By Christoph Börgers
This e-book is meant as a textual content for a one-semester path on Mathematical and Computational Neuroscience for upper-level undergraduate and beginning graduate students of arithmetic, the traditional sciences, engineering, or laptop science. An undergraduate advent to differential equations is greater than enough mathematical historical past. just a narrow, excessive school-level history in physics is thought, and none in biology.
Topics comprise versions of person nerve cells and their dynamics, versions of networks of neurons coupled through synapses and hole junctions, origins and features of inhabitants rhythms in neuronal networks, and versions of synaptic plasticity.
An vast on-line choice of Matlab courses producing the figures accompanies the publication.
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Additional resources for An Introduction to Modeling Neuronal Dynamics
C) Re-compute Fig. 4 μA/cm2 to convince yourself that the resulting voltage trace is very close to that of Fig. 4. Chapter 6 The Classical Hodgkin-Huxley PDEs The model proposed by Hodgkin and Huxley in 1952 is not a set of ODEs, but a set of PDEs — the dependent variables are not only functions of time, but also of space. This dependence will be neglected everywhere in this book, except in the present chapter. You can therefore safely skip this chapter, unless you are curious what the PDE-version of the Hodgkin-Huxley model looks like, and how it arises.
In the early 1850s, the transatlantic cable project motivated the great Scottish physicist William Thomson, nowadays known as Lord Kelvin, to study the physics of leaky cables immersed in water. He showed that the voltage would diﬀuse along the length of the cable; thus he derived the term proportional to vzz that appears in the Hodgkin–Huxley PDE. 1. 8), we used that lim Δz→0 v(z + Δz, t) − 2v(z, t) + v(z − Δz, t) ∂2v = 2 (z, t) = vzz (z, t). 2 Δz ∂z Explain why this is true using l’Hospital’s rule or, better, Taylor’s theorem.
Chapter 6 The Classical Hodgkin-Huxley PDEs The model proposed by Hodgkin and Huxley in 1952 is not a set of ODEs, but a set of PDEs — the dependent variables are not only functions of time, but also of space. This dependence will be neglected everywhere in this book, except in the present chapter. You can therefore safely skip this chapter, unless you are curious what the PDE-version of the Hodgkin-Huxley model looks like, and how it arises. When there is no piece of silver wire threaded through the axon, that is, when there is no space clamp, the membrane potential v, as well as the gating variables m, h, and n, become dependent on the position on the neuronal membrane.
An Introduction to Modeling Neuronal Dynamics by Christoph Börgers