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

By Gerasimos G. Rigatos

This ebook presents a whole research on neural buildings showing 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 when it comes to electric circuits and examines their balance houses with using dynamical platforms concept.

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

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Extra resources for Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons

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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].

The aggregate resistance of the circuit is Nk Rk D rk . 2 Equivalent Circuits of the Cell’s Membrane 7 Fig. VM ENa / C where A denoted membrane’s surface (Fig. 4). 29) Therefore, in case that there is no external current, the membrane’s voltage VM in the steady state is the weighted sum of the equilibrium potentials ECl , EK , and ENa (according to Nernst’s equation) for the three ionic channels. t / D I0 between time instant t D 0 and t D T . 30) D 4 2 0 otherwise Using Eq. t / one obtains the change in time of the membrane’s potential which is described by Eq.

8 is obtained: In the latter case there are stable trajectories along eigenvector v1 and unstable trajectories along eigenvector v2 of matrix A. 0; 0/ is said to be a saddle point. When 1 > 2 > 0 then one has the phase diagrams of Fig. 4 Phase Diagrams and Equilibria of Neuronal Models Fig. 7 Phase diagram of initial state variables x1 , x2 of a second order linear autonomous system with negative eigenvalues, where 1 < 2 <0 39 10 8 6 4 x2 2 0 −2 −4 −6 −8 −10 −15 Fig. 5 −2 −30 x1 Case 2: Complex eigenvalues: Typical phase diagrams in the case of stable complex eigenvalues are given in Fig.

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