{"title": "Self-organization in real neurons: Anti-Hebb in 'Channel Space'?", "book": "Advances in Neural Information Processing Systems", "page_first": 59, "page_last": 66, "abstract": "", "full_text": "Self-organisation in real neurons: \nAnti-Hebb in 'Channel Space'? \n\nAnthony J. Bell \n\nAI-lab, \n\nVrije U niversiteit Brussel \n\nPleinlaan 2, B-I050 Brussels \n\nBELGIUM, (tony@arti.vub.ac.be) \n\nAbstract \n\nIon channels are the dynamical systems of the nervous system. Their \ndistribution within the membrane governs not only communication of in(cid:173)\nformation between neurons, but also how that information is integrated \nwithin the cell. Here, an argument is presented for an 'anti-Hebbian' rule \nfor changing the distribution of voltage-dependent ion channels in order \nto flatten voltage curvatures in dendrites. Simulations show that this rule \ncan account for the self-organisation of dynamical receptive field properties \nsuch as resonance and direction selectivity. It also creates the conditions \nfor the faithful conduction within the cell of signals to which the cell has \nbeen exposed. Various possible cellular implementations of such a learn(cid:173)\ning rule are proposed, including activity-dependent migration of channel \nproteins in the plane of the membrane. \n\n1 \n\nINTRODUCTION \n\n1.1 NEURAL DYNAMICS \n\nNeural inputs and outputs are temporal, but there are no established ways to think \nabout temporal learning and dynamical receptive fields. The currently popular sim(cid:173)\nple recurrent nets have only one kind of dynamical component: a capacitor, or time \nconstant. Though it is possible to create any kind of dynamics using capacitors and \nstatic non-linearities, it is also possible to write any program on a Turing machine. \n59 \n\n\f60 \n\nBell \n\nBiological evolution, it seems, has elected for diversity and complexity over unifor(cid:173)\nmity and simplicity in choosing voltage-dependent ion channels as the 'instruction \nset' for dynamical computation. \n\n1.2 \n\nION CHANNELS \n\nAs more ion channels with varying kinetics are discovered, the question of their \ncomputational role has become more pertinent. Figure 1, derived from a model \nthalamic cell, shows the log time constants of 11 currents, plotted against the voltage \nranges over which they activate or inactivate. The variety of available kinetics is \nprobably under-represented here since a combinatorial number of differences can be \nobtained by combining different protein sub-domains to make a channel [6]. \nGiven the likelihood that channels are inhomogenously distributed throughout the \ndendrites [7], one way to tackle the question of their computational role is to search \nfor a self-organisational principle for forming this distribution. Such a 'learning \nrule' could be construed as operating during development or dynamically during \nthe life of an organism, and could be considered complementary to learning involv-\ning synaptic changes. \nresulting distribution and mix! \nof channels would then be, in] \nsome sense, optimal for integrat- : \ning and communicating the par- ~ \nticular high-dimensional spatia-\n10-1 \ntemporal inputs which the cell \nwas accustomed to receiving. \n\nThe'U 1....--.-----,--,-----\u00b7---,----\u00b7 \n\n\",.-... \n\n/ \n\n\"-.'\" \n\nII' M \n\n~ \n\n1 \n\n10-2 \n\nFigure 1: Diversity of ion chan(cid:173)\nnel kinetics. \nThe voltage(cid:173)\ndependent equilibrium log time \nconstants of 11 channels are plot-\nted here for the voltage ranges \nfor which their activation (or \ninactivation) variables go from \n0.1 ~ 0.9 (or 0.9 ~ 0.1). The \nchannel kinetics are taken from a \nmodel by W.Lytton [10]. Notice W' \nthe range of speeds of operation \nfrom the spiking N a+ channel \naround O.lms, to the J{M chan-\nnel in the Is (cognitive) range. \n\n10-3 \n\n............. ......... \n\n\"'. No act. \n\n-100 \n\nVR\u00a3ST \n\n-50 \n\no \n\n50 \n\nMembrane potential (mV) \n\n2 THE BIOPHYSICAL SUBSTRATE \n\nThe substrate for self-organisation is the standard cable model for a dendrite or \naxon: \n\n(1 ) \n\n\fAnti-Hebb in 'Channel Space'? \n\n61 \n\nIn this Go represents the conductance along the axis of the cable, C is the capac(cid:173)\nitance and the two sums represent synaptic (indexed by j) and intrinsic (indexed \nby k) currents. G is a maximum conductance (a channel density or 'weight'), 9 is \nthe time-varying fraction of the conductance active, and E is a reversal potential. \nThe system can be summarised by saying that the current flow out of a segment of \na neuron is equal to the sum of currents input to that segment, plus the capacitive \ncharging of the membrane. \n\nThis leads to a simpler form: \n\ni = L:9j ij + L:9kik \n\nj \n\nk \n\n(2) \n\nHere, i = 02V lox2, gj = Gj IG a , ij = gj(V -Ej) and C is considered as an intrinsic \nconductance whose 9k and ik are CIGa and oV lot respectively. In this form, it \nis more clear that each part of a neuron can be considered as a 'unit', diffusively \ncoupled to its neighbours, to which it passes its weighted sum of inputs. The weights \n\nexcitatory \nchannels \n\ninhibitory \nchannels \n\nleakage \nrohannels \n\nsynaptic \n\ncapacitive \naeDibrane \ncharging \n\nBlectro(cid:173)\ndiffusive \nspread \n\n~ \n\nFigure 1: A compartment of a neuron, shown schematically and as a circuit. The \ncable equation is just Kirchoff's Law: current in = current out \n\n9k' representing the Go-normalised densities of channel species k, are considered to \nspan channel space, as opposed to the 9j weights which are our standard synaptic \nstrength parameters. Parameters determining the dynamics of gk's specify points \nin kinetics space. Neuromodulation [8], a universally important phenomenon in real \nnervous systems, consists of specific chemicals inducing short-term changes in the \nkinetics space co-ordinates of a channel type, resulting, for example, in shifts in the \ncurves in Figure 1. \n\n3 THEARGUMENTFORANT~HEBB \n\nLearning algorithms, of the type successful in static systems, have not been con(cid:173)\nsidered for these low-level dynamical components (though see [2] for approaches to \nsynaptic learning in realistic systems). Here, we address the issue of unsupervised \nlearning for channel densities. In the neural network literature, unsupervised learn(cid:173)\ning consists of Hebbian-type algorithms and information theoretic approaches based \non objective functions [1]. In the absence of a good information theoretic frame(cid:173)\nwork for continuous time, non-Gaussian analog systems where noise is undefined, \nwe resort to exploring the implications of the effects of simple local rules. \n\n\f62 \n\nBell \n\nThe most obvious rule following from equation 2 would be a correlational one of \nthe following form, with the learning rate f positive or negative: \n\n~9k = fiki \n\n(3) \nWhile a polarising (or Hebbian) rule (see Figure 3) makes sense for synaptic chan(cid:173)\nnels as an a method for amplifying input signals, it makes less sense for intrinsic \nchannels. Were it to operate on such channels, statistical fluctuations from the \nuniform channel distribution would give rise to self-reinforcing 'hot-spots' with no \nunderlying 'signal' to amplify. For this reason, we investigate the utility of a recti(cid:173)\nfying (or anti-Hebbian) rule. \n\nFigure 3: A schematic dis(cid:173)\nplay showing contingent positive \nand negative voltage curvatures \n(\u00b1i) above a segment of neu(cid:173)\nron, and inward and outward \ncurrents (\u00b1ik), through a par(cid:173)\nticular channel type. \nIn situ(cid:173)\nations (a) and (b), a Hebbian \nversion of Equation 3 will raise \nthe channel density (9k T), and \nin (c) and (d) an anti-Hebbian \nrule will do this. In the first two \ncases, the channels are polar(cid:173)\nising the membrane potential, \ncreating high voltage curvature, \nwhile in the latter two, they are \nrectifying (or flattening) it. De(cid:173)\npending on the sign of f, equa(cid:173)\ntion 3 attempts to either max(cid:173)\nimise or minimise (8 2V /8x 2 )2. \n\n4 EXAMPLES \n\n\",,~ \n-?' I -ve '---\n\n(a) gJ if E Is +ve \n\ni k \n\n-ve \n\nI k+ ve \n\n~ \n\n(b) '9J if E Is +ve \n\n(c) '9J if E \n\nis -ve \n\n(d) ;J If E \n\nis -ve \n\nFor the purposes of demonstration, linear RLC electrical components are often used \nhere. These simple 'intrinsic' (non-synaptic) components have the most tractable \nkinetics of any, and as shown by [11] and [9], the impedances they create capture \nsome of the properties of active membrane. The components are leakage resistances, \ncapacitances and inductances, whose 9k'S are given by 1/ R, C and 1/ L respectively. \nDuring learning, all 9k's were kept above zero for reasons of stability. \n\n4.1 LEARNING RESONANCE \n\nIn this experiment, an RLC 'compartment' with no frequency preference was stim(cid:173)\nulated at a certain frequency and trained according to equation 3 with f negative. \nAfter training, the frequency response curve of the circuit had a resonant peak at \nthe training frequency (Figure 4). This result is significant since many auditory \nand tactile sensory cells are tuned to certain frequencies, and we know that a major \ncomp onent of the tuning is electrical, with resonances created by particular balances \nof ion channel populations [13]. \n\n\f..NVV \nsin 0.4t \n\nI.' ... ... \n'.' \n.. , \n... \n... \n'.-\n\nk \n\nIt \n+ \n\nAnti-Hebb in 'Channel Space'? \n\n63 \n\n... \n\nf \n\no\u00b7 \n\nFigure 4: Learning resonance. The curves show the frequency-response curves of \nthe compartment before and after training at a frequency of 0.4. \n\n4.2 LEARNING CONDUCTION \n\nAnother role that intrinsic channels must play within a cell is the faithful transmis(cid:173)\nsion of information. Any voltage curvatures at a point away from a synapse signify \na net cross membrane current which can be seen as distorting the signal in the cable. \nThus, by removing voltage curvatures, we preserve the signal. This is demonstrated \n5 .:!.-1' _____ -= ~tnlli \n4 - -~ \n3 _ =-----\n- _ :'Ni.f\\V.0S(~\\lf\\~ \n2 _ ~_ ~f.\\V.l\\V.r.J!.f\\\\lf\\~ \nj: =~=:WJ\\~W~\\(~ \n, ~ \n\n) LJ LJ LJ l/ ~t) l/ LJ l/ l/ l~t \nFigure 5: Learning conduction. The cable consists of a chain of compartments, \nwhich only conduct the impulse after they acquire active channels. \n\n, , \n, \n, \n, , \n, \n, , , \n\nin the following example: 'learning to be an axon'. A non-linear spiking compart(cid:173)\nment with Morris-Lecar Cal J{ kinetics (see [14]) is coupled to a long passive cable. \nBefore learning, the signal decays passively in the cable (Figure 5). The driving \ncompartment ?i-vector, and the capacitances in the cable are then clamped to stop \nthe system from converging on the null solution (g -+ 0). All other g's (including \nspiking conductances in the cable) can then learn. The first thing learnt was that \nthe inward and outward leakage conductances (?it and ?i\"l) adjusted themselves to \nmake the average voltage curvature in each compartment zero (just as bias units in \nerror correction algorithms adjust to make the average error zero). Then the cable \nfilled out with Morris-Lecar channels (9Ca and gK) in exactly the same ratios as the \ndriving compartment, resulting in a cable that faithfully propagated the signal. \n\n\f64 \n\nBell \n\n4.3 LEARNING PHASE-SHIFTING (DIRECTION SELECTIVITY) \n\nThe last example involves 4 'sensory' compartments coupled to a 'somatic' compart(cid:173)\nment as in Figure 6. All are similar to the linear compartments in the resonance \nexample except that the sensory ones receive 'synaptic' input in the form of a si(cid:173)\nnusoidal current source. The relative phases of the input were shifted to simulate \nleft-to-right motion. After training, the 'dendritic' components had learned, using \ntheir capacitors and inductors, to cancel the phase shifts so that the inputs were \nsynchronised in their effect on the 'soma'. This creates a large response in the \ntrained direction, and a small one in the 'null' direction, as the phases cancelled \neach other. \n----------------------- --~ .. \n\" \u00b7 \" \u00b7 , \u00b7 \" \n\n:,: \n,-- -- - -- - -- - - - -- ---- - ------. \n\n, \n\n!- .... _ ...... -\n\n\u00b7 . \n, ' \u00b7 : \n\n- - - - - - . - . direction \n... ___ ___ null \n\ntrained \n\ndirection \n\nFigure 6: Learning direction selectivity. After training on a drifting sine wave, the \noutput compartment oscillates for the trained direction but not for the null direction \n(see the trace, where the direction of motion is reversed halfway). \n\n5 DISCUSSION \n\n5.1 CELLULAR MECHANISMS \n\nThere is substantial evidence in cell biology for targeting of proteins to specific \nparts of the membrane, but the fact that equation 3 is dependent on the correla(cid:173)\ntion of channel species' activity and local voltages leaves only 4 possible biological \nim plementations: \n\n1. the cellular targeting machinery knows what kind of channel it is delivering, \n\nand thus knows where to put it \n\n2, channels in the wrong place are degraded faster than those in the right place \n\n3. channels migrate to the right place while in the membrane \n\n4. the effective channel density is altered by activity-dependent neuromodulation \n\nor channel-blockage \n\nThe third is perhaps the most intriguing. The diffusion of channels in the plane \nof the membrane, under the influence of induced electric fields has received both \ntheoretical [4, 12] and empirical [7, 3] attention. To a first approximation, the \n\n\fAnti-Hebb in 'Channel Space'? \n\n6S \n\nevolution of channel densities can be described by a Smoluchowski equation: \n\nay\" = a a2g\" + b~ (g aV) \nax \" ax \nat \n\nax 2 \n\n(4) \n\nwhere a is the coefficient of thermal diffusion and b is the coefficient of field induced \nmotion. This system has been studied previously [4] to explain receptor-clustering \nin synapse formation, but if the sign of b is reversed, then it fits more closely with \nthe anti-Hebbian rule discussed here. The crucial requirement for true activity(cid:173)\ndependence, though, is that b should be different when the channel is open than \nwhen it is closed. This may be plausible since channel gating involves movements of \ncharges across the membrane. Coefficients of thermal diffusion have been measured \nand found not to exceed 10- 9 cm/sec. This would be enough to fine-tune channel \ndistributions, but not to transport them all the way down dendrites. \n\nThe second method in the list is also an attractive possibility. The half-life of \nmembrane proteins can be as low as several hours [3], and it is known that proteins \ncan be differentially labeled for recycling [5]. \n\n5.2 ENERGY AND INFORMATION \n\nThe anti-Hebbian rule changes g\" 's in order to minimise the square membrane cur(cid:173)\nrent density, integrated over the cell in units of axial conductance. This corresponds \nin two senses to a minimisation of energy. From a circuit perspective, the energy \ndissipated in the axial resistances is minimised. From a metabolic perspective, the \nATP used in pumping ions back across the membrane is minimised. The compu(cid:173)\ntation consists of minimising the expected value of this energy, given particular \nspatiotemporal synaptic input (assuming no change in 9j'S). More precisely, it \nsearches for: \n\n(5) \n\nThis search creates mutual information between input dynamics and intrinsic dy(cid:173)\nnamics. In addition, since the Laplacian (\\7; V = 0) is what a diffusive system seeks \nto converge to anyway, the learning rule simply configures the system to speed this \nconvergence on frequently experienced inputs. \n\nSimple zero-energy solutions exist for the above, for example the 'ultra-leaky' com(cid:173)\npartment (gl - l (0) and the 'point' (or non-existent) compartment (g\" - l 0, Vk), \nfor compartments with and without synapses respectively. The anti-Hebb rule alone \nwill eventually converge to such solutions, unless, for example, the leakage or capac(cid:173)\nitance are prevented from learning. Another solution (which has been successfully \nused for the direction selectivity example) is to make the total available quantity of \neach g\" finite. The g\" can then diffuse about between compartments, following the \nvoltage gradients in a manner suggested by equation 4. The resulting behaviour is \na finite-resource version of equation 3. \n\nThe next goal of this work is to produce a rigorous information theoretic account \nof single neuron computation. This is seen as a pre-requisite to understanding both \nneural coding and the computational capabilities of neural circuits, and as a step \non the way to properly dynamical neural nets. \n\n\f66 \n\nBell \n\nAcknowledgements \n\nThis work was supported by a Belgian government IMPULS contract and by ES(cid:173)\nPRIT Basic Research Action 3234. Thanks to Prof. L. Steels for his support and \nto Prof T. Sejnowski his hospitality at the Salk Institute where some of this work \nwas done. \n\nReferences \n\n[1] Becker S. 1990. 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Methods in Neuronal Modeling, MIT \nPress \n\n\f", "award": [], "sourceid": 545, "authors": [{"given_name": "Anthony", "family_name": "Bell", "institution": null}]}