By Erik De Schutter
A consultant to computational modeling equipment in neuroscience, overlaying a diversity of modeling scales from molecular reactions to giant neural networks.
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Extra resources for Computational Modeling Methods for Neuroscientists (Computational Neuroscience)
PDEs are even more di‰cult to solve. For the kinds of PDEs in neuroscience, such as models for dendrites and axons, the easiest way to solve them is to divide them into small isopotential compartments and then solve the resulting large system of ODEs (see chapter 11, section 1). This is precisely what programs like NEURON and GENESIS do to simulate reconstructed neurons (see the software appendix). ODEs Here we brieﬂy discuss methods for solving ODEs using a one-dimensional model as an example: u 0 ¼ Gðu; tÞ: ð1:38Þ The easiest way to approximate this is to recall the ﬁnite-di¤erence approximation: Du ¼ Gðu; tÞ; Dt ð1:39Þ which says that the change in u is approximately equal to Du A DtGðuðtÞ; tÞ: ð1:40Þ Let us (for notational convenience) denote Dt by the parameter h, which we will call the step size of the approximation.
6d but this time f ðx 0 Þ < f ðx1 Þ, so a further point x 00 has to be tested. Since f ðx 0 Þ < f ðx 00 Þ, we do not try further and x3 is replaced by x 0 . 6e, the opposite situation occurs: f ðx 0 Þ > f ðx3 Þ, which drives to a new test point x 00 in the middle between x3 and xc . 6f shows the successive simplexes found by the algorithm that nicely converge toward the minimum, located at ð0; 0Þ. 3 Deterministic Global Parameter Search Methods Global Searches: The Exploration–Exploitation Balance The methods we have seen so far are called local because they search for a minimum of the ﬁtness function in the vicinity of the initial point(s).
7 for k ¼ 10; 12, and 14, respectively. We can therefore answer the ﬁrst question: the best curve of the three is the dotted one (k ¼ 12). 7. We also wanted to know if there is a range of k values giving good sigmoids. Our ﬁtness function (w 2 or w 2 =d:o:f) can tell us if one sigmoid (a model) is better than another one, but the limit between ‘‘good’’ and ‘‘bad’’ models is always a matter of choice. html#csq). 5 or 2 would have been good criteria too, but a value of 10 is without doubt out of bounds.