{"title": "Coupled Dynamics of Fast Neurons and Slow Interactions", "book": "Advances in Neural Information Processing Systems", "page_first": 447, "page_last": 454, "abstract": null, "full_text": "Coupled Dynamics of Fast Neurons and \n\nSlow Interactions \n\nA.C.C. Coolen \n\nR.W. Penney \n\nD. Sherrington \n\nDept. of Physics - Theoretical Physics \n\nUniversity of Oxford \n\n1 Keble Road, Oxford OXI 3NP, U.K. \n\nAbstract \n\nA simple model of coupled dynamics of fast neurons and slow inter(cid:173)\nactions, modelling self-organization in recurrent neural networks, \nleads naturally to an effective statistical mechanics characterized \nby a partition function which is an average over a replicated system. \nThis is reminiscent of the replica trick used to study spin-glasses, \nbut with the difference that the number of replicas has a physi(cid:173)\ncal meaning as the ratio of two temperatures and can be varied \nthroughout the whole range of real values. The model has inter(cid:173)\nesting phase consequences as a function of varying this ratio and \nexternal stimuli, and can be extended to a range of other models. \n\n1 A SIMPLE MODEL WITH FAST DYNAMIC \n\nNEURONS AND SLOW DYNAMIC INTERACTIONS \n\nAs the basic archetypal model we consider a system of Ising spin neurons (J'i E \n{-I, I}, i E {I, ... , N}, interacting via continuous-valued symmetric interactions, \nIij, which themselves evolve in response to the states of the neurons. The neurons \nare taken to have a stochastic field-alignment dynamics which is fast compared with \nthe evolution rate of the interactions hj, such that on the time-scale of Iii-dynamics \nthe neurons are effectively in equilibrium according to a Boltzmann distribution, \n\n(1) \n\n447 \n\n\f448 \n\nCooien, Penney, and Sherrington \n\nwhere \n\nHVoj} ({O\"d) = - L JijO\"iO\"j \n\ni<j \n\n(2) \n\nand the subscript {Jij} indicates that the {Jij} are to be considered as quenched \nvariables. In practice, several specific types of dynamics which obey detailed balance \nlead to the equilibrium distribution (1), such as a Markov process with single-spin \nflip Glauber dynamics [1]. The quantity /3 is an inverse temperature characterizing \nthe stochastic gain. \nFor the hj dynamics we choose the form \n\nd \n\n1 \n\nT' dthj = N(O\"iO\"j)V,i} - jjJij + viirJij(t) \n\n1 \n\n(i < j) \n\n(3) \n\nwhere ( .. ')ViJ} refers to a thermodynamic average over the distribution (1) with \nthe effectively instantaneous {Jij}, and TJij (t) is a stochastic Gaussian white noise \nof zero mean and correlation \n\n(TJij(t)TJkl(t')) = 2T'ffi- 1o(ij),(kl)O(t - t') \n\nThe first term on the right-hand side of (3) is inspired by the Hebbian process in \nneural tissue in which synaptic efficacies are believed to grow locally in response to \nthe simultaneous activity of pre- an~ post-synaptic neurons [2]. The second term \nacts to limit the magnitude of hj; f3 is the characteristic inverse temperature of \nthe interaction system. (A related interaction dynamics without the noise term, \nequivalent to ffi = 00, was introduced by Shinomoto [3]; the anti-Hebbian version of \nthe above coupled dynamics was studied in layered systems by Jonker et al. [4, 5].) \nSubstituting for (O\"iO\"j) in terms of the distribution (1) enables us to re-write (3) as \n\nd \n\nNT' dthj = - aJij 11. ({Jij}) + VNTJij(t) \n\na \n\n(4) \n\n(5) \n\nwhere the effective Hamiltonian 11. ({ hj}) is given by \n\n1 ~ 2 \n11. ( { Jij }) = - /3 In Z {3 ( { Jij } ) + 2 jjN ~<. Jij \n\n1 \n\nl \n\nJ \n\nwhere Z{3 ({ hj}) is the partition function associated with (2): \n\n2 COUPLED SYSTEM IN THERMAL EQUILIBRIUM \n\nWe now recognise (4) as having the form of a Langevin equation, so that the equilib(cid:173)\nrium distribution of the interaction system is given by a Boltzmann form. Hence(cid:173)\nforth, we concentrate on this equilibrium state which we can characterize by a \npartition function Z t3 an d an associated 'free energy' F t3: \n\nZ {3 = J P dJij [Z{3 ({ Jij}) r exp [-~ ffijjN ~ Ji~] \n\nS<J \n\nS<J \n\n-\n\nF{3 = -f3 \n\n- -1 -\n\nIn Z{3 \n\n(6) \n\n\fCoupled Dynamics of Fast Neurons and Slow Interactions \n\n449 \n\nwhere n _ ~/j3. We may use Z~ as a generating functional to produce thermody(cid:173)\nnamic averages of state variables <I> ( {O\"d; {Jij}) in the combined system by adding \nsuitable infinitesimal source terms to the neuron Hamiltonian (2): \nHP.j}({O\"d) -+ Hp.j} ({O\"d) + A<p({ud;{Jij}) \n\nop-\nlim \u00a3:J: = (<p({O\"d;{Jij})){J } \nA-+O UA \n\n'J \n\n_ IfL<j dhj (<p({O\"d; {Jij}))plj}e-~1l(Plj}) \n\nIfL<jdhj e-~1l({J\u00b7j}) \n\n(7) \n\nwhere the bar refers to an average over the asymptotic {hj} dynamics. \nThe form (6) with n -+ 0 is immediately reminiscent of the effective partition \nfunction which results from the application of the replica trick to replace In Z by \nlimn-+o ~(zn - 1) in dealing with a quenched average for the infinite-ranged spin(cid:173)\nglass [6], while n = 1 relates to the corresponding annealed average, although we \nnote that in the present model the time-scales for neuron and interaction dynamics \nremain completely disparate. These observations correlate with the identification \nof n with fi / j3, which implies that n -+ 0 corresponds to a situation in which the \ninteraction dynamics is dominated by the stochastic term T)ij (t), rather than by the \nbehaviour of the neurons, while for n = 1 the two characteristic temperatures are \nthe same. For n -+ 00 the influence of the neurons on the interaction dynamics \ndominates. In fact, any real n is possible by tuning the ratio between the two {3's. \nIn the formulation presented in this paper n is always non-negative, but negative \nvalues are possible if the Hebbian rule of (3) is replaced by an anti-Hebbian form \nwith (UiO\"j) replaced by - (O\"iO\"j) (the case of negative n is being studied by Mezard \nand co-workers [7]). \nThe model discussed above is range-free/infinite-ranged and can therefore be an(cid:173)\nalyzed in the thermodynamic limit N -+ 00 by the replica mean-field theory as \ndevised for the Sherrington-Kirkpatrick spin-glass [6, 8, 9]. This can be developed \nprecisely for integer n [6, 8, 9, 10] and analytically continued. In the usual manner \nthere enters a spin-glass order parameter \n\n(, f- b) \n\nwhere the superscripts are replica labels. q\"{6 is given by the extremum of \n\nF({q1'6})=_LL:[q1'O]2+ ln Tr exp [ ~ L:O\"1'q1' OO\"O] \n\n2J-ln2 1'<6 \n\n{O\"1'} \n\nJ-ln2 1'<6 \n\nwhile Z~ is proportional to exp [NextrF ({q1'6})]. In the replica-symmetric region \n(or ansatz) one assumes q1'O = q. \nWe will first choose as the independent variables nand j3 and briefly discuss the \nphase picture of our model (full details can be found in [11]). The system exhibits \na transition from a paramagnetic state (q = 0) to an ordered state (q > 0) at a \ncritical j3c(n). For n ::; 2 this transition is second order at j3c = 1, down to the SK \n\n\f450 \n\nCoolen, Penney, and Sherrington \n\n0, but for n > 2 the coupled dynamics leads to a qualitative, \nspin-glass limit, n -\nas well as quantitative, change to first order. Replica symmetry is stable above a \ncritical value n c(!3), at which there is a de Almeida-Thouless (AT) transition (c.f. \nKondor [12]). As expected from spin-glass studies, n c(f3) goes to zero as {3 ! 1 \nbut rises for larger /3, having a maximum of order 0.3 at {3 of order 2. Thus, for \nn > nc(max) ::::: 0.3 there is no instability against small replica-symmetry breaking \nfluctuations, while for smaller n there is re-entrance in this stability. The transition \nfrom a paramagnetic to an ordered state and the onset of local RS instability for \nvarious temperatures is shown in Figure 1. \n\n3 EXTERNAL FIELDS \n\nSeveral simple modifications of the above model are possible. One consists of adding \nexternal fields to the spin dynamics and/or to the interaction dynamics, by making \nthe substitutions \n\nHV,j} ({O\"d) ~ HV'J} ({O\"d) - LOiO\"i \n\n1\u00a3 ( {Jii }) ~ 1\u00a3 ( {Jij }) - L hi Kij \n\ni<i \n\ni \n\nin (2) and (5) respectively. These external fields may be viewed as generating fields \nin the sense of (7); for example \n\nFor neural network models a natural first choice for the external fields would be \nOi = hei and Kij = Keiej, ei E {-I, I}, where the ei are quenched random vari(cid:173)\nables corresponding to an imposed pattern. Without loss of generality all the ei \ncan be taken as +1, via the gauge transformation O\"i ~ O\"iei, Jii ~ Jiieiei. Hence(cid:173)\nforth we shall make this choice. The neuron perturbation field h induces a finite \n'magnetization' characterized by a new order parameter \n\nm a = (O\"f) \n\nwhich is independent of Q: in the replica-symmetric assumption (which turns out \nto be stable with respect to variation in this parameter). As in the case of the \nspin-glass, there is now a critical surface in (h, n, {3) space characterizing the onset \nof replica symmetry breaking. In introducing the interaction perturbation field K \nwe find that K/ J-l is the analogue of the mean exchange Jo in the SK spin-glass \nmodel, ]2 = ({3nJ-l )-1 being the analogue of the variance. If large enough, this field \n\nleads to a spontaneous 'ferromagnetic' order. \n\nAgain we find further examples of both second and first order transitions (details \ncan be found in [11]). For the paramagnetic (P; m = 0, q = 0) to ferromagnetic \n(F; m I=- 0, q I=- 0) case, the transition is second order at the SK value f3Ja = 1 so \nlong as ({3])-2 ~ 3n - 2. Only when ({3])-2 < 3n - 2 do the interaction dynamics \n\n\fCoupled Dynamics of Fast Neurons and Slow Interactions \n\n451 \n\n1.2 \nll--_P_A_RAM __ A_GN_ET _____ -.-.-------.. ~ .\u2022 -\u00b7,...\u00b7 \n\n0.8 \nT \n0.6 \n\n0.2 \n\nWA'M'IS GLASS \n\nSPIN GLASS \n\n1 \n\n2 \n\nn \n\n3 \n\nFigure 1: Phasediagrarn for j = 1. Dotted line: first order transition, solid line: \nsecond order transition. The separation between Mattis-glass and spin-glass phase \nis defined by the de Almeida-Thouless instability \n\n\f452 \n\nCoolen, Penney, and Sherrington \n\ninfluence the transition, changing it to first order at a lower temperature. Regarding \nthe ferromagnetic to spin-glass (SGj m = 0, q \"# 0) transition, this exhibits both \nsecond order (lower .70) and first order (higher Jo) sections separated by a tricritical \npoint for n less than a critical value of the order of 3.3. This tricritical point exhibits \nre-entrance as a function of n. \n\n4 COMPARISON BETWEEN COUPLED DYNAMICS \n\nAND SK MODEL \n\nIn order to clarify the differences, we will briefly summarize the two routes that \nlead to an SK-type replica theory: \n\nCoupled Dynamics: \nFast Ising spin neurons + slow dynamic interactions, \n\nd \n-J .. = -((J'(J'){J .} + - - IIJ .. + GWN \ndt lJ \n\n1 \nN \n\nK \nN \n\nlJ \n\n'J \n\nI \n\nJ \n\nr \n\nFree energy: \n\nDefine: \n\nThermodynamics: \n\n-\nf - --_-logZ, \n\n-\n\n1 \nf3N \n\nz = fIT dhj e-1ht({J,j}) \n\ni<j \n\nio = K/ /-t, \n\nN-+oo: \n\nf = - f3n extr G ({q'Y }; {m'Y}) + const. \n-\n\n1 \n\nD \n\nSK spin-glass: \nIsing spins + fixed random interactions, \n\nP(Jij) = [27rJ2]-~e-~[J;j-Jo]2/J2 \n\nFree energy: \n\nf \n\nSelt-averaging: \n\nPhysical scaling: \n\nThermodynamics: \n\n] \n=--logZ=--hm- Z -1 \n\n1 . l[n \nf3N n-O n \n\n1 \nf3N \n\nJo = Jo/N, \n\nJ = J/Vii \n\nf = - lim 131 extr G ({q'YD}j {m'Y}) + const. \n\nn_O n \n\n\fCoupled Dynamics of Fast Neurons and Slow Interactions \n\n453 \n\n5 DISCUSSION \n\n\\Ve have obtained a solvable model with which a coupled dynamics of fast stochas(cid:173)\ntic neurons and slow dynamic interactions can be studied analytically. Furthermore \nit presents the replica method from a novel perspective, provides a direct inter(cid:173)\npretation of the replica dimension n in terms of parameters controlling dynamical \nprocesses and leads to new phase transition characters. As a model for neural learn(cid:173)\ning the specific example analyzed here is however only a first step, with hand K \nas introduced corresponding to only a single pattern. Its adaptation to treat many \npatterns is the next challenge. \n\nOne type of generalization is to consider the whole system as of lower connectivity \nwith only pairs of connected sites being available for interaction upgrade. For \nexample, the system could be on a lattice, in which case the corresponding coupled \npartition function will have the usual greater complication of a finite-dimensional \nsystem, or randomly connected with each bond present with a probability C IN, \nin which case there results an analogue of the Viana-Bray [13] spin-glass. In each \nof these cases the explicit factors involving N in the {hj} dynamics (3) should \nbe removed (their presence or absence being determined by the need for statistical \nrelevance and physical scaling). \n\nYet another generalization is to higher order interactions, for example to p-neuron \nones: \n\nHp} ({O\"d) = - L Ji l , ... ,i 1'O\"i 1 0\"i 2 \u2022\u2022 . 00i 1' \n\ni l, \u00b7 . . ,i l' \n\nwith corresponding interaction dynamics \n\nd \ndt \n\nr-J\u00b7 \n\n1 \n-(0\"' \n\n- N Sl'\" \n\n. -\n\n11 ,\" .t 1' \n\n0\"' ){J} -\n\n11' \n\nIIJ\u00b7 \nr \n\ntl , .. \u00b7,l1' \n\n. + -T)' \n\n1 \nffi Sl,\u00b7 .. ,l1' \n\n. (t) \n\nor to more complex neuron types. \nIf the symmetry-breaking fields Kij in the interaction dynamics are choosen at \nrandom, we obtain a curious theory in which we find replicas on top of replicas (the \nreplica trick would be used to deal with the quenched disorder of the K ij , for a \nmodel in which replicas are already present. due to the coupled dynamics). \n\nFinally, our approach can in fact be generalized to any statistical mechanical system \nwhich in equilibrium is described by a Boltzmann distribution in which the Hamilto(cid:173)\nnian has (adiabatically slowly) evolving parameters. By choosing these parameters \nto evolve according to an appropriate Langevin process (involving the free energy \nof the underlying faRt system) one always arrives at a replica theory describing the \ncoupled system in equilibrium. \n\nAcknowledgements \n\nFinancial support from the U.K. Science and Engineering Research Council under \ngrants 9130068X and GR/H26703, from the European Community under grant \nSISCI *915121, and from Jesus College, Oxford, is gratefully acknowledged. \n\n\f454 \n\nCoolen, Penney, and Sherrington \n\nReferences \n\n[1] Glauber R.J. (1963) J. Math. Phys. 4 294 \n[2] Hebb D.O. (1949) 'The Organization of Behaviour' (Wiley, New York) \n[3] Shinomoto S. (1987) J. Phys. A: Math. Gen. 20 L1305 \n[4] Jonker II.J.J. and Cool en A.C.C. (1991) J. Phys. A: Math. Gen. 24 4219 \n[5] Jonker H.J.J., Coolen A.C.C. and Denier van der Gon J.J. (1993) J. Phys. A: \n\nMath. Gen. 26 2549 \n\n[6] Sherrington D. and Kirkpatrick S. (1975) Phys. Rev. Lett. 35 1792 \n[7] Mezard M. prit1ate communication \n[8] Kirkpatrick S. and Sherrington D. (1978) Phys. Rev. B 17 4384 \n[9] Mezard M., Parisi G. and Virasoro M.A. (1987) 'Spin Glass Theory and Be(cid:173)\n\nyond' (World Scientific, Singapore) \n\n[10] Sherrington D. (1980) J. Phys. A: Math. Gen. 13 637 \n[11] Penney R.W., Cool en A.C.C. and Sherrington D. (1993) J. Phys. A: Math. \n\nGen. 26 3681-3695 \n\n[12] Kondor I. (198~{) J. Phys. A: Math. Gen. 16 L127 \n[13] Viana L. and Bray A.J. (1983) J. Phys. C 16 6817 \n\n\f", "award": [], "sourceid": 775, "authors": [{"given_name": "A.C.C.", "family_name": "Coolen", "institution": null}, {"given_name": "R.", "family_name": "Penney", "institution": null}, {"given_name": "D.", "family_name": "Sherrington", "institution": null}]}