By Gerasimos G. Rigatos
This booklet presents an entire learn on neural constructions showing nonlinear and stochastic dynamics, elaborating on neural dynamics via introducing complicated types 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 conception.
It is appropriate for researchers and postgraduate scholars engaged with neural networks and dynamical structures theory.
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Extra info for Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons
If all of the eigenvalues have positive real parts, then the hyperbolic fixed point is called an unstable node or source. If the eigenvalues are purely imaginary, then one has an elliptic fixed point which is said to be a center. Case 1: Both eigenvalues of matrix A are real and unequal, that is 1 ¤ 1 ¤0. For 1 < 0 and 2 < 0 the phase diagram for z1 and z2 is shown in Fig. 7: In case that 2 is smaller than 1 , the term e 2 t decays faster than e 1 t . For 1 > 0 > 2 the phase diagram of Fig. 8 is obtained: In the latter case there are stable trajectories along eigenvector v1 and unstable trajectories along eigenvector v2 of matrix A.
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].
Moreover, using as an example, the model of biological neurons the following properties are analyzed: phase diagram, isoclines, attractors, local stability, bifurcations of fixed points, and chaotic dynamics. 1 Characteristics of the Dynamics of Nonlinear Systems Main features characterizing the stability of nonlinear dynamical systems are defined as follows [92, 209]: 1. Finite escape time: It is the finite time within which the state-vector of the nonlinear system converges to infinity. 2. Multiple isolated equilibria: A linear system can have only one equilibrium to which converges the state vector of the system in steady-state.