By Gerasimos G. Rigatos

This booklet presents an entire examine on neural constructions showing nonlinear and stochastic dynamics, elaborating on neural dynamics through 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 thought.

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

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

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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.

A nonlinear system, under periodic excitation can generate oscillations with frequencies which are several times smaller (subharmonic) or multiples of the frequency of the input (harmonic). 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.

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. 2; 0/. 39) The eigenvalues of the system are 1 D 2 and 2 D 2. 0; 0/ is an unstable one. 2; 0/ is analyzed. The associated Jacobian matrix is computed again.

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