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 heritage. just a slender, excessive school-level historical past in physics is believed, and none in biology.
Topics contain versions of person nerve cells and their dynamics, versions of networks of neurons coupled by way of synapses and hole junctions, origins and capabilities of inhabitants rhythms in neuronal networks, and versions of synaptic plasticity.
An wide on-line selection of Matlab courses producing the figures accompanies the e-book.
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Additional info for An Introduction to Modeling Neuronal Dynamics
11. [LIF_VOLTAGE_TRACE] If I > 1/τm , the time it takes for v(t) to reach 1 equals T = τm ln τm I . 9) The frequency is the reciprocal of the period T . We always assume that time is measured in ms. Therefore 1/T is the number of spikes per ms. However, while milliseconds are a natural unit of time in neuroscience, the customary and natural unit of frequency is not the reciprocal millisecond, but the reciprocal second, namely the hertz (Hz). If f denotes the frequency of the LIF neuron in Hz, the relation between f and T (the period in ms) is f= 1000 .
1. 2. 3. 4. 5. 6. 7. 1 αm + βm for the RTM neuron as a function of v ∈ [−100, 50]. You can use the code that generates the red, solid curves in Fig. 1 as a starting point. (∗) Using Matlab, plot h + n for a solution of the RTM model equations. You can use the code generating Fig. 2 as a starting point. 1. (∗) In the code used to generate Fig. 2, make the modiﬁcation of setting h = 1−n instead of allowing h to be governed by its own diﬀerential equation. ) Plot the analogue of Fig. 2 with this modiﬁcation.
A) Using Taylor’s theorem, show that there exists a constant C, independent of Δt, so that y(t) − y(t − Δt) − y (t − Δt) ≤ CΔt Δt for all Δt with 0 < Δt ≤ Δtmax , where y = dy/dt. ˆ independent (b) Using Taylor’s theorem, show that there exists a constant C, of Δt, so that y(t) − y(t − Δt) ˆ 2 − y (t − Δt/2) ≤ CΔt Δt for all Δt with 0 < Δt ≤ Δtmax , where again y = dy/dt. 4. Using separation of variables, ﬁnd a solution of the initial-value problem dy = y2, dt y(0) = 1. Show that the limit of y(t) as t → 1 from the left is ∞.
An Introduction to Modeling Neuronal Dynamics by Christoph Börgers