{"title": "A Model for Resolution Enhancement (Hyperacuity) in Sensory Representation", "book": "Advances in Neural Information Processing Systems", "page_first": 444, "page_last": 450, "abstract": null, "full_text": "444 \n\nA MODEL FOR RESOLUTION ENHANCEMENT \n\n(HYPERACUITY) IN SENSORY REPRESENTATION \n\nJun Zhang and John P. Miller \n\nNeurobiology Group, University of California, \n\nBerkeley, California 94720, U.S.A. \n\nABSTRACT \n\nHeiligenberg (1987) recently proposed a model to explain how sen(cid:173)\n\nsory maps could enhance resolution through orderly arrangement of \n\nbroadly tuned receptors. We have extended this model to the general \n\ncase of polynomial weighting schemes and proved that the response \n\nfunction is also a polynomial of the same order. We further demon(cid:173)\n\nstrated that the Hermitian polynomials are eigenfunctions of the sys(cid:173)\n\ntem. Finally we suggested a biologically plausible mechanism for sen(cid:173)\n\nsory representation of external stimuli with resolution far exceeding the \n\ninter-receptor separation. \n\n1 \n\nINTRODUCTION \n\nIn sensory systems, the stimulus continuum is sampled at discrete points \n\nby receptors of finite tuning width d and inter-receptor spacing a. In order \n\nto code both stimulus locus and stimulus intensity with a single output, \nthe sampling of individual receptors must be overlapping (i. e. a < d). \n\nThis discrete and overlapped sampling of the stimulus continuum poses a \n\nquestion of how then the system could reconstruct the sensory stimuli with \n\n\fResolution Enhancement in Sensory Representation \n\n445 \n\na resolution exceeding that is specified by inter-receptor spacing. This is \n\nknown as the hyperacuity problem (Westheimer,1975). \n\nHeiligenberg (1987) proposed a model in which the array of receptors (with \n\nGaussian-shaped tuning curves) were distributed uniformly along the entire \n\nrange of stimulus variable x. They contribute excitation to a higher order in(cid:173)\n\nterneuron, with the synaptic weight of each receptor's input set proportional \n\nto its rank index k in the receptor array. Numerical ~lmulation and subse(cid:173)\n\nquent mathematical analysis (Baldi and Heiligenberg, 1988) demonstrated \nthat, so long as a <:: d, the response function f( x) of the higher order neu(cid:173)\n\nron was monotone increasing and surprisingly linear. The smoothness of this \n\nfunction offers a partial explanation of the general phenomena of hyperacu(cid:173)\n\nity (see Baldi and Heiligenberg in this volumn). Here we consider various \n\nextensions of this model. Only the main results shall be stated below; their \n\nproof is presented elsewhere (Zhang and Miller, in preparation). \n\n2 POLYNOMIAL WEIGHTING FUNCTIONS \n\nFirst, the model can be extended to incorporate other different weighting \n\nschemes. The weighting function w( k) specifies the strength of the excita(cid:173)\n\ntion from the k-th receptor onto the higher order interneuron and therefore \n\ndetermines the shape of its response f( x). In Heiligenberg's original model, \nthe linear weighting scheme w( k) = k is used. A natural extension would \n\nthen be the polynomial weighting schemes. Indeed, we proved that, for suf(cid:173)\n\nficiently large d, \na) If w(k) = k2m , \n\nthen: \n\n\f446 \n\nZhang and Miller \n\nIf w( k) = k2m+l , \n\nthen: \n\nf( ) \n\nX = alX + a3X + ... + a2m+IX \n\n3 \n\n2m+1 \n\nwhere m = 0,1,2, ... , and ai are real constants. \nNote that for w(k) = kP , \nfunction for odd interger p and even function for even interger p. The case \n\nf(x) has parity (-I)P , that is, it is an odd \n\nof p = 1 reduces to the linear weighting scheme in Heiligenberg's original \n\nmodel. \nb) If w(k) = Co + clk + c2k2 + ... + cpkP , then: \n\nNote that this is a direct result of a), because f( x) is linearly dependent on \nw( k). The coefficients Ci and ai are usually different for the two polynomials. \n\nOne would naturally ask: what kind of polynomial weighting function then \n\nwould yield an identical polynomial response function? This leads to the \n\nimportant conclusion: \n\nc) If w(k) = Hp(k) \n\nis an Hermitian polynomial, \n\nthen f(x) = Hp(x) , \n\nthe same Hermitian polynomial. \n\nThe Hermitian polynomial Hp(t) is a well-studied function in mathematics. \n\nIt is defined as: \n\n2 \nHp(t) = (-I)Pet -d e-t \n\n2 dP \ntP \n\nFor reference purpose, the first four polynomials are given here: \n\nHo(t) \n\nHI(t) \n\nH2(t) \n\nH3(t) \n\n1\u00b7 , \n2t\u00b7 , \n4t2 - 2\u00b7 , \n8t3 - 12t\u00b7 , \n\n\fResolution Enhancement in Sensory Representation \n\n447 \n\nThe conclusion of c) tells us that Hermitian polynomials are unique in the \n\nsense that they serve as eigenfunctions of the system. \n\n3 REPRESENTATION OF SENSORY STIMULUS \n\nHeiligenberg's model deals with the general problem of two-point resolution, \n\ni. e. how sensory system can resolve two nearby point stimuli with a reso(cid:173)\n\nlution exceeding inter-receptor spacing. Here we go one step further to ask \n\nourselves how a generalized sensory stimulus g( x) is encoded and represented \n\nbeyond the receptor level with a resolution exceeding the inter-receptor spac(cid:173)\n\ning. We'll show that if, instead of a single higher order interneuron, we have \n\na group or layer of interneurons, each connected to the array of sensory \n\nreceptors using some different but appropriately chosen weighting schemes \n\nwn(k), then the representation of the sensory stimulus by this interneuron \ngroup (in terms of In , each interneuron's response) is uniquely determined \nwith enhanced resolution (see figure below). \n\nINTERNEURON GROUP \n\n. . . \n\n. . . \n\nRECEPTOR ARRAY \n\n\f448 \n\nZhang and Miller \n\nSuppose that 1) each interneuron in this group receives input from the re(cid:173)\n\nceptor array, its weighting characterized by a Hermitian polynomial H1'(k) ; \n\nand that 2) the order p of the Hermitian polynomial is different for each \n\ninterneuron. We know from mathematics that any stimulus function g( x) \n\nsatisfying certain boundary conditions can be decomposed in the following \n\nway: \n\n00 \n\ng(x) = ~ cnHn(x)e-X2 \n\nn=O \n\nThe decomposition is unique in the sense that Cn completely determines g(x). \nHere we have proved that the response /1' of the p-th interneuron (adopting \nH1'(k) as weighting scheme) is proportional to c1' : \n\nThis implies that g( x) can be uniquely represented by the response of this \n\nset of interneurons {/1'}' Note that the precision of representation at this \n\nhigher stage is limited not by the receptor separation, but by the number of \n\nneurons available in this interneuron group. \n\n4 EDGE EFFECTS \n\nSince the array of receptors must actually be finite in extent, simple weight(cid:173)\n\ning schemes may result in edge-effects which severely degrade stimulus reso(cid:173)\n\nlution near the array boundaries. For instance, the linear model investigated \n\nby Heiligenberg and Baldi will have regions of degeneracy where two nearby \n\npoint stimuli, if located near the boundary defined by receptor array cover(cid:173)\n\nage, may yield the same response. We argue that this region of degeneracy \n\ncan be eliminated or reduced in the following situations: \n\n1) If w( k) approaches zero as k goes to infinity, then the receptor array \n\n\fResolution Enhancement in Sensory Representation \n\n449 \n\ncan still be treated as having infinite extent since the contributions by the \n\nlarge index receptors are negligibly small. We proved, using Fourier analysis, \n\nthat this kind of vanishing-at-infinity weighting scheme could also achieve \n\nresolution enhancement provided that the tuning width of the receptor is \n\nsufficiently larger than the inter-receptor spacing and meanwhile sufficiently \n\nsmaller than the effective width of the entire weighting function. \n\n2) If the receptor array \"wraps around\" into a circular configuration, then it \n\ncan again be treated as infinite (but periodic) along the angular dimension. \n\nThis is exactly the case in the wind-sensitive cricket cercal sensory system \n\n(Jacobs et al,1986; Jacobs and Miller,1988) where the population of direc(cid:173)\n\ntional selective mechano-receptors covers the entire range of 360 degrees. \n\n5 CONCLUSION \n\nHeiligenberg's model, which employs an array of orderly arranged and broadly \n\ntuned receptors to enhance the two-point resolution, can be extended in a \n\nnumber of ways. We first proved the general result that the model works \n\nfor any polynomial weighting scheme. We further demonstrated that Her(cid:173)\n\nmitian polynomial is the eigenfunction of this system. This leads to the new \n\nconcept of stimulus representation, i. e. a group of higher-order interneurons \n\ncan encode any generalized sensory stimulus with enhanced resolution if they \n\nadopt appropriately chosen weighting schemes. Finally we discussed possible \n\nways of eliminating or reducing the \"edge-effects\". \n\nACKNOWLEDGMENTS \n\nThis work was supported by NIH grant # ROI-NS26117. \n\n\f450 \n\nZhang and Miller \n\nREFERENCES \n\nBaldi, P. and W. Heiligenberg (1988) How sensory maps could enhance res(cid:173)\n\nolution through ordered arrangements of broadly tuned receivers. Bioi. \n\nCybern. 59: 314-318. \n\nHeiligenberg, W. (1987) Central processing of the sensory information in \n\nelectric fish. J. Compo Physiol. A 161: 621-631. \n\nJacobs, G.A. and J.P. Miller (1988) Analysis of synaptic integration using \n\nthe laser photo-inactivation technique. Experientia 44: 361- 462. \n\nJacobs, G.A., Miller, J.P. and R.K. Murphey (1986) Cellular mechanisms \n\nunderlying directional sensitivity of an identified sensory interneuron. \n\nJ.Neurosci. 6: 2298-2311. \n\nWestheimer, G. (1975) Visual acuity and hyperacuity. Invest. Ophthalmol. \n\nVis. 14: 570-572. \n\n\f", "award": [], "sourceid": 178, "authors": [{"given_name": "Jun", "family_name": "Zhang", "institution": null}, {"given_name": "John", "family_name": "Miller", "institution": null}]}