Advanced Models of Neural Networks: Nonlinear Dynamics and by Gerasimos G. Rigatos

By Gerasimos G. Rigatos

This ebook offers a whole examine on neural buildings displaying nonlinear and stochastic dynamics, elaborating on neural dynamics via introducing complicated versions of neural networks. It overviews the most findings within the modelling of neural dynamics by way of electric circuits and examines their balance homes with using dynamical structures thought.

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

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30) For different values of c one has the following isoclines diagram depicted in Fig. 4. 2 Computation of Isoclines 33 Fig. 0 c=1/2 c=1/3 c=1/4 4 3 2 x 2 1 0 −1 −2 −3 −4 −5 −5 0 x Fig. x/ D x2 5 1 5 c1=−3/4 c2=−5/6 c3=−1 c4=−3/2 c5=1/2 c6=0 c7=−1/6 c8=−1/4 4 3 2 x2 1 0 −1 −2 −3 −4 −5 −5 0 x 5 1 Example 2. x1 / 0:5Cc For different values of parameter c the isoclines are depicted in Fig. 5. 3 Systems Theory and Neurodynamics Basic features that are important in the study of neurodynamics are (1) equilibria (fixed points), (2) limit cycles, (3) phase diagrams, (4) periodic orbits, and (5) bifurcations of fixed points [92, 209].

As gm grows then Hopf bifurcations may appear. With reference to the dynamic model of Eq. V Ek / D . 3 1 Modelling Biological Neurons in Terms of Electrical Circuits Inward Rectifier This current appears when the neuron becomes hyperpolarized. 95) Typical values of the parameters appearing in the above model are Vth D 85 mV and K D 5 mV. 5 Voltage Sags After hyperpolarization of the neuron takes place then an inward current appears with values between 43 and 0 mV. 6 Currents and Concentration of Ions The assumption that the concentration of ions in the inner and outer part of the membrane remains constant is in general acceptable, apart from the case of Ca2C ions.

However, there may be more ion channels across the neuron’s membrane which are described as follows [16, 60, 65]. 85) where m and h are variables taking values between 0 and 1, p and q are non-negative integers, and V is the membrane’s potential. In general, the more the membrane’s potential grows, the more parameter h gets smaller (inactivation) and the more parameter m grows (activation). Sodium channels of the Hodgkin–Huxley model have both activation and inactivation. Channel KC of the Hodgkin–Huxley model, and channels Ca2C and KC of the Morris–Lecar model do not exhibit inactivation.

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