By Shinji Doi, Junko Inoue, Zhenxing Pan, Kunichika Tsumoto (auth.)
Biological structures inherently own a lot ambiguity or uncertainty. Computational electrophysiology is the only quarter, from one of the monstrous and swiftly growing to be self-discipline of computational and structures biology, during which computational or mathematical types have succeeded. This booklet offers a realistic and quickly advisor to either computational electrophysiology and numerical bifurcation research. Bifurcation research is the most important device for the research of such hugely nonlinear organic structures. Bifurcation thought offers how to examine the impact of a parameter switch on a procedure and to realize a severe parameter price whilst the qualitative nature of the procedure adjustments. incorporated during this paintings are many examples of numerical computations of bifurcation research of varied versions in addition to mathematical types with diverse abstraction degrees from neuroscience and electrophysiology. This quantity will profit graduate and undergraduate scholars in addition to researchers in diversified fields of science.
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Extra resources for Computational electrophysiology: dynamical systems and bifurcations
Panel (b) is the waveforms of the gating variables m, n and h. Total membrane current, Na current, K current and leak current are shown in (c)–(f), respectively. 3) are nonlinear differential equations with four variables and apparently look very complicated. 3b–d) which describe the dynamics of gating variables, however, share a simple common structure. v/, x D m; n; h depend on the membrane potential v and thus vary temporally with the temporal change of v. t/ D exp. 2 0 0 c 5 10 15 20 25 d 2 G (µA/cm ) 0 600 150 100 400 50 0 –50 5 10 15 20 25 5 10 15 20 25 5 10 15 20 25 INa (µA/cm2) 250 200 e 43 200 0 0 5 10 15 20 25 2 IK (µA/cm ) 0 2 IL (µA/cm ) 5 0 –5 –10 –15 –20 –25 –200 –400 –600 –800 0 0 f 5 10 15 t (ms) 20 25 0 t (ms) Fig.
7) (Krinskii and Kokoz 1973; Rinzel 1985) is essentially the same as the one used by FitzHugh to derive the BVP model (FitzHugh 1961). 7) by Krinskii and Kokoz (1973) and by Rinzel (1985) is derived by some logical process while the BVP model is derived a priori. 7) rather than in the BVP model. For more systematic reduction of general HH-type models, see Abbott and Kepler (1990), Golomb et al. (1993), and Kepler et al. (1992). 5 Bifurcation Analysis In this subsection, we consider the dependence of the HH equations’ behavior on the parameter Iext ; the constant-current-transfer characteristic of the HH neuron (Rinzel 1978; Guttman et al.
In the case of Na channel (solid curve), when t D 0:02, it does not flow much current (almost flat). After suitable time elapses, it shows large nonlinear characteristics and then finally becomes almost flat again. The combination of these two current–voltage relations makes the I –V relation of the total current in Fig. 8. 3). The relation when t D 10:0 is considered as a steady-state current–voltage relation. 0 0 0 –1000 –1000 –2000 –2000 –3000 –3000 –4000 –4000 –40 0 40 80 120 –40 Fig. 3 Nonlinear Dynamical Analysis of the Original HH Equations Fig.