{"title": "Order Reduction for Dynamical Systems Describing the Behavior of Complex Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 55, "page_last": 61, "abstract": null, "full_text": "Order Reduction for Dynamical Systems \n\nDescribing the Behavior of Complex Neurons \n\nThomas B. Kepler \nBiology Dept. \n\nL. F. Abbott \nPhysics Dept. \n\nEve Marder \nBiology Dept. \n\nBrandeis University \nWaltham, MA 02254 \n\nAbstract \n\nWe have devised a scheme to reduce the complexity of dynamical \nsystems belonging to a class that includes most biophysically realistic \nneural models. The reduction is based on transformations of variables \nand perturbation expansions and it preserves a high level of fidelity to \nthe original system. The techniques are illustrated by reductions of the \nHodgkin-Huxley system and an augmented Hodgkin-Huxley system. \n\nINTRODUCTION \n\nFor almost forty years, biophysically realistic modeling of neural systems has followed \nthe path laid out by Hodgkin and Huxley (Hodgkin and Huxley, 1952). Their seminal \nwork culminated in the accurately detailed description of the membrane currents \nexpressed by the giant axon of the squid Loligo, as a system of four coupled non-linear \ndifferential equations. Soon afterward (and ongoing now) simplified, abstract models \nwere introduced that facilitated the conceptualization of the model's behavior, e.g. \n(FitzHugh, 1961). Yet the mathematical relationships between these conceptual models \nand the realistic models have not been fully investigated. Now that neurophysiology is \ntelling us that most neurons are complicated and subtle dynamical systems, this situation \nis in need of change. We suggest that a systematic program of simplification in which \na realistic model of given complexity spawns a family of simplified meta-models of \nvarying degrees of abstraction could yield considerable advantage. In any such scheme, \nthe number of dynamical variables, or order, must be reduced, and it seems efficient and \nreasonable to do this first. This paper will be concerned with this step only. A sketch \nof a more thoroughgoing scheme proceeding ultimately to the binary formal neurons of \n\n55 \n\n\f56 \n\nKepler, Abbott, and Marder \n\nHopfield (Hopfield, 1982) has been presented elsewhere (Abbott and Kepler, 1990). \nThere are at present several reductions of the Hodgkin-Huxley (HH) system (FitzHugh, \n1961; Krinskii and Kokoz, 1973; Rose and Hindmarsh, 1989) but all of them suffer to \nvarying degrees from a lack of generality and/or insufficient realism. \n\nWe will present a scheme of perturbation analyses which provide a power-series \napproximation of the original high-order system and whose leading term is a lower-order \nsystem (see (Kepler et ai., 1991) for a full discussion). The techniques are general and \ncan be applied to many models. Along the way we will refer to the HH system for \nconcreteness and illustrations. Then, to demonstrate the generality of the techniques and \nto exhibit the theoretical utility of our approach, we will incorporate the transient outward \ncurrent described in (Connor and Stevens, 1972) and modeled in (Connor et aI., 1977) \nknown as IA\u2022 We will reduce the resulting sixth-order system to both third- and second(cid:173)\norder systems. \n\nEQUIVALENT POTENTIALS \n\nMany systems modeling excitable neural membrane consist of a differential equation \nexpressing current conservation \n\ndV edi + I(V, {Xi)) = I~t) \n\n(1) \n\nwhere V is the membrane potential difference, C is the membrane capacitance and I(V,x) \nis the total ionic current expressed as a function of V and the ~, which are gating \nvariables described by equations of the form \n\ndxi \ndi = kiO')(xj(J')-xJ. \n\n-\n\n(2) \n\nproviding the balance of the system's description. The ubiquity of the membrane \npotential and its role as \"command variable\" in these model systems suggests that we \nmight profit by introducing potential-like variables in place of the gating variables. We \ndefine the equivalent potential (EP) for each of the gating variables by \n\n-\nV, = Xi \n\n-1 \n\n) \n\n(Xi' \n\n(3) \n\nIn realistic neural models, the function i is ordinarily sigmoid and hence invertible. The \nchain rule may be applied to give us new equations of motion. Since no approximations \nhave yet been made, the system expressed in these variables gives exactly the same \nevolution for V as the original system. The evolution of the whole HH system expressed \nin EPs is shown in fig. 1. There is something striking about this collection of plots. The \ntransformation to EPs now suggests that of the four available degrees of freedom, -only \ntwo are actually utilized. Specifically, V m is nearly indistinguishable from V, and Vh and \nV n are likewise quite similar. This strongly suggests that we form averages and \ndifferences of EPs within the two classes. \n\n\fOrder Reduction for Dynamical Systems \n\n57 \n\nV \n\n60 \n\n30 \n\n0 \n\n-30 \n\n\"-'\" \n\n;:-\na -60 \n... J::: \n.::! \n... 0 \nv \n\n60 \n\n30 \n\n~ \n\nV \nm \n\n0 \n\n-30 \n-60 ~/l/\\/V \n20 25 \n\n10 \n\n0 \n\n5 \n\n,-. ) \nt (msec) \n\nVI>. \n\nV \nn \n\n5 \n\n10 \n\n15 20 25 \n\nt (msec) \n\nFigure 1: Behavior of equivalent potentials in repete(cid:173)\ntive firing mode of Hodgkin-Huxley system. \n\nPERTURBATION SERIES \n\nthe \n\nIn the general situation, the EPs \nmust be segregated into two or \nmore classes. One class will \ncontain \ntrue membrane \npotential V. Members of this \nclass will be subscripted with \ngreek letters /L, JI, etc. while the \nothers will be subscripted with \nlatin indices i, j, etc. We make, \nwithin each class, a change of \nvariables to 1) a new representa(cid:173)\ntive EP taken as a weighted \naverage over all members of the \nclass, and 2) differences between \neach member and their average. \nThe transformations and their \ninverses are \n\nand \n\n~ = L u\"V\" \na = v - \n\nII \n\nv \n\n\" \n\n\" \n\nt = LU,Vi \n\nI \n\n~, = V, - \n\nVI' = ~ - L uv~v + 3\" \n\nv \n\nVi = t - E U J~ J + 3,. \n\nJ \n\n(4) \n\n(5) \n\nWe constrain the ai and the al'to sum to one. The a's will not be taken as constants, \nbut will be allowed to depend on 4> and t/t. We expect, however, that their variation will \nbe small so that most of the time dependence of 4> and t/t will be carried by the V's. We \ndifferentiate eqs.(4), use the inverse transformations of eq.(5) and expand to first order \nin the o's to get \n\n--;ji = ~ u, k,(t~x~t> - x,(.\u00bb + 0(3) \ndt ~ \n\n-\n\n-\n\nand the new current conservation equation, \n\nc~ + uol(cj),{i',,(t)},(i,(t)}) = l~t) + O(~). \n\n(6) \n\n(7) \n\nThis is still a current conservation equation, only now we have renormalized the \ncapacitance in a state-dependent way through ao' The coefficient the of o's in eq.(6) will \nbe small, at least in the neighborhood of the equilibrium point, as long as the basic \npremise of the expansion holds. No such guarantee is made about the corresponding \n\n\f58 \n\nKepler, Abbott, and Marder \n\ncoefficient in eq.(7). Therefore we will choose the a's to make the correction term \nsecond order in the a's by setting the coefficient of each ai and al\u00a3 to zero. For the al\u00a3 we \nget, \n\nfor JL ~ 0, where \n\nal -( \n1,) == -x) \nax) \n\nand we use the abbreviation A - E I,,,. And for JL = 0, \n\nCX~ - u/'O - CE cxyky + Cil.o = 0 \n\nv.o \n\n(8) \n\n(9) \n\n(10) \n\nNow the time derivatives of the a's vanish at the equilibrium point, and it is with the \nneighborhood of this point that we must be primarily concerned. Ignoring these terms \nyields surprisingly good results even far from equilibrium. This choice having been \nmade, we solve for ao, as the root of the polynomial \n\nuoA - 1'0 - CE kl,,,[cxoA + Ck\"r1 = 0 \n\n11..0 \n\n(11) \n\nwhose order is equal to the number of EPs combining to form q,. The time dependence \nof Vt is given by specifying the ai' This may be done as for the al\u00a3 to get \n\n(12) \n\nEXAMPLE: HODGKIN-HUXLEY + IA \n\nFor the specific cases in which the HH system is reduced from fourth order to second, \nby combining V and V m to form q, and combining V h and V n to form Vt, the plan outlined \nabove works without any further meddling, and yields a very faithful reduction. Also \nstraightforward is the reduction of the sixth-order system given by Connor et al. (Connor \net aI., 1977) in which the HH system is supplemented by IA (HH + A) to third order. In \nthis reduction, the EP for the IA activation variable, a, joins V h and V n in the formation \nof Vt. Alternatively, we may reduce to a second order system in which V.joins with V \nand V m to form q, and the EPs for n,h and the IA inactivation variable, b, are combined \nto form Vt. This is not as straightforward. A direct application of eq.(12) produces a \ncurve of singularities where the denominator vanishes in the expression for dVt/dt; on one \nside dVt/dt has the same sign as q, - Vt, (which it should) and on the other side it does not. \nSome additional decisions must be made here. We may certainly take this to be an \nindication that the reduction is breaking down, but through good fortune we are able to \nsalvage it. This matter is dealt with in more detail elsewhere (Kepler et al., 1991). The \nreduced models are related in that the first is recovered when the maximum conductance \n\n\fOrder Reduction for Dynamical Systems \n\n59 \n\nof IA is set to zero in either of the other two. \n\n30 \n\n60 \n\n0 \n\n;:: \nIII \n\n(\") \nC \n\n'\"'l .., \n\no \n\n10 \n\n20 \nhme (mS) \n\n~----~------~------~--~ \n\n\" \n\" \n\" \" \n\n, ' \u2022 \n\" , , , , , , : \nI ' . \u00b7 . \u00b7 . \n\n(1) \n::l \n20 r+ \n..-.. \no 3 \n-20 2::-\n\nFigure 2: Response of full HH+A (solid line), 3Td order \n(dashed) and zuI order systems to current step, showing \nlatency to Jiring. \n\n-----\n> \nS \n'-\" 30 \nto \n:;:; \nC \n<:> \n..... \n0 \n0. \nIII -30 \nc \n/dt \n= 0 nullcline has acquired a \nIn order to get \nsecond branch. \nfrom the initial (hyperpolarized) \nlocation, the phase point must \ncrawl over this obstacle, and the \nlower it starts, the farther it has \nto climb. \n\n\u00b7 , , , , , , , , , , \n\nlatency \n\n, , . , , . I \n\n-20 \n\n-40 \n\n-60 \n\n........... \n:> \nE \n\nto \n\nFigure 4 shows \n\nthe \nfiring frequency as a function of \nthe injected current, for the full \nHH and HH+A sytems (solid \nlines), \nthe HH second order \nHH + A \n(dashed \nlines) and HH + A second order \n\nthird order \n\n, , , , \nI , , , , , , \n\n-BO \n\n-90 \n\n\u00b7 \u00b7 \u00b7 \u00b7 \u00b7 \n\n-GO \n\nI \n\nI \n\nI \nI \n\nI \nI \n\n0 \n-30 \nrp (m V) \n\n30 \n\n60 \n\nFigure 3: Phase portrait of event shown in Jig. 3 for \nzuI order reduced system. \n\n\f60 \n\nKepler, Abbott, and Marder \n\n(dotted line).- Note that the first reduction matches the full system quite well in both \ncases. The second reduction, however, does not do as well. It does get the qualitative \nfeatures right, though. The expansion of the dynamic range for the frequency of firing \nis still present, though squeezed into a much smaller interval on the current axis. The \nbifurcation occurs at nearly the right place and seems to have the proper character, i. e. , \nsaddle rather than Hopf, though this has not been rigorously investigated. \n\n0.3 ....----.--.,---,---,---.,..-------, \n\n......... \nN :\u00a7 0 .2 \n>. \n() \n~ \nQ) \n~ 01 \ncr \n. \nQ) \n..... \nI-. \n\nHH \n\no \n\n0 .0 '-----\"-_---'-_--'-_-L-_-'------' \n60 \n\n10 \n50 \ninjected current (/.LA) \n\n20 \n\n30 \n\n40 \n\nFigure 4: Firing frequency as a function of injected \ncurrent. Solid: full systems I dashed: rt order HH &: \n3Td order HH + A, dotted: ztd order HH + A. (From \nKepler et al. I 1991) \n\nCONCLUSION \n\nThe reduced systems \nare intended to be dynamically \nrealistic, to respond accurately \nto the kind of time-dependent \nexternal currents that would be \nencountered in real networks. \nTo put this to the test, we ran \nsimulations in which Icxtemal(t) \nwas given by a sum of sinusoids \nof equal amplitude and randomly \nchosen frequency and phase. \nFigure 5 illustrates the remark(cid:173)\nable match between the full \nHH + A system and the third(cid:173)\norder reduction, when such an \nirregular (quasiperiodic) current \nsignal is used to drive them. \n\nWe have presented a systematic approach to the reduction of order for a class of \ndynamical systems that includes the Hodgkin-Huxley system, the Connor et al. IA \nextension of the HH system, and many other realistic neuron models. As mentioned at \nthe outset, these procedures are merely the first steps in a more comprehensive program \nof simplification. In this way, the conceptual advantage of abstract models may be joined \nto the biophysical realism of physiologically derived models in a smooth and tractable \nmanner, and the benefits of simplicity may be enjoyed with a clear conscience. \n\n-For purposees of comparison, the HH system used here is as \nmodified by (Connoret al., 1977), but withIA removed and the leakage \nreversal potential adjusted to give the same resting potential as the \nHH+A cell. \n\n\fOrder Reduction for Dynamical Systems \n\n61 \n\n0 ~ ______ ~ ______ ~ ______ ~ ______ ~ ______ ~ \n\n_ \n> to \nS \n'-\"\"'0 \nro \n-\nC\") \n...... \n....., \nc: 0 \n....., \nQ) \no o..g \nc: ro \ns... \n.D \n6 \n6 \n\n0 \nto \nI \n\nQ) \n\nQ) \n\nI \n\n..... \n::; \n....... \n(!) \n() \nr(cid:173)\n(!) \n0.. \n\n() \nC \n'\"\"l \n'\"\"l \n(!) \n::; \n~------+-------~-------+------~------~ W r-\n0 . -\n1:: 0> \n\no \n\n100 \n\n200 \n\n300 \n\n400 \n\n500 \n\ntime (mS) \n\nFigure 5: Response of HH+A system to i\"eguiar cu\"ent injection. Solid line:full \nsystem, dashed line: r order reduction. \n\nAcknowledgment \n\nThis work was supported by National Institutes of Health grant T32NS07292 (TBK), \nDepartment of Energy Contract DE-AC0276-ER03230 (LFA) and National Institutes of \nMental Health grant MH46742 (EM). \n\nREFERENCES \n\nL.F.Abbottand T.B.Kepler, 1990 in Proceedings of the XI Sitges Conference on Neural \n\nNetworks in press \n\nJ.A.Connor and C.F.Stevens, 1971 J.Physiol.,Lond. 213 31 \nJ.A.Connor, D.Walter and R.McKown, 1977 Biophys.J. 18 81 \nR.FitzHugh, 1961 Biophys. J. 1445 \nA.L.Hodgkin and A.F.Huxley, 1952 J. Physiol. 117,500 \nJ.J.Hopfield, 1982 Proc.Nat.Acad.Sci. 792554 \nT.B.Kepler, L.F. Abbott and E.Marder, 1991 submitted to Bioi. Cyber. \nV.I.Krinskii and Yu.M.Kokoz, 1973 Biojizika 18506 \nR.M.Rose and J.L.Hindmarsh, 1989 Proc.R.Soc.Lond. 237267 \n\n\f", "award": [], "sourceid": 427, "authors": [{"given_name": "Thomas", "family_name": "Kepler", "institution": null}, {"given_name": "L.", "family_name": "Abbott", "institution": null}, {"given_name": "Eve", "family_name": "Marder", "institution": null}]}