Abstract
The human nervous system is constantly responding to stimuli as we interact with our surrounding environment. Information pertaining to a stimulus is encoded in electrochemical signals transmitted by specialized nerve cells called neurons. If the intensity of a stimulus reaches a certain threshold, a signal is generated and transmitted as the result of an action potential or spike. This describes the basis for communication between the central nervous system and the rest of the body. This chapter is devoted to the presentation and implementation of several spiking neuron models beginning with the integrate-and-fire model. We take the reader through a short treatment of physical structure and electrical properties of a neuron. This will provide the background needed to understand model parameters as well as how to implement them in Python and how to run simulations of their own.
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Notes
- 1.
We will use the term spike for the remainder of the chapter, which is synonymous with the firing of an action potential.
- 2.
In a biological neuron, the membrane resistance varies over the surface area, and thus for simplicity, R m is considered to be constant and represents the average resistance across the neuronal surface.
- 3.
Leakage channels are responsible for maintaining the resting membrane potential. This is because the reversal potential, E L, serves as V rest in many simulations.
- 4.
The arrow notation ← should be thought of as assigning a particular voltage value, in this case V spike and V reset, to the value V , at time t. For example, if the membrane potential reaches the threshold value, V th set to -54 mV at time t = 3 ms, then V (t) = V (3) = V spike and V (t + 1) = V (4) = V reset. The idea of assigning these values allows us to think about the evolution of changing voltage values of a neuron over time.
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Acknowledgements
This work is based upon work supported, in part, by the U.S. Army Research Office and the DEVCOM U.S. Army Research Laboratory under grant #W911NF-21-1-0192. This work was partially supported by the Army Research Office (Grant W911NF2110192).
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Bohling, M.E., Udeigwe, L.C. (2022). The Spiking Neuron Model. In: Goldwyn, E.E., Ganzell, S., Wootton, A. (eds) Mathematics Research for the Beginning Student, Volume 2. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-08564-2_5
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