Reservoir computing uses the rich dynamics of randomly connected networks as a representation on which online computation can be performed. Inhibitory synaptic plasticity may tune networks into a state of detailed balance where strong excitation is counterbalanced by strong inhibition. The resulting network patterns exhibit similarities with cortical data.

Oscillations are present in multiple brain areas, and at various frequencies. Oscillations in networks of coupled model neurons can be mathematically characterized as an instability of the stationary state of irregular firing (cf. Chs. 13 and 14 ) or as a stable limit cycle where all neurons fire in synchrony. The stability of perfectly synchronized oscillation is clarified by the locking theorem: A synchronous oscillation is stable if the spikes are triggered during the rising phase of the input potential which is the summed contribution of all presynaptic neurons. Stable synchronous oscillations can occur for a wide range of parameters and both for excitatory and inhibitory couplings.

Phase models describe neurons in the oscillatory state. If a stimulus is given while the neuron is at a certain phase, its phase shifts by an amount predicted by the phase response curves and the size of the stimulus.

Oscillatory activity has been linked to numerous brain diseases, in particular Parkinson’s. Modern protocols of DBS aim at exploiting the interaction between phase response curves, oscillations, and synaptic plasticity so as to reduce the motor symptoms of Parkinson’s disease.

The potential computational use of the rich network dynamics of randomly connected networks has been emphasized in the framework of ‘liquid computing’ ( 313 ) and ‘echo state networks’ ( 241 ) . The network dynamics can be influenced by a variety of optimization algorithms ( 241; 312; 502; 225 ) and the resulting networks can be analyzed with principles from dynamical systems ( 168; 503 ) . The theory of random neural networks has been developed around the Eigenvalue spectrum of connectivity matrices ( 413 ) and the notion of chaos ( 487 ) .

Synchronization is a traditional topic of applied mathematics ( 555; 282 ) . For pulse-coupled units, synchronization phenomena in pulse-coupled units have been widely studied in a non-neuronal context, such as the synchronous flashing of tropical fireflies ( 81 ) , which triggered a whole series of theoretical papers on synchronization of pulse-coupled oscillators, e.g., Mirollo and Strogatz ( 348 ) . The locking theorem ( 179 ) is formulated for SRM neurons which cover a large class of neuronal firing patterns and includes the leaky integrate-and-fire model as a special case (see Ch. 6 ). The more traditional mathematical theories are typically formulated in the phase picture ( 555; 282; 397 ) and have found ample applications in the mathematical neurosciences ( 201; 87 ) .

Oscillations in the visual system and the role of synchrony for feature binding has been reviewed by Singer ( 479, 480 ) . Oscillations in sensory systems have been reviewed by Ritz and Sejnowski ( 441 ) and, specifically in the context of the olfactory system, by Laurent ( 292 ) , and the hippocampus by O’Keefe and Recce ( 375 ) ; Buzsaki ( 85 ) . For oscillations in EEG, see ( 42 ) .

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