The Settlers - who had to use machines to compute
John Moore's view and experience with numerical experiments (with
a reference to my 1962 studies of the HH model on the Duke IBM 7072)
Machine coded SEAC calculations
Kacy Cole soon arranged to have the Hodgkin-Huxley equations calculated on the
National Bureau of Standards new digital computer - Standards Eastern Automatic Computer
(SEAC). The equations were written in code and took about 1/2 hour to calculate 5mS of axon
time. Kacy quickly noted that SEAC could be rated at 16 "Huxley power"! From the
printed output I plotted both the voltage-vs-time and the phase-plane (voltage-vs-dV/dt). We found a
"saddle point" - where phase-plane trajectories diverged - and this was taken as the "threshold"
for a patch of HH membrane. Soon after Kacy and I left NMRI moved across the
street to the Nationl Institutes of Health, where Kacy chose a position of Chief of the Lab of Biophysics
(with less responsibility and lower salary), Richard (Dick) FitzHugh joined our lab. Because he was familiar with
phase-plane analysis of equations, Dick challenged the validity of a saddle point,
knowing that the threshold for the HH equations
was not a point but a moving target in phase space. There was a standoff between the computer
results and FitzHugh's analysis until he and I found a discontinuity in a rate constant at a
voltage at which the rate constant value was indeterminate. The programmer, deep in machine
code, had simply overlooked the fact that there were several voltages at which divisions by zero
were called for and FitzHugh was vindicated
(FitzHugh & Antociewicz, 1959)!
Analog Computer Simulations
After George Bekey (of Berkeley Instrument Co.) showed that the HH equations for a
membrane action potential could be solved on an analog computer in a few seconds, Dick
and I convinced Kacy to buy one of their machines for us to run a variety of simulations. This
analog computer used state-of-the-art line segment function generators (for the voltage-sensitive
rate constants), multiple pot servo multipliers, and highest quality capacitors for integration. A
plotter could be connected to any variable for rapid recording of its value with time or plotting
against any other variable. Thus not only were the simulation times reduced enormously but
also the tedious process of transfer from printed numerical output to plots was eliminated.
entirely. We used this computer to simulate our (Cole & Moore, 1960)
observations of the long
delay in the onset of the potassium current upon depolarization from strongly hyperpolarized
Upon my move to Duke Univ. in 1961, an Electronic Associates TR48 analog
computer was used to continue my simulations. This computer used a new and much faster, yet accurate,
multiplier ("time-division" multiplier designed by Art Vance at the RCA Lab in Princeton, NJ) and
thus was able to display its output on an oscilloscope at a repetition rate in excess of 30/sec so
that the eye was not aware of flicker! In addition to guiding my experimental program and
simulating its results, I was able to quickly find the flaw in the interpretation of "ramp
clamp" data which had been proposed as a much simpler way to gather current and conductance information
family of voltage steps to various levels used in the conventional method (See
Chap. 3, Professional Style).
Fortran on mainframes & Focal on minis
My associates: Frank Starmer, an engineering student in my first course at Duke on the Physiology of
Excitable membranes, became interested and, on his own initiative, carried out Fortran simulations of HH
membrane action potentials and refractory periods on Duke's IBM 360, the only digital computer
on our campus in 1962.
In my lab, HH simulations were run on Digital Equipment Co. (DEC) PDP8s and Linc8s, written in DEC's
proprietary language FOCAL,
and others were run in Fortran on DEC PDP 9s and PDP11s.
Later my collaborators (see Chap. 4 on Neuron's history) developed Fortran programs to solve
for propagating impulses. It is astonishing that they were able to solve the partial differential
equations on a DEC PDP8 with only 28Kb of memory, far smaller than the operating systems in
Several Others were at work during this period, for example:
Fred Dodge: As a graduate student with Bernard Frankenhaeuser, Fred developed
a reliable voltage clamp for nodes of Ranvier in myelinated nerve fibers and carried out many experiments on nodes in frogs.
(1958; J. Physiol. 143:76 and 1959; J. Physiol. 148:188). In his thesis at Rockefeller Institute (1963),
he analyzed these data in the framework of the Hodgkin & Huxley equations and carried out simulations of nodal action
Fred also was the first to publish digital computer solutions of the partial differential equations for propagating impulse in an axon
also famous for his work with Tukey
on fast Fourier Transforms.
Dodge and Cooley extended this work at IBM to the first simulations of the generation and propagation
of impulses in a neuron with an active soma and axon but passive dendrites.
B. I. Kodorov of the Academy of Medical Sciences in
Moscow, his students, and several Russian collaborators were
also work on both experimental and computational approaches to
excitability and propagation in nerves. They were expecially interested
in the effects of geometry and branch points on propagation. Their work
in summarized in a volume "The Problem of Excitability: Electrical
excitability and ionic permeability of the nerve membrane", published
by Plenum Publishing Corp. in 1974. the translation into English was
edited by Fred Dodge.
Wilfred Rall began his mathematical analysis of linear
systems, especially passive dendrites, at the Naval Medical Research
Institute. He moved to NIH about 1960 and continued this landmark work
ever since. In addition, Will published seminal calculations on the
effects of diameter change on the propagation of action potential. (xx)
Increasing Numbers of Investigators entered the field in the late 1970s and 1980s.
The list below is illustrative only, and far from comprehensive. It
deals with those who have tried to simulate biological systems
realistically and omits mathematical treatments and "artificial neural
networks" (ANNs). To me, much of the ANN work represents the solution
of linear simultaneous equations in parallel. Art Vance, at RCA, led a
group who used a network of operatioal amplifiers to solve a set of
10-by-10 simultaneous equations in the 1940s. As a member of that
group, I was proud of this achievement but aware of the technology was
ahead of the times as far as problems went. I don't think that
this work was ever published outside of the company. Maybe it would
have been if someone had thought up such a provocative term as
artificial nerve networks!
Rand Corp: Don Perkel, now deceased, and Ron McGregor, Univ. Colorado
MIT: T. Poggio, and Vincent Torre, now at Genoa, Italy, Christof Kock, now at Cal. Tech.,
and Steve Waxman, now at Yale.
Art Vance, who had developed not only negative feedback concepts and operational amplifiers but also
fast nonlinear analog computing elements,
proposed in the 1940s that by coupling the speed of
analog integration with the accuracy of digital arithmetic operations one could achieve
a very fast yet accurate simulation, In the early 1970s I
purchased a hybrid compurter consisting of a DEC PDP15 digital computer and an Electronic Associates 580 analog computer
(digitally controlled), interfaced with analog-to-digital and
Mike Hines was able to program this system to produce HH impulses
propagating at blinding speeds. Although this system simulated axons with
nonuniform diameters, there was no reasonable way to handle the branching of
dendritic trees. At about the same time, the speed of pure digital machines was
beginning to increase rapidly and we switched to using minicompouters such as the DEC PDP11.
Simulations on Machines of the '80s and '90s
Workstations, running Unix, brought further
increases in power and speed of computers at reasonable prices for labs or individuals
and they are now extensively used for simulations. Lately the power and speed of PCs
has taken extraordinary leaps to where they rival workstations. In the early '90s, they
are found in almost all labs because their prices have declined so dramatically. The porting of
NEURON to Microsoft Windows 3.1 for PCs provides simulation power rivaling that of more
Membrane Action Potentials. NEURON incorporates two features which provide fast and accurate simulations:
With these features, it is now possible for
the time required for simulation of a membrane action potential to approach that
in the axon. For example 1 mSec of membrane time can be simulated, using an integration time step of .05 mSec,
on a 100 MHz Pentium in:
- Rate tables for the HH alphas and betas - - their values
for any voltage can be found by interpolation much more quickly than by algebraic calculations
- Implicit integration, stable at large time steps (dts), much more accurate and faster than explicit
numerical integration methods.
A graph of the membrane voltage as a function of
time (with the default value of 40 points plotted per millisecond) usually takes another 0.5 - 1 second to draw.
With Pentiums running at the now available 133 MHz, the Linux time to would be reduced to only 5 mSec per mSec of nerve time.
- 10 mSec with the MS Windows version of NEURON 3.0
- 6.7 mSec with the Linux version with the same parameters.
With advanced 64bit CPUs, running at much higher clock speeds,
it should be possible soon for the simulation
speed to match and then surpass that of the membrane!
Propagating Action Potentials. Because NEURON uses Mike Hines new algorithm for "efficient"computation
of branched nerve equations,
the time required for solution of a multicompartment cell is
proportional to only the number of compartments rather than the square
of that number used in conventional solutions of the matrix equations.
Thus simulation of a full nerve cell with 200 compartments and HH
channels may take only about 1 sec on these Pentiums. However
generation of the spatial distribution of the membrane potential often
requires 50-fold the simulation time. For such displays very fast
accelerated graphic cards become
very valuable because all of the points have to be plotted and then
erased at each time step.
Several other simulation tools were and are being developed in this era as well, most of which are designed to operate
under Unix.Super Computers.
Mike Hines has ported NEURON to a Cray machine for batch process such as
parameter and threshold searches.
As of this writing in the early 1990s, a few simulations of cells and networks have been
done on parallel machines.
When NEURON is ported to these architectures, very rapid
parameter searches will become possible and simulations of networks of
cells will become quite fast.
In 1992 Kevin Martin ported a NEURON
simulation of a stylized motorneuron to a MasPar parallel computer but,
because of the memory limitation for
had to restrict the number of compartments. As such limitations are
removed and software for parallel and distributed simulations become
available, NEURON should be readily transportable and fulfil its
potential as a full network simulator.