# A Method to Obtain Parameters of One-Column Jansen–Rit Model Using Genetic Algorithm and Spectral Characteristics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeled Signal

_{1}, P

_{2}, and P

_{3}blocks can be presented using the equations:

_{1}, C

_{2}, C

_{3}, and C

_{4}are numbers of synapses established between particular neuron populations, bounded by equation:

_{0}and v

_{0}are directly responsible for the activation function’s shape, i.e., the inflection point. The parameter r is responsible for steepness (slope). These parameters were being modified in a course of this study, since the postsynaptic potential transforms into a firing rate not exactly in the same way from a population to a population [10,26].

#### 2.2. Model Parameter Obtaining Process

_{0}, v

_{0}, r, and p’s lower limit and range. These parameters mostly affect the form of the signal. Due to the relatively high number of arguments in the minimized function, the genetic algorithm (GA) was used for the minimization process. Along with other heuristics, it is often used for parameter estimation [27].

#### 2.2.1. Function Minimized in This Study

#### 2.2.2. The Genetic Algorithm

#### 2.2.3. Computation Complexity

#### 2.3. Repeatability of the Method

#### 2.4. Accuracy of the Method

#### 2.5. Reference Signals Generation Process

#### 2.6. Real, Measured Signal

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The one-column Jansen–Rit model. The P

_{1}and P

_{2}blocks are the functions (he) presented in Equation (1), P

_{3}is the function (hi) presented in Equation (2). The S

_{1}, S

_{2}, and S

_{3}(sigm) blocks are the sigmoidal functions presented in Equation (5), p is noise, while the C

_{1}–C

_{4}are the coefficients of connections between particular neural structures.

**Figure 3.**Visualization of the function minimized in this study. Plots (

**a**–

**c**) show spectra of the signals: original, modeled, and their difference, respectively. Note, that plot (

**c**) is an absolute value of the difference.

**Figure 4.**Visualization of the accuracy study. Note, that this process was conducted 100 times, with different reference parameters.

**Figure 5.**Plots of the measured signal: (

**a**) full signal, (

**b**) spectrogram of the full signal, (

**c**) analyzed fragment with the most apparent alpha-waves, (

**d**) the discrete Fourier transform of the (

**c**) plot.

**Figure 6.**The convergence curve of the genetic algorithm. Minimized function, so the score, is dimensionless.

**Table 1.**Results of the preliminary study comparing common heuristics in task of obtaining Jansen Rit (J-R) model parameters. Values of time and optimum are normalized to the highest value.

Algorithm | Time | Found Optimum |
---|---|---|

Genetic algorithm | 1.000 | 0.161 |

Simulated annealing | 0.440 | 1.000 |

Particle swarm | 0.978 | 0.189 |

Surrogate optimization | 0.236 | 0.275 |

Parameter | Lower Search Range | Upper Search Range |
---|---|---|

A (mV) | 2.25 | 4.25 |

B (mV) | 12 | 32 |

C | 70 | 675 |

v0 (mV) | 5 | 7 |

e0 (s^{−1}) | 2 | 3 |

r | 0.5 | 0.6 |

Lower noise limit (pps) | 50 | 300 |

Noise range (pps) | 200 | 1000 |

Parameter | Accuracy (Mean) | Accuracy (Std) | ICC |
---|---|---|---|

A (mV) | 0.794 | 0.272 | 0.897 |

B (mV) | 0.724 | 0.343 | 0.910 |

C | 0.853 | 0.189 | 0.978 |

v0 (mV) | 0.756 | 0.306 | 0.865 |

e0 (s^{−1}) | 0.662 | 0.403 | 0.676 |

r | 0.700 | 0.384 | 0.733 |

Lower noise limit (pps) | 0.761 | 0.320 | 0.877 |

Noise range (pps) | 0.863 | 0.191 | 0.966 |

**Table 4.**Values of parameters obtained for the measured signal. Values in column “default” were used in original works.

Parameter | Lower Range | Upper Range | Default Value | Obtained Value |
---|---|---|---|---|

A (mV) | 2.25 | 4.25 | 3.25 | 4.21 |

B (mV) | 12 | 32 | 22 | 12.06 |

C | 70 | 675 | 135 | 406.50 |

v0 (mV) | 5 | 7 | 6 | 6.60 |

e0 (s^{−1}) | 2 | 3 | 2.5 | 2.92 |

r | 0.5 | 0.6 | 0.56 | 0.6 |

Lower noise limit (pps) | 50 | 300 | 120 | 278.77 |

Noise range (pps) | 200 | 1000 | 200 | 897.34 |

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**MDPI and ACS Style**

Łysiak, A.; Paszkiel, S.
A Method to Obtain Parameters of One-Column Jansen–Rit Model Using Genetic Algorithm and Spectral Characteristics. *Appl. Sci.* **2021**, *11*, 677.
https://doi.org/10.3390/app11020677

**AMA Style**

Łysiak A, Paszkiel S.
A Method to Obtain Parameters of One-Column Jansen–Rit Model Using Genetic Algorithm and Spectral Characteristics. *Applied Sciences*. 2021; 11(2):677.
https://doi.org/10.3390/app11020677

**Chicago/Turabian Style**

Łysiak, Adam, and Szczepan Paszkiel.
2021. "A Method to Obtain Parameters of One-Column Jansen–Rit Model Using Genetic Algorithm and Spectral Characteristics" *Applied Sciences* 11, no. 2: 677.
https://doi.org/10.3390/app11020677