A Hodgkin-Huxley Type Neuron Model 
That Learns Slow Non-Spike Oscillation 
Kenji Doya* 
Allen I. Selverston 
Department of Biology 
University of California, San Diego 
La Jolla, CA 92093-0357, USA 
Peter F. Rowat 
Abstract 
A gradient descent algorithm for parameter estimation which is 
similar to those used for continuous-time recurrent neural networks 
was derived for Hodgkin-Huxley type neuron models. Using mem- 
brane potential trajectories as targets, the parameters (maximal 
conductances, thresholds and slopes of activation curves, time con- 
stants) were successfully estimated. The algorithm was applied to 
modeling slow non-spike oscillation of an identified neuron in the 
lobster stomatogastric ganglion. A model with three ionic currents 
was trained with experimental data. It revealed a novel role of 
A-current for slow oscillation below -50 mV. 
I INTRODUCTION 
Conductance-based neuron models, first formulated by Hodgkin and Huxley [10], 
are commonly used for describing biophysical mechanisms underlying neuronal be- 
havior. Since the days of Hodgkin and Huxley, tens of new ionic channels have 
been identified [9]. Accordingly, recent H-H type models have tens of variables and 
hundreds of parameters [1, 2]. Ideally, parameters of H-H type models are deter- 
mined by voltage-clamp experiments on individual ionic currents. However, these 
experiments are often very difficult or impossible to carry out. Consequently, many 
parameters must be hand-tuned in computer simulations so that the model behavior 
resembles that of the real neuron. However, a manual search in a high dimensional 
*current address: The Salk Institute, CNL, P.O. Box 85800, San Diego, CA 92186-5800. 
566 
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 567 
Figure 1: A connectionist's view of the H-H neuron model. 
parameter space is very unreliable. Moreover, even if a good match is found be- 
tween the model and the real neuron, the validity of the parameters is questionable 
because there are, in general, many possible settings that lead to apparently the 
same behavior. 
We propose an automatic parameter tuning algorithm for H-H type neuron models 
[5]. Since a H-H type model is a network of sigmoid functions, multipliers, and 
leaky integrators (Figure 1), we can tune its parameters in a manner similar to the 
tuning of connection weights in continuous-time neural network models [0, 12]. By 
training a model from many initial parameter points to match the experimental 
data, we can systematically estimate a region in the parameter space, instead of a 
single point. 
We first test if the parameters of a spiking neuron model can be identified from the 
membrane potential trajectories. Then we apply the learning algorithm to a model 
of slow non-spike oscillation of an identified neuron in the lobster stomatogastric 
ganglion [7]. The resulting model suggests a new role of A-current [3] for slow 
oscillation in the membrane potential range below -50 mV. 
2 STANDARD FORM OF IONIC CURRENTS 
Historically, different forms of voltage dependency curves have been used to repre- 
sent the kinetics of different ionic channels. However, in order to derive a simple, 
efficient learning algorithm, we chose a unified form of voltage dependency curves 
which is based on statistical physics of ionic channels [11] for all the ionic currents 
in the model. 
The dynamics of the membrane potential v is given by 
Ci '- I- EIj, I.i = g.ia?bjq. J(v - vrj), (1) 
J 
where C is the membrane capacitance and I is externally injected current. The j-th 
ionic current Ij is the product of the maximum conductance gj, activation variable 
568 Doya, Selverston, and Rowat 
aj, inactivation variable bj, and the difference of the membrane potential v from 
the reversal potential vrj. The exponents pj and qj represent multiplicity of gating 
elements in the ionic channels and are usually an integer between 0 and 4. Variables 
aj and bj are assumed to obey the first order differential equation 
 = k(v). (-x + x(v)), (: aj, bj). (2) 
Their steady states aj and bj are sigmoid functions of the membrane potential 
1 
 () = ( =  ), (3) 
1 + e-(-) ' ' 
where v and s represent the threshold and slope of the steady state curve, re- 
spectively. The rate coefficients k(v) and ka(v) have the voltage dependence 
[] 
1 cosh s(v - v) (x = a, b), (4) 
():  2 
where t is the time constant. 
3 ERROR GRADIENT CALCULUS 
Our goal is to minimize the average error over a cycle with period T: 
E: 7 (v(t)- *(t))"t, 
where v*(t) is the target membrane potential trajectory. 
(5) 
i> = OF OF 
OX Y + OOi' 
.' = F(X;...,Oi,...), 
is evaluated by the variation equation 
i: kx(*(t)). (- + (*(t)))(: a,b). 
We use (6) in place of (2) during training. 
The effect of a small change in a parameter Oi of a dynamical system 
(x  a ') (7) 
(  a'), (8) 
which is an n-dimensional linear system with time-varying coefficients [6, 12]. In 
general, this variation calculus requires O(n 2) arithmetics for each parameter. How- 
ever, in the case of H-H model with teacher forcing, (8) reduces to a first or second 
order linear system. For example, the effect of a small change in the maximum 
conductance gj on the membrane potential v is estimated by 
Ct = -G(t)y - aj(7)PJ bj(7 )qJ (v(7) -- Vrj ), (9) 
(6) 
We first derive the gradient of E with respect to the model parameters (..., Oi, ...) = 
(..., gj, va,, sa, ta, ...). In studies of recurrent neural networks, it has been shown 
that teacher forcing is very important in training autonomous oscillation patterns [4, 
6, 12, 13]. In H-H type models, teacher forcing drives the activation and inactivation 
variables by the target membrane potential v* (t) instead of v(t) as follows. 
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 569 
where G(t) :  ga(t)Pkb(t) qk is the total membrane conductance. 
the effect of the activation threshold va is estimated by the equations 
C = -G(t)y - gipjaj(t)PJ-bj(t)qJ(v(t) - vrj) z, 
}: --kay(t)[Z + S{aj(t)+aj(t)- 2aj(t)aj(t)}] . 
The solution y(t) represents the perturbation in v at time t, namely 
error gradient is then given by 
OE 1 for 
ooi = 7 
Similarly, 
(10) 
The 
(11) 
4 PARAMETER UPDATE 
Basically, we can use arbitrary gradient-based optimization algorithms, for example, 
simple gradient descent or conjugate gradient descent. The particular algorithm we 
used was a continuous-time version of gradient descent on normalized parameters. 
Because the parameters of a H-H type model have different physical dimensions and 
magnitudes, it is not appropriate to perform simple gradient descent on them. We 
represent each parameter by the default value Oi and the deviation i as below. 
gj=jeJ, 
Then we perform gradient descent on the normalized parameters i. 
Instead of updating the parameters in batches, i.e. after running the model for T 
and integrating the error gradient by (11), we updated the parameters on-line using 
the running average of the gradient as follows. 
Ov(t) 00 
+ 
Oi -- --/X,, (13) 
where Ta is the averaging time and e is the learning rate. This on-line scheme was 
less susceptible to 2T-periodic parameter oscillation than batch update scheme and 
therefore we could use larger learning rates. 
5 PARAMETER ESTIMATION OF A SPIKING MODEL 
First, we tested if a model with random initial parameters can estimate the pa- 
rameters of another model by training with its membrane potential trajectories. 
The default parameters Oi of the model was set to match the original H-H model 
[10] (Table 1). Its membrane potential trajectories at five different levels of current 
injection (I = 0, 15,30, 45, and 60yA/cm ) were used alternately as the target v*(t). 
We ran 100 trials after initializing i randomly in [-0.5,+0.5]. In 83 cases, the error 
became less than 1.3 mV rms after 100 cycles of training. Figure 2a is an exam- 
ple of the oscillation patterns of the trained model. The mean of the normalized 
570 Doya, Selverston, and Rowat 
Table 1: Parameters of the spiking neuron model. Subscripts L, Na and K speci- 
fies leak, sodium and potassium currents, respectively. Constants: C=lluF/cm 2, 
VN,=55mV, vK=-72mV, v=-50mV, pN,=3, qN,=l, pK=4, qK=p=q=O, 
Av=20mV, e=0.1, T, = 5T. 
gL 
gNa 
VaNa 
$aNa 
7aN a 
VbNa 
$bNa 
7bNa 
gK 
YaK 
8aK 
7 a K 
default value 
0.3 mS/cm 
120.0 mS/cm 
-36.0 mV 
0.1 1/mY 
0.5 l'nsec 
-62.0 mV 
-0.09 1/mV 
12.0 msec 
40.0 mS/cm 
-50.0 mV 
0.06 1/mV 
5.0 msec 
ti after learning 
mean s.d. 
-0.017 0.252 
-0.002 0.248 
0.006 0.033 
-0.052 0.073 
-0.103 0.154 
0.012 0.202 
-0.010 0.140 
0.093 0.330 
0.050 0.264 
-0.021 0.136 
-0.061 0.114 
-0.073 0.168 
a_Na tbNa 
sbNa 
b_Na taNa 
saNa 
gNa 
0 10 20 30 
time (ms) 
gL gNavaNasaNataNavbNasbNatbNa gK yak saK taK 
(a) (b) 
Figure 2: (a) The trajectory of the spiking neuron nodel at I = 30t. tA/cm . v: 
membrane potential (-80 to +40 mV). a and b: activation and inactivation variables 
(0 to 1). The dotted line in v shows the target trajectory v*(t). (b) Covariance 
matrix of the normalized parameters i after learning. The black and white squares 
represent negative and positive covariances, respectively. 
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 571 
Table 2: Parameters of the DG cell model. Constants: C=lF/cm 2, VA=-80mV, 
VH= -10mV, v=-50mV, pA=3, qA=l, pH=l, qH=p=q=O, Av=20mV, e=0.1, 
T, = 2T. 
tuned 0 i 
0.01 0.025 mS/cm 2 
50 41.0 mS/cm 2 
-12 -11.1 mV 
0.04 0.022 1/mV 
7.0 7.0 msec 
-62 -76 mV 
-0.16 -0.19 1/mV 
300 292 msec 
0.1 0.039 mS/cm 2 
-70 -75.1 mV 
-0.14 -0.11 1/mV 
3000 4400 msec 
gL 
VaA 
$aA 
taA 
VbA 
8bA 
7bA 
gH 
Wall 
Sail 
7 aH 
0 10000 20000 30000 40000 50000 
time(ms) 
Figure 3: Oscillation pattern of the DG cell model. v: membrane potential (-70 to 
-50 mV). a and b: activation and inactivation variables (0 to 1). I: ionic currents 
(-1 to +1 tA/cm2). 
parameters i were nearly zero (Table 1), which implies that the original parame- 
ter values were successfully estimated by learning. The standard deviation of each 
parameter indicates how critical its setting is to replicate the given oscillation pat- 
terns. From the covariance matrix of the parameters (Figure 2b), we can estimate 
the distribution of the solution points in the parameter space. 
6 MODELING SLOW NON-SPIKE OSCILLATION 
Next we applied the algorithm to experimental data from the "DG cell" of the lob- 
ster stomatogastric ganglion [7]. An isolated DG cell oscillates endogenously with 
the acetylcholine agonist pilocarpine and the sodium channel blocker TTX. The 
oscillation period is 5 to 20 seconds and the membrane potential is approximately 
between -70 and -50 mV. From voltage-clamp data from other stomatogastric neu- 
rons [8], we assumed that A-current (potassium current with inactivation) [3] and 
H-current (hyperpolarization-activated slow inward current) are the principal active 
currents in this voltage range. The default parameters for these currents were taken 
from [2] (Table 2). 
572 Doya, Selverston, and Rowat 
2 
-2 
ionic currents 
! 
! 
! 
-60 -40 -20 0 20 40 
v (mY) 
--I_L 
.... I_A 
-- I_H 
Figure 4: Current-voltage curves of the DG cell model. Outward current is positive. 
Figure 3 is an example of the model behavior after learning for 700 cycles. The 
actual output v of the model, which is shown in the solid curve, was very close 
to the target output v*(t), which is shown in the dotted curve. The bottom three 
traces show the ionic currents underlying this slow oscillation. Figure 4 shows the 
steady state I-V curves of three currents. A-current has negative conductance in 
the range from -70 to -40 mV. The resulting positive feedback on the membrane 
potential destabilizes a quiescent state. If we rotate the I-V diagram 180 degrees, it 
looks similar to the I-V diagram for the H-H model; the faster outward A-current 
in our model takes the role of the fast inward sodium current in the H-H model and 
the slower inward H-current takes the role of the outward potassium current. 
7 DISCUSSION 
The results indicate that the gradient descent algorithm is effective for estimating 
the parameters of H-H type neuron models from membrane potential trajectories. 
Recently, an automatic parameter search algorithm was proposed by Bhalla and 
Bower [1]. They chose only the maximal conductances as free parameters and used 
conjugate gradient descent. The error gradient was estimated by slightly changing 
each of the parameters. In our approach, the error gradient was more efficiently de- 
rived by utilizing the variation equations. The use of teacher forcing and parameter 
normalization was essential for the gradient descent to work. 
In order for a neuron to be an endogenous oscillator, it is required that a fast pos- 
itive feedback mechanism is balanced with a slower negative feedback mechanism. 
The most popular example is the positive feedback by the sodium current and the 
negative feedback by the potassium current in the H-H model. Another common 
example is the inward calcium current counteracted by the calcium dependent out- 
ward potassium current. We found another possible combination of positive and 
negative feedback with the help of the algorithm: the inactivation of the outward 
A-current and the activation of the slow inward H-current. 
A Hodgkin-Huxley Type Neuron Model That Learns Slow Non-Spike Oscillation 573 
Acknowledgement s 
The authors thank Rob Elson and Thom Cidand for providing physiological data 
from stomatogastric cells. This study was supported in part by ONR grant N00014- 
91-J-1720. 
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