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Download e-book for kindle: Advanced Models of Neural Networks: Nonlinear Dynamics and by Gerasimos G. Rigatos

By Gerasimos G. Rigatos

ISBN-10: 3662437635

ISBN-13: 9783662437636

ISBN-10: 3662437643

ISBN-13: 9783662437643

This publication offers a whole learn on neural constructions showing nonlinear and stochastic dynamics, elaborating on neural dynamics through introducing complex types of neural networks. It overviews the most findings within the modelling of neural dynamics when it comes to electric circuits and examines their balance homes with using dynamical structures idea.

It is acceptable for researchers and postgraduate scholars engaged with neural networks and dynamical structures theory.

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Additional info for Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons

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Tk //. VR ; ÁR / D 0. The closed trajectories are associated with periodic solutions. tk C T //. Another useful parameter is the nullclines. V; Á/ D 0. V; Á/ D 0. The fixed points (or equilibria) are found on the intersection of nullclines. 1 Local Linearization A manner to examine stability in nonlinear dynamical systems is to perform local linearization around an equilibrium. x0 ; u/ D 0 that is x0 is the local equilibrium. 36) @fn @un The eigenvalues of matrix A define the local stability features of the system: Example 1.

5 The FitzHugh–Nagumo Model of Neurons The FitzHugh–Nagumo model is a two-dimensional simplified form of the Hodgkin–Huxley model and in several cases represents efficiently the membrane’s voltage dynamics in neurons [16, 55]. The model captures all the qualitative properties of the Hodgkin–Huxley dynamics, in the sense that it results in the same bifurcation diagram (loci of fixed points with respect to model’s parameters). In Eq. 54) variable V stands for the membrane’s voltage, while variable w is known as recovery variable and is associated with the ionic currents of the membrane.

6 Cable equivalent circuit of the neuron’s membrane Fig. x; t / Assuming that along the cable (dendrite) there are parallel ion channels, which are noted by the resistance RM , for the variation of the current along the neuron’s axis it holds (see Fig. 7). 41) q M and M D cM rM . 0/. The solution of the partial differential equation given in Eq. x; t /. e. when @V@tM D 0, the partial differential equation becomes an ordinary differential equation of the form 2 d 2 Vss dx2 From Eq. 1. x/ D Time constant rL I0 ˛2 e affects the variation of voltage.

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Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons by Gerasimos G. Rigatos

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