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Current Thinking in Neuronal Modeling:
Does it Embrace Systems Biology?
Matthew S. Craig ([email protected])
Abstract - Advances in computational power and neuronal
modeling techniques in the past decade have led to the
development of specialized software to model neurons. Such
software (NEURON, GENESIS and others) uses the common
techniques of equivalent cylinder modeling and
compartmental modeling, which reduce complex neuron
structures, which can not be practically modeled, into
mathematical idealizations, which can be simulated. This
paper will discuss the level of physiological fidelity possible in
these two techniques. Intuitively, one might think that models
with as much morphological complexity as possible, ie. a
systems biology approach, would be best in any situation.
However, parameters inherent to the models become
increasingly difficult to measure on smaller scales. Moreover,
a consideration of the particular behaviour being studied may
obviate the need for finer details.
Index Terms – systems biology, equivalent cylinder model,
compartmental model, dendritic branching, tapering
I. INTRODUCTION
Systems biology seeks to describe a biological system as a
complete set of components; all working interactively,
dynamically, and all inter-affecting. System biology’s main
premise is that only when we understand this totality of
components, causes, and effects can we correctly explain and
predict biological outcomes. To understand how this principle
applies to the modeling of neurons, we can investigate it on
different scales.
On a high level, a neuron can be described as a set of
dendrites connected to the soma, which is connected to an
axon which, in turn, leads to a synapse. At this level, we can
only understand the interactions between these components in
a very simplistic way: a nerve impulse is inputted into a
dendrite, it travels through the soma, is transmitted down the
axon, and exits through the synapse. From this portrayal, we
cannot relate the binding of neurotransmitter to receptors at the
dendrites, nor can we discuss how the action potential travels
down the axon, because our high level depiction made no
mention of neurotransmitters, receptors or ion channels.
Because of its crude description, this model could not predict
anything more than the simplest neural responses.
Alternately, in an arbitrarily low-level description, the same
neuron is described as an infinitely complex collection of
molecules, each dynamic and interrelated, each responding to
infinitely complex inputs and releasing equally intricate
outputs. The sum of all of these relations will then determine
the output from the neuron.
This definition is clearly a much more systems biology
view, because it seeks to consider every possible component
and interaction. From this second portrayal, however, we can
only explain basic functions of the neuron, such as
neurotransmitter release, by understanding fantastically
complicated and infinitely numerous molecular interactions.
For this reason, this second model is more impractical than the
first. It is doubtful that all of the computational power on the
Earth combined could process such vast amounts of
information, and even the simplest neural impulses would be
out of reach.
Current neuronal modeling takes place in the region
between these two extremes. The level of complexity in the
model will depend on the scale of investigation, be it modeling
ion channels, modeling the axon, modeling an entire neuron, or
modeling assemblies of neurons. So, we must address the
questions: Given an experimental scale, how complex a model
do we need to accurately describe it? When are systems
biology views essential to neuronal modeling? When can they
be relaxed?
This paper will outline two computational neuronal
modeling techniques, and discuss the extent that these
techniques follow the principles of systems biology, that is,
how well they can represent complicated neuronal structures.
Following that will be a discussion showing that the systems
biology ideal is not always necessary or practical in this field.
The two techniques to be discussed are the multiple equivalent
cylinder model and the compartmental model. Both of these
models are standards in the field and are recognized as being
essential elements in most realistic neuronal modeling [8] [12].
II. THE MULTIPLE EQUIVALENT CYLINDER MODEL
The dendritic structure in neurons has been observed to be
extremely complex and diverse, leading neurophysiologists to
believe that this structure might play an extremely important
role in neural signal processing. Van Pelt and Schierwagen
[18] were the first to show that morphologically realistic
dendrites showed different electronic and functional properties
than unrealistic ones. Such unrealistic models typically
simplified dendrites into symmetric trees rather than their nonsymmetric, “realistic” partners.
In light of systems biology, neurophysiologists wish to
incorporate such complex morphology. The problem is that
complicated and extensively branching dendrites do not lend
themselves to computational modeling and numerical
solutions.
In 1962, Rall proposed a model of a nerve cell to deal with
the intricate structure of the dendritic tree: the totality of
branches could be mathematically reduced into a single,
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equivalent cylinder [13]. The model was later modified to
include a shunt at the soma, so that it could reproduce
electrical potential decays in neurons that been observed
experimentally [4].
The model, although it made an analysis of dendritic
structure possible, was markedly unrealistic because the
collapsing of many complicated, branching dendrites into a
single equivalent cylinder requires very limiting assumptions:
a very high symmetry between each dendritic branch, uniform
dendritic length, and uniform dendritic diameter (cylindrical
shape), to name a few [6].
The multiple equivalent cylinder model is an extension of
this, that allows the dendritic trees to be represented by many
cylinders. Any number of dendritic trees can be reduced into a
smaller number of equivalent cylinders. A fully-branched
model represents the extreme and most work intensive version:
there is no branch reduction so that each and every branch is
represented by a cylinder. Fig 1.
multiple equivalent cylinder model through the work of Major
et al. [10] and Poznanski [12].
Physiological faithfulness of the multiple cylinder model
Continual developments in the equivalent cylinder model
make it increasingly realistic while keeping the model simpler
than a fully-branched one.
The original multiple equivalent cylinder model contained
many of the unrealistic assumptions of single-equivalent model
such as uniform dendrite length and uniform dendrite
diameter. A fully branched model could account for any
arbitrary level of complexity by allowing the branch length, l
 0, but this would defeat the purpose of the multipleequivalent cylinder model; to simplify computations. The fact
remains that a dendrite is simply not a cylinder, and this
assumption severely limits the model’s realism.
In this regard, much work over the past few years has
focused on increasing this realism. Specifically, the
improvements try to account for three things that the original
did not: 1) non-uniform diameters, 2) non-uniform membrane
resistivity, and 3) nonuniform lengths.
A) NON-UNIFORM DIAMETER IMPROVEMENTS
Non-uniform diameter models, referred to as “tapering”, are
models that allow the dendrite diameter to change with
distance from the origin. For dendrites, the diameter, in
general, gets thinner as you move farther away from the origin.
Tapering was first modeled using exponential and linear
functions for the taper, and has since been modeled to allow
non-uniform cross sections [5]. The type of function to be used
for the taper will depend on the type of neuron and its
morphology.
B) NON-UNIFORM RESISTIVITY IMPROVEMENTS
Experiments have shown that dendrites in different parts of
the neuron have different electrical resistivities through their
membrane, as well as different resistivities than the soma’s
membrane. This variation has recently been included to the
Fig 1.[12] Various cylindrical model complexities of the same neuron. (A)
Typical dendrite physiology. (B) A fully-branched equivalent cylinder model.
(C) A multiple-equivalent cylinder model. (D) Rall’s single equivalent
cylinder model.
C) NON-UNIFORM DENDRITE LENGTH
IMPROVEMENTS
The uniform dendrite lengths that were required by the
single-cylinder model, and early versions of the multiple
cylinder model, were shown to give erroneous electronic
properties, and overestimate dendrite length [7]. Major et al.
[10] were the first to derive equations and solutions allowing
arbitrary dendrite lengths.
Although the equivalent cylinder model is a useful
analytical tool, at its current state it is not appropriate to deal
with complex situations such as arbitrarily branching trees,
dynamic membrane properties, voltage-dependant
conductances and multiple stimulation sites [6]. These
problems will undoubtedly be the focus of future research in
order to make the model more closely mimic the neuron’s
physiology.
In the case of arbitrarily branching trees, it is intractable to
solve the continuous differential equations generated by the
model. The compartmental approach reduces these equations
into a set of ordinary differential equations which are more
easily numerically solvable.
III. THE COMPARTMENTAL MODEL
The compartmental model of the neuron seeks to
decompose a neuron into a grid of compartments. Each
compartment is then treated as an isopotential element, so that
a chain of separate compartments can approximate a
continuous neuronal structure [15]. This simplifies the math
considerably, and it allows individual compartments to have
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very non-uniform characteristics. Since their conception in the
1960’s, compartmental models have been a backbone in
neuronal computation and modeling [12].
The compartmental model is a means of embodying in depth
information about the physiology, dynamic interactions, and
synaptic input of the neuron. Based on this definition, the
model is at the very heart of systems biology: it allows us to
take all the information of neuronal structure and interaction
and put it into a form that not only explains scientific
experimentation, but allows us to explore ideas about current
flows, voltage perturbations and input-output relations [15.
Because of the mathematical simplifications made in the
model, “realistic” computer simulations become feasible, and
have spawned the development of specialized neuronal
simulation software, such as NEURON, GENESIS and
NODUS, three widely-used packages [15].
Physiological faithfulness of the compartmental model
The compartmental model allows for almost arbitrarily
complex dendritic and axonal structures. To consider such
morphology, one needs only to add more compartments.
Because each compartment is isopotential, non-uniformities
occur between the cells, not inside them.
Although the compartmental approach allows for extremely
complex structures, it is inherently incapable of dealing with
other neuronal morphology, in particular, the branch angles
between dendrites. The branch angles are angles between
adjacent dendrites taken in all three dimensions. Without these
angles, the compartmental model restricts neurons to a single
dimension rather than three. In such a model, Costa et al. [2]
acknowledge that because the model relies on only the average
diameters and lengths of dendrite segments, sets of neurons
with completely different shapes can potentially yield identical
models.
In compartmental models, the soma is usually represented
by an isopotential sphere. The myelinated axon is assumed to
have a negligible electronic impact on the soma, so it is often
left out of such models entirely. The legitimacy of these two
assumptions is not yet known, however, and they will require
additional study [15].
IV. ISSUES OF MODEL COMPLEXITY
From a systems biology standpoint, we would best
understand the neuron by accounting for all of its complex
morphology. That is, by representing a neuron with as many
compartments or equivalent cylinders as possible, while
keeping it mathematically and computationally feasible. This
reasoning is not wholly practical in light of two considerations:
parameter estimation problems, and the scale of investigation.
Parameter Estimation
The formulation of the compartmental and equivalent
cylinder models involves specific, essential parameters,
electrical and geometrical, that must be experimentally
determined, and very little of this data currently exists. In very
detailed models, the number of these parameters is colossal
[20]. Such parameters typically include the specific membrane
resistivity (Rm), the cytoplasmic resistivity (Ri), and the
specific membrane capacitance (Cm) [15]. Informed estimates
of these parameters are crucial to any neuronal modeling
project, because without them, modeled data cannot be
matched to experimental data. To make matters worse, most
neuronal models are sensitive to very small changes in these
parameters [1].
Current microscopic techniques allow us to get extremely
detailed morphological data, and therefore determine some of
the geometrical parameters quite well. Other geometrical
parameters are not well defined. For example, channel
distributions are generally not known and assumed to be
uniform. Some channel densities, such as Na+ , can be
estimated using benzofuran isophthalate imaging [19], but K+
and Ca2+ channel densities remain out of experimental reach.
The electrical parameters are much harder to obtain,
particularly in very fine branching structures. Consequently, as
modelers increase the level of complexity they are faced with
increasingly difficult, and increasingly numerous, parameter
estimation. In fact, in many studies, parameters are chosen by
trial and error, and are adjusted, after simulations are
performed, to match the experimental data [9]. On this topic,
Evans [4] remarked that using multi-compartment models to
model neuronal complexity introduces too many degrees of
freedom, and that it is preferable to reduce these degrees of
freedom by building simpler models that rely on key,
measurable parameters.
Scale of investigation
When determining the required complexity in a model, we
should also consider the scale of investigation. To show this,
we discuss the modeling of low and high scales, specifically,
channel dynamics and small neuronal networks.
A typical model of ion channel dynamics in a neuron will
certainly involve a study of channel densities, spatial
distributions, and of channels types. A single-compartment
model would be completely inappropriate in this case. None of
these three studies could be performed, and all of the channel
currents simply sum linearly into a total current through the
membrane [9]. To effectively simulate this interplay between
channels, the modeler would require many compartments of
much smaller size. For example, Destexhe et al. [3] in order to
study dendritic currents mapped the dendritic structure to an
accuracy of 0.1 micrometers and simulated the neurons with
80 or 200 compartments.
Conversely, in biological neural nets, modeling is often
based on the assumption that the network’s behaviour is
governed mainly by the connections between neurons, not on
the interactions within the individual cell, so fewer
compartments per neuron are needed, and often only one is
used.
Never was this tradeoff between compartment number and
parameter estimation more clear than in the modeling of
guinea pig hippocampal neural networks. In 1991, Traub et al
[16] created nineteen compartment models of the cells, using
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NEURON, to demonstrate various facets of rhythmogenesis.
They understood that the success of their model in matching
experimental data relied on their estimation of parameters for
ion channel densities. The data would match only if the soma
and dendrites were given drastically different densities. A few
years later, in a simpler model, Pinsky and Rinzel [11] showed
that the same behaviour could be demonstrated in a network of
cells of only two compartments each: one to represent the
soma and proximal dendrites, the other representing the distal
dendrites. For the particular behaviour being studied, the
second, and much further reduced, model is easier to
understand in terms of only a few parameters, and is therefore
highly preferred over the first model. A year later, however,
while examining different behaviour in the same hippocampal
neurons, Migiliore et al. [19], noted that the Traub model was
too simplistic to account for the cells’ atypical bursting
characteristics and [Ca2+] dynamics. Using the Traub model as
a starting point, they modeled the six types of hippocampal
neurons with much more complexity: 129 compartments for
the simplest, and 385 for the most detailed.
V. CONCLUSIONS
In the two models discussed, there is much room for more
realism and enhancement. In particular, we can expect that the
parameter estimation problem will be improved as future
experimentation techniques allow us to measure the electrical
parameters at levels that are currently out of reach. This will
allow us to model single neurons with much more complexity.
This will certainly be useful in low level models, but in high
level models, it may be unnecessary and only serve to make
the models intuitively out of reach, for it is much easier for the
human mind to understand neurons with few compartments
and a few main parameters, than to understand multi-hundredcompartmental neurons with hundreds of parameters.
From a systems biology standpoint, we would strive to
describe a system with as much detail as possible. But, as we
have seen, this scheme is not always practical in neuron
modeling. Currently, there exists no ideal model for a neuron,
and no standard method to know how far a model should be
reduced in complexity. Some models may be adequate at one
level and worthless at another because they are either oversimplified or too complicated for the situation. Hence, it is
instructive to devise ways to determine how complex a neural
model needs to be.
One way to do so is to perform the same simulation at
different levels of detail. If we systematically decrease the
level of complexity until an essential element of the
experimental behaviour is no longer observed, we know that
the model is too simple. If we systematically increase the level
complexity of the model, the numerical results will eventually
converge, telling us that the compartment size and cylinder
detail are sufficient [9] [15]. Certainly, a more efficient
method of determining how complex such models need to be,
for a given situation, would be valuable.
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