Download Computational Modeling Methods for Neuroscientists by Erik De Schutter PDF

By Erik De Schutter

A consultant to computational modeling tools in neuroscience, masking 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)

Example text

Consider the passive cable (with time constant, t and length constant, l) of length l (see chapter 10, section 1): t qV q2V ¼ ÀV ðx; tÞ þ l 2 2 qt qx ð1:53Þ Di¤erential Equations 29 V ð0; tÞ ¼ Va qV ðl; tÞ ¼ 0: qx We consider the potential at N þ 1 discrete locations spaced an amount k ¼ l=N apart, and define Vj ðtÞ ¼ V ð jDx; tÞ where j ¼ 0; . . ; N. We approximate the spatial derivative by the di¤erence: q 2 V Vjþ1 À 2Vj þ VjÀ1 A qx 2 Dx 2 ð1:54Þ and obtain the set of ODEs: tVj0 ¼ ÀVj þ ðl 2 =Dx 2 ÞðVjþ1 À 2Vj þ VjÀ1 Þ: ð1:55Þ At j ¼ 0 and j ¼ N, we need to take the boundary conditions into account.

In a sense, the local search methods we have seen so far are methods for which the exploration–exploitation balance bends to the exploitation side only. This section describes deterministic global methods and the next one is devoted to stochastic ones. In both, we will keep a careful eye on the exploration–exploitation balance. The Brute Force Method The simplest way to explore a parameter space is to systematically scan it. This is done with a grid of M equally distant points in each of the N dimensions of the problem.

36 P. Achard, W. Van Geit, and G. 4 Electrophysiological traces and phase-plane trajectories. Simple spiking activity (a) leads to a sharp trajectory in the phase plane (b) while bursts (c) populate a wider region of the phase plane (d). A simpler solution has been proposed by LeMasson (LeMasson and Maex, 2001) with a fitness function based on the phase-plane trajectory. 4b. The derivative dV ðtÞ=dt is simply the slope of the voltage trace V ðtÞ and can easily be approximated by dV ðtÞ DV ðtÞ V ðt þ 1Þ À V ðt À 1Þ A ¼ ; dt Dt 2DT ð2:5Þ where DT stands for the time interval between two voltage measurements.

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