Dynamical reservoir properties as network effects

It has been proposed that chaos can serve as a reservoir providing an infinite number of dynamical states [1, 2, 3, 4, 5]. These can be interpreted as different behaviors, search actions or computational states which are selectively adequate for different tasks. The high flexibility of chaotic regimes has been noted, as well as other advantages over regular regimes. However, the model neurons used to demonstrate these ideas could be criticized as lacking physical or biological realism. In the present paper we show that the same kind of rich behavior displayed by the toy models can be found with a more realistic neural model [6]. Furthermore, much of the complex behavior arises from network properties often overlooked in the literature. 1 Chaotic spatiotemporal neural chaos and its use Following the discovery of putative chaotic regimes in electrical signals from the brain, and much scientific speculation as to the possible roles of chaos in cognition, actual computational models were proposed [1, 2, 3, 4, 5]. These models arguably explain some of what could be going on in the brain, but they also point to possible artificial devices taking advantage of the dynamical richness of chaos. To this end, a continuous-time setting is adopted and nonlinear network properties are investigated. Knowledge of general properties of nonlinear oscillators, as well as of generic networks, turns out to be very useful and can prompt a first approach to dynamical neural network modeling. This is the case of the references above, where e.g. Ginzburg-Landau and Rossler oscillators are meant to capture the essential oscillatory features of neurons. Particularly in Refs. [1, 2, 4], a full network setting is presented mimicking cortical architecture. Thus an actual spatiotemporal dynamics is unveiled, overcoming the limitations and criticism that result from working with single-unit or otherwise very small networks [4]. Unstable Periodic Orbits (UPOs) can be stabilized from within chaos, very fast and with minimum perturbation of the original system. The original chaotic attractor contains an infinite number of such dynamical modes, which could be stabilized at will according to the requirements of computational ∗The author acknowledges the partial support of Fundação para a Ciência e a Tecnologia and EU FEDER via the Center for Logic and Computation and the project ConTComp (POCTI/MAT/45978/2002), and also via the project PDCT/MAT/57976/2004. tasks. In Ref. [4], this is applied to the processing of spatiotemporal visual input patterns with different symmetries. The units (or “neurons”) are topologically arranged in a network, and the simultaneous monitoring of their state variables reveals such spatiotemporal regimes as standing and rotating waves of different symmetries, or a complex mixture thereof. 2 Nonlinear oscillators: from out-of-the-box units to more realistic neurons The mathematical models mentioned above face some criticism when comparisons are made with actual neurons. Neurons are not oscillators, even if certain cells can display autonomous rhythmic firing (a behavior we are not addressing here, but certainly nothing like an actual oscillator, especially if the membrane potential is the monitored variable). However, groups of neurons can show oscillating electrical activity, sustained by the exchange of excitation and inhibition. Neural coupling is far from the simple linear connectivity of diffusive type considered in Refs. [1, 2, 4] and other studies of networks of the reaction-diffusion type. Rather, neurons are connected via highly nonlinear transfer functions such as sigmoids. Finally, in real life there is an unavoidable delay in signal transmission between all neurons, which is usually not considered as an intrinsic property of the model networks. This includes the networks mentioned above. Hence a more realistic model is sought. The purpose is to attain just the “right” level of biological or physical plausibility, while still having a manageable model for dynamical exploration. Although proposed in a different context, the model in Ref. [6] provides a good compromise. The individual unit is a slightly more complex version of the leaky integrator, and is also called the single-compartment neuron. Passive as well as active membrane properties are considered, along with highly nonlinear coupling and delays in signal transmission between neurons. 2.1 Deriving the neuron model For lack of space, only an abbreviated account can be given here. Figure 1 illustrates the starting point: the electrical equivalent of the neural membrane.