Looking for mathematical models of biological neurons

Joern Erbguth jnerbgut at cip.informatik.uni-erlangen.de
Mon Jun 21 10:43:37 EST 1993

I finally found those email-messages which I thought of being accidently deleted.
So here is a more complete summary: 

In article <1vda2iEbp7 at uni-erlangen.de> I wrote:
>We are working on a neural net simulator with a wide range of net-models.
>My part is to implement some models which should simulate the functioning
>of real (biological) neurons.
>Do you know of any mathematic models that could be implemented?
>Do you know of any books or articles which deal with real neurons and
>offer a mathematic model to simulate their behavior.

Thank you for your replies to my questions. 

I got about 10 replies. I thought I accidently deleted some of them, but
I did not. So here are the references.
Mitchell Maltenfort (mgm at nwu.edu):

Ronald J. MacGregor, "Neural and Brain Modeling"
     Acad. Press 1987, (Addison Wesley 1988?)

has a model that is quasi-realistic (integrate-and-fire plus
afterhyperpolarization conductance and optional accomodation and dendritic
model).  I've implemented the point-neuron model into a C program for my own
simulation work.  

	There are more realistic models out, but they are also more
complicated - the strength of the MacGregor model is that known behaviors
can be modeled based on known physiological parameters (passive conductance,
membrane time constant, etc.).  

Paul Fawcett (paulf at manor.demon.co.uk):

Look no further! Get yourself a copy of

Trends in Neurosciences, Vol. 15, No. 11 1992.

This is a special edition on 'Modeling the Nervous System'.
Here is part of the introduction:

'.....The papers in this issue were selected to demonstrate the
considerable diversity of model based approaches currently in use in
neurobiology. The majority of these articles describe computer-based 
modeling efforts intent on replicating stimulus-induced neuronal
response or neurally controlled behavior.......'

It is worth reading just for the references, most of the classical
and contemporary computational neuroscience literature is listed.

Jacques Brisson (x042.hec.ca) pointed to the same issue
"As for articles, a good start would be to take a look at ... . You
might particularly be interested in NEURON and GENESIS, two neural networks
simulation packages that are based on biologically plausible considerations. 

The subject you want to cover is HUGE. You might want to check those
books :
- An Introduction to the Mathematics of Neurons
  FC Hoppensteadt, Cambridge U Press, 1986
- Corticonics; Neural Circuits of the Cerebral Cortex
  M Abeles, Cambridge U Press,  1991
- Modeling Brain Function
  DJ Amit, Cambridge U Press, 1989
and of course...
- Methods in Neuronal Modeling; from Synapses to Networks
  C Koch, I Segev (eds), MIT Press, 1989

me: some other people pointed to the same book, 
e.g. Joseph Devlin (jdevlin at cs.usc.edu):
Not only does it describe the models in some detail it also has an excellent
list of references which I've found very helpful.

Gene Wallenstein (wallenstein at walt.ccs.fau.edu):

There are several books:

1) The physiology of excitable cells by Aidley; Cambridge University
2) Ionic channels of excitable membranes by Bertil; Sinauer Press.

These are biophysical in nature - they are NOT texts on building
perceptrons or backpropagation nets. If you are interested in building
a general purpose simulator, you should be aware that several extremely
sophisticated packages are already available. For instance GENESIS
from Caltech can model single neurons (multpile compartments) or circuits,
etc. There is also a package called NEURON written by M. Hines which is 
very functional. It is not capable of circuit analysis but is much smaller
than GENESIS (which has about 60,000 lines of C code).

I saw a post in comp.ai.neural-nets
From: georgiou at silicon.csci.csusb.edu (George M. Georgiou)
Subject: Re: Neural Nets and Neurophysiology

Another book that that deals to a great extent with the relation of
biological and artificial NNs is:

"An introduction to the Modeling of Neural Networks," Pierre Peretto,
Cambridge University Press, 1992.

The level is introductory graduate. It includes chapters/sections on the
anatomy of the nervous system, neurophysiology, and a critique of
artificial vs biological NNs.  The bulk of the book is on artificial
Tobias Fabio Christen <tchriste at iiic.ethz.ch>

if you would like to have a look at a nice simulator get the SWIM from
thalamus.sans.kth.se  in /pub/  by anonymous ftp

At the moment we are modelling synapses ...
From: Michael Kisley <kisley at ucsu.Colorado.EDU>

I am a graduate student studying the mathematical modeling of biological
neural networks.  My advisor, Ron MacGregor, has published a lot of
material on the topic.  Probably the book he's written that looks like
the best for you is:

Neural and Brain Modeling (1987) Academic Press (see above)

This book presents a review of modelling attempts up to 1987, and then
an extensive section of MacGregor's original models.  He starts by
modelling a single soma only, and then adds dendrites and etc.  He
also just had a new book released which is most all his own work (no

Theoretical Mechanics of Biological Neural Networks (1993) Academic Press

This is a neat book but is much more theory than modelling.  If you can't
find these books, some of his material has been published in journals.
Unfortunately I don't have most of the references with me right now, but
I do have one:

Cross-talk theory of memory capacity in neural networks (1991) Biol
Cyber, vol. 65, pp.351-355.

This one refers to other articles which refer to other articles which ...

From: Richard Wood (rwood%hertz.UUCP%dirac.UUCP at uunet.UU.NET)

Hoppensteadt, F.C., "Signal processing by model neural networks",
SIAM Review, 34(3), pp. 426-444, Sept. 1992.

He works with an electronic circuit model called the VCON (voltage
controlled oscillator neuron).  He studies the phase locking
properties in networks of VCONS that are firing together.  This phase
locking results in spike synchrony.

He presents (from the abstract) several networks that:

        "fire bursting patterns similar to neural circuits in the
        thalamus and reticular complex of mammalian brains; they
        reproduce searchlight behavior that is speculated to be a
        mechanism by which a brain focuses attention on one among
        many competing stimuli; they convert a temporal signal
        into a spatial pattern of phase locked firing, similar to
        a tonotopic mapping in mammalian auditory systems; they
        store frequency-encoded information in autocorrelation
        filters that are similar to neurotransmitter synapses at
        chemical equilibrium; and they recognize stored signals
        by cross-correlation with new inputs."

A journal nearby to SIAM Review had an article on phase locking:

Stensby, J., "Lock detection in phase-locked loops", SIAM Journal
of Applied Math, 52(5), pp 1469-1475, October 1992.

Thomas Chimento <chimento at ursa.arc.nasa.gov>

There are at least these three books worth looking at:

Single Neuron Computation, T. McKenna, J. Davis S.F. Zornetzer
Academic Press Boston 1992


Electric Current Flow in Excitable Cells, Julian J. Jack, D. Noble, R.W. Tsien
Oxford: Clarendon 1975, 1985; Oxford Science Publications 1983

If you dont already know it, this is a rapidly growing and already
enormous field. There are several journals dedicated to it.

Neural Networks, Pergamon Press
IEEE Transactions on Neural Networks
  and many others.

Colin Prepscius <cp5m at broca.med.virginia.edu>

Well... what you ask is actually quite a tall order.  When we need to
truly simulate a neuron, what is used is the Hodgkin-Huxley model of
neuron.  In fact, there exists a program called NEURON which
implements this model.  Unfortunately, it is very complex, and too
slow to implement in a network.  When we try to act "biologically
plausible" with a network, what is usually used is some variant of
the Integrate-and-Fire model.  This model is basically a capacitor,
integrating its afferent stimuli until it fires, at which point the
built-up charge can be reset to resting potential or not.  There are
many variations of this model.  In general, they do a pretty good job
simulating a biological neuron, but there is, of course, controversy,
and these models are very much just simplifications.

"The distribution of the intervals between neural impulses in the maintained
discharges of retinal ganglion cells" by M.W. Levine
Biological Cybernetics (Springer) 65, 459-467 (1991)

I haven't looked at all those articles and books yet. So far I think the
Special Issue of "Trends in Neurosciences" is the best way to start.


Joern Erbguth   email: jnerbgut at cip.informatik.uni-erlangen.de 
Change of       snail: Helmstrasse 6, W8510 <90762> Fuerth, Germany 
Address!        phone: +49 911 7419528 

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