By Christoph Börgers
This publication is meant as a textual content for a one-semester direction on Mathematical and Computational Neuroscience for upper-level undergraduate and beginning graduate students of arithmetic, the ordinary sciences, engineering, or laptop science. An undergraduate advent to differential equations is greater than enough mathematical historical past. just a narrow, excessive school-level historical past in physics is thought, and none in biology.
Topics contain types of person nerve cells and their dynamics, types of networks of neurons coupled by way of synapses and hole junctions, origins and services of inhabitants rhythms in neuronal networks, and versions of synaptic plasticity.
An wide on-line choice of Matlab courses producing the figures accompanies the publication.
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Additional info for An Introduction to Modeling Neuronal Dynamics
As n falls to values near 0 following a spike, the potassium conductance follows less rapidly when the exponent is 2, not 4. This has the eﬀect that the hyperpolarization following a spike is deeper. 3. Surprisingly, even though the potassium current is hyperpolarizing, and g K n2 is greater than g K n4 for 0 < n < 1, ﬁring is slightly faster with g K n2 than with gK n4 . In essence, this is explained by the fact that the narrower action potentials in Fig. 4 allow less time for deep inactivation of the sodium current (Exercise 8).
2. 6). Show: (a) v(t) = 1 − e−t/τm τm I. (b) v(t) eventually reaches 1 if and only if τm I > 1. (c) If τm I > 1, then the time it takes for v(t) to reach 1 equals T = τm ln τm I . 3. Let T be the period of the LIF neuron, as given in exercise 2(c). Show that T ∼ 1 I as I → ∞. ) Thus the membrane time constant τm becomes irrelevant for very large I. 4. 6). 95 and v(T ) = 1. 95. Your formula should reveal that (T − T˜)/T is a function of T /τm only. Plot it as a function of T /τm , and discuss what the plot tells you.
Both kinds are found in many neurons in the brain. They have different properties. In particular, M-currents are active even before the neuron ﬁres, while calcium-dependent AHP currents are ﬁring-activated. 4. Adaptation currents often make the inter-spike interval increase (at least approximately) monotonically; see Fig. 1 for an example. However, it is not clear that the increase in the inter-spike interval must be monotonic. One might think that a longer inter-spike interval would give the adaptation current more time to decay substantially, and might therefore be followed by a shorter inter-spike interval, which would give the adaptation less time to decay and therefore be followed by a longer inter-spike interval again, etc.