Tsuda examines the potential contribution of nonlinear dynamical systems, with many degrees of freedom, to understanding brain function. We offer suggestions concerning symmetry and transients to strengthen the physiological motivation and theoretical consistency of this novel research direction: Symmetry plays a fundamental role, theoretically and in relation to real brains. We also highlight a distinction between chaotic "transience" and "itineracy"

Breakspear, M

Friston, K

English

UCL Discovery

Behavioral and Brain Scienceshttp://journals.cambridge.org/BBSAdditional services for Behavioral and Brain Sciences:Email alerts: Click hereSubscriptions: Click hereCommercial reprints: Click hereTerms of use : Click hereSymmetries and itineracy in nonlinear systems with many degrees of freedomMichael Breakspear and Karl FristonBehavioral and Brain Sciences / Volume 24 / Issue 05 / October 2001, pp 813 ­ 813DOI: 10.1017/S0140525X01250092, Published online: 15 November 2002Link to this article: http://journals.cambridge.org/abstract_S0140525X01250092How to cite this article:Michael Breakspear and Karl Friston (2001). Symmetries and itineracy in nonlinear systems with many degrees of freedom. Behavioral and Brain Sciences,24, pp 813­813 doi:10.1017/S0140525X01250092Request Permissions : Click hereDownloaded from http://journals.cambridge.org/BBS, IP address: 144.82.107.151 on 13 Aug 2012on. For example, Kazanovich and Borisyuk (1994; 1999) studiedoscillatory networks with a central element and the applications ofthis network to modelling attention focus formation. The mecha-nism for controlling the system dynamics is based on synchroni-sation of neural activity. It has been shown that the regime of par-tial synchronisation is very promising for the description ofneurodynamics. In this regime, some oscillators work synchro-nously with a central element forming a temporally existing at-tractor. Makarenko and Llinas (1998) have applied the synchroni-sation principle to study phase synchronisation of chaotic systemsand model the activity of inferior olivary neurons.Conclusion: The chaotic neurodynamics seems a very intriguingand promising mathematical technique. Further research shouldbe done in mathematics and neuroscience to understand themeaning of chaotic dynamics for modelling of information pro-cessing in the brain.Symmetries and itineracy in nonlinearsystems with many degrees of freedomMichael Breakspeara and Karl FristonbaDepartment of Mathematics, University of Surrey, Guildford, GU2 5XH,United Kingdom and Brain Dynamics Centre, Westmead Hospital,Westmead, NSW, 2450, Australia; bThe Wellcome Department of CognitiveNeurology, Institute of Neurology, London, WC1N 3BG,United Kingdom.mbreak@physics.usyd.edu.au k.friston@fil.ion.ucl.ac.ukAbstract: Tsuda examines the potential contribution of nonlinear dynam-ical systems, with many degrees of freedom, to understanding brain func-tion. We offer suggestions concerning symmetry and transients tostrengthen the physiological motivation and theoretical consistency of thisnovel research direction: Symmetry plays a fundamental role, theoreticallyand in relation to real brains. We also highlight a distinction betweenchaotic “transience” and “itineracy.”Attractor networks and brain-like neural systems: Symmetryis the missing link. Symmetry has been central to the conceptualdevelopment of the dynamics of high dimensional nonlinear sys-tems, but is a notable absentee from Tsuda’s target article. Basinriddling was first described in the context of dynamical systemswith symmetry (Alexander et al. 1992, not Grebogi et al. 1987).Symmetrical systems remain the focus of research into basin rid-dling (e.g., Heagy et al. 1994) and attractor networks (Buono 2000;Kaneko 1997). Symmetries play the crucial role of enforcing thedynamical invariance of low-dimensional linear manifolds. Whenthese manifolds support chaotic sets, Milnor attractors and basinriddling can arise naturally (unlike the “exceptional” examples thatoriginally motivated Milnor [1985] such as depicted in Fig. 5).Basin riddling occurs when these sets also contain a dense set ofperiodic orbits which are repelling in the transverse direction.However, typical orbits (of full measure) are transversely attract-ing. Hence, the “natural” transverse Lyapunov exponent associ-ated with the chaotic set is negative (not positive as erroneouslystated in para. 6 of sect. 3.4). It is only low-order periodic orbits(that have zero measure) that are associated with positive Lya-punov exponents (and hence “connections” to other attractors).Approaching high dimensional systems from the perspective ofsymmetry thus permits a clear understanding of the mechanismsof “weak” instability. In addition, it is possible to exploit the differ-ent degrees of symmetry exhibited by the attractors to construct arigorous classification and ordering of the network (e.g., Ashwinet al. 1992). This permits an improvement on the vague notion ofattractors arbitrarily distributed throughout phase space, as de-picted in Figure 4.Symmetries arise naturally in systems of coupled nonlinear os-cillators (Field et al. 1996). Brain-like neural systems are charac-terised by networks of coupled nonlinear oscillators – from thescale of the neuron, up to the scale of the macrocolumn. In thesesystems dense local excitatory and inhibitory interconnectionsconstruct individual “nodes,” which are coupled into larger en-sembles by sparser long-range excitatory connections. Thus, theorganisation of the brain motivates a study of coupled nonlinearsystems and, hence, symmetry. Moreover, the attractors of sym-metrical systems represent synchronous oscillations among clus-ters of nodes of different sizes (Kaneko 1997) which strengthensthis motivation. Attractor networks in symmetrical systems havebeen used to model normal olfactory perception (Breakspear2000), visual hallucinations (Bressloff 2001) and animal gaits(Buono et al. 2000). In contrast, systems with skew-product struc-ture (as considered by Tsuda) are not well motivated, becausenearly all brain interactions are reciprocal (even the LGN of thethalamus sends many projections to the retina). Symmetry con-siderations may strengthen the relevance of Tsuda’s interestingand original proposals.Saddles, chaotic transients, and noise: The need for clarity.Tsuda is correct in pointing out that it may be more relevant tostudy transient or itinerant behaviour rather than attractors in dy-namic systems where inputs and parameters change relentlessly(Friston 1997). However, it is important to ensure clarity and con-sistency in the use of the terms “transience” and “itineracy.” Tra-ditionally, the term “chaotic transience” was applied in the fol-lowing way (Greborgi et al. 1983): A chaotic attractor (A), subjectto some parameter perturbation, “collides” with its own basinboundary. Subsequently, orbits on the attractor are mapped intoanother basin and subsequently onto another attractor. Put an-other way, A is no longer an invariant of the dynamic. However, alarge set of initial conditions will still approach the region of A(now an attractor “ruin”) and transiently mimic the behaviour ofthe former attractor, before collapsing onto the alternative attrac-tor. After this collapse, the transient is not seen again unless thesystem’s parameters are tuned back in the opposite direction. Ifthis is the case, attractors may constantly be “ruined” and then “re-built.” Such relatively rapid changes in attractors may be effectedby NMDA-receptor mediated changes in the underlying controlparameters (Friston 1997).On the other hand, the process of chaotic itineracy – whichTsuda exploits – occurs by a different mechanism. A chaotic at-tractor, A, is subject to a parameter perturbation that weakens itstransverse stability. At some critical point (the blowout bifurca-tion), the transverse Lyapunov exponent for the attractor (the nat-ural measure) becomes positive (Ashwin et al. 1996). A is then asaddle, not an attractor ruin. Note that A is still an invariant of thedynamic, but will attract only a zero measure set of initial condi-tions. However, if the phase space contains many such saddles, itmay be that typical orbits relentlessly shadow these saddles.Hence the evolution of the system is characterised by irregularswitching between different types of itinerant chaos correspond-ing to the shadowing of different saddles. This tuning of the dy-namics into a regime of saddle networks may be achieved by en-during monoamine-mediated changes in functional synapticcoupling (Breakspear 2000).In summary, there are two types of “transient chaos” with po-tentially distinct neurophysiological mechanisms. “Chaotic tran-sience” induced by dynamically changing control parameters and“Chaotic itineracy” due to an invariant but complex manifold (dis-cussed as engendering type 1 and type 2 complexity in Friston2000). As brain science calls more upon dynamical systems theory,it is important to keep such distinctions clear.Summary. The progression from autonomous, low-dimensionalstrange attractors to systems with noise and many degrees of free-dom represents an important advance in the theory of neural sys-tems (Wright 2000). The present paper by Tsuda outlines manypotential computational benefits of this progression. Yet, it is crit-ically important that due respect is paid to both neurophysiologyand nonlinear theory, before another magic “man in the machine”in cognitive neuroscience research takes shape.Commentary/Tsuda: Toward an interpretation of dynamic neural activity in terms of chaotic dynamical systemsBEHAVIORAL AND BRAIN SCIENCES (2001) 24:5 813

Symmetries and itineracy in nonlinear systems with many degrees of freedom

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