Control of Collective Network Chaos
Control of Collective Network Chaos
Under certain conditions, the collective behavior of a large globally-coupled heterogeneous network of coupled oscillators, as quantified by the macroscopic mean field or order parameter, can exhibit low-dimensional chaotic behavior. Recent advances describe how a small set of “reduced” ordinary differential equations can be derived that captures this mean field behavior. Here, we show that chaos control algorithms designed using the reduced equations can be successfully applied to imperfect realizations of the full network. To systematically study the effectiveness of this technique, we measure the quality of control as we relax conditions that are required for the strict accuracy of the reduced equations, and hence, the controller. Although the effects are network-dependent, we show that the method is effective for surprisingly small networks, for modest departures from global coupling, and even with mild inaccuracy in the estimate of network heterogeneity.
Control of Traveling Waves in Mammalian Cortex
Neural activity can propagate as waves in the brain. Such waves of activity may be important in processing of sensory information when awake, are present during deep sleep, and may be involved with spread of epileptic seizures. In this present paper, we predicted from a mathematical model of wave propagation, then confirmed experimentally, that externally applied electrical fields can slow such waves sufficiently to stop them. Specifically, we demonstrated that by using electric fields to modulate neuronal excitability, we can speed up, slow down and even halt propagation of seizure-like waves of activity in rat brain slices. An important application of such control over the propagation of waves of activity in human brain would allow for the development of implantable seizure control electrical devices that can be used to contain seizure activity within a small localized region and thereby prevent such seizures from spreading throughout the brain.
Abstract: We experimentally confirmed predictions that modulation of neuronal threshold with electrical fields can speed up, slow down, and even block traveling waves in neocortical slices. The predictions are based on a Wilson-Cowan type integrodifferential equation model of propagating neocortical activity. Wave propagation could be modified quickly and reversibly within targeted regions of the network. To the best of our knowledge, this is the first example of direct modulation of threshold to control wave propagation in a neural systems.
Adaptive Electric Field Control of Epileptic Seizures
We describe a novel method of adaptively controlling epileptic seizure-like events in hippocampal brain slices using electric fields. Extracellular neuronal activity is continuously recorded during field application through differential extracellular recording techniques, and the applied electric field strength is continuously updated using a computer-controlled proportional feedback algorithm. This approach appears capable of sustained amelioration of seizure events in this preparation when used with negative feedback. Seizures can be induced or enhanced by using fields of opposite polarity through positive feedback. In negative feedback mode, such findings may offer a novel technology for seizure control. In positive feedback mode, adaptively applied electric fields may offer a more physiological means of neural modulation for prosthetic purposes than previously possible.
Electric
Field Suppression of Epileptiform Activity in Hippocampal Slices
- 1. The effects of relatively small external DC electric fields on synchronous activity in CA1 and CA3 from transverse and longitudinal type hippocampal slices were studied.
- 2. To record neuronal activity during significant field changes, differential DC amplification was employed with a reference electrode aligned along an isopotential with the recording electrode.
- 3. Suppression of epileptiform activity was observed in 31 of 33 slices independent of region studied and type of slice but was highly dependent on field orientation with respect to the apical dendritic-somatic axis.
- Modulation of neuronal activity in these experiments was readily observed at field strengths <5-10 mv/mm. Suppression was seen with the field oriented (positive to negative potential) from the soma to the apical dentrites.
- In vivo application of these results may be feasible.
Controlling Chaos in the Brain
In a spontaneously bursting neuronal network in vitro, chaos can be demonstrated by the presence of unstable fixed-point behaviour. Chaos control techniques can increase the periodicity of such neuronal population bursting behaviour. Periodic pacing is also effective in entraining such systems, although in a qualitatively different fashion. Using a strategy of anticontrol such systems can be made less periodic. These techniques may be applicable to in vivo epileptic foci.
Control and synchronization of chaos in high dimensional systems: Review of some recent results
Tracking Unstable Periodic Orbits in Nonstationary High-Dimensional Chaotic Systems: Method and Experiment
We consider the adaptive control of chaos in nonstationary high-dimensional dynamical systems. In particular, we propose and experimentally implement a technique to stabilize and track unstable periodic orbits based on the use of time series. In our technique, the position of the periodic orbit and other parameters in the controller are continually updated from recent measurements of the systems state and perturbation histories, while the environment, simulated by one or several of the system's parameters, drifts independent of the control algorithm. We demonstrate the effectiveness of the technique computationally for the Henon map, a chemical reaction model, and a coupled driven Duffing oscillator, and experimentally for a magnetoelastic ribbon system.
Controlling Chaos in High Dimensions: Theory and Experiment
The main contribution of this work is the development of a high-dimensional chaos control method that is effective, robust against noise, and easy to implement in experiment. Assuming no knowledge of the model equations, the method achieves control by stabilizing a desired unstable periodic orbit with any number of unstable directions, using small time-dependent perturbations of a single system parameter. Specifically, our major results are as follows. First, we derive explicit control laws for time series produced by discrete maps. Second, we show how to apply this control law to continuous-time problems by introducing straightforward ways to extract from a continuous-time series a discrete time series that measures the dynamics of some Poincare map of the original system. Third, we illustrate our approach with two examples of high-dimensional ordinary differential equations, one autonomous and the other periodically driven. Fourth, we present the result on our successful control of chaos in a high-dimensional experimental system, demonstrating the viability of the method in practical applications.
Controlling Chaos using Time Delay Coordinates via Stabilization of Periodic Orbits
Using time delay coordinates, we propose a method that stabilizes a desired periodic orbit with an arbitrary number of unstable directions. Similar to the original OGY control algorithm, the stabilization is done via small time dependent perturbations of an accessible control parameter. However, since the system in time delay coordinates will in general depend on past parameteric variations, our parameteric control law is constructed based on the combined dynamics of the "state-plus-parameter" system.
Observing Chaos: Deducing and Tracking the State of a Chaotic System from Limited Observation
A method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar. It is assumed that an accurate mathematical description of the system is available. Noise effects on the procedure are investigated using as an example a kicked mechanical system which results in a four-dimensional dissipative map.
Observing Chaos: Deducing and Tracking the State of a Chaotic System from Limited Observation
Assuming that an accurate mathematical description of the system is available, a method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar function of the system state. Noise effects on the procedure are investigated using an example a kicked mechanical system which results in a four-dimensional dissipative map.
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