By Gerasimos G. Rigatos
This e-book presents a whole learn on neural constructions showing nonlinear and stochastic dynamics, elaborating on neural dynamics by means of 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 houses 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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Additional info for Advanced Models of Neural Networks: Nonlinear Dynamics and Stochasticity in Biological Neurons
T /jj < , 8t 0. This means that if the state vector of the system starts in a disc of radius ı then as time advances it will remain in the same disc, as shown in Fig. 6. t /jj D x0 D 0, then the system is globally asymptotically stable. Example 1. x1 D 0; x2 D 0/. 0; 0/. x/ D 2x1 xP 1 C 2x2 xP 2 D 2x1 x2 C 2x2 . 0; 0/. Example 2. 45) The equilibrium point is x1 D 0; x2 D 0. 0; 0/. 3 Stability Analysis of the Morris–Lecar Nonlinear Model Local stability of the Morris–Lecar model can be studied round the associated equilibria.
X/ D Time constant rL I0 ˛2 e affects the variation of voltage. 44) 12 1 Modelling Biological Neurons in Terms of Electrical Circuits Fig. x; t / along dendrites’ axis. 47) where gk is the conductance of the KC channel, gNa is the conductance of the NaC channel, and gL is the conductance of the leakage channel (Fig. 8). Through an identification procedure, Hodgkin and Huxley derived specific relations for the conductances in the ionic channels KC and NaC . 48) where gN K and gN Na are the maximum values for the conductances and n, m, h are gating variables which take values between 0 and 1.
G. 1007/978-3-662-43764-3__2, © Springer-Verlag Berlin Heidelberg 2015 27 28 2 Systems Theory for the Analysis of Biological Neuron Dynamics Fig. 1 Pendulum performing oscillations frequencies which are not necessarily multiples of a basis frequency (almost periodic oscillations). 5. Chaos: A nonlinear system in steady-state can exhibit a behavior which is not characterized as equilibrium, periodic oscillation, or almost periodic oscillation. This behavior is characterized as chaos. As time advances the behavior of the system changes in a random-like manner, and this depends on the initial conditions.