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An improved neuronal energy model that better captures of dynamic property of neuronal activity

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Abstract

Among the theories of neural information coding, the neural energy coding is more accessible to global coding features than traditional neural encoding. According to the shortcomings existing in the neuronal energy model, that is, the non-smooth nature of the energy curve, we proposed an improved neuronal energy model in this paper. The modified energy model is a good choice for establishment of the global model of brain function. And it is also the basis of energy calculation for functional cognitive neural networks in the future.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under (Grant Nos. 11232005, 11472104, 61633010 and 61473110).

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Authors and Affiliations

  1. School of Computer Science, Hangzhou Dianzi University, Hangzhou, China

    Rubin Wang

  2. Institute for Cognitive Neurodynamics, East China University of Science and Technology, Shanghai, 200237, China

    Yanyan Wang & Rubin Wang

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  1. Yanyan Wang
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  2. Rubin Wang
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Corresponding author

Correspondence to Rubin Wang.

Appendix

Appendix

$$\begin{aligned} d_{1m}= & {} C_m ^{2}\left( {1+\frac{r_{3m} }{r_m }} \right) \\&\quad \times \left( {r_{0m} +r_{2m} +r_{3m} +\frac{r_{3m} \left( {r_{0m} +r_{2m} } \right) }{r_m }} \right) \\ d_{2m}= & {} C_m \left( {2r_{0m} +r_{2m} +r_{3m} +\frac{2r_{0m} r_{3m} +r_{2m} r_{3m} }{r_m }} \right) \\&\times \left( {Ke^{-at}-\frac{r_{2m} }{L_m }e^{-at}\int _{t_0 }^t {I_m e^{at}\mathrm{d}t} } \right) \\&{-\,2C_m \left( {1+\frac{r_{3m} }{r_m }} \right) \left( {r_{0m} +r_{2m} } \right) I_m } \\ d_{3m}= & {} C_m \left[ 1+2r_{0m} \left( {1+\frac{r_{3m} }{r_m }} \right) \frac{r_m +r_{1m}+r_{2m} }{r_m r_{1m} } \right. \\&\left. +\frac{r_m r_{2m} +r_m r_{3m} +r_{2m} r_{3m} }{r_m ^{2}}\left( {2+\frac{r_m +r_{2m} }{r_{1m} }} \right) \right] \\ d_{4m}= & {} \frac{r_m +r_{1m} +r_{2m} }{r_m ^{2}r_{1m} }\\&\quad \times \left( {\frac{r_{0m} }{r_{1m} }\left( {r_m +r_{1m} +r_{2m} } \right) +r_m +r_{2m} } \right) \\ d_{5m}= & {} \frac{2r_{0m} \left( {r_m +r_{1m} +r_{2m} } \right) +r_{1m} \left( {r_m +r_{2m} } \right) }{r_m r_{1m} }\\&\quad \times \left( {Ke^{-at}-\frac{r_{2m} }{L_m }e^{-at}\int _{t_0 }^t {I_m e^{at}\mathrm{d}t} } \right) \end{aligned}$$
$$\begin{aligned}&{-\frac{2r_{0m} \left( {r_m +r_{1m} +r_{2m} } \right) +r_{2m} \left( {r_m +2r_{1m} +r_{2m} } \right) }{r_m r_{1m} }\times I_m } \\ d_{6m}= & {} \left[ r_{0m} \left( {Ke^{-at}-\frac{r_{2m} }{L_m }e^{-at}\int _{t_0 }^t {I_m e^{a\tau }\mathrm{d}\tau } } \right) \right. \\&\left. -\,(r_{0m} +r_{2m} )I_m \right] \left( {Ke^{-at}-\frac{r_{2m} }{L_m }e^{-at}\int _{t_0 }^t {I_m e^{a\tau }\mathrm{d}\tau } -I_m } \right) \\&\quad \times \int _{t_0 }^t {I_m e^{a\tau }\mathrm{d}\tau } =\sum _{j=0}^n {\Gamma _j } +\sum _{j=0}^n {J_{mj} \frac{e^{at}-e^{at_j }}{a}} \\&-\,i_{0m\left( n\right) } \frac{e^{at}-e^{at_n }}{a}\sin \varphi _n \\&+\frac{i_{0m\left( n \right) } }{a^{2}+\omega _{m\left( n \right) } ^{2}}\left( ae^{at}\sin \left( {\omega _{m\left( n \right) } \left( {t-t_n } \right) +\varphi _n } \right) \right. \\&\left. -\,ae^{at_n }\sin \varphi _n \right) \\&+\frac{i_{0m\left( n \right) } }{a^{2}+\omega _{m\left( n \right) } ^{2}}\left( -\omega _{m\left( n \right) } e^{at}\cos \left( {\omega _{m\left( n \right) } \left( {t-t_n } \right) +\varphi _n } \right) \right. \\&\left. +\,\omega _{m\left( n \right) } e^{at_n }\cos \varphi _n \right) \end{aligned}$$

where

$$\begin{aligned} y_2= & {} \frac{i_{0m\left( n \right) } \omega _{m\left( n \right) } ^{2}r_{2m} \left( {al_1 +l_3 } \right) }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \\&+\frac{i_{0m\left( n \right) } \lambda _m \left[ {ar_{2m} l_3 +a^{2}l_4 L_m +\omega _{m\left( n \right) } ^{2}\left( {l_4 L_m -l_1 r_{2m} } \right) -\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \omega _{m\left( n \right) } l_2 L_m } \right] }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \\ y_3= & {} \frac{-\lambda _m i_{0m\left( n \right) } \omega _{m\left( n \right) } r_{2m} \left( {al_1 +l_3 } \right) }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \\&+\frac{i_{0m\left( n \right) } \omega _{m\left( n \right) } \left[ {ar_{2m} l_3 +a^{2}l_4 L_m +\omega _{m\left( n \right) } ^{2}\left( {l_4 L_m -l_1 r_{2m} } \right) -\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \omega _{m\left( n \right) } l_2 L_m } \right] }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \end{aligned}$$
$$\begin{aligned} C_n= & {} \frac{r_{2m} \left( {al_1 +l_3 } \right) \left( {\omega _{m\left( n \right) } \cos \varphi _n -a\sin \varphi _n } \right) }{\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {a+\lambda _m } \right) }\\&-\frac{l_3 r_{2m} -l_1 r_{2m} \lambda _m +l_4 L_m \left( {a+\lambda _m } \right) }{\lambda _m \left( {a+\lambda _m } \right) }\sin \varphi _n \\&+\frac{\omega _{m\left( n \right) } r_{2m} \left( {al_1 +l_3 } \right) \left( {\omega _{m\left( n \right) } \sin \varphi _n -\lambda _m \cos \varphi _n } \right) }{\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \\&-\frac{\omega _{m\left( n \right) } ^{2}l_1 r_{2m} -ar_{2m} l_3 +L_m \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\omega _{m\left( n \right) } l_2 -l_4 } \right) }{\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) }\\&\quad \left( {\lambda _m\sin \varphi _n +\omega _{m\left( n \right) } \cos \varphi _n } \right) \\ D_n= & {} \frac{\left( {al_1 +l_3 } \right) KL_m }{\left( {a+\lambda _m } \right) }-\frac{r_{2m} \left( {al_1 +l_3 } \right) e^{-\lambda _m t_n }}{a\lambda _m }\sum _{k=0}^n {J_{mk} } \end{aligned}$$
$$\begin{aligned}&+\frac{r_{2m} \left( {al_1 +l_3 } \right) e^{-(a+\lambda _m )t_n }}{\left( {a+\lambda _m } \right) }\sum _{k=0}^n {\left( {J_{mk} \frac{e^{at_k }}{a}-\Gamma _k } \right) } \\&-\frac{l_1 r_{2m} -l_4 L_m }{\lambda _m }\sum _{j=0}^n {J_{mj} e^{-\lambda _m t_j }} \\&+\frac{r_{2m} \left( {al_1 +l_3 } \right) }{\left( {a+\lambda _m } \right) }\sum _{j=1}^n \sum _{k=0}^{j-1} {\left( {\Gamma _k -J_{mk} \frac{e^{at_k }}{a}} \right) } \\&\quad \times \left( {e^{-(a+\lambda _m )t_j }-e^{-(a+\lambda _m )t_{j-1} }} \right) \\&+\frac{r_{2m} \left( {al_1 +l_3 } \right) }{a\lambda _m }\sum _{j=1}^n {\sum _{k=0}^{j-1} {J_{mk} } \left( {e^{-\lambda _m t_j }-e^{-\lambda _m t_{j-1} }} \right) } \\&+\,r_{2m} \left( {al_1 +l_3 } \right) \sum _{j=1}^n \frac{i_{0m\left( {j-1} \right) } }{\left( {a^{2}+\omega _{m\left( {j-1} \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( {j-1} \right) } ^{2}} \right) }\\&\times \left( {a\mathrm{A}_j +\omega _{m\left( {j-1} \right) } \mathrm{B}_j } \right) \\&+\left( {l_4 L_m -l_1 r_{2m} } \right) \sum _{j=1}^n {\frac{i_{0m\left( {j-1} \right) } }{\lambda _m ^{2}+\omega _{m\left( {j-1} \right) } ^{2}}A_j } -l_2 L_m \end{aligned}$$
$$\begin{aligned}&\quad \sum _{j=1}^n {\frac{i_{0m\left( {j-1} \right) } \omega _{m\left( {j-1} \right) } }{\lambda _m ^{2}+\omega _{m\left( {j-1} \right) } ^{2}}} A_j \\&+\frac{r_{2m} \left( {al_1 +l_3 } \right) }{\left( {a+\lambda _m } \right) }\sum _{j=1}^n \frac{i_{0m\left( {j-1} \right) } }{a^{2}+\omega _{m\left( {j-1} \right) } ^{2}}\\&\quad \times \left( {e^{-(a+\lambda _m )t_j +at_{j-1} }-e^{-\lambda _m t_{j-1} }} \right) \\&\quad \times \left( {\omega _{m\left( {j-1} \right) } \cos \varphi _{j-1} -a\sin \varphi _{j-1} } \right) \\&+\frac{l_1 r_{2m} -l_4 L_m }{\lambda _m }\sum _{j=1}^n i_{0m\left( {j-1} \right) } \\&\quad \times \left( {e^{-\lambda _m t_j }-e^{-\lambda _m t_{j-1} }} \right) \sin \varphi _{j-1} \\&-\frac{r_{2m} \left( {al_1 +l_3 } \right) }{a\lambda _m }\sum _{j=1}^n i_{0m\left( {j-1} \right) } \sin \varphi _{j-1} \\&\quad \times \left( {e^{-\lambda _m t_j }-e^{-\lambda _m t_{j-1} }} \right) \\&+\frac{r_{2m} \left( {al_1 +l_3 } \right) }{a\left( {a+\lambda _m } \right) }\sum _{j=1}^n i_{0m\left( {j-1} \right) } \sin \varphi _{j-1} \end{aligned}$$
$$\begin{aligned}&\quad \times \left( {e^{-(a+\lambda _m )t_j +at_{j-1} }-e^{-\lambda _m t_{j-1} }} \right) \\ \mathrm{A}_j= & {} \lambda _m e^{-\lambda _m t_j }\sin \left( {\omega _{m\left( {j-1} \right) } \left( {t_j -t_{j-1} } \right) +\varphi _{j-1} } \right) +\omega _{m\left( {j-1} \right) }\\&\quad \times e^{-\lambda _m t_j }\cos \left( {\omega _{m\left( {j-1} \right) } \left( {t_j -t_{j-1} } \right) +\varphi _{j-1} } \right) \\&-\,\lambda _m e^{-\lambda _m t_{j-1} }\sin \varphi _{j-1} -\omega _{m\left( {j-1} \right) } e^{-\lambda _m t_{j-1} }\cos \varphi _{j-1}\\ \mathrm{B}_j= & {} \omega _{m\left( {j-1} \right) } e^{-\lambda _m t_j }\sin \left( {\omega _{m\left( {j-1} \right) } \left( {t_j -t_{j-1} } \right) +\,\varphi _{j-1} } \right) \\&-\,\lambda _m e^{-\lambda _m t_j }\cos \left( {\omega _{m\left( {j-1} \right) } \left( {t_j -t_{j-1} } \right) +\varphi _{j-1} } \right) \\&-\,\omega _{m\left( {j-1} \right) } e^{-\lambda _m t_{j-1} }\sin \varphi _{j-1} +\lambda _m e^{-\lambda _m t_{j-1} }\cos \varphi _{j-1} \\ g_0= & {} \frac{r_{2m} l_3 +al_4 L_m }{2L_m d_{1m} \lambda _m ^{2}a}\left( {\sum _{k=0}^n {J_{mk} } -i_{0m\left( n \right) } \sin \varphi _n } \right) \\ \end{aligned}$$
$$\begin{aligned} g_2= & {} -\frac{K\left( {al_1 +l_3 } \right) }{2d_{1m} \left( {\lambda _m ^{2}-a^{2}} \right) } +\frac{r_{2m} \left( {al_1 +l_3 } \right) }{2L_m d_{1m} \left( {\lambda _m ^{2}-a^{2}} \right) }\\&\quad \times \left[ {\sum _{k=0}^n {\Gamma _k } -\frac{1}{a}\sum _{k=0}^n {J_{mk} e^{at_k }} +\left( {\frac{\omega _{m\left( n \right) } \cos \varphi _n -a\sin \varphi _n }{a^{2}+\omega _{m\left( n \right) } ^{2}}+\frac{\sin \varphi _n }{a}} \right) i_{0m\left( n \right) } e^{at_n }} \right] \\ g_3= & {} \frac{i_{0m\left( n \right) } \left[ {ar_{2m} l_3 +a^{2}l_4 L_m +\omega _{m\left( n \right) } ^{2}\left( {l_4 L_m -l_1 r_{2m} } \right) -\left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \omega _{m\left( n \right) } l_2 L_m } \right] }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \\ g_4= & {} \frac{-r_{2m} \left( {al_1 +l_3 } \right) i_{0m\left( n \right) } \omega _{m\left( n \right) } }{2L_m d_{1m} \left( {a^{2}+\omega _{m\left( n \right) } ^{2}} \right) \left( {\lambda _m ^{2}+\omega _{m\left( n \right) } ^{2}} \right) } \end{aligned}$$

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Wang, Y., Wang, R. An improved neuronal energy model that better captures of dynamic property of neuronal activity. Nonlinear Dyn 91, 319–327 (2018). https://doi.org/10.1007/s11071-017-3871-9

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