Table Of ContentN E R V O US S Y S T EM
T H E O RY
An Introductory Study
K. N. LEIBOVIC
CENTER FOR THEORETICAL BIOLOGY
AND
DEPARTMENT OF BIOPHYSICS
STATE UNIVERSITY OF NEW YORK AT BUFFALO
AMHERST, NEW YORK
ACADEMIC PRESS New York and London 1972
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PREFACE
As a science, nervous system theory must be based on experimental data
and must relate to the actual characteristics of structure, function, and mode
of operation of the nervous sytem. But it must be more that a mere collection
of observations. It must be concerned with the responses of single cells and
the interactions of groups of cells and how these may form the basis of per
ceptual and motor functions. It must be concerned with the principles of
operation of the vast and intricate cell assemblies which form a nervous system.
As scientific theory, it should be a distillation of a body of observations of
precisely formulated data from which one can make logical inferences. But
often such data are not available in the form or to the extent required by
such a theory. This is due not only to the experimental difliiculties of ob
taining the data but even more to the rudimentary form of our understanding:
in fact, data abound in this as in other fields of biology, but the question is
what to look for. In such a situation it is essential to attempt to develop
theoretical formulations.
The balance between the appropriate amount of speculation and the
earthy digest of experimental records is a delicate one and can only be based
on individual judgement. One mix may appeal to some and not to others. In
this volume the stress is on the means for understanding the nature of the bio
logical system rather than on the elaboration of mathematical theories. An
attempt has been made to stay close to the experimental data but also to con
sider speculative models which are judged to be valid in the sense that they
may provide insight into the relationships between structure and function,
into the principles of operation, and into the neural bases of behavioral phen
omena. Naturally, only a beginning can be made in this respect. This book
is not intended to be a complete account of nervous system theory or even an
ix
χ Preface
exhaustive treatment of the topics considered. It contains not only my per
sonal work but includes that of many others. On the other hand, in regard to
theories, I have given preference to those I have fathered together with my
collaborators.
In the pages which follow, single-cell responses are treated first. This is
followed by a discussion of sensory information processing which leads into
models of perceptual processes and their possible neural bases. Much of the
material here refers to the visual pathway, since experimentally this has been
studied more than any other sensory pathway and is thus the most accessible
for theoretical investigations. In the last part of the book some general prin
ciples are considered. Thus, we proceed from the units that make up a
nervous system, through a sensory pathway, to central structures. Since the
peripheral stimuli can be described accurately, we may hope to understand
something about the operations of the brain by following the flow of informa
tion through it. But the brain is not there simply to receive sensory infor
mation. One of its most important functions is concerned with motorsensory
integration. This is briefly considered in the last part of the book. In addition,
there are larger problems of a more philosophical nature that I have dis
cussed elsewhere and have not included in this volume.
ACKNOWLEDGMENTS
The material of this book is based on graduate courses of lectures in bio
physics and theoretical biology which I have given for several years at the State
University of New York at Buffalo and at the University of California at
Berkeley during the summer of 1969, the latter sponsored by the Division of
Medical Physics. My thanks are due to Drs. J. F. Danielli and F. M. Snell at
Buffalo for their support on several occasions, and to Dr. Cornelius Tobias
for extending his hospitality and making my stay at the Donner Laboratories
in Berkeley such a stimulating experience. Thanks are also due to my students
at Berkeley and Buffalo for their lively responses which helped to improve
the presentation of this material and to my colleagues and collaborators
without whose help and enthusiasm much of the work reported here would
not have been done.
Last, but not least, I have pleasure in expressing my appreciation to Drs.
Nassir Sabah and Richard Srebro for reading and commenting on portions
of the manuscript. It goes without saying that any errors or shortcoming
remain entirely my own.
The help and expertise of the staff of Academic Press in producing this
volume and of Mrs. Kathryn Ward, who has done most of the typing, have
been of great value. Mrs Evelyn Wood drew some of the figures, and Mrs.
Ruth Harvey also typed some portions of the manuscript. The latter were
supported by Grant NASA NGR 33015002, while some of our work referred
to in the text was done in part with support from Grants NIH NB 06682
and NASA NGR 33015002.
XI
Chapter 1
INTRODUCTION
The nerve cell is distinguished from other cells by having a number of
long, fibrous processes emanating from the cell body and by its ability to
generate and transmit signals. Of these the axonal signals are of constant
amplitude and variable frequency, whereas the dendritic ones, as a rule, decay
in amplitude as they spread along the fiber. In both cases the signals are in
the form of changes of membrane potential. In the resting state the potential
inside the cell is some 50-90 mV below that outside the cell. This potential
difference is regulated by differences of ionic concentration.
The cell membrane is 50-100 A thick and separates two aqueous solutions
of which the one outside is rather more electroconductive than the one inside.
The sodium and chloride ion concentrations in the external medium are about
10 times as high as those inside the cell, whereas the potassium ion concentra
tion inside the cell is about 30 times that outside. The permeabiUty of the cell
membrane is low but the K"^ and CI" ions move through it much more
readily than the Na"^ ions, whose normal leakage into the cell is counteracted
by the metabolically driven "sodium pump."
If the ionic permeabilities of the membrane are fixed, then a change of
polarization produced at some point on a fiber will spread and decay as it is
conducted " electrotonically," as in a passive, leaky cable. In an active fiber, on
the other hand, the ionic permeabilities depend on the state of polarization
of the membrane, and a pulse of depolarization can be regenerated as it is
conducted along the fiber. Thus, in an axon, when the resting potential of the
initial segment is raised above a certain threshold value, the depolarization is
rapidly amplified, producing a "spike," or action potential, which travels
along the axon with constant amplitude. This is based on the following
mechanism (Hodgkin and Huxley, 1952). The depolarization of the membrane
4 Part One: Single Cell Responses
increases Na"^ permeability, whereupon external Na^ ions rapidly enter the
cell and amplify the initial depolarization. With only a brief delay, however,
the K"^ permeability is increased and K"^ flows out of the cell, while the Na"^
permeability is reduced to its former value, returning the membrane potential
toward the resting level. As the spike develops, the membrane immediately
ahead is depolarized and the process is repeated along the length of the ñber,
ensuring the propagation of the action potential. The process can be com
pared to conduction along a leaky cable with repetitive boosting. But during
and shortly after the action potential, the fiber goes through a refractory
period in which the generation of new impulses is suppressed. In nonmy
elinated axons the action potential travels continuously along the fiber.
Myelinated axons are surrounded by insulating sheaths with periodic inter
ruptions, the so-called nodes of Ranvier, and the action potential jumps from
node to node and conduction is more rapid.
Some neuron characteristics are illustrated quantitatively by the following
typical figures. The internal resistivity is of the same order of magnitude as,
though up to three times higher than, the external resistivity, both being
within the range of a few tens to a few hundreds of ohm centimeters. The
axonal membrane resistance is of the order of a few thousand ohms per
square centimeter and membrane capacitance is typically 1 //F/cm^. During
the nerve impulse in a squid giant axon, some 3-4 χ 10" mole of Na"^ ions
may be taken up, and the same amount of K"^ ions may be lost, per square
centimeter of fiber. This represents only a small fraction of the ionic content
of the squid axon—in the case of K"^ it is of the order of 10"^—and the
original concentrations are restored quickly by metabolic activity.
Although the system consisting of membrane, external medium, and
internal medium is electrically neutral on the macroscopic scale, the inside
of the cell membrane carries a slight negative charge which is balanced by an
equal positive charge on the outside. This capacitative charge is of the order
of 6 X 10" ® C/cm^, corresponding to a capacitance of 1 μ¥ and a potential
difference of, say, 60 mV. From this it can be seen that relatively large changes
of potential can arise from quite small charge displacements. For example, a
20% change of potential would involve a change of 1.2 χ 10"® C/cm^,
corresponding to a displacement of some 7.5 χ 10^^ univalent ions.
Although the foregoing figures are fairly representative, it should be
remembered that the properties of neurons vary widely. For example, fiber
lengths may vary between several microns and a meter or more, fiber diam
eters between a micron or less and as much as a millimeter. In general, the
larger the diameter, the higher the conduction velocity and the shorter the
spike duration and refractory period. The variables are, however, inter
related in a complex fashion. Conduction velocity, for example, depends
among other factors on the rate at which Na"^ permeability increases and
Chapter 1: Introduction 5
then decreases, the rate and timing with respect to Na"^ of the K"^ perme
ability changes, and the spike amplitude and the ionic mobilities within the
fiber.
The axon represents the output line of the neuron. It usually ends in a
number of branches making contact with the dendrites or cell body of other
neurons. The dendrites represent input lines, and the numerous signals
impinging on them are integrated within the dendritic tree and the cell soma.
Although dendrite responses are not as well understood as the axonal
potential, they clearly form a very important part of neuron function. The
extent and the ramifications of a dendritic tree are often most impressive in
relation to the rest of the cell. In many dendrites there may only be electro-
tonic conduction, but in others spikes are generated, and clearly involve an
active membrane.
Communication between neurons can take place through chemical
transmitters at synaptic junctions, through direct electrical transmission at
tight junctions, or through changes in the ionic composition of the inter
cellular space produced by activity of neighboring cells. The chemical trans
mitter is stored in synaptic vesicles and released across the synaptic gap in
quantal fashion. The ionic permeabilities of the postsynaptic membrane are
altered selectively, depending on the transmitter and the membrane properties.
As a result, the postsynaptic membrane may be either depolarized or hyper-
polarized.
Properties of nerve cells have been described extensively as, for example, in
the following publications, where additional references can be found: Cole
(1968); Eccles (1964); Hodgkin and Huxley (1952); Katz (1966); Lorente de
No (1947); Mountcastle (1968); Quarton et al. (1967); Ruch et ai (1965).
The topics mentioned in this introduction will be taken up in more detail
in the following chapters, starting with the resting potential, going on to the
transmission of signals in axons and dendrites, and concluding with synaptic
communication and some reduced models of neuron function. It will be found
that different problems necessitate the formulation of different models. For
example, in the treatment of the resting potential the ionic flows are divided
into components due to diffusion and the electric potential, whereas in the
treatment of the cable properties of a nerve fiber all the flows are considered
as electric current dependent on conductance, capacitance, and potential.
Again in the treatment of active membranes the ionic flows are separated into
species components, each with its own conductance and driving force expressed
in terms of potentials. Clearly there is a connection between different formu
lations, but the moral of all this is that it is necessary to maintain some
flexibility in formulating a model. This is dictated by convenience of the
mathematical treatment of experimental results, and it is developed further
in the final chapters of this part of the book.
Part One: Single Cell Responses
REFERENCES
COLE, K. S. (1968). *' Membranes, Ions and Impulses." Univ. of California Press, Berkeley,
California.
EccLES, J. C. (1964). "The Physiology of Synapses." Springer-Verlag, New York.
HODGKIN, A. L., and HUXLEY, A. F. (1952). J. Physiol. 116, 449-506; 117, 500-544.
KATZ, B. (1966). "Nerve, Muscle and Synapse." McGraw-Hill, New York.
LoRENTE DE NO, R. (1947). A Study of Nerve Physiology, Stud, Rockefeller Inst. Med.
Res. 131, 132.
MouNTCASTLE, V. B. (Ed.) (1968). "Medical Physiology," Vol. II. Mosby, St. Louis,
Missouri.
QuARTON, G. C, MELNECHUK, R., and SCHMITT, F. O. (Eds.) (1967). "The Neurosciences."
Rockefeller Univ. Press, New York.
RUCH, T C, PATTON, H. D., WOODBURY, J. W., and TOWE, A. L. (1965). "Neurophysi
ology." Saunders, Philadelphia, Pennsylvania.
Chapter 2
THE RESTING POTENTIAL
As already mentioned, the resting potential of the nerve cell is due to the
differences of ionic concentration which are maintained on the two sides of
the membrane by metabolic processes. If undisturbed, the system remains in a
steady state, but it is not in thermodynamic equilibrium. Nevertheless, the
following argument gives reasonable agreement with experimental data, for
the departure from equilibrium is not too great. In addition, it is worth
comparing the equilibrium treatment with the subsequent one, which takes
into account ionic exchanges across the membrane.
When a molecule is dissociated into ions in a solution, and the latter is
divided by a semipermeable membrane into two compartments with unequal
ionic concentrations, then an electric potential difference exists between the
compartments. The relationship between potential and concentration differ
ences at equilibrium can be derived as follows (Moelwyn-Hughes, 1966).
Consider two compartments 1 and 2 at different electric potentials
ΦkΛ = 1,2. Suppose a small quantity of material δη moles carrying an electric
charge önzF is transferred reversibly from compartment 2 to compartment 1 at
constant temperature Τ and pressure p. F is the Faraday constant and ζ is
the valence of the material. In general, the internal energy U, entropy S, and
volume Κ per mole of the substance will be different in the two compartments.
The change of internal energy δη AU of the transferred material will be due
to a change of heat energy δηΤΑΞ, a change of work energy -δηρ A{pV),
and a change of electric energy —δηζΓΑφ, where AX=X2 — is the
difference of a quantity X between the two compartments. Thus,
δηΑυ = δηΤΑΞ - δη A(pV) - δηζΓ Αφ
that is,
O = Δ(/ + A{pV) - TAS + ζΓΑφ, O = AG + zFΑφ (2.1)
