Computational Modeling Methods for Neuroscientists by Erik De Schutter

By Erik De Schutter

A advisor to computational modeling equipment in neuroscience, protecting a diversity of modeling scales from molecular reactions to giant neural networks.

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Additional info for Computational Modeling Methods for Neuroscientists (Computational Neuroscience)

Sample text

Writing so-called adaptive step-size solvers is not as easy as the fixed-step integrators, but they can dramatically increase the speed of solving ODEs. There are many adaptive step-size packages available, so most users need not worry about this. Accuracy is one important issue in solving ODEs. There is another important issue, stability, which roughly describes whether the iteration scheme will converge to a solution near the true solution. Consider x 0 ¼ Àax; xð0Þ ¼ 1; ð1:42Þ where a > 0 is large.

Let t1 ¼ t0 þ h and more generally, tn ¼ t0 þ nh. Then we can approximate the function uðtÞ at these values, u0 ¼ uðt0 Þ; u1 ¼ uðt1 Þ; . . ; un ¼ uðtn Þ, by the iteration: unþ1 ¼ un þ hGðun ; tn Þ: ð1:41Þ This iterative scheme, called Euler’s method, is the easiest to implement and is therefore quite popular although it is one of the most ine‰cient methods; in order to be accurate, h must be very small. At each step, the approximation gives an error that is roughly of size h 2 , so that at the end of a simulation the error is roughly proportional to h.

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. We have calculated the w 2 =d:o:f of 100 curves with a k value between 8 and 18. 2a. With this figure, we can define a range of good k values: 10:9 < k < 14:6. We can also find the best possible value: kbest ¼ 12:6.

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