How to define energy function for memristive oscillator and map

Abstract

During the release and propagation of intracellular and extracellular ions, electromagnetic field is induced accompanying with propagation of energy flow. The firing mode is dependent on the energy level, and external energy injection will induce distinct mode transition. Exact energy function for a neuron developed from a neural circuit can be obtained directly by applying scale transformation for the physical field energy. For generic neuron models, dimensionless Hamilton energy function can be obtained by using Helmholtz theorem, and this energy function can be considered as a specific Lyapunov function. In this review, approach of energy function for memristive neuron is discussed by designing equivalent neural circuit coupled by two kinds of memristors, which are dependent on the magnetic flux and charge flux, respectively. A scheme is suggested to get equivalent energy function for memristive neuron in the form of map by introducing a scale parameter. The memristive map reduced from the memristive neuron can produce similar attractors and firing modes under applying the same parameters, and the average Hamilton energy for the map neuron is decreased because of regulation from the scale parameter. On the other hand, a memristive map is replaced by an equivalent memristive oscillator for finding an equivalent Hamilton energy function according to the Helmholtz theorem. The energy scheme can be helpful for further investigating energy property of artificial neurons, maps and discrete memristors. It also provides evidence that maps are more suitable for investigating neural activities than neuron oscillators.

Appendix

$$ \left\{ {\begin{array}{*{20}l} {A_{{11}} = \frac{{i_{s}^{\prime } + (a^{\prime } + b^{\prime } z^{2} )(x - e_{1} ) - (c^{\prime } + d^{\prime } w^{2} )(x - e_{2} ) + x - \frac{1}{2}x^{2} - \frac{1}{3}x^{3} + \frac{1}{2}(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} ) + \frac{1}{2}c^{\prime } (x - e_{2} )}}{{x + \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}w}}} \hfill \\ {A_{{22}} = \frac{{\alpha ^{2} [e - \xi y - \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) - \frac{1}{2}w - \frac{{3b^{\prime } }}{{2a^{\prime } }}z^{2} (x - e_{1} ) - \frac{1}{2}(e_{2} - e_{1} )]}}{y}} \hfill \\ {A_{{33}} = \frac{{ - 2e_{1} - (a^{\prime } z + b^{\prime } z^{3} ) - w + 2\frac{y}{{a^{\prime } }}}}{{(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} )}}} \hfill \\ {A_{{44}} = \frac{{ - 2c^{\prime } e_{2} + 2d^{\prime } w^{2} (x - e_{2} ) - c^{\prime } (a^{\prime } z + b^{\prime } z^{3} ) - c^{\prime } w - 2y}}{{(x - e_{2} )}}} \hfill \\ \end{array} } \right. $$

(A1)

$$ \begin{aligned} \left( \begin{gathered} {\dot{x}} \hfill \\ {\dot{y}} \hfill \\ {\dot{z}} \hfill \\ {\dot{w}} \hfill \\ \end{gathered} \right) = & F_{c} + F_{d} = \left( {\begin{array}{*{20}l} { - y - \frac{1}{2}(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} ) - \frac{1}{2}c^{\prime } (x - e_{2} )} \hfill \\ {\alpha x + \frac{1}{2}\alpha (a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}\alpha w + \frac{{3\alpha b^{\prime } }}{{2a^{\prime } }}z^{2} (x - e_{1} ) + \frac{1}{2}\alpha (e_{2} - e_{1} )} \hfill \\ {x + \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}w - \frac{y}{{a^{\prime } }}} \hfill \\ {c^{\prime } x + \frac{1}{2}c^{\prime } a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}c^{\prime } w + y} \hfill \\ \end{array} } \right) \\ + & \left( {\begin{array}{*{20}l} {i_{s}^{\prime } + (a^{\prime } + b^{\prime } z^{2} )(x - e_{1} ) - (c^{\prime } + d^{\prime } w^{2} )(x - e_{2} ) + x - \frac{1}{2}x^{2} - \frac{1}{3}x^{3} + \frac{1}{2}(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} ) + \frac{1}{2}c^{\prime } (x - e_{2} )} \hfill \\ {\alpha e - \alpha \xi y - \frac{1}{2}\alpha (a^{\prime } z + b^{\prime } z^{3} ) - \frac{1}{2}\alpha w - \frac{{3\alpha b^{\prime } }}{{2a^{\prime } }}z^{2} (x - e_{1} ) - \frac{1}{2}\alpha (e_{2} - e_{1} )} \hfill \\ { - e_{1} - \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) - \frac{1}{2}w + \frac{y}{{a^{\prime } }}} \hfill \\ { - c^{\prime } e_{2} + d^{\prime } w^{2} (x - e_{2} ) - \frac{1}{2}c^{\prime } (a^{\prime } z + b^{\prime } z^{3} ) - \frac{1}{2}c^{\prime } w - y} \hfill \\ \end{array} } \right); \\ = & \left( \begin{gathered} 0\quad - \alpha \quad {\kern 1pt} \; - 1\quad - c^{\prime } \hfill \\ \alpha \quad \;\;0\quad \;\;\frac{\alpha }{{a^{\prime } }}{\kern 1pt} \;\quad - \alpha \hfill \\ 1\quad - \frac{\alpha }{{a^{\prime } }}{\kern 1pt} \;\quad 0\quad \quad 0 \hfill \\ c^{\prime } \quad \alpha \quad \quad 0\;\quad \;\;0\quad \hfill \\ \end{gathered} \right)\left( \begin{gathered} x + \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}w \hfill \\ \frac{y}{\alpha } \hfill \\ \frac{1}{2}(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} ) \hfill \\ \frac{1}{2}(x - e_{2} ) \hfill \\ \end{gathered} \right)\;\; + \left( \begin{gathered} A_{{11}} \quad 0\quad {\kern 1pt} \;\;0\quad \;\;0 \hfill \\ 0\quad \;\;A_{{22}} \quad \;0\quad \;0 \hfill \\ 0\quad \quad 0{\kern 1pt} \;\quad A_{{33}} \quad 0 \hfill \\ 0\quad \quad 0\quad \quad 0\;\;\;A_{{44}} \quad \hfill \\ \end{gathered} \right)\left( \begin{gathered} x + \frac{1}{2}(a^{\prime } z + b^{\prime } z^{3} ) + \frac{1}{2}w \hfill \\ \frac{y}{\alpha } \hfill \\ \frac{1}{2}(x - e_{1} )(a^{\prime } + 3b^{\prime } z^{2} ) \hfill \\ \frac{1}{2}(x - e_{2} ) \hfill \\ \end{gathered} \right)\quad ; \\ \end{aligned} $$

(A2)