CN106709569A - Parameter estimation method for FitzHugh-Nagumo neuron system - Google Patents

Abstract

The invention discloses a parameter estimation method for a FitzHugh-Nagumo neuron system, which comprises the steps of performing model transformation on the FitzHugh-Nagumo neuron system to acquire a linear regression model, acquiring input-output data, building an output matrix and an information matrix, further acquiring an innovation vector, and finally estimating system parameters under zero-mean white noise interference by using an innovation projection algorithm. The parameter estimation method has the advantages of high convergence rate, small calculation amount, no need of calculating a covariance matrix and the like.

Description

FitzHugh-Nagumo neuron system parameter estimation method

Background

In recent years, inspired by biological research results, biological neuron models are becoming one of the research hotspots. The biological neuron model has abundant calculation characteristics, and the characteristics bring wide application prospects for the biological neuron model. Research shows that the FitzHugh-Nagumo neuron system can show a chaotic state when parameters are properly selected, but system parameters are not necessarily known in practice, and although some parameter estimation methods of the chaotic biological neuron system exist at present, such as an adaptive synchronization algorithm and the like, the calculation amount is large, the realization is complex, and certain limitations exist.

The innovation projection algorithm is a generalization of the projection algorithm. The projection algorithm only uses the current data and the information to correct the parameter estimation value, and the information projection algorithm not only uses the current data and the information, but also fully uses the past data and the information, so the convergence speed is higher compared with the projection algorithm, and the parameter estimation accuracy is higher. The invention provides a novel parameter estimation method aiming at a FitzHugh-Nagumo neuron system by combining with an innovation projection algorithm.

Disclosure of Invention

The invention aims to estimate parameters of a FitzHugh-Nagumo neuron system by using an innovation projection algorithm under zero-mean white noise interference. The FitzHugh-Nagumo neuron system is obtained by simplifying a classical Hodgkin-Huxley neuron model, and the mathematical equation is as follows:

wherein x is₁，x₂For system state variables, a and f are the amplitude and frequency, ω, of the external stimulus current, respectively₁(t) and ω₂(t) is zero mean, variance is σ²R and b are the system parameters to be estimated.

The technical scheme adopted by the invention comprises the following steps:

step 1: and (5) model transformation. The FitzHugh-Nagumo neuron model was transformed into the following estimated standard form:

wherein,

θ＝[b r]^T。

step 2: and (6) parameter estimation. To estimate the parameter vector θ, the following recursive innovation projection algorithm is used:

(a) and initializing parameters. Setting the parameter estimation value asTaking the maximum iteration number as t_maxT is 0, the innovation length is taken as p, and the initial parameter is estimated asWherein 1 ═ 11 … 1]^T。

(b) Collecting data, and constructing output matrix Y (p, t) and information matrix

Y(p，t)＝[Y^T(t) Y^T(t-1) … Y^T(t-p+1)]^T

(c) The innovation vector E (p, t) is calculated as follows:

(d) updating parameter estimates

(e) Let t: t +1, if t < t_maxTurning to the step (b), and continuing to perform recursive calculation;otherwise, go to step 3.

And step 3: stopping calculation to obtain the final estimated parameter value

Compared with the existing estimation algorithm, the method has the following advantages: the innovation projection algorithm finds a compromise between convergence speed and computational effort. Compared with a projection algorithm, the convergence speed of the innovation projection algorithm is higher, and the convergence speed can be improved by changing the innovation length of the innovation projection algorithm. Compared with the least square method, the innovation projection algorithm has small calculation amount and does not need to calculate a covariance matrix.

Drawings

Fig. 1 is a flow chart of parameter estimation based on the scheme of the present invention.

Fig. 2 is a graph of an iterative process of parameter estimation based on the scheme of the present invention in an example.

Detailed Description

In order to better understand the technical solution of the present invention, the following embodiments are further described in detail.

Example (b): consider a specific FitzHugh-Nagumo neuron system model:

wherein x is₁，x₂For the system state variables, a-0.1 and f-0.129 are the amplitude and frequency, respectively, of the external stimulus current, ω₁(t) and ω₂(t) is zero mean, variance is σ²0.01 ═ 0.01White noise, r and b are system parameters to be estimated. When the system parameter is r-10 and b-1, the neuron system will exhibit chaos.

The working flow of the method of the invention is shown in fig. 1, and the specific implementation mode can be divided into the following steps:

step 1: and (5) model transformation. The FitzHugh-Nagumo neuron model was transformed into the following estimated standard form:

wherein,

θ＝[b r]^T。

step 2: and (6) parameter estimation. To estimate the parameter vector θ, the following recursive innovation projection algorithm is used:

(a) and initializing parameters. Setting the parameter estimation value asTaking the maximum iteration number as t_max1500, t is 0, the innovation length is p is 5, and the initial parameter is estimated asWherein 1 ═ 11 … 1]^T。

(b) Collecting data (obtained by solving a neuron system equation by adopting a fourth-order Runge-Kutta method), and constructing an output matrixY (p, t) and information matrix

Y(p，t)＝[Y^T(t) Y^T(t-1) … Y^T(t-p+1)]^T

(c) The innovation vector E (p, t) is calculated as follows:

(d) updating parameter estimates

(e) Let t: t +1, if t < t_maxTurning to the step (b), and continuing to perform recursive calculation; otherwise, go to step 3.

And step 3: stopping calculation to obtain the final estimated parameter value

Fig. 2 shows a parameter estimation iteration process curve. As can be seen, all parameter estimates converge to true values approximately after 1000 iterations.

Claims (1)

1. A FitzHugh-Nagumo neuron system parameter estimation method is characterized by comprising the following steps:

step 1: and (5) model transformation. Consider the FitzHugh-Nagumo neuronal system:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1=-r{x}_1^3+(1+r)x_1^2-x_1-x_2+\frac{a}{2\mathrm{\&pi;}f}cos2\mathrm{\&pi;}ft+{\mathrm{\&omega;}}_1\left(t\right)\\ {\stackrel{\&CenterDot;}{x}}_2=b{x}_1+{\mathrm{\&omega;}}_2\left(t\right)\end{array}\right.$

wherein x is₁，x₂For system state variables, a and f are the amplitude and frequency, ω, of the external stimulus current, respectively₁(t) and ω₂(t) is zero mean, variance is σ²R and b are the system parameters to be estimated.

The FitzHugh-Nagumo neuron model was transformed into the following estimated standard form:

wherein,

$Y\left(t\right)=\left[\begin{array}{c}{\stackrel{\&CenterDot;}{x}}_1+x_1-x_1^2+x_2-\frac{a}{2\mathrm{\&pi;}f}cos2\mathrm{\&pi;}f\\ {\stackrel{\&CenterDot;}{x}}_2\end{array}\right],\mathrm{\&omega;}\left(t\right)=\left[\begin{array}{c}{\mathrm{\&omega;}}_1\left(t\right)\\ {\mathrm{\&omega;}}_2\left(t\right)\end{array}\right],$

θ＝[b r]^T。

step 2: and (6) parameter estimation. To estimate the parameter vector θ, the following recursive innovation projection algorithm is used:

(a) and initializing parameters. Setting the parameter estimation value asTaking the maximum iteration number as t_maxT is 0, the innovation length is taken as p, and the initial parameter is estimated asWherein 1 ═ 11 … 1]^T。

(b) Collecting data, and constructing output matrix Y (p, t) and information matrix

Y(p，t)＝[Y^T(t) Y^T(t-1) … Y^T(t-p+1)]^T

(c) The innovation vector E (p, t) is calculated as follows:

(d) updating parameter estimates

(e) Let t: t +1, if t < t_maxTurning to the step (b), and continuing to perform recursive calculation; otherwise, go to step 3.

And step 3: stopping calculation to obtain the final estimated parameter value