Additive Noise Tunes the SelfOrganization in Complex Systems
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Glossary
 Additive noise

Random fluctuations that add to the phase space flow of model systems.
 Center manifold theorem

Mathematical theorem describing the slaving principle in complex systems.
 Slaving principle

Units in a complex system that interact nonlinearly with other units evolve on different time scales. Close to instability points, fast units obey the dynamics of slow units and are enslaved by them. Such units may be spatial modes in spatially extended systems or neural ensembles in neural populations.
Introduction
The dynamics of natural systems is complex, e.g., due to various processes and their interactions on different temporal and spatial scales. Several of such processes appear to be of random nature, i.e., they cannot be predicted by known laws. In this context, it is not necessary to know whether these processes are random in reality or whether we just do not know their deterministic law and they appear to be random. The insight that unknown laws of processes may be replaced or modelled by laws for random processes is helpful in modelling complex systems. Examples for such a replacement are manifold, and we mention model parametrization in meteorology (Noilhan and Planton 1989) and stimulus parametrization in biology (Doiron et al. 2004).
Considering random processes (or noise) in dynamical models, it is important how they are included. If the randomness is taken into account in multiplicative factors, e.g., parametrizing the unknown underlying dynamics of the factor, we call this multiplicative noise. Its effect has been studied extensively for the last decades in physics and mathematics, e.g., see the books of Horsthemke and Lefever (1984) and GarciaOjalvo and Sancho (1999). Conversely, additive noise is included in a model when the randomness is just added to the phase space flow. For instant, considering a model of differential equations in time additive noise is just added to the temporal deviation over time. For a long time, it has been known that multiplicative noise easily shifts the stability of systems, i.e., may shift bifurcations, whereas additive noise does not. This paradigm has been challenged recently in the studies of spatially extended systems (Hutt et al. 2007, 2008; Hutt 2008) and delayed systems (Lefebvre et al. 2012; Lefebvre and Hutt 2013; Hutt et al. 2012; Hutt and Lefebvre 2016). These studies show that additive noise may induce bifurcation shifts close to bifurcation points. This recent finding is illustrated and explained in a later section. Moreover, additive noise may not only affect the stability of systems close to instability points, but may also tune intrinsic time scales. We show in a later section that this effect occurs close and far from the bifurcation point.
Taking a close look at the complex systems subjected to additive noise, one learns that additive noise affects the coupling of the systems elements. Such elements may be spatial modes in spatially extended systems or microscopic elements, such as single neurons, whose interactions generate novel macroscopic order parameter modes, such as the macroscopic population dynamics. To illustrate such an interaction before we apply the concept to complex problems, we present briefly the major elements of the slaving principle in synergetics (Haken 1996, 2004) in the subsequent section and put it into relation to its mathematical equivalent, the center manifold theorem.
Slaving Principle and Center Manifold Theorem
We start with the illustration of the major concept of the center manifold theorem and finally put the concept into a physical context to explain the slaving principle.
At the Stability Threshold
For the moment, we assume α = 0. Then close to the fixed point y evolves in the stable subspace spanned by the eigenvector (0, 1)^{ t } of the linearized system with corresponding eigenvalue λ_{2} < 0 and x evolves in the center subspace. We observe that x = 0 is an invariant manifold which is a stable manifold of the origin since dx/dt = 0 and dy/dt < 0. Hence for initial points (0, y _{0})^{ t } the system evolves on the stable manifold.
In physical terms, the variable x evolves on a much larger time scale than y since the time scales are inversely proportional to the corresponding eigenvalues of the linearized system. Moreover, the variable y obeys the slow variable x on the center manifold. In other words, the slow variable x enslaves the fast variable y and determines the dynamics of the full system. This prominent role of x is the reason why it is called an order parameter. Hence at bifurcation points, the slow variables enslave the fast variables. This slaving concept applies at all bifurcations that fulfil the rather general conditions of the center manifold and allows to describe most bifurcations observed in nature (Haken 1983), be oscillatory instabilities in the laser (Haken 1985) or human motorcoordination phase transitions in the brain (Fuchs et al. 1992; Jirsa et al. 1995). By virtue of the generality of this concept, it is called slaving principle. It is often formulated equivalently by an adiabatic approach in which the fast slaved variable decays rapidly and follows the slow order parameter dynamics (Haken 1996; Schanz and Pelster 2003; Schoener and Haken 1986).
About the Stability Threshold
This solution extends naturally the case α = 0.
The latter discussion assumes deterministic dynamics, while stochastic dynamics on center manifolds close to bifurcation points can be studied as well. This is shown in the subsequent section.
Additive Noise in LowDimensional Models: Stochastic Center Manifold Theory
The effects of additive noise emerge in multidimensional systems, e.g., in lowdimensional nondelayed systems or in infinitedimensional delayed systems. The subsequent sections consider both cases.
Nondelayed Systems
Equation (3) and Fig. 1 reveal that the bifurcation point of the order parameter u _{ c } is shifted by additive noise in the slaved mode u _{0}. The underlying mechanism is known from multiplicative noise and can be understood as follows: the fast mode u _{0} is stochastic and nonlinear coupling \( {u}_c{u}_0^n \) with even order n yields an effective noise shift since \( \left\langle {u}_0^n\right\rangle \ne 0 \), whereas nonlinear coupling at odd order does not yield a shift since \( \left\langle {u}_0^n\right\rangle =0 \). In sum, additive noise in a mode that is nonlinearly coupled at even order to the order parameter dynamics acts like multiplicative noise and hence tunes the bifurcation.
Delayed Systems
Additive Noise in Discrete Network Models
To extend the gained results of additive noise to large and more realistic systems, now we consider network models evolving far from bifurcation points.
Neural Mass Network
The last equation expresses the assumption that all columns and rows are statistically independent from each other.
Analysis of the Global Synchronization
Global Mode
Fluctuation Modes
Merging Global and Fluctuation Dynamics
Equation (30) describes the meanfield evolution of the global mode and permits to illustrate coherent structures. If ū = 0, then network elements are not coherent, whereas ū ≠ 0 reflects coherent activity. In the following examples, we will see that coherence emerges in certain frequency bands dependent of the external noise level σ.
Hence the stability and timescale of solutions about the stationary state depends on the nonlinear gain. If the level of external noise increases (decreases), the transfer function becomes more flat (steep) and γ at upper and lower stationary state in Fig. 4 γ increases (decreases). In case of an oscillatory stable stationary state, the noise level determines the frequency of solutions (Hutt et al. 2016).
Coupling Induces SelfOrganization in the Presence of Noise and Noise Affects System Frequency
We compute the timedependent spectral power distribution and the phaselocking value (PLV) (Lachaux et al. 1999) in the course of time. The spectral power is computed by a windowed Fourier Transform with a 4swindow width, the PLV is computed as the circular variance of phases of 30 randomly chosen elements for each timefrequency pair. The phases result from a Morlet wavelet transform. The maximum value PLV = 1 reflects complete synchronization in the network, whereas the minimum value PLV = 0 reflects vanishing synchrony in the network. Figure 5 shows that the network elements do not synchronize at low coupling strengths since power and PLV are low. However, synchronization emerges with larger coupling expressed by large power and large PLV at ν = 45 Hz. It is wellknown that complex systems selforganize if the interaction between subunits are large enough. This is seen in our simple example. Analytically, the stationary state u _{0} is a stable focus when power and PLV are low. When synchronization sets in at stronger coupling strength, the stable focus becomes unstable, the system oscillates along a limit cycle, and power is much stronger.
Until now, the noise level has been kept constant. Now removing the noise while retaining the coupling strength, cf. Fig. 5, lower panel for t > T = 20s, the PLV jumps to very high values while the maximum power jumps to lower values. Interestingly, the oscillation frequency with maximum power drops to ν = 40 Hz that represents the systems endogenous oscillatory rhythm in the absence of noise. This drop clearly demonstrates that systems’ frequency observed may depend heavily on the intrinsic noise level.
Analytically, this behavior can be understood by Eqs. (30) and (32) and Fig. 4: additive noise tunes the effective transfer function, thus determines the stationary state and its stability and consequently the amplitude and frequency of the systems dynamics.
Noise Can Destruct SelfOrganization While It Changes the System Frequency
Synchronization in a Spiking Neural Network
Future Directions
Additive noise may have a strong impact on complex systems. The previous sections have shown corresponding conditions and mathematical techniques. Additive noise may shift instability thresholds and tunes frequency and amplitude of rhythmic activity. We showed that additive noise in lower levels, e.g., in neurons or neural ensemble patches, may destroy synchronization in an upper level, e.g., in neural populations or populations of ensemble patches.
The insight, that additive noise affects endogenous brain activity, indicates impact of electric brain stimulation on the behavior of subjects. Corresponding experiments have been performed in the last decades, both for rhythmic stimulation (Herrmann et al. 2013) and noise stimulation (Terney et al. 2008). Perceptual learning under noise stimulation has been shown (Fertonani et al. 2011) to improve considering highfrequency noise (>100 Hz). Understanding how noise stimulation affects neural activity and how it enhances the perceptual learning is one of the great challenges in future years.
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