For the nervous system to work at all, a delicate balance of excitation and
inhibition must be achieved. However, when such a balance is sought by global
strategies, only few modes remain balanced close to instability, and all other
modes are strongly stable. Here we present a simple model of neural tissue in
which this balance is sought locally by neurons following `anti-Hebbian'
behavior: {\sl all} degrees of freedom achieve a close balance of excitation
and inhibition and become "critical" in the dynamical sense. At long
timescales, the modes of our model oscillate around the instability line, so an
extremely complex "breakout" dynamics ensues in which different modes of the
system oscillate between prominence and extinction. We show the system develops
various anomalous statistical behaviours and hence becomes self-organized
critical in the statistical sense

Cecchi, Guillermo A.

Magnasco, Marcelo O.

Piro, Oreste

English

arXiv

PACS: 84.35.+i, 05.65.+b, 64.70.qj, 87.18.VfIt is widely recognized that balancing excitation and inhibition is important in the nervous system. When such a balance is sought by global strategies, few modes remain poised close to instability, and all other modes are strongly stable. Here we present a simple abstract model in which this balance is sought locally by units following "anti-Hebbian" evolution: all degrees of freedom achieve a close balance of excitation and inhibition and become "critical" in the dynamical sense. At long time scales, a complex "breakout" dynamics ensues in which different modes of the system oscillate between prominence and extinction; the model develops various long-tailed statistical behaviors and may become self-organized critical.Supported in part by MCI project CGL2008-06245-
C02-02 (O. P.).Peer reviewe

Digital.CSIC

Self-Tuned Critical Anti-Hebbian NetworksMarcelo O. Magnasco,1 Oreste Piro,2 and Guillermo A. Cecchi31Laboratory of Mathematical Physics, Rockefeller University, 1230 York Avenue, New York, New York 10065, USA2Departament de Fı´sica and IFISC(CSIC-UIB), Universitat de les Illes Balears, 07122 Palma de Mallorca, Spain3IBM Research, T. J. Watson Laboratory, 1101 Kitchawan Road, Yorktown Heights, New York 10598, USA(Received 28 August 2008; published 22 June 2009)It is widely recognized that balancing excitation and inhibition is important in the nervous system.When such a balance is sought by global strategies, few modes remain poised close to instability, and allother modes are strongly stable. Here we present a simple abstract model in which this balance is soughtlocally by units following ‘‘anti-Hebbian’’ evolution: all degrees of freedom achieve a close balance ofexcitation and inhibition and become ‘‘critical’’ in the dynamical sense. At long time scales, a complex‘‘breakout’’ dynamics ensues in which different modes of the system oscillate between prominence andextinction; the model develops various long-tailed statistical behaviors and may become self-organizedcritical.DOI: 10.1103/PhysRevLett.102.258102 PACS numbers: 84.35.+i, 05.65.+b, 64.70.qj, 87.18.VfDynamical systems theory holds that systems of interestshould be structurally stable: their behavior should notdrastically change with small perturbations of the definingdynamics [1]. Thus high-order criticality, the simultaneouspresence of several critical features such as Hopf bifurca-tions, is not expected to be ever observed in a naturalsystem. However, natural systems lacking such structuralstability are not infrequent: within neuroscience examplesinclude dynamically critical systems such as line attractors[2] in motor control [3] and decision making [4], self-tunedHopf bifurcations in the auditory periphery [5] and olfac-tory system [6], and ‘‘regulated criticality’’ models [7].There are also examples in neuroscience of statisticalcriticality [8]: spontaneous heavy-tailed or scale-free fluc-tuations typical of critical phase transitions, such as neuro-nal avalanches [9], anomalous correlations in the retina[10,11] and in functional imaging [12]; models based onthe nonlinear dynamics of spiking elements display ava-lanchelike statistical criticality [13,14]. There is no realunderstanding of a relation between these different con-cepts of criticality; developed turbulence, a well-studiedexample, displays both statistical criticality [15] and dy-namical criticality (extensive number of zero Lyapunovs[16]), but a relationship between them is far from clear.We present a simple abstract model, an anti-Hebbian[17] network which spontaneously poises itself at a dy-namically critical state: an extensive number of degrees offreedom approach Hopf bifurcations, becoming arbitrarilysensitive to external perturbations. As the dynamics con-trolling this state has itself a marginal fixed point, theeigenvalues do not converge but fluctuate, close to theimaginary axis; when they become slightly unstable, thecorresponding mode ‘‘breaks out’’ and becomes moreprominent, and as they become slightly stable the modeslowly damps out. This breakout dynamics displays ava-lanchelike activity bursts whose sizes may be power-lawdistributed. Within these epochs the neurons of our modelare slightly correlated; yet, as the number of small butsignificant correlations is high, the model has stronglycorrelated network states [10]. This system is, on the shorttime scale, sensitive in bulk to any outside input, even ifapplied only to a small subset of the neurons. However, itdoes not learn—being anti-Hebbian, it constantly forgets.We can achieve learning by adding another plasticity term‘‘positively’’ Hebbian to directed correlations, i.e., thosecausal in the sense of Granger [18]. Then the network maylearn ‘‘predictable’’ stimuli and will display timing-dependent synaptic changes reminiscent of spike-timingdependent plasticity [19]).The scope of this Letter is to demonstrate that a simpledynamics based on activity-dependent plasticity may beused, not to implement memory, but to create a largenumber of states with very special sensitivity properties;such states may be the substrate of interesting computa-tional strategies. It is meant as an ‘‘existence proof’’: weexhibit one exemplar of models displaying such behavior,and conjecture this may be just one in a large family ofsuch models. Our scope is not to reproduce in detail anyneural function; our model is simplified so only the behav-ior of interest is reproduced.We now present our model. The activities of a set ofabstract neurons, encoded in the vector x, evolve under thesynaptic connectivity matrix W; meanwhile W itselfevolves, at a slower pace , under an anti-Hebbian rule._x ¼ Wx (1)_W ¼ ðI xx>Þ; (2)where the matrixW encodes the synaptic connections,  isthe speed of synaptic evolution, assumed slow, and I is theidentity matrix. In a realistic model, to the right-hand side(rhs) of (1) would be added inputs iðtÞ, neuronal noise ðtÞ,and nonlinear limiting terms such as x3, but we shall notneed them for our purposes. For sufficiently small , _WPRL 102, 258102 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending26 JUNE 20090031-9007=09=102(25)=258102(4) 258102-1  2009 The American Physical Societyintegrates the rhs over a long period, and so we couldreplace xx> with a local running average hxx>i: the matrixW would stop evolving when the components of x haveunit variance and are uncorrelated to one another on epochsOð1=Þ. The initial conditions at t ¼ 0, Wð0Þ and xð0Þ,have elements given by independent Gaussian randomvariables with unit variance.The evolution of this system is surprisingly complex andgenerates several different time scales, as shown in Figs. 1and 2. On a time scale t  1, the eigenmode e whoseeigenvalue has the largest positive real part diverges, in-curring a large penalty _W  ee> eliminating it fromW, causing the real part to become negative; all othereigenvalues with positive real parts follow suit until noeigenvalue has a positive real part. A second dynamicalregime ensues in which the real parts of eigenvalues in-crease at a rate  and approach zero. Finally, the eigenval-ues have migrated to a strip around the imaginary axis,where they oscillate around their equilibrium positions(Fig. 2).As the rhs of (2) is symmetric, the antisymmetric com-ponent of the matrix W is an invariant of the motion; onlyits symmetric component evolves. Calling S and A thesymmetric or antisymmetric components of W_x ¼ ðAþ SÞx (3)_S ¼ ðI  xx>Þ (4)_A ¼ 0: (5)Take the time derivative of Eq. (4)€S ¼ ddt_S ¼  ddtxx> ¼ ð _xx> þ x _x>Þ: (6)Use Eq. (3) to substitute _x, Eq. (4) to solve xx> ¼ I _S=, and the symmetry of S and A to get€S ¼ 2Sþ ½A; _S þ fS; _Sg: (7)Equation (7) lives in a higher-dimensional space thanEqs. (1) and (2), and so the solutions to Eqs. (1) and (2)are embedded in this larger-dimensional space [20]. Forexample, Eq. (7) has a fixed point SðtÞ  0 which is notattainable in the original dynamics.A detailed analysis of Eq. (7) is beyond our scope; wenote that it contains the time derivative of_S ¼ ½A; S ¼ i½iA; S;the Heisenberg equation for the evolution of operator Sunder the (Hermitian) Hamiltonian iA; S therefore hasone component that oscillates at frequencies given by A;numerically, the amplitude of this component is S  OðÞ.Equation (7) also contains the time-reversed Heisenbergequation 2S ¼ ½A; _S, which causes much slower oscil-lations of frequencies given by  and A. Finally, Eq. (7)contains€Sþ 2S ¼ 0;a set of uncoupled undamped harmonic oscillators withfrequencyﬃﬃﬃﬃﬃﬃ2p. There is therefore a new time scale givenby 2=ﬃﬃﬃﬃﬃﬃ2p(Fig. 2). The nonlinear term f _S; Sgmakes largeinitial values of Sðt ¼ 0Þ decay until Sðt 1=Þ Oð ﬃﬃﬃﬃp Þ. In Ref. [21] we show that adding white noise toEq. (1) decoheres the dynamics a small amount but doesnot destroy this time scale. The oscillation frequencyﬃﬃﬃﬃﬃﬃ2pis approximately the geometric mean between the neuronaloscillation time scale (in this Letter, 1) and the synapticupdate time scale . In a real neuron, the oscillatory timescale is bound to be in the 10–120 Hz frequency bands,while the synaptic update time scale would be in the orderof several minutes. The geometric mean between these,FIG. 1 (color). Relaxation of the real parts of the eigenvaluesof W. For clarity of illustration, N ¼ 20. At short times (1) alleigenvalues with positive real parts relax to having negative realparts; they typically overshoot and flip sign in doing so. On ascale given by  ¼ 103, all real parts relax to the vicinity of thereal axis. Beyond this scale, all eigenvalues fluctuate around thereal axis.720 740 760 780 800 820 840 860 880 900−0.06−0.04−0.0200.020.040.06timeRe [ eig(A) ]FIG. 2 (color). Zooming in the rightmost portion of Fig. 1.Starting from an antisymmetric matrix, the eigenvalues fluctuatearound the instability line with a time scale  ﬃﬃﬃﬃp .PRL 102, 258102 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending26 JUNE 2009258102-2marking the scale in which any given mode would sponta-neously activate and deactivate, would be in the secondsrange, bridging these two scales; this time scale marks theability of our model to ‘‘switch task’’ and correspondsroughly to the time scale of thought. ‘‘Slow’’ synaptic up-date, in conjunction with the population dynamics close toa dynamically critical state, creates this relatively fast timescale: minute alterations in synaptic strength across apopulation of neurons may change dynamical behavior ofthe network in a mere fraction of the time required to swinga single synapse from low to high strength.Figure 3 displays the time evolution of our systemand some of its statistical features, for two different con-nectivities. In the top row, Eqs. (1) and (2) are imple-mented verbatim, and every matrix element may havenonzero values. In the bottom row, the ‘‘1D3NN’’ model,neurons on a line are connected to their first, second, andthird neighbors: W is heptadiagonal. Thus the top case is‘‘infinite-dimensional,’’ while the lower one is one-dimensional. Different features of the system become sta-tistically critical in these two extreme cases. Avalanchesare defined as a cluster of contiguous times during which atleast one neuron is activated above a given threshold;avalanche size is defined as the number of ‘‘pixels’’ inthe spatiotemporal plot above threshold during the ava-lanche. The 1D case shows a propensity to have a largenumber of violent, system-wide large avalanches; the 1Dcase shows power-law-like probability of ‘‘avalanche’’sizes, though due to the difficulty in evolving these equa-tions the size of our system could not be made much larger.The 1D case shows ‘‘regular’’ statistics for probability ofsimultaneous firing as well as Gaussian marginal distribu-tion of the values of x, while the1D case shows anomalousprobabilities of simultaneous firing, even though individualneurons display small correlations, as in [10], and non-Gaussian marginals.In an attractor neural net, such as a Hopfield net, theantisymmetric component A of W is either null or small,and learning is carried out by using a Hebbian rule, whichthen encodes the learned objects in the symmetric part S ofW. In our case, anti-Hebbian dynamics takes control of Sand uses it to create the critical, highly resonant state wehave described. Meanwhile A is evidently untouched byEq. (2) and is the only degree of freedom available forlearning in our system. The xx> term is an instantaneousdensity of correlation, which Eq. (2) integrates in time dueto the smallness of ; rewrite it asðxx>ÞijðtÞ ¼ZðsÞxiðtþ sÞxjðt sÞds: (8)An antisymmetric analog of this correlation density in therhs of Eq. (2) would be given by partial directed correla-tions, which attempt to isolate influences between time-series embodying Granger causality [18]. Such correlationfunctions are obtained through a kernel which is antisym-metric in time, causing the correlation density to becomeantisymmetric in the neuron indexes; the simplest analogof Eq. (8) is the Hilbert transformCij ¼Z 1sxiðtþ sÞxjðt sÞds: (9)The Hilbert transform kernel 1=s is divergent at both short5000 5200 5400 5600 5800 600010203040500 10 20 30 4010−410−310−210−110010010110210−410−310−210−1100−10 −5 0 5 1010−810−610−410−20 2 4 6 810−410−310−210−11005000 5200 5400 5600 5800 600020406080100100 101 102 10310−610−410−2−4 −2 0 2 410−410−310−2FIG. 3 (color). Statistical criticality in our model. Top row: Globally coupled (unrestricted W). Bottom row: Nodes arranged in onedimension with periodic boundary conditions; only entries of W up to third nearest neighbor are allowed to be nonzero. Firstcolumn: A display of the spatiotemporal dynamics. Second column: The distribution of the number of simultaneously active units inthe dynamics (blue) and in surrogate data (red); compare to [10], where the argument was made this distribution indicates that eventhough the two-point correlations between individual neurons are small, the system has globally correlated states. Third column: Sizesof avalanches (blue) versus surrogate data (red); note in the 1D case the power-law distribution of avalanche sizes, while the globallycoupled (1D) case shows a piece of a power law followed by a large lump of rather large avalanches (as clearly visible in thespatiotemporal plot). Fourth column: Marginal distribution of the values of x (invariant under surrogation). Surrogation is carried outby scrambling the pixels of the spatiotemporal plot.PRL 102, 258102 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending26 JUNE 2009258102-3and long time scales, and should be cutoff according to thefastest and longest time scales which the system can use forits evaluation; the fast time scale controls the transitionbetween increasing synaptic strength when the presynapticneuron leads the postsynaptic one, to decreasing it in theopposite case, and reflects the accuracy with which thesystem can compute simultaneity. The slow time scalecontrols how much memory is kept of previous activity,i.e., over which range time intervals the presynaptic andpostsynaptic activities are evaluated. When the Hilbertkernel is cutoff according to these two time scales, thesynaptic rule left looks qualitatively like spike-timing de-pendent plasticity [19].Our model proposes a view of neural systems as showingcoexistence and superposition of different modes of neuro-nal activity, which can be simultaneously long-lived interms of the time scales of electrical activity, yet extremelyfast in terms of synaptic update time scales. The funda-mental distinguishing factor between each of the activatedmodes is the different phase relationship of each neuronwith respect to the underlying oscillation. Since the dy-namics consists of the activation and deactivation of modesof behavior given by eigenvectors which are in generaldelocalized, the dynamics of our net is resilient to stochas-ticity or even failure in individual units. Detailed analysisof this resilience shall be carried out elsewhere. Similarly,because the dynamical modes are delocalized, the systemis sensitive to the topological structure of the underlyingnetwork on scales much longer than individual connectionsor plaquettes. This extended spatial sensitivity mirrors theextended temporal behavior discussed above and will beexplored elsewhere.We have presented a simple model in which an under-lying anti-Hebbian dynamics permits the system to use thesymmetric components of its synaptic connectivity to poiseitself at a dynamically critical state and becomes infinitelysusceptible to inputs which, once applied, can reverberatefor long times. In the absence of inputs, this state evolvesby the eigenvalues oscillating around the stability line, sodifferent modes (eigenvectors) break out and then extin-guish haphazardly, with a time scale which bridges theelectrical and synaptic time scales. We have shown thatlearning can be encoded in the antisymmetric componentof the synaptic connectivity, driven by a term antisymmet-ric both in space as well as time—only inputs which areGranger-causal and time-symmetry broken can be learnedby this system. We have analyzed the statistics of oursystem to show that it can generate anomalous, heavy-tailed distributions, as well as power-law avalanches,showing explicitly a connection between criticality in thedynamical and statistical senses. Finally, our model isintended to provide a scaffold to explore the implicationsof the reverberating circuit theory introduced byLorente de No´ and furthered by Lashley and Hebb [22]which, for all their influence in physiology and behaviorscience, have not found consistent formal expressions.Supported in part by MCI project CGL2008-06245-C02-02 (O. P.).[1] J. Guckenheimer and P. Holmes, Nonlinear Oscillations,Dynamical Systems, and Bifurcations of Vector Fields(Springer, New York, 2002); Kotik K. Lee, Lectures onDynamical Systems, Structural Stability and TheirApplications (World Scientific, Singapore, 1992).[2] H. S. Seung, Neural Networks 11, 1253 (1998).[3] H. S. Seung, D. Lee, B. Reis, and D. Tank, Neuron 26, 259(2000).[4] C. K. Machens, R. Romo, and C.D. Brody, Science 307,1121 (2005).[5] S. Camalet, T. Duke, F. Julicher, and J. Prost, Proc. Natl.Acad. Sci. U.S.A. 97, 3183 (2000); V.M. Eguı´luz,M. Ospeck, Y. Choe, A. J. Hudspeth, and M.O.Magnasco, Phys. Rev. Lett. 84, 5232 (2000); L. Moreauand E. Sontag, Phys. Rev. E 68, 020901 (2003).[6] W. J. Freeman and M.D. Holmes, Neural Networks 18,497 (2005).[7] E. Bienenstock and D. Lehmann, Adv. Complex Syst. 1,361 (1998).[8] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59,381 (1987).[9] J.M. Beggs and D. Plenz J. Neurosci. 23, 11 167(2003); C. Haldeman and J.M. Beggs, Phys. Rev. Lett.94, 058101 (2005); E. D. Gireesh and D. Plenz, Proc. Natl.Acad. Sci. U.S.A. 105, 7576 (2008).[10] E. Schneidman, M. J. Berry II, R. Segev, and W. Bialek,Nature (London) 440, 1007 (2006).[11] M.H. Hennig, C. Adams, D. Willshaw, and E. Sernagor,J. Neurosci. 29, 1077 (2009).[12] V.M. Eguı´luz, D. R. Chialvo, G. A. Cecchi, M. Baliki, andA.V. Apkarian, Phys. Rev. Lett. 94, 018102 (2005).[13] M. Lin and T.-L. Chen, Phys. Rev. E 71, 016133(2005).[14] A. Levina, J.M. Herrmann, and T. Geisel, Nature Phys. 3,857 (2007).[15] A. N. Kolmogorov, J. Fluid Mech. 13, 82 (1962); B.Castaign et al., J. Fluid Mech. 204, 1 (1989).[16] T. Bohr et al., Dynamical Systems Approach to Turbulence(Cambridge University Press, Cambridge, England, 2005),pp. 63ff.[17] K. P. Lamsa et al., Science 315, 1262 (2007); A. Destexheand E. Marder, Nature (London) 431, 789 (2004).[18] C.W. J. Granger, Econometrica 37, 424 (1969).[19] W.B. Levy and O. Stewart, Neuroscience (Oxford) 8, 791(1983); H. Markram, J. Lu¨bke, M. Frotscher, andB. Sakmann, Science 275, 213 (1997); G. Bi andM. Poo, J. Neurosci. 18, 10 464 (1998).[20] J. H. E. Cartwright, M.O. Magnasco, O. Piro, and I. Tuval,Phys. Rev. Lett. 89, 264501 (2002).[21] See EPAPS Document No. E-PRLTAO-103-005928 forsupplementary material. For more information on EPAPS,see http://www.aip.org/pubservs/epaps.html.[22] J. Orbach, The Neurophysiological Theories of Lashleyand Hebb (University Press of America, Lanham, MD,1998).PRL 102, 258102 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending26 JUNE 2009258102-4

Dynamical and Statistical Criticality in a Model of Neural Tissue

Abstract

Similar works

Full text

Available Versions

Digital.CSIC