{"title": "A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation", "book": "Advances in Neural Information Processing Systems", "page_first": 566, "page_last": 573, "abstract": "", "full_text": "A Hodgkin-Huxley Type Neuron Model \nThat Learns Slow Non-Spike Oscillation \n\nKenji Doya* \n\nAllen I. Selverston \nDepartment of Biology \n\nUniversity of California, San Diego \n\nLa Jolla, CA 92093-0357, USA \n\nPeter F. Rowat \n\nAbstract \n\nA gradient descent algorithm for parameter estimation which is \nsimilar to those used for continuous-time recurrent neural networks \nwas derived for Hodgkin-Huxley type neuron models. Using mem(cid:173)\nbrane potential trajectories as targets, the parameters (maximal \nconductances, thresholds and slopes of activation curves, time con(cid:173)\nstants) were successfully estimated. The algorithm was applied to \nmodeling slow non-spike oscillation of an identified neuron in the \nlobster stomatogastric ganglion. A model with three ionic currents \nwas trained with experimental data. It revealed a novel role of \nA-current for slow oscillation below -50 mY. \n\n1 \n\nINTRODUCTION \n\nConductance-based neuron models, first formulated by Hodgkin and Huxley [10], \nare commonly used for describing biophysical mechanisms underlying neuronal be(cid:173)\nhavior. Since the days of Hodgkin and Huxley, tens of new ionic channels have \nbeen identified [9]. Accordingly, recent H-H type models have tens of variables and \nhundreds of parameters [1, 2]. Ideally, parameters of H-H type models are deter(cid:173)\nmined by voltage-clamp experiments on individual ionic currents. However, these \nexperiments are often very difficult or impossible to carry out. Consequently, many \nparameters must be hand-tuned in computer simulations so that the model behavior \nresembles that of the real neuron. However, a manual search in a high dimensional \n\n*current address: The Salk Institute, CNL, P.O. Box 85800, San Diego, CA 92186-5800. \n\n566 \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n567 \n\nI \n\nFigure 1: A connectionist's view of the H-H neuron model. \n\nparameter space is very unreliable. Moreover, even if a good match is found be(cid:173)\ntween the model and the real neuron, the validity of the parameters is questionable \nbecause there are, in general, many possible settings that lead to apparently the \nsame behavior . \n\nWe propose an automatic parameter tuning algorithm for H-H type neuron models \n[5]. Since a H-H type model is a network of sigmoid functions, multipliers, and \nleaky integrators (Figure 1), we can tune its parameters in a manner similar to the \ntuning of connection weights in continuous-time neural network models [6, 12]. By \ntraining a model from many initial parameter points to match the experimental \ndata, we can systematically estimate a region in the parameter space, instead of a \nsingle point. \n\nWe first test if the parameters of a spiking neuron model can be identified from the \nmembrane potential trajectories. Then we apply the learning algorithm to a model \nof slow non-spike oscillation of an identified neuron in the lobster stomatogastric \nganglion [7]. The resulting model suggests a new role of A-current [3] for slow \noscillation in the membrane potential range below -50 m V. \n\n2 STANDARD FORM OF IONIC CURRENTS \n\nHistorically, different forms of voltage dependency curves have been used to repre(cid:173)\nsent the kinetics of different ionic channels. However, in order to derive a simple, \nefficient learning algorithm, we chose a unified form of voltage dependency curves \nwhich is based on statistical physics of ionic channels [11] for all the ionic currents \nin the model. \n\nThe dynamics of the membrane potential v is given by \n\nGil = I - LIj, \n\nj \n\n(1) \n\nwhere G is the membrane capacitance and I is externally injected current. The j-th \nionic current Ij is the product of the maximum conductance 9j, activation variable \n\n\f568 \n\nDoya, Selverston, and Rowat \n\naj, inactivation variable bj , and the difference of the membrane potential v from \nthe reversal potential Vrj. The exponents Pi and qj represent multiplicity of gating \nelements in the ionic channels and are usually an integer between 0 and 4. Variables \naj and bj are assumed to obey the first order differential equation \n\nTheir steady states ajoo and bjoo are sigmoid functions of the membrane potential \n\n(2) \n\nxoo(v) = \n\n1 \n()' (x=aj,bj ), \n\n1 + e-~'\" v-v\", \n\n(3) \n\nwhere Vx and Sx represent the threshold and slope of the steady state curve, re(cid:173)\nspectively. The rate coefficients ka \u00b7 (v) and kb \u00b7 (v) have the voltage dependence \n[11] \n\n] ] \n\nk ( ) - 1 \n-\nx v -\ntx \nwhere tx is the time constant. \n\nh sx( v - vx) \n, \n\ncos \n\n2 \n\n3 ERROR GRADIENT CALCULUS \n\nOur goal is to minimize the average error over a cycle with period T: \n\nE = ~ iT ~(v(t) - v*(t\u00bb2dt, \n\n(4) \n\n(5) \n\nwhere v*(t) is the target membrane potential trajectory. \nWe first derive the gradient of E with respect to the model parameters ( ... , Oi, ... ) = \n( ... , 9j, va], Saj' taj' ... ). In studies of recurrent neural networks, it has been shown \nthat teacher forcing is very important in training autonomous oscillation patterns [4, \n6, 12, 13]. In H-H type models, teacher forcing drives the activation and inactivation \nvariables by the target membrane potential v*(t) instead of vet) as follows. \n\nx = kx(v*(t\u00bb\u00b7 (-x +xoo(v*(t\u00bb) \n\n(x = aj,bj ). \n\nWe use (6) in place of (2) during training. \n\nThe effect of a small change in a parameter Oi of a dynamical system \n\nx = F(X; ... , Oi, ... ), \n\nis evaluated by the variation equation \n\n. of \nof \ny = oX y + OOi' \n\n(6) \n\n(7) \n\n(8) \n\nwhich is an n-dimensional linear system with time-varying coefficients [6, 12]. In \ngeneral, this variation calculus requires O(n 2 ) arithmetics for each parameter. How(cid:173)\never, in the case of H-H model with teacher forcing, (8) reduces to a first or second \norder linear system. For example, the effect of a small change in the maximum \nconductance 9j on the membrane potential v is estimated by \n\n(9) \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n569 \n\nwhere GCt) = l:k 9kak(t)Pkbk(t)Qk is the total membrane conductance. Similarly, \nthe effect of the activation threshold va] is estimated by the equations \n\nGiJ = -G(t)y - 9jpjaj(t)pj- 1bj (t)Qj(v(t) - Vrj) Z, \n\nZ = -kaj(t) [z + 8;j {aj(t) + ajoo(t) - 2aj(t)aj oo(t)}] . \n\n(10) \n\nThe solution yet) represents the perturbation in v at time t, namely 8;b~). The \nerror gradient is then given by \n\n1 fT \n\n.. \n\naE \nOBi = T Jo (v(t) - v (t)) OBi dt. \n\nav(t) \n\n(11) \n\n4 PARAMETER UPDATE \n\nBasically, we can use arbitrary gradient-based optimization algorithms, for example, \nsimple gradient descent or conjugate gradient descent. The particular algorithm we \nused was a continuous-time version of gradient descent on normalized parameters. \n\nBecause the parameters of a H-H type model have different physical dimensions and \nmagnitudes, it is not appropriate to perform simple gradient descent on them. We \nrepresent each parameter by the default value Oi and the deviation Bi as below. \n\n(12) \n\nThen we perform gradient descent on the normalized parameters Bi . \nInstead of updating the parameters in batches, i.e. after running the model for T \nand integrating the error gradient by (11), we updated the parameters on-line using \nthe running average of the gradient as follows. \n\n. \n\n1 . . av(t) OBi \nTa.D. o; = -.D.o, + T(v(t) - v (t)) OBi oBi' \n\n(13) \nwhere Ta is the averaging time and \u20ac \nis the learning rate. This on-line scheme was \nless susceptible to 2T-periodic parameter oscillation than batch update scheme and \ntherefore we could use larger learning rates. \n\nBi = -\u20ac.D. o, , \n\n5 PARAMETER ESTIMATION OF A SPIKING MODEL \n\nFirst, we tested if a model with random initial parameters can estimate the pa(cid:173)\nrameters of another moqel by training with its membrane potential trajectories. \nThe default parameters Bi of the model was set to match the original H-H model \n[10] (Table 1). Its membrane potential trajectories at five different levels of current \ninjection (I = 0,15,30,45, and 60J..lA/cm2 ) were used alternately as the target v*(t). \nWe ran 100 trials after initializing Bi randomly in [-0.5,+0.5]. In 83 cases, the error \nbecame less than 1.3 m V rms after 100 cycles of training. Figure 2a is an exam(cid:173)\nple of the oscillation patterns of the trained model. The mean of the normalized \n\n\f570 \n\nDoya, Selverston, and Rowat \n\nTable 1: Parameters of the spiking neuron model. Subscripts L, Na and K speci(cid:173)\nfies leak, sodium and potassium currents, respectively. Constants: C=1J.lF/cm2 , \nvNa=55mV, vK=-72mV, vL=-50mV, PNa=3, QNa=l, PK=4, QK=PL=qL=O, \nLlv=20mV, (=0.1, Ta = 5T. \n\ndefault value iii mean \n-0.017 \n-0.002 \n0.006 \n-0.052 \n-0.103 \n0.012 \n-0.010 \n0.093 \n0.050 \n-0.021 \n-0.061 \n-0.073 \n\n0.1 \nl/mV \n0.5 msec \n-62.0 mV \n-0.09 \nl/mV \n12.0 msec \n40.0 mS/cm2 \n-50.0 mV \n0.06 \nl/mV \n5.0 msec \n\n()i after learning \ns.d. \n0.252 \n0.248 \n0.033 \n0.073 \n0.154 \n0.202 \n0.140 \n0.330 \n0.264 \n0.136 \n0.114 \n0.168 \n\n0.3 mS/cm \n120.0 mS/cm2 \n-36.0 mV \n\ngL \ngNa \nVaNa \nSaNa \ntaNa \nVbNa \nSbNa \ntbNa \ngK \nVaK \nSaK \ntaK \n\nv[ \n\na_No [ \n\nb_Na[ ________ \n\na_K [-------.....-\n\ntaX \n\nsaK \n\nvaK \n\ngK \n\nIbNa \n\nsbNa \n\nvbNa \n\ntaNa \n\nsaNa \n\nvaNa \n\ngNa \n\ngL \n\no \n\n10 \n\ntime (ms) \n\n20 \n\n30 \n\ngL gNa vaNa saNa taNa vbNasbNa IbNa gK vaK saK \n\ntaK \n\n(a) \n\n(b) \n\nFigure 2: (a) The trajectory of the spiking neuron model at I = 30J.lA/cm2 \u2022 v: \nmembrane potential (-80 to +40 mY). a and b: activation and inactivation variables \n(0 to 1). The dotted line in v shows the target trajectory v*(t). (b) Covariance \nmatrix of the normalized parameters Oi after learning. The black and white squares \nrepresent negative and positive covariances, respectively. \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n571 \n\nTable 2: Parameters of the DG cell model. Constants: C=1J.lF/cm2 , vA=-80mV , \nVH= -lOmV, vL=-50mV, PA=3, qA=l, PH=l, QH=PL=qL=O, ~v=20mV, (=0.1, \nTa = 2T. \n\niJ\u00b7 \nt \n0.01 \n50 \n-12 \n0.04 \n7.0 \n-62 \n-0.16 \n300 \n0.1 \n-70 \n-0.14 \n3000 \n\ngL \ngA \nVaA \nSaA \ntaA \nVbA \nSbA \ntllA \ngH \nVaH \nSaH \ntaH \n\ntuned (}i \n\n0.025 mS cm \n41.0 mS/cm2 \n-11.1 mV \n0.022 1/mV \n7.0 msec \n-76 mV \n\n-0.19 1/mV \n292 msec \n\nv[ \n\na~[ \n\nb~[ \n\n'_H[ \n\n---\n\n-\n\n0.039 mS/cm2 \n-75.1 mV \n\n-0.11 1/mV ~ ~ \"\"\"'\" \n\n4400 msec \n\nI_L \n\nI _A \n\n10000 \n\n20000 \n\ntlme(msl \n\n30000 \n\n40000 \n\n50000 \n\nFigure 3: Oscillation pattern of the DG cell model. v: membrane potential (-70 to \n-50 mY). a and b: activation and inactivation variables (0 to 1) . I: ionic currents \n(-1 to +1 pAlcm2 ). \n\nparameters iii were nearly zero (Table 1), which implies that the original parame(cid:173)\nter values were successfully estimated by learning. The standard deviation of each \nparameter indicates how critical its setting is to replicate the given oscillation pat(cid:173)\nterns. From the covariance matrix of the parameters (Figure 2b), we can estimate \nthe distribution of the solution points in the parameter space. \n\n6 MODELING SLOW NON-SPIKE OSCILLATION \n\nNext we applied the algorithm to experimental data from the \"DG cell\" of the lob(cid:173)\nster stomatogastric ganglion [7]. An isolated DG cell oscillates endogenously with \nthe acetylcholine agonist pilocarpine and the sodium channel blocker TTX. The \noscillation period is 5 to 20 seconds and the membrane potential is approximately \nbetween -70 and -50 m V. From voltage-clamp data from other stomatogastric neu(cid:173)\nrons [8], we assumed that A-current (potassium current with inactivation) [3] and \nH-current (hyperpolarization-activated slow inward current) are the principal active \ncurrents in this voltage range. The default parameters for these currents were taken \nfrom [2] (Table 2). \n\n\f572 \n\nDoya, Selverston, and Rowat \n\nionic currents \n\n. / \n\n~ ~ \n\n../ \n\n.. .. .... ~ \n\nV ..-\n\n~ \n\n2 \n\n,r .. \n.. \n1 \n, \n~' \no \nW \nIf \" \n\n-2 \n\n-60 -40 -20 \n\n0 \n\n20 \n\n40 \n\nv \n\n(mV) \n\nFigure 4: Current-voltage curves of the DG cell model. Outward current is positive. \n\nFigure 3 is an example of the model behavior after learning for 700 cycles. The \nactual output v of the model, which is shown in the solid curve, was very close \nto the target output v*(t), which is shown in the dotted curve. The bottom three \ntraces show the ionic currents underlying this slow oscillation. Figure 4 shows the \nsteady state I-V curves of three currents. A-current has negative conductance in \nthe range from -70 to -40 m V. The resulting positive feedback on the membrane \npotential destabilizes a quiescent state. If we rotate the I-V diagram 180 degrees, it \nlooks similar to the I-V diagram for the H-H model; the faster outward A-current \nin our model takes the role of the fast inward sodium current in the H-H model and \nthe slower inward H-current takes the role of the outward potassium current. \n\n7 DISCUSSION \n\nThe results indicate that the gradient descent algorithm is effective for estimating \nthe parameters of H-H type neuron models from membrane potential trajectories. \n\nRecently, an automatic parameter search algorithm was proposed by Bhalla and \nBower [1]. They chose only the maximal conductances as free parameters and used \nconjugate gradient descent . The error gradient was estimated by slightly changing \neach of the parameters. In our approach, the error gradient was more efficiently de(cid:173)\nrived by utilizing the variation equations. The use of teacher forcing and parameter \nnormalization was essential for the gradient descent to work. \n\nIn order for a neuron to be an endogenous oscillator, it is required that a fast pos(cid:173)\nitive feedback mechanism is balanced with a slower negative feedback mechanism. \nThe most popular example is the positive feedback by the sodium current and the \nnegative feedback by the potassium current in the H-H model. Another common \nexample is the inward calcium current counteracted by the calcium dependent out(cid:173)\nward potassium current. We found another possible combination of positive and \nnegative feedback with the help of the algorithm: the inactivation of the outward \nA-current and the activation of the slow inward H-current. \n\n\fA Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation \n\n573 \n\nAcknowledgements \n\nThe authors thank Rob Elson and Thom Cleland for providing physiological data \nfrom stomatogastric cells. This study was supported in part by ONR grant N00014-\n91-J-1720. \n\nReferences \n[1] U. S. Bhalla and J. M. Bower. Exploring parameter space in detailed single \nneuron models: Simulations of the mitral and granule cells of the olfactory \nbulb. Journal of Neurophysiology, 69:1948-1965, 1993. \n\n[2] F. Buchholtz, J. Golowasch, I. R. Epstein, and E. Marder. Mathematical model \nof an identified stomatogastric ganglion neuron. Journal of Neurophysiology, \n67:332-340, 1992. \n\n[3] J. A. Connor, D. Walter, and R. McKown. Neural repetitive firing, modifi(cid:173)\n\ncations of the Hodgkin-Huxley axon suggested by experimental results from \ncrustacean axons. Biophysical Journal, 18:81-102, 1977. \n\n[4] K. Doya. Bifurcations in the learning of recurrent neural networks. In Proceed(cid:173)\nings of 1992 IEEE International Symposium on Circuits and Systems, pages \n6:2777-2780, San Diego, 1992. \n\n[5] K. Doya and A. I. Selverston. 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Journal of \nPhysiology, 117:500-544, 1952. \n\n[11] H. Lecar, G. Ehrenstein, and R. Latorre. Mechanism for channel gating in \nexcitable bilayers. Annals of the New York Academy of Sciences, 264:304-313, \n1975. \n\n[12] P. F. Rowat and A.I. Selverston. Learning algorithms for oscillatory networks \n\nwith gap junctions and membrane currents. Network, 2:17-41, 1991. \n\n[13] R. J. Williams and D. Zipser. Gradient based learning algorithms for recurrent \nconnectionist networks. Technical Report NU-CCS-90-9, College of Computer \nScience, Northeastern University, 1990. \n\n\f", "award": [], "sourceid": 783, "authors": [{"given_name": "Kenji", "family_name": "Doya", "institution": null}, {"given_name": "Allen", "family_name": "Selverston", "institution": null}, {"given_name": "Peter", "family_name": "Rowat", "institution": null}]}