Parameter Estimation of a Class of Neural Systems with Limit Cycles
Abstract
:1. Introduction
1.1. Background
1.2. Parameter Estimation in Neural Model
1.3. Contributions
- We formulate the FHN neuron system as an identification model based on the explicit forward Euler method.
- We propose a recursive least-squares algorithm and a stochastic gradient algorithm to estimate the unknown parameters of the model.
- We extend the innovation concept in [24], and explore the multiinnovation recursive least-squares algorithm and multiinnovation stochastic gradient algorithm for parameter estimation of the FHN neuron system.
- We show that a faster convergence rate and better accuracy can be achieved using the innovation and repeated available data.
1.4. Organization
2. The Spiking Neuron Model
3. The Identification Model of Spiking Neurons
4. Parameter Estimation of the Spiking Neurons
4.1. Least-Squares Estimation Algorithms
Algorithm 1 RLS algorithm |
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Algorithm 2 MIRLS algorithm |
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4.2. Stochastic Gradient Estimation Algorithms
Algorithm 3 SG algorithm |
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Algorithm 4 MISG algorithm |
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5. Simulations
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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k | J | |||||||
---|---|---|---|---|---|---|---|---|
10 | 3.7906 | −0.7233 | 0.6077 | 2.9444 | 2.6491 | 0.3230 | 98.2022 | |
20 | 1.6180 | 4.6890 | −6.8895 | 2.7357 | 1.8070 | 0.5639 | 97.0998 | |
50 | 94.7731 | 102.8236 | 8.3295 | 47.5574 | 0.9996 | 0.6066 | 5.9548 | |
100 | 99.1037 | 108.8017 | 9.7432 | 49.5408 | 1.0207 | 0.5576 | 1.0100 | |
150 | 99.2001 | 108.8906 | 9.7232 | 49.5928 | 1.0401 | 0.5390 | 0.9256 | |
200 | 99.5227 | 109.3771 | 9.8946 | 49.7620 | 1.0404 | 0.5346 | 0.5272 | |
10 | 19.0174 | −2.3066 | 7.1256 | 10.3718 | 4.2456 | 0.4154 | 91.6760 | |
20 | 11.4943 | 28.0891 | −11.5647 | 8.2026 | 2.0516 | 0.9504 | 82.3618 | |
50 | 93.5511 | 100.3815 | 7.4373 | 47.0843 | 1.2176 | 0.9968 | 7.7788 | |
100 | 98.8246 | 108.3058 | 9.5814 | 49.3799 | 1.0948 | 0.6696 | 1.4012 | |
150 | 98.9660 | 108.4155 | 9.5199 | 49.4549 | 1.1355 | 0.6290 | 1.2950 | |
200 | 99.7218 | 109.5163 | 9.8858 | 49.8592 | 1.1210 | 0.5971 | 0.3861 | |
True values | 100.0000 | 110.0000 | 10.0000 | 50.0000 | 1.0000 | 0.5000 |
k | J | |||||||
---|---|---|---|---|---|---|---|---|
10 | 7.9850 | −1.6092 | 2.5997 | 4.8664 | 2.4115 | 0.3541 | 96.5310 | |
20 | 3.4710 | 8.9666 | −7.0001 | 3.6340 | 1.7240 | 0.5638 | 94.2987 | |
50 | 98.1459 | 107.4963 | 9.4579 | 49.1372 | 0.9894 | 0.6051 | 2.0867 | |
100 | 99.6327 | 109.4124 | 9.8139 | 49.8019 | 1.0195 | 0.5559 | 0.4751 | |
150 | 99.5767 | 109.3555 | 9.7988 | 49.7747 | 1.0364 | 0.5407 | 0.5280 | |
200 | 99.7563 | 109.6468 | 9.9260 | 49.8776 | 1.0393 | 0.5327 | 0.2896 | |
10 | 33.4907 | −4.5706 | 13.6947 | 17.0438 | 3.7755 | 0.4402 | 86.9092 | |
20 | 24.8108 | 41.9383 | −8.4833 | 14.4792 | 1.9920 | 0.9177 | 69.3805 | |
50 | 96.1344 | 104.1168 | 8.4045 | 48.2817 | 1.1906 | 0.9819 | 4.7325 | |
100 | 99.3021 | 108.8667 | 9.6535 | 49.6153 | 1.0919 | 0.6652 | 0.9166 | |
150 | 99.3244 | 108.8691 | 9.6029 | 49.6278 | 1.1276 | 0.6338 | 0.9145 | |
200 | 99.9346 | 109.7613 | 9.9147 | 49.9651 | 1.1189 | 0.5926 | 0.1935 | |
True values | 100.0000 | 110.0000 | 10.0000 | 50.0000 | 1.0000 | 0.5000 |
k | J | |||||||
---|---|---|---|---|---|---|---|---|
500 | 13.3084 | −7.8497 | 13.6492 | 11.1506 | 1.0467 | 0.4382 | 96.3408 | |
1000 | 21.9925 | −4.8275 | 9.1816 | 21.6546 | 1.0218 | 0.5516 | 90.1498 | |
5000 | 61.4462 | 38.2353 | 2.5148 | 31.3570 | 1.0511 | 0.5234 | 53.3865 | |
10,000 | 79.7290 | 72.4560 | 5.9073 | 40.7280 | 1.0606 | 0.5298 | 27.9030 | |
15,000 | 88.3858 | 90.8851 | 7.0928 | 46.6020 | 1.0202 | 0.5696 | 14.5130 | |
20,000 | 94.5236 | 99.8808 | 8.9198 | 47.4423 | 0.9398 | 0.3133 | 7.5321 | |
500 | 15.7472 | −7.8692 | 14.3078 | 8.5323 | 1.0953 | 0.3400 | 95.9265 | |
1000 | 22.1267 | −4.0507 | 8.4011 | 23.2820 | 1.0782 | 0.6592 | 89.5044 | |
5000 | 61.7189 | 39.9100 | 1.8494 | 34.0418 | 1.1217 | 0.5866 | 52.0776 | |
10,000 | 81.1552 | 73.8568 | 5.7945 | 40.9238 | 0.8981 | 0.5475 | 26.7046 | |
15,000 | 90.2873 | 91.2729 | 7.7195 | 45.1400 | 1.0227 | 0.6214 | 13.8507 | |
20,000 | 95.0617 | 100.5981 | 8.9675 | 47.9177 | 0.8727 | 0.0131 | 6.9244 | |
True values | 100.0000 | 110.0000 | 10.0000 | 50.0000 | 1.0000 | 0.5000 |
k | J | |||||||
---|---|---|---|---|---|---|---|---|
500 | 19.1634 | −11.8220 | 23.2437 | 16.5106 | 1.1374 | 0.4206 | 95.8047 | |
1000 | 35.0060 | −8.8759 | 18.5691 | 37.0688 | 1.0426 | 0.5921 | 86.7670 | |
5000 | 85.1399 | 57.0311 | −0.6404 | 42.6241 | 1.0992 | 0.5409 | 35.9599 | |
10,000 | 94.1694 | 90.4475 | 6.3213 | 47.8311 | 1.0994 | 0.5577 | 13.2635 | |
15,000 | 97.3389 | 103.1461 | 8.3135 | 49.9311 | 1.0338 | 0.6045 | 4.8003 | |
20,000 | 99.1619 | 107.5047 | 9.5947 | 49.7100 | 0.9374 | 0.2108 | 1.7150 | |
500 | 23.1916 | −11.9815 | 24.6474 | 12.4417 | 1.2785 | 0.2591 | 95.2371 | |
1000 | 35.3666 | −7.5615 | 17.1582 | 39.9121 | 1.1419 | 0.7780 | 85.7224 | |
5000 | 85.5678 | 59.2415 | −0.6955 | 44.1461 | 1.2350 | 0.6827 | 34.4612 | |
10,000 | 95.2375 | 91.8121 | 6.4314 | 47.7921 | 0.9014 | 0.6650 | 12.2575 | |
15,000 | 98.4425 | 103.4602 | 8.5951 | 49.2526 | 1.0869 | 0.7208 | 4.3983 | |
20,000 | 99.6694 | 108.1012 | 9.7009 | 50.1496 | 0.9396 | −0.2527 | 1.3341 | |
True values | 100.0000 | 110.0000 | 10.0000 | 50.0000 | 1.0000 | 0.5000 |
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Lou, X.; Cai, X.; Cui, B. Parameter Estimation of a Class of Neural Systems with Limit Cycles. Algorithms 2018, 11, 169. https://doi.org/10.3390/a11110169
Lou X, Cai X, Cui B. Parameter Estimation of a Class of Neural Systems with Limit Cycles. Algorithms. 2018; 11(11):169. https://doi.org/10.3390/a11110169
Chicago/Turabian StyleLou, Xuyang, Xu Cai, and Baotong Cui. 2018. "Parameter Estimation of a Class of Neural Systems with Limit Cycles" Algorithms 11, no. 11: 169. https://doi.org/10.3390/a11110169