Abstract
The brain is without doubt the most complex adaptive system known to humanity, arguably also a complex system about which we know very little. Throughout this book we have considered and developed general guiding principles for the understanding of complex networks and their dynamical properties; principles and concepts transcending the details of specific layouts realized in real-world complex systems. We follow the same approach here, considering the brain as a prominent example of what is called a cognitive system, a specific instance of what one denotes, cum grano salis, a living dynamical system.
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Notes
- 1.
- 2.
Humans can distinguish cognitively about 10–12 objects per second.
- 3.
See Edelman and Tononi (2000).
- 4.
- 5.
See Crick and Koch (2003).
- 6.
Note that neuromodulators are typically released in the intercellular medium from where they physically diffuse towards the surrounding neurons.
- 7.
This is a standard result for so-called Hopfield neural networks, see e.g. Ballard (2000).
- 8.
A neural network is denoted “recurrent” when loops dominate the network topology.
- 9.
For a mathematically precise definition, a memory is termed fading when forgetting is scale-invariant, viz having a power law functional time dependence.
- 10.
We note that general n-point interactions could be generated additionally when eliminating the interneurons. “n-point interactions” are terms entering the time evolution of dynamical systems depending on (n − 1) variables. Normal synaptic interactions are 2-point interactions, as they involve two neurons, the presynaptic and the postsynaptic neuron. When integrating out a degree of freedom, like the activity of the interneurons, n-point interactions are generated generally. The postsynaptic neuron is then influenced only when (n − 1) presynaptic neurons are active simultaneously. n-point interactions are normally not considered in neural networks theory. They complicate the analysis of the network dynamics considerably.
- 11.
Here we use the term “transient attractor” as synonymous with “attractor ruin”, an alternative terminology from dynamical system theory.
- 12.
A possible mathematical implementation for the reservoir functions, with α = w, z, is \(f_{\alpha }(\varphi )\ =\ f_{\alpha }^{(\min )}\, +\, \left (1 - f_{\alpha }^{(\min )}\right ) \frac{\mathrm{atan}[(\varphi -\varphi _{c}^{(\alpha )})/\Gamma _{\varphi }]-\mathrm{atan}[(0-\varphi _{c}^{(\alpha )})/\Gamma _{\varphi }]} {\mathrm{atan}[(1-\varphi _{c}^{(\alpha )})/\Gamma _{\varphi }]-\mathrm{atan}[(0-\varphi _{c}^{(\alpha )})/\Gamma _{\varphi }]}\). Suitable values are \(\varphi _{c}^{(z)} = 0.15\), \(\varphi _{c}^{(\text{w})} = 0.7\) \(\Gamma _{\varphi } = 0.05\), f w (min) = 0. 1 and \(f_{z}^{(\min )} = 0\).
- 13.
A Kohonen network is an example of a neural classifier via one-winner-takes-all architecture, see e.g. Ballard (2000).
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Gros, C. (2013). Elements of Cognitive Systems Theory. In: Complex and Adaptive Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36586-7_8
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