Identification of biological neurons using adaptive observers

Abstract

This paper is to investigate the use of adaptive observers for the modeling of biological neurons and networks. Assuming that a neuron can be modeled as a continuous-time nonlinear system, it is possible to determine its unknown parameters using adaptive observer, based on the concept of adaptive synchronization. The same technique can be extended for the identification of an entire biological neural network. Some conventional observer designs are studied in this paper and satisfactory results are obtained, yet with some restrictions. To further extend the applicability of adaptive observers for the modeling process, a new design is suggested. It is based on a combination of linear feedback control approach and the dynamical minimization algorithm. The effectiveness of the designed adaptive observer is confirmed with simulations.

Appendix

$$ \begin{gathered} \dot{\hat{x}}_{1} (t) = a\,\hat{x}_{1}^{2} - \hat{x}_{1}^{3} - r_{1} \,\hat{y}_{1} - r_{6} \,\hat{z}_{1} - \sum\limits_{j = 1}^{5} {g\,\hat{c}_{1j} \,\sigma_{{V_{\rm s} }} \,(\hat{x}_{1} )\,r_{{v,\theta_{\rm s} }} (\hat{x}_{j} )} + K_{1} (x_{1} - \hat{x}_{1} ) \hfill \\ \dot{\hat{y}}_{1} (t) = b\,\hat{x}_{1}^{2} - f\,\hat{y}_{1} \hfill \\ \dot{\hat{z}}_{1} (t) = c\,\hat{x}_{1} - d\,\hat{z}_{1} + e \hfill \\ \dot{\hat{x}}_{2} (t) = a\,\hat{x}_{2}^{2} - \hat{x}_{2}^{3} - l\hat{y}_{2} - r_{2} \hat{z}_{2} - \sum\limits_{j = 1}^{5} {g\,\hat{c}_{2j} \,\sigma_{{V_{\rm s} }} \,(\hat{x}_{2} )\,r_{{v,\theta_{\rm s} }} \,(\hat{x}_{j} )} + K_{2} (x_{2} - \hat{x}_{2} ) \hfill \\ \dot{\hat{y}}_{2} (t) = b\,\hat{x}_{2}^{2} - r_{7} \,\hat{y}_{2} \hfill \\ \dot{\hat{z}}_{2} (t) = c\,\hat{x}_{2} - d\,\hat{z}_{2} + e \hfill \\ \dot{\hat{x}}_{3} (t) = a\,\hat{x}_{3}^{2} - \hat{x}_{3}^{3} - l\,\hat{y}_{3} - h\,\hat{z}_{3} - \sum\limits_{j = 1}^{5} {g\,\hat{c}_{3j} \,\sigma_{{V_{\rm s} }} \,(\hat{x}_{3} )\,r_{{v,\theta_{\rm s} }} (\hat{x}_{j} )} + K_{3} (x_{3} - \hat{x}_{3} ) \hfill \\ \dot{\hat{y}}_{3} (t) = b\,\hat{x}_{3}^{2} - r_{3} \,\hat{y}_{3} \hfill \\ \dot{\hat{z}}_{3} (t) = r_{8} \,\hat{x}_{3} - d\,\hat{z}_{3} + e \hfill \\ \dot{\hat{x}}_{4} (t) = a\,\hat{x}_{4}^{2} - \hat{x}_{4}^{3} - l\hat{y}_{4} - h\,\hat{z}_{4} - \sum\limits_{j = 1}^{5} {g\,\hat{c}_{4j} \,\sigma_{{V_{\rm s} }} \,(x_{4} )\,r_{{v,\theta_{\rm s} }} (x_{j} )} + K_{4} (x_{4} - \hat{x}_{4} ) \hfill \\ \dot{\hat{y}}_{4} (t) = b\,\hat{x}_{4}^{2} - r_{9} \,\hat{y}_{4} \hfill \\ \dot{\hat{z}}_{4} (t) = c\,\hat{x}_{4} - r_{4} \,\hat{z}_{4} + e \hfill \\ \dot{\hat{x}}_{5} (t) = a\,\hat{x}_{5}^{2} - \hat{x}_{5}^{3} - r_{10} \,\hat{y}_{5} - h\,\hat{z}_{5} - \sum\limits_{j = 1}^{5} {g\,\hat{c}_{5j} \,\sigma_{{V_{\rm s} }} (x_{5} )\,r_{{v,\theta_{\rm s} }} (x_{j} )} + K_{5} (x_{5} - \hat{x}_{5} ) \hfill \\ \dot{\hat{y}}_{5} (t) = b\,\hat{x}_{5}^{2} - f\,\hat{y}_{5} \hfill \\ \dot{\hat{z}}_{5} (t) = r_{5} \,\hat{x}_{5} - d\,\hat{z}_{5} + e \hfill \\ \end{gathered} $$

$$ \dot{\hat{c}}_{ij} = - k_{ij} \,g\,\sigma_{{V_{\rm s} }} (\hat{x}_{i} )\,r_{{v,\theta_{\rm s} }} (\hat{x}_{j} )\,(x_{i} - \hat{x}_{i} ) $$

$$ \begin{array}{*{20}c} \begin{gathered} \dot{r}_{1} = - \delta_{1} \hat{y}_{1} (x_{1} - \hat{x}_{1} ) \hfill \\ \dot{r}_{2} = - \delta_{2} \hat{z}_{2} (x_{2} - \hat{x}_{2} ) \hfill \\ \dot{r}_{3} = \delta_{3} \hat{y}_{3} (x_{3} - \hat{x}_{3} ) \hfill \\ \dot{r}_{4} = \delta_{4} \hat{z}_{4} (x_{4} - \hat{x}_{4} ) \hfill \\ \dot{r}_{5} = - \delta_{5} \hat{x}_{5} (x_{5} - \hat{x}_{5} ) \hfill \\ \end{gathered} & \begin{gathered} \dot{r}_{6} = - \delta_{6} \hat{z}_{1} (x_{1} - \hat{x}_{1} ) \hfill \\ \dot{r}_{7} = \delta_{7} \hat{y}_{2} (x_{2} - \hat{x}_{2} ) \hfill \\ \dot{r}_{8} = - \delta_{8} \hat{x}_{3} (x_{3} - \hat{x}_{3} ) \hfill \\ \dot{r}_{9} = \delta_{9} \hat{y}_{4} (x_{4} - \hat{x}_{4} ) \hfill \\ \dot{r}_{10} = - \delta_{10} \hat{y}_{5} (x_{5} - \hat{x}_{5} ) \hfill \\ \end{gathered} \\ \end{array} . $$

The simulation is based on the following parameter settings:

$$ g = 0.33,\;K_{i} = 30,\;k_{ij} = 2,\;\theta = [\begin{array}{*{20}c} 1 & 1 & 1 & {0.001} & {0.009} & 1 & 1 & {0.009} & 1 & 1 \\ \end{array} ], $$

$$ [\begin{array}{*{20}c} a & b & c & d & e & f & l & h \\ \end{array} ] = [\begin{array}{*{20}c} {2.8} & {4.4} & {0.009} & {0.001} & {0.005} & 1 & 1 & 1 \\ \end{array} ] \quad \text{and\ the\ initial\ values} $$

$$ \begin{gathered} \hat{c}_{ij} (0) = 1.3,\;x_{i} (0) \in [ - 1.02, - 1.04],\;y_{i} (0) \in [ - 0.04,0.02],\;z_{i} (0) \in [ - 0.03,0.05], \hfill \\ [\begin{array}{*{20}c} {\hat{x}_{i} (0)} & {\hat{y}_{i} (0)} & {\hat{z}_{i} (0)} \\ \end{array} ] = [\begin{array}{*{20}c} { - 1.01} & {0.01} & {0.01} \\ \end{array} ], \hfill \\ \end{gathered} $$

$$ \begin{gathered} \hat{\theta }(0) = [\begin{array}{*{20}c} {r_{1} (0)} & {r_{2} (0)} & {r_{3} (0)} & {r_{4} (0)} & {r_{5} (0)} & {r_{6} (0)} & {r_{7} (0)} & {r_{8} (0)} & {r_{9} (0)} & {r_{10} (0)} \\ \end{array} ], \\ = [\begin{array}{*{20}c} 2 & 2 & 2 & {0.003} & {0.003} & 2 & 2 & {0.003} & 2 & 2 \\ \end{array} ] \\ \end{gathered} $$

$$ \begin{gathered} [ {\begin{array}{*{20}c} {\delta_{1} } & {\delta_{2} } & {\delta_{3} } & {\delta_{4} } & {\delta_{5} } & {\delta_{6} } & {\delta_{7} } & {\delta_{8} } & {\delta_{9} } & {\delta_{10} } \\ \end{array} } ] \\ = [ {\begin{array}{*{20}c} {10} & {10} & {10} & {0.01} & {0.01} & {10} & {10} & {0.01} & {10} & {10} \\ \end{array} } ]. \\ \end{gathered} $$