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 leastsquares 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 leastsquares 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. LeastSquares Estimation Algorithms
Algorithm 1 RLS algorithm 

Algorithm 2 MIRLS algorithm 

4.2. Stochastic Gradient Estimation Algorithms
Algorithm 3 SG algorithm 

Algorithm 4 MISG algorithm 

5. Simulations
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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${\mathit{\sigma}}^{2}$  k  $\mathit{\mu}$  $(\mathit{a}+\mathit{b})\mathit{\mu}$  $\mathit{ab}\mathit{\mu}$  J  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  $\mathit{\delta}\phantom{\rule{4pt}{0ex}}(\%)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ 

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  
$0.{2}^{2}$  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  
$0.{5}^{2}$  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 
${\mathit{\sigma}}^{2}$  k  $\mathit{\mu}$  $(\mathit{a}+\mathit{b})\mathit{\mu}$  $\mathit{ab}\mathit{\mu}$  J  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  $\mathit{\delta}\phantom{\rule{4pt}{0ex}}(\%)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ 

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  
$0.{2}^{2}$  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  
$0.{5}^{2}$  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 
${\mathit{\sigma}}^{2}$  k  $\mathit{\mu}$  $(\mathit{a}+\mathit{b})\mathit{\mu}$  $\mathit{ab}\mathit{\mu}$  J  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  $\mathit{\delta}\phantom{\rule{4pt}{0ex}}(\%)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ 

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  
$0.{2}^{2}$  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  
$0.{5}^{2}$  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 
${\mathit{\sigma}}^{2}$  k  $\mathit{\mu}$  $(\mathit{a}+\mathit{b})\mathit{\mu}$  $\mathit{ab}\mathit{\mu}$  J  ${\mathit{c}}_{1}$  ${\mathit{c}}_{2}$  $\mathit{\delta}\phantom{\rule{4pt}{0ex}}(\%)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}$ 

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  
$0.{2}^{2}$  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  
$0.{5}^{2}$  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