hand, through the electrode position arrangement optimization module, combined with the convex optimization methods of genetic algorithms and particle swarm optimization algorithms, the electric field intensity of the target brain region under different electrode arrangement schemes is verified and optimized. This optimization method based on the forward lead field matrix can accurately calculate the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs, ensuring precise adjustment of the current parameters. The electrical stimulation module responds to the electrical stimulation instruction and outputs a waveform for the configured stimulation parameter through a communication protocol and a hardware device to achieve precise electrical stimulation of the target brain region. This precise adjustment of the current parameters and electrical stimulation output significantly improves the accuracy of the treatment, thereby improving the treatment effect and safety.

[0069] The present application also provides a computer-readable medium, on which program instructions are stored. When the program instructions are executed by a processor, the closed-loop neural regulation method based on electroencephalogram and time-domain interference electrical stimulation provided by each of the above method embodiments is implemented.

[0070] The present application also provides a computer program product containing instructions. When it runs on a computer, it enables the computer to execute the closed-loop neural regulation method based on electroencephalogram and time-domain interference electrical stimulation of each of the above method embodiments.

[0071] Those of ordinary skill in the art can understand that all or part of the processes of implementing the methods in the above embodiments can be completed by instructing relevant hardware through a computer program. The program for closed-loop neural regulation based on electroencephalogram and time-domain interference electrical stimulation can be stored in a computer-readable storage medium. When the program is executed, it can include the processes of the embodiments of the above methods. Among them, the storage medium of the program for closed-loop neural regulation based on electroencephalogram and time-domain interference electrical stimulation can be a magnetic disk, an optical disc, a read-only memory, or a random access memory, etc.

[0072] The above-disclosed are only the preferred embodiments of the present application. Of course, the scope of the rights of the present application cannot be limited thereby. Therefore, equivalent changes made according to the claims of the present application still fall within the scope covered by the present application.

Claims

1. A nerve regulation and intervention system based on time-domain interference electrical stimulation, characterized in that, The system includes: A magnetic resonance image registration module, configured to receive and register the magnetic resonance images of the target object imported into the client, and obtain the MRI image to be processed; A head model tissue segmentation module, configured to perform head model tissue segmentation on the MRI image to be processed by using a Gaussian mixture model and a Markov random field model, and obtain the tissue segmentation structure of the head model; A stimulation simulation module, configured to perform time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge based on the tissue segmentation structure of the head model, and obtain the forward lead field matrix of the head model; An intervention coordinate selection module, configured to determine the coordinate position of the target brain region that needs to be stimulated and intervened in response to a position selection instruction for the MRI image to be processed, and put the target brain region corresponding to the coordinate position of the target brain region into the running sequence; An electrode position arrangement optimization module, configured to verify and optimize the electric field intensity of the target brain region under different electrode arrangement schemes based on the forward lead field matrix, in combination with the convex optimization methods of the genetic algorithm and the particle swarm optimization algorithm, so as to determine the optimal placement position of the electrodes and the amplitude parameters of the electrode pairs and display them. In response to a fine-tuning instruction for the displayed information, fine-tune the displayed information, obtain the electrode configuration parameter result and display it; A parameter import module, configured to receive the selected electrode configuration parameters and the imported preset electrical stimulation parameters as the configured stimulation parameters in response to a parameter selection instruction for the displayed electrode configuration parameter result; An electrical stimulation module, configured to output a waveform for the configured stimulation parameters through a communication protocol and a hardware device in response to an electrical stimulation instruction, so as to perform electrical stimulation on the target brain region of the target object.

2. The system according to claim 1, wherein The system further includes: An initialization and user login module, configured to load and initialize computing resources and visualization components into the memory; receive a user login request, and perform verification based on the user name and password carried in the login request; after successful verification, display the actual operation page of the software; A hardware device connection module, configured to automatically scan the port numbers of the currently connected serial port devices; establish a connection with them according to the scanned port numbers of the serial port devices; after the connection is successfully established, periodically send a heartbeat request based on the communication protocol to the serial port devices; when receiving a heartbeat packet returned by the hardware device, display a parameter configuration interface; A data storage and reuse module, configured to store the tissue segmentation structure of the head model and the configured stimulation parameters when the electrical stimulation is completed, so as to be reused in the next electrical stimulation.

3. The system according to claim 1, characterized in that, The performing head model tissue segmentation on the MRI image to be processed by using a Gaussian mixture model and a Markov random field model, and obtaining the tissue segmentation structure of the head model includes: Adopting the Gaussian mixture model to perform probability modeling on the gray value of each pixel in the MRI image to be processed, and obtaining the probability distribution of the pixel gray value; Initializing the parameter set of the Gaussian mixture model, where the parameter set includes a mixing coefficient, a mean value, and a variance; Calculating the posterior probability that each pixel in the MRI image to be processed belongs to each Gaussian component according to the probability distribution of the pixel gray value, in combination with the EM algorithm; Generate a three-dimensional model after tissue segmentation and an MRI model with segmentation labels according to the posterior probability and the Markov random field model as the tissue segmentation structure of the head model.

4. The system according to claim 3, wherein The generating of a three-dimensional model after tissue segmentation and an MRI model with segmentation labels according to the posterior probability and the Markov random field model includes: Update the mixing coefficient, mean, and variance according to the posterior probability; Continue to execute the step of calculating the posterior probability that each pixel in the to-be-processed MRI image belongs to each Gaussian component by combining the EM algorithm according to the probability distribution of the pixel gray values until the parameter update no longer changes, then the parameter iteration ends. Based on the posterior probability that each pixel belongs to each Gaussian component currently, assign each pixel to a tissue type to obtain the label category classification based on GMM; Use the Markov random field to smooth the label category classification based on GMM, and process it based on the data term and the smooth term included in the energy function to obtain a three-dimensional model after tissue segmentation and an MRI model with segmentation labels; wherein, the data term measures the matching degree between the pixel label and its gray value, and the smooth term encourages adjacent pixels to have the same label.

5. The system according to claim 1, wherein The performing of time-domain interference electrical stimulation simulation on the set electrode positions in the form of dipole discharge according to the tissue segmentation structure of the head model to obtain the forward lead field matrix of the head model includes: Mesh the tissue segmentation structure of the head model to create a mesh model composed of nodes and elements; Construct a conductivity stiffness matrix according to the nodes and node distances in the mesh model; Use the conductivity stiffness matrix and a preset control equation for finite element simulation to calculate the steady-state current distribution of the meshed head model; Based on the steady-state current distribution of the head model, perform stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge to obtain the forward lead field matrix of the head model.

6. The system according to claim 5, characterized in that The preset control equation for the steady-state current distribution is the Laplace equation or the Poisson equation; wherein, The Laplace equation or Poisson equation is as follows: , where is the conductivity distribution, is the potential distribution, is the current source density, is the gradient operator, which is a vector differential operator used to represent the gradient of a scalar function or the divergence of a vector field in multivariable calculus.

7. The system according to claim 5, wherein The performing of stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge based on the steady-state current distribution of the head model to obtain the forward lead field matrix of the head model includes: During the process of performing stimulation simulation on each electrode position set according to the 10-10 system in the stimulation intervention in the form of dipole discharge, define a shape function according to the steady-state current distribution on each finite element unit; Expand the potential and test function during the stimulation simulation using the shape function to obtain an algebraic equation at the element level; For the elements in the mesh model, obtain the stiffness matrix; Assemble the stiffness matrices of all elements into a global stiffness matrix, and in the assembly process, map the stiffness matrix to the corresponding position of the global matrix according to the global number of the nodes; Calculate the electric field and current density according to the algebraic equation at the element level and the global stiffness matrix; Based on the electric field and current density, a pre-lead field matrix is constructed, and the pre-lead field matrix describes the relationship between the current source and potential measurement.

8. The system according to claim 7, wherein The algebraic equations at the unit level include: Among them, is the stiffness matrix, is the potential vector of the element nodes, is the element load vector.

9. The system according to claim 1, wherein Receiving and registering the magnetic resonance image of the target object imported into the client to obtain the MRI image to be processed, including: Receiving the magnetic resonance image of the target object imported into the client; Based on a preset spatial transformation matrix, performing format conversion and spatial pose adjustment on the magnetic resonance image; Taking the adjusted magnetic resonance image as the MRI image to be processed.

10. The system according to claim 3, characterized in that, The probability distribution calculation formula of the pixel gray value is: Among them, is the probability distribution of the gray value of the -th pixel in the MRI image to be processed, is the gray value of the -th pixel in the MRI image to be processed, is the total number of all pixels, is the number of Gaussian components in the Gaussian mixture model, is the mixing coefficient of the -th Gaussian component, satisfying , is the mean of the -th Gaussian component, is the variance of the -th Gaussian component, is the Gaussian distribution with mean and variance .

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