Order Reduction for Dynamical Systems 
Describing the Behavior of Complex Neurons 
Thomas B. Kepler 
Biology Dept. 
L. F. Abbott 
Physics Dept. 
Brandeis University 
Waltham, MA 02254 
Eve Marder 
Biology Dept. 
Abstract 
We have devised a scheme to reduce the complexity of dynamical 
systems belonging to a class that includes most biophysically realistic 
neural models. The reduction is based on transformations of variables 
and perturbation expansions and it preserves a high level of fidelity to 
the original system. The techniques are illustrated by reductions of the 
Hodgkin-Huxley system and an augmented Hodgkin-Huxley system. 
INTRODUCTION 
For almost forty years, biophysically realistic modeling of neural systems has followed 
the path laid out by Hodgkin and Huxley (Hodgkin and Huxley, 1952). Their seminal 
work culminated in the accurately detailed description of the membrane currents 
expressed by the giant axon of the squid Loligo, as a system of four coupled non-linear 
differential equations. Soon afterward (and ongoing now) simplified, abstract models 
were introduced that facilitated the conceptualization of the model's behavior, e.g. 
(FitzHugh, 1961). Yet the mathematical relationships between these conceptual models 
and the realistic models have not been fully investigated. Now that neurophysiology is 
telling ns that most neurons are complicated and subtle dynamical systems, this situation 
is in need of change. We suggest that a systematic program of simplification in which 
a realistic model of given complexity spawns a family of simplified meta-models of 
varying degrees of abstraction could yield considerable advantage. In any such scheme, 
the number of dynamical variables, or order, must be reduced, and it seems efficient and 
reasonable to do this first. This paper will be concerned with this step only. A sketch 
of a more thoroughgoing scheme proceeding ultimately to the binary formal neurons of 
55 
56 Kepler, Abbott, and Marder 
Hopfield (Hopfield, 1982) has been presented elsewhere (Abbott and Kepler, 1990). 
There are at present several reductions of the Hodgkin-Huxley (HH) system (FitzHugh, 
1961; Krinskii and Kokoz, 1973; Rose and Hindmarsh, 1989) but all of them suffer to 
varying degrees from a lack of generality and/or insufficient realism. 
We will present a scheme of perturbation analyses which provide a power-series 
approximation of the original high-order system and whose leading term is a lower-order 
system (see (Kepler et al., 1991) for a full discussion). The techniques are general and 
can be applied to many models. Along the way we will refer to the HH system for 
concreteheSS and illustrations. Then, to demonstrate the generality of the techniques and 
to exhibit the theoretical utility of our approach, we will incorporate the transient outward 
current described in (Connor and Stevens, 1972) and modeled in (Connor et al., 1977) 
known as I A. We will reduce the resulting sixth-order system to both third- and second- 
order systems. 
EQUIVALENT POTENTIALS 
Many systems modeling excitable neural membrane consist of a differential equation 
expressing current conservation 
c ar . ,r(g, {xJ) -- 
(1) 
where V is the membrane potential difference, C is the membrane capacitance and I(V,x) 
is the total ionic current expressed as a function of V and the x i, which are gating 
variables described by equations of the form 
- kt(l/)((V)-x.,). (2) 
providing the balance of the system's description. The ubiquity of the membrane 
potential and its role as "command variable" in these model systems suggests that we 
might profit by introducing potential-like variables in place of the gating variables. We 
define the equivalent potential (EP) for each of the gating variables by 
;'i -- (3) 
In realistic neural models, the function g is ordinarily sigmoid and hence invertible. The 
chain rule may be applied to give us new equations of motion. Since no approximations 
have yet been made, the system expressed in these variables gives exactly the same 
evolution for V as the original system. The evolution of the whole HH system expressed 
in EPs is shown in fig. 1. There is something striking about this collection of plots. The 
transformation to EPs now suggests that of the four available degrees of freedom,-only 
two are actually utilized. Specifically, V m is nearly indistinguishable from V, and V h and 
V,, are likewise quite similar. This strongly suggests that we form averages and 
differences of EPs within the two classes. 
Order Reduction for Dynamical Systems 57 
6O 
3O 
0 
-30 
-60 
6O 
3O 
0 
-30 
-60 
0 
v 
5 10 .3 2O 25 
t (msec) 
V 
5 10 15 20 25 
t (msec) 
Figure 1: Behavior of equivalent potentials in repete- 
tive firing mode of Hodgkin-Huxley system. 
PERTURBATION SERIES 
In the general situation, the EPs 
must be segregated into two or 
more classes. 
contain the 
potential V. 
class will be 
One class will 
true membrane 
Members of this 
subscripted with 
greek letters/, v, etc. while the 
others will be subscripted with 
latin indices i, j, etc. We make, 
within each class, a change of 
variables to l) a new representa- 
tive EP taken as a weighted 
average over all members of the 
class, and 2) differences between 
each member and their average. 
The transformations and their 
inverses are 
(4) 
and 
We constrain the ot i and the % to sum to one. The ot's will not be taken as constants, 
but will be allowed to depend on b and . We expect, however, that their variation will 
be small so that most of the time dependence of b and  will be carried by the V's. We 
differentiate eqs. (4), use the inverse transformations of eq. (5) and expand to first order 
in the 's toget 
(6) 
and the new current conservation equation, 
db d(b, ii'.(b)},{()}) -- l,,.a(t) + 0(). 
+ 
(7) 
This is still a current conservation equation, only now we have renormalized the 
capacitance in a state-dependent way through u 0. The coefficient the of 's in eq.(6) will 
be small, at least in the neighborhood of the equilibrium point, as long as the basic 
premise of the expansion holds. No such guarantee is made about the corresponding 
58 Kepler, Abbott, and Marder 
coefficient in eq.(7). Therefore we will choose the ot's to make the correction term 
second order in the/J's by setting the coefficient of each/Ji and/Jr, to zero. For the/Jr, we 
get, 
for   O, where 
(8) 
and we use the abbreviation A -- 1 I,,. And for  = 0, 
aA - 0/,0 - C avkv + C& 0 -- 0 (10) 
Now the time derivatives of the ot's vanish at the equilibrium point, and it is with the 
neighborhood of this point that we must be primarily concerned. Ignoring these terms 
yields surprisingly good results even far from equilibrium. This choice having been 
made, we solve for ot 0, as the root of the polynomial 
%a - - kd,.[%a  ck.l -t; 0 
(11) 
whose order is equal to the number of EPs combining to form 4'. The time dependence 
of k is given by specifying the ot i. This may be done as for the %, to get 
(12) 
EXAMPLE: HODGKIN-HUXLEY + I, 
For the specific cases in which the HH system is reduced from fourth order to second, 
by combining V and V m to form qb and combining V h and V,, to form , the plan outlined 
above works without any further meddling, and yields a very faithful reduction. Also 
straightforward is the reduction of the sixth-order system given by Connor et al. (Connor 
et al., 1977) in which the HH system is supplemented by I^ (I-IH+A) to third order. In 
this reduction, the EP for the I^ activation variable, a, joins Vh and V,, in the formation 
of . Alternatively, we may reduce to a second order system in which V a joins with V 
and V m to form qb and the EPs for n,h and the I^ inacfivation variable, b, are combined 
to form . This is not as straightforward. A direct application of eq.(12) produces a 
curve of singularities where the denominator vanishes in the expression for d/dt; on one 
side dk/dt has the same sign as qb - k, (which it should) and on the other side it does not. 
Some additional decisions must be made here. We may certainly take this to be an 
indication that the reduction is breaking down, but through good fortune we are able to 
salvage it. This matter is dealt with in more detail elsewhere (Kepler et al., 1991). The 
reduced models are related in that the first is recovered when the maximum conductance 
Order Reduction for Dynamical Systems 59 
of I A is set to zero in either of the other two. 
6O 
3O 
-3O 
-6O 
-90 
I 
lO 20 30 
time (mS) 
Figure 2: Response of full HH+A (solid line), 
(dashed) and 2 na order systems to current step, 
latency to ilring. 
-20 > 
3 ra order 
showing 
Figure 2 shows 
the voltage trace of a 
HH+A cell that is first 
hyperpolarized and then 
suddenly depolarized to 
above threshold. Trac- 
es from all three sys- 
tenas (full, 3 a order, 
2 na order) are shown 
superim.nosed. This 
example focuses on the 
phenomenon of post 
inhibitory latency to 
firing. When a HH cell 
is depolarized suffi- 
ciently to produce 
firing, the onset of the 
first action potential is 
immediate and virtually 
independent of the degree of hyperpolarization experienced immediately beforehand. In 
contrast, the same cell with an I A now shows a latency to firing which depends mono- 
tonically on the depth to which it had been 
depolarization. 
This is most clearly seen in fig. 3 0 
showing the phase portrait of the 
second-order system. The d/dt 
= 0 nullcline has acquired a 
second branch. In order to get 
from the initial Oayperpolarized) > 
 -40 
location, the phase point must 
crawl over this obstacle, and the 
lower it starts, the farther it has -60 
to climb. 
Figure 4 shows the 
firing frequency as a function of 
the injected current, for the full 
HH and HH+A sytems (solid 
lines), the HH second order 
HH+A third order (dashed 
lines) and HH+A second order 
hyperpolarized immediately prior to 
-8O 
-90 60 
-00 -30 0 30 
 (mV) 
Figure 3: Phase portrait of event shown in fig. 3 for 
2 na order reduced system. 
60 Kepler, Abbott, and Marder 
(dotted line).* Note that the first reduction matches the full system quite well in both 
cases. The second reduction, however, does not do as well. It does get the qualitative 
features right, though. The expansion of the dynamic range for the frequency of firing 
is still present, though squeezed into a much smaller interval on the current axis. The 
bifurcation occurs at nearly the right place and seems to have the proper character, i.e., 
saddle rather than Hopf, though this has not been rigorously investigated. 
0.3 
0.2 
0.1 
0.0 
HH 
IH+ 
0 10 20 30 40 50 60 
injected curren (/A) 
Figure 4: Firing frequency as a function of injected 
current. Solid:full systems, dashed: 2 n order HH & 
3 ra order HH+A, dotted: 2 na order HH+A. (From 
Kepler et al., 1991) 
The reduced systems 
are intended to be dynamically 
realistic, to respond accurately 
to the kind of time-dependent 
external currents that would be 
encountered in real networks. 
To put this to the test, we ran 
simulations in which Ia(0 
was given by a sum of sinusoids 
of equal amplitude and randomly 
chosen frequency and phase. 
Figure 5 illustrates the remark- 
able match between the full 
HH+A system and the third- 
order reduction, when such an 
irregular (quasiperiodic) current 
signal is used to drive them. 
CONCLUSION 
We have presented a systematic approach to the reduction of order for a class of 
dynamical system.q that includes the Hodgkin-Huxley system, the Connor et al. I^ 
extension of the HH system, and many other realistic neuron models. As mentioned at 
the outset, these procedures are merely the first steps in a more comprehensive program 
of simplification. In this way, the conceptual advantage of abstract models may be joined 
to the biophysical realism of physiologically derived models in a smooth and tractable 
manner, and the benefits of simplicity may be enjoyed with a clear conscience. 
*For purposees of comparison, the HH system used here is as 
modified by (Connor et al., 1977), but with I A removed and the leakage 
reversal potential adjusted to give the same resting potential as the 
HH + A cell. 
Order Reduction for Dynamical Systems 61 
0 100 200 300 400 500 
time (mS) 
Figure 5: Response of HH+A system to irregular current injection. Solid line:full 
system, dashed line: 3 'a order reduction. 
Acknowledgment 
This work was supported by National Institutes of Health grant T32NS07292 (TBK), 
Department of Energy Contract DE-AC0276-ER03230 (LFA) and National Institutes of 
Mental Health grant MH46742 (EM). 
REFERENCES 
L.F.Abbott and T.B.Kepler, 1990 in Proceedings of the XI Sitges Conference on Neural 
Networks in press 
J.A. Connor and C.F.Stevens, 1971 J. Physiol. ,Lond. 213 31 
J.A. Connor, D.Walter and R.McKown, 1977 Biophys. J. 18 81 
R.FitzHugh, 1961 Biophys. J. 1 445 
A.L.Hodgkin and A.F.Huxley, 1952 J. Physiol. 117, 500 
J.J.Hopfield, 1982 Proc. Nat. Acad. Sci. 79 2554 
T.B.Kepler, L.F. Abbott and E.Marder, 1991 submitted to BioL Cyber. 
V.I.Krinskii and Yu.M.Kokoz, 1973 Biofizika 18 506 
R.M.Rose and J.L.Hindmarsh, 1989 Proc. R.$oc. Lond. 237 267 
