{"title": "Analog Computation at a Critical Point: A Novel Function for Neuronal Oscillations?", "book": "Advances in Neural Information Processing Systems", "page_first": 137, "page_last": 144, "abstract": null, "full_text": "Analog Computation at a Critical Point: A Novel \n\nFunction for Neuronal Oscillations? \n\nLeonid Kruglyak and Willianl Bialek \n\nDepart.ment of Physics \n\nUniversity of California at Berkeley \n\nBerkeley, California 94720 \n\nand NEC Research Institute\u00b7 \n\n4 Independence vVay \n\nPrinceton, New Jersey 08540 \n\nAbstract \n\n\\Ve show that a simple spin system bia.sed at its critical point can en(cid:173)\ncode spatial characteristics of external signals, sHch as the dimensions of \n\"objects\" in the visual field. in the temporal correlation functions of indi(cid:173)\nvidual spins. Qualit.ative arguments suggest that regularly firing neurons \nshould be described by a planar spin of unit lengt.h. and such XY models \nexhibit critical dynamics over a broad range of parameters. \\Ve show how \nto extract these spins from spike trains and then mea'3ure t.he interaction \nHamilt.onian using simulations of small dusters of cells. Static correla(cid:173)\ntions among spike trains obtained from simulations of large arrays of cells \nare in agreement with the predictions from these Hamiltonians, and dy(cid:173)\nnamic correlat.ions display the predicted encoding of spatial information. \n\\Ve suggest that this novel representation of object dinwnsions in temporal \ncorrelations may be relevant t.o recent experiment.s on oscillatory neural \nfiring in the visual cortex. \n\n1 \n\nINTRODUCTION \n\nPhysical systems at a critical point exhibit long-range correlations even though \nthe interactions among the constituent partides are of short range . Through the \nfluct.uation-dissipation theorem this implies that the dynamics at one point in the \n\n\u00b7Current address. \n\n137 \n\n\f138 \n\nKruglyak and Bialek \n\nsystem are sensitive t.o external pert.urbat.ions which are applied very far away. If \nwe build a.ll analog computer poised precisely at such a critical point it should be \npossible to evaluate highly non-local funct.ionals of the input signals using a locally \ninterconnected architecture. Such a scheme would be very useful for visual compu(cid:173)\ntations, especially those which require comparisons of widely separated regions of \nthe image. From a biological point of view long-range correlat.ions at a critical point \nmight provide a robust scenario for \"responses from beyond the classical receptive \nfield\" [1]. \n\nIn this paper we present. an explicit model for analog computation at a critical \npoint and show that this model has a remarkable consequence: Because of dynamic \nscaling, spatial properties of input. signals are mapped into temporal correlat.ions \nof the local dynamics. One can, for example, measure t.he size and t.opology of \n\"object.s\" in a scene llsing only the temporal correlations in t.he output of a single \ncomputational unit (neuron) locat.ed within the object. We then show that our \nabst.ract model can be realized in networks of semi-realistic spiking neurons. The \nkey to this construction is that. neurons biased in a regime of regular or oscillatory \nfiring can be mapped to XY or planar spins [2,3]' and two-dimensional arrays of \nthese spins exhibit a broad range of parameters in which the system is generically \nat a critical point. Non-oscillatory neurons cannot, in general, be forced to operate \nat a critical point. without delicate fine tuning of the dynamics, fine tuning which \nis implausible both for biology and for man-made analog circuits. We suggest t.hat \nthese arguments may be relevant to the recent observations of oscillatory firing in \nthe visual cortex [4,5,6]. \n\n2 A STATISTICAL MECHANICS MODEL \n\n\\Ve consider a simple two-dimensional array of spins whose stat.es are defined by unit \ntwo-vect.ors Sn. These spins interact. with their neighbors so that the total energy of \nthe syst.em is H = -J L Sn ,Sm, with the sum restricted to nearest neighbor pairs. \nThis is the XY model, which is interesting in part because it possesses not a critical \npoint but rather a critical line [7] . At a given temperature, for all J > J c one finds \nthat correlations among spins decay algebraically, (Sn ,Sm) ex l/lrn - rm 117 , so that \nthere is no characteristic scale or correlation length; more precisely the correlation \nlength is infinite. In contrast, for J < J c we have (Sn,Sm) ex exp[-Irn -\nrml/{], \nwhich defines a finite correlation length {. \n\nIn the algebraic phase the dynamics of t.he spins on long length scales are rigorously \ndescribed by the spin wave approximation, in which one assumes that fluctuations \nin the angle between neighboring spins are small. In this regime it. makes sense to \nuse a continuum approximation rather than a lattice, and the energy of the system \nbecomes H = J J ({l ,z'lv 4>(x)l2, where \u00a2(x) is the orientation of the spin at position \nx. The dynamics of the syst.em are determined by the Langevin equation \n\niJ\u00a2(x,t) \n\not \n\n') \n\n= J'V-4>(x, t) + 1J(x, t), \n\nwhere I] is a Gaussian t.hermal noise source with \n\n(1J(x, i)-I](x', t')} = 2k B TcS(x - x')cS(t - I'). \n\n(1) \n\n(2) \n\n\fAnalog Computation at a Critical Point \n\n139 \n\n\\Ve can then show that the time correlation function of the spin at a single sit,(> x \nis given by \n\n(S(x. t)\u00b7S(x. 0)) = exp [-2kn TJ ~~' J (J22~k)'? 1.)- e;~:t4l. \n\n7r - w- + -' \n\n'l.ir \n\n(:3 ) \n\nIn fact Eq. 3 is valid only for an infinite array of spins. Imagine that external signals \nto this array of spins can \"activate\" and \"deactivate\" t.he spins so that one must \nreally solve Eq. 1 on finite rpgions or clusters of active spins. Then we can writ.e \nthe analog of Eq. :3 as \n\n(S(x, t)\u00b7S(x. 0)) = exp - - L-1~'71(x)I--(1- e-\n\n(4) \n\n[ knT ~ , \n\nJ \n\n7l \n\n? 1 \nAll \n\nJ>.. I I 1 \n\nn t) \u2022 \n\nwhere 1/'71 and All are the eigenfunctions and associated eigenvalues of (- v 0) the interaction is ferromagnetic, as expected (sf'e Fig. 1). The \nHamiltonian takes other interesting forms for inhibitory, delayed, and nonreciprocal \nsynapses. By simulating small clusters of cells we find that interactions other than \nnearest neighbor are negligible. This leads us to predict that the entire network is \ndesc.ribed by the effective Hamiltonian H = Lij Hij(\u00a2i - \u00a2j), where Hij(\u00a2i -