Abstract
This exposition shows that the potassium ion-channels and the sodium ion-channels that are distributed over the entire length of the axons of our neurons are in fact locally-active memristors. In particular, they exhibit all of the fingerprints of memristors, including the characteristic pinched hysteresis Lissajous figures in the voltage-current plane, whose loop areas shrink as the frequency of the periodic excitation signal increases. Moreover, the pinched hysteresis loops for the potassium ion-channel memristor, and the sodium ion-channel memristor, from the Hodgkin-Huxley axon circuit model are unique for each periodic excitation signal. An in-depth circuit-theoretic analysis and characterizations of these two classic biological memristors are presented via their small-signal memristive equivalent circuits, their frequency response, and their Nyquist plots. Just as the Hodgkin-Huxley circuit model has stood the test of time, its constituent potassium ion-channel and sodium ion-channel memristors are destined to be classic examples of locally-active memristors in future textbooks on circuit theory and bio-physics.
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Notes
- 1.
We assume \( \left| {G(x_{1} ,x_{2} , \ldots ,x_{n} ;v)} \right| < \infty \) and \( \left| {R(x_{1} ,x_{2} , \ldots ,x_{n} ;i)} \right| < \infty \, \,{\text{for }}\left\| {R(x_{1} ,x_{2} , \ldots ,x_{n} )} \right\| < \infty \).
- 2.
Assuming each lobe is enclosed by a simple closed curve without self-intersections except at the origin.
- 3.
This external excitation signal summarizes the external stimulations from neighboring neurons via the neuron’s dendrites.
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Acknowledgements
This research project was supported by an AFOSR grant no. FA 9550-13-1-0136.
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Sah, M.P., Kim, H., Chua, L. (2019). Brains Are Made of Memristors. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_11
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