Controlling chaos in the brain
Nature; Aug 25, 1994; 370, 6491; ProQuest Central pg. 615
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ACKNOWLEDGEMENTS. We thank D. W. Sciama, P. Salucci, C. J. Hogan, C. J. Isham and, particularly, P. J. E. Peebles for comments; and M. Plionis for preparing Fig. 2 and for comments. P.C. received a SERC Advanced Research fellowship. We both thank the SERC for support under the QMW Visitor's Programme, and G.F.R.E. thanks the FRD (South Africa) for financial support.
ARTICLES
Controlling chaos In the brain
Steven J. Schiff*, Kristin Jerger*, Due H. Duong*, Taeun Chang*, Mark L. Spanof & William L. Ditto"
* Department of Neurosurgery, Children's National Medical Center and The George Washington University School of Medicine, Washington DC 20010, USA t Naval Surface Warfare Center, White Oak Laboratory, Silver Spring, Maryland 20903, USA i School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
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.
Following the recent theoretical prediction that chaotic physical systems might be controllable with small perturbations1,2, there has been rapid and successful application of this technique to mechanical systems3, electrical circuits", lasers5 and chemical reactions6'7. Following the demonstration of the control of chaos in arrhythmic cardiac tissue8, there are no longer any technical barriers to applying these techniques to neural tissue.
One of the hallmarks of the human epileptic brain during periods of time in between seizures is the presence of brief bursts of focal neuronal activity known as interictal spikes. Often such spikes emanate from the same region of brain from which the seizures are generated but the relationship between the spike patterns and seizure onsets remains unclear9'10. Several types
NATURE ■ VOL 370 ■ 25 AUGUST 1994
of in vitro brain slice preparations, usually after exposure to convulsant drugs that reduce neuronal inhibition, exhibit population burst-firing activity that in many ways seems analogous to the interictal spike". One of these preparations is the high potassium concentration ([K+]) model, where slices from the hippocampus of the temporal lobe of the rat brain (a frequent site of epileptogenesis in the human) are exposed to artificial cerebrospinal fluid containing 6.5-10 mM [K+]12. After exposure to high [K+], spontaneous bursts of synchronized neuronal activity originate in a region known as the third part of the cornu ammonis or CA3 (ref. 13). Impulses from the CA3 bursts are propagated through a recurrent collateral fibre tract (the Schaffer collateral fibres) from CA3 to CA1, where electrographic  seizure-like  discharges  can   frequently be
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MíneuEs
observed'*. A detailed computer model of the high {K1] burst discharges in CA3 successfully replicates many of the experimental findings'* (reviewed in ref. 16). Although it is difficult to identify determinism in long time series of such bursting activity using nonlinear prediction techniques, some evidence of determinism was recently identified17'". We sought to determine whether such neuronal bursting activity was amenable to control.
There have been substantial efforts to influence neuronal activity--with electric fields and currents. Regarding the in vitro hippocampai slice, brief direct current (d.c.) currents from non-polarizabte electrodes in the perfusion bath can influence the evoked excitability of pyramidal cells in normal [K*1 when the electric fields are suitably oriented1*. Apparently similar effects can be achieved with brief currents from monopolar polarizable microelectrodes placed directly into the tissue'0,21. To our knowledge the use of electrical stimulation to entrain spontaneous burst discharges from CA3 in high [K/] has not been attempted,
whether indirectly with electric fields or with direct stimulation of the nerve ceils themselves. Direct stimulation can be accomplished for CA3 neurons by stimulating the Mossy fibre neurons that synapse on the CA3 pyramidal cells (orthodromic stimulation) or by stimulating branches of the CA3 pyramidal cell axons (Schaffer collateral fibres) and allowing the action potentials to propagate into the pyramidal cells in a retrograde fashion (an t idroniic stimula t ion).
With the observation thai it was feasible to entrain burst discharges from CA3 with both orthodromic and antidromic stimulation (KJ. and S.J.S.. manuscript in preparation), we were in a position to ask the foilowing questions: (1) is there evidence for deterministic chaotic behaviour in this preparation; and (2) could such activity be controlled?
If one observes the timing of events from a chaotic physical system, those events are aperiodic. The timing of events evolves from one unstable periodicity to another. Furthermore the approach to these unstable periodicities shows recurring patterns
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FIG. i a, Schematic diagram of the transverse hip-pocampai slice, and arrangement of recording electrodes. Female Sprague-Dawley rats weighing 125-150 g were anaesthetfced with diethyt-ether and decapitated. Transverse slices 400 am thick were prepared from the hippocampus with a tissue chopper and placed in art Interface-type perfusion cftember at 32-35 °C. Slices were perfused with artificial cerebrospinal fluid (ACSF) flowing at 2ml mm"1  and composed  of 165mM Ha", 136 mM a-.       3.5 mM K*.       1.2 mM Ca2*, l^mMMg**,   1.25mMPOj", 24mMHC%', 1.2 mMSOf". and 10mM dextrose. After 90min of ioctirjation. slices were tested for viability by recordin 2 inV unitary population spike in the stratum pyrami response to stimulation of Schaffer collateral fibres in th turn with 100 us constant current 50-150 p,A squa delivered at 0,1 Hi through tungsten mieroelectroPes. f made with 2-4-Mft glass needle electrodes filled with With confirmation of viability, the perfusate was sw containing 8.5 mM [K' I and 141 mM p']. After 15--20 perfusion, spontaneous burst firing could be recorder CA3b. Recordings were digitized across 12 hits at 5 kH; 1200 analogue to digital converter (Axon instruments), personal computer using Axotape 2.0 (Axon Instruments intervals were measured with Datapac H (Run Technoloj purposes, a separate computer system, digitizing ac 1 kHz, identified spontaneous bursts from CA3 using ; peak amplitude detection strategy. This computer trigge (Model SS800, Grass Corp.) linked to a photoelectric si unit (Model SIU7, Grass Corp,), to deliver 100-us e square-wave pulses to the Schaffer collateral fibres tnn microelectrode. At times, double pulses consisting of pulses with 150-jiS interpulse intervals were used, b, Sr
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NATURE • VOL 370 • 25 AUGUST 1904
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ARTICLES
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which can be quantitatively understood by examining the relationship between the timing of sequential events. This can be visualized using a plot which is a type of a return map. Such
a map plots the present interval between events, /„, versus the /th previous interval, Periodicities (of period j) arc revealed on such a plot as intersections with the line of identity, J„ = /„...,•. In a chaotic system these intersections are known as unstable fixed points, These unstable fixed points are determinislically approached from a direction called the stable direction or manifold and exponentially diverge from these points along the unstable direction or manifold. This type of local geometry has the shape of a saddle. Chaos control'"'* consists of the identification and characterization of these saddles, followed by a control intervention which exploits the local geometry of the saddle to increase the periodic behaviour of the system.
We demonstrate here three separate approaches to control: periodic pacing, an implementation of chaos control theory, and the inverse of chaos control which we term anticontrol.
NATURE ■ VOL 370 - 25 AUGUST 1994
Brain slice preparation
Hippocampal slices were prepared using standard techniques'8. Glass micropipette electrodes in CA1 and CA3 were used to record neuronal activity, and a computer calculated and delivered appropriately timed control pulses. The anatomy of the transverse hippocampal slice and arrangement of electrodes are illustrated in Fig. la, and details of the experiinental preparation arc summarized in the figure legend.
Chaos Identification and control
A key feature of chaotic systems is that they contain an infinite
number of unstable periodic fixed points-". That these spontaneously active neuronal networks are chaotic was supported by evidence of unstable fixed points in the return maps with the characteristics shown in Fig. la, b. These candidate unstable fixed points met four criteria. First, the sequence of points approaches the unstable fixed-point candidate along a stable direction and diverges from it along an unstable direction,
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are simply the slopes of the respective manifolds, it is important to note that we found that the stable manifolds had negative slopes with magnitudes less than 1 and that the unstable manifolds had negative slopes with magnitudes greater than 1, further supporting the presence of chaos in these data (see Fig. 2).
Our chaos control technique begins with a learning phase consisting of the identification of unstable fixed points and the performance of local linear least-square fits to obtain the stable and unstable directions along with the rates of approach and divergence along them. The control phase consisted of waiting until the system executed a close approach to the unstable fixed point (within a small radius e) along the stable direction, followed by an intervention that modified the timing of the predicted next interval in order to place it back onto the stable manifold. In this way we use the saddle structure inherent in the chaotic dynamics to bring the system back to the desired unstable fixed point with minimal intervention.
NATURE • VOL 370 • 25 AUGUST 1994
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TABLE :?,  Summary of experiments
litxlk- p			i i i '..•! using bo	ns ol' chaos control, h single and double		Chaos control Single Double		Periodic Single	pacing Double	Anticontrol Single Double	
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ARTICLES
square deviation) and interburst interval but found no correlations. The plot of the full series of these intervals before and during control is seen in the first half of the plot in Fig. 3.
Figure lc shows 100 s of recording from the same experimental run before and after the initiation of periodic pacing. Note that now pulses immediately entrain the population bursts in the fifth tracing, but by the sixth tracing there is occasional escape behaviour evident when the population burst-fires just before the delivered pulses. The plot of the full series of these intervals before and during periodic pacing is seen in the second half of the plot in Fig. 3.
Figure 3 illustrates burst intervals before, during and after both single-pulse chaos control and single-pulse periodic pacing. Note that there is a qualitative difference in the control achieved with each method, as was seen from the traces of raw data shown in Fig. lb, c.
Figure 4 compares the effects of anticontrol, double-pulse chaos control and single-pulse chaos control for another experiment. Here, the effect of anticontrol in reducing the periodicity of the preparation is clearly seen. We also demonstrated an increase in the effectiveness of double over single pulsing for this experiment.
Figure 5a illustrates an experiment where double-pulse chaos control and periodic pacing were compared, but the degree of control with .chaos control was not as good as with periodic pacing. This experiment was exceptional, because most of the double-pulse experiments achieved very tight control (see Fig. 4, and summary in Table 1). Note that even with the increased stimulation of double pulsing, chaos control was not simply overdrive pacing, in that there were frequent escapes and recaptures of control consistent with the interactive control hypothesis.
Figure 56 illustrates the full gamut of results obtained with single-pulse chaos control. The first control trial demonstrated good control. Note that during this control sequence, the quality of the control improved. We then allowed the algorithm to re-learn the fixed point and it chose one with a longer interval, again with good control. Whether this is a manifestation of the non-stationary nature of the system, or truly reflects the presence of multiple unstable period 1 fixed points is unclear. Relearning a third time, the algorithm selected different manifolds without substantially changing the value of the fixed point, but the control is not as good. This third control run exhibited what we call partial control in the results shown in Table 1; in partial control, manifold selection was not optimal, and time intervals longer than those of the chosen fixed-point interval appear eliminated. Relearning a fourth time, a fixed point is chosen at a longer interval than the previous three attempts. This was a control failure. Nevertheless, relearning a fifth time with an even longer
fixed-point interval gave partial control. This sequence demonstrates that the quality of control is not simply a function of the frequency chosen, the control is critically dependent on the quality of the fixed points and manifolds selected.
Discussion
This is the second attempt at achieving control of a chaotic biological system8 with a derivative method of Ott, Grebogi and Yorke1. The observation of small-scale structure and the identification of stable and unstable manifolds near unstable fixed points for many of these burst-firing slices demonstrated the presence of deterministic chaos in this simple neuronal system. For this preparation, complicated control theory is not required just to achieve relatively fast periodic behaviour. Above certain frequencies, periodic pacing was effective in entraining the spontaneous burst discharges in CA3. However, the quality of control with periodic pacing was not equivalent with the chaos control method; furthermore, the chaos control method has the advantage over overdrive periodic pacing in terms of its ability to identify and track fixed points over time. In addition, the control of chaos strategy offers the ability to break up fixed-point behaviour with anticontrol. The anticontrol method used here uses a minimal number of stimuli needed to prevent periodic behaviour.
Although it was easy to observe the unstable manifolds from this preparation, it was difficult to place accurately the stable manifold. This is because the system has many degrees of freedom (that is, is high dimensional), and the expectation of fully disentangling its dynamics with a two-dimensional embedding is simplistic. In addition, increasing amounts of noise in a system will increase the frequency of escapes from control and eventually render control impossible1. Despite these difficulties, good or partial control could frequently be achieved with our implementation of chaos control. The application of new theoretical techniques recently developed for the control of high dimensional systems23 could improve the reliability of control for such neuronal preparations.
We experimented with several variants of stimulation delivery. In addition to single pulses, double pulses were used and were often more effective in achieving higher quality control. We also explored limiting the delivery of control pulses by prohibiting consecutive control pulses without an intervening spontaneous burst. This latter method was less effective than permitting consecutive pulsing.
Because this neuronal preparation shares similar characteristics with epileptic interictal spike foci, we believe these methods may be applied to such foci. Although it is impossible to predict what effect increasing the periodicity of epileptic foci will have, the opposite effect of breaking up fixed-point periodic behaviour with anticontrol could be a more useful intervention. □
Received 11 May; accepted 5 July 1994.
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ACKNOWLEDGEMENTS. We thank P. G. Aitken for his invaluable teaching and advice and S. Mahan for research assistance. We gratefully acknowledge support from an NIMH grant (SJ.S.), the Children's Research Institute (S.J.S. and T.C.), an Office of Naval Research Young Investigator Award (W.LD.), the Naval Surface Warfare Center's Independent Research Program (M.LS.)and the Physics Division of the Office of Naval Research (M.LS. and W.LD.) The contribution of software from Manugistics Inc. is greatly appreciated.
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