[MURG] Communication in Neuronal Networks

[Eugen Leitl]

http://www.sciencemag.org/cgi/content/full/301/5641/1870

Communication in Neuronal Networks
Simon B. Laughlin1 and Terrence J. Sejnowski2,3*

Brains perform with remarkable efficiency, are capable of prodigious
computation, and are marvels of communication. We are beginning to understand
some of the geometric, biophysical, and energy constraints that have governed
the evolution of cortical networks. To operate efficiently within these
constraints, nature has optimized the structure and function of cortical
networks with design principles similar to those used in electronic networks.
The brain also exploits the adaptability of biological systems to reconfigure
in response to changing needs.

1 Department of Zoology, University of Cambridge, Downing Street, Cambridge
CB2 3EJ, UK.
2 Howard Hughes Medical Institute, Salk Institute for Biological Studies, La
Jolla, CA 92037, USA.
3 Division of Biological Sciences, University of California, San Diego, La
Jolla, CA 92093, USA.

* To whom correspondence should be addressed. E-mail: terry at salk.edu

Neuronal networks have been extensively studied as computational systems, but
they also serve as communications networks in transferring large amounts of
information between brain areas. Recent work suggests that their structure
and function are governed by basic principles of resource allocation and
constraint minimization, and that some of these principles are shared with
human-made electronic devices and communications networks. The discovery that
neuronal networks follow simple design rules resembling those found in other
networks is striking because nervous systems have many unique properties.

To generate complicated patterns of behavior, nervous systems have evolved
prodigious abilities to process information. Evolution has made use of the
rich molecular repertoire, versatility, and adaptability of cells. Neurons
can receive and deliver signals at up to 105 synapses and can combine and
process synaptic inputs, both linearly and nonlinearly, to implement a rich
repertoire of operations that process information (1). Neurons can also
establish and change their connections and vary their signaling properties
according to a variety of rules. Because many of these changes are driven by
spatial and temporal patterns of neural signals, neuronal networks can adapt
to circumstances, self-assemble, autocalibrate, and store information by
changing their properties according to experience.

The simple design rules improve efficiency by reducing (and in some cases
minimizing) the resources required to implement a given task. It should come
as no surprise that brains have evolved to operate efficiently. Economy and
efficiency are guiding principles in physiology that explain, for example,
the way in which the lungs, the circulation, and the mitochondria are matched
and coregulated to supply energy to muscles (2). To identify and explain
efficient design, it is necessary to derive and apply the structural and
physicochemical relationships that connect resource use to performance. We
consider first a number of studies of the geometrical constraints on packing
and wiring that show that the brain is organized to reduce wiring costs. We
then examine a constraint that impinges on all aspects of neural function but
has only recently become apparent.energy consumption. Next we look at
energy-efficient neural codes that reduce signal traffic by exploiting the
relationships that govern the representational capacity of neurons. We end
with a brief discussion on how synaptic plasticity may reconfigure the
cortical network on a wide range of time scales.


Geometrical and Biophysical Constraints on Wiring

Reducing the size of an organ, such as the brain, while maintaining adequate
function is usually beneficial. A smaller brain requires fewer materials and
less energy for construction and maintenance, lighter skeletal elements and
muscles for support, and less energy for carriage. The size of a nervous
system can be reduced by reducing the number of neurons required for adequate
function, by reducing the average size of neurons, or by laying out neurons
so as to reduce the lengths of their connections. The design principles
governing economical layout have received the most attention.

Just like the wires connecting components in electronic chips, the
connections between neurons occupy a substantial fraction of the total
volume, and the wires (axons and dendrites) are expensive to operate because
they dissipate energy during signaling. Nature has an important advantage
over electronic circuits because components are connected by wires in
three-dimensional (3D) space, whereas even the most advanced VLSI (very large
scale integration) microprocessor chips use a small number of layers of
planar wiring. [A recently produced chip with 174 million transistors has
seven layers (3).] Does 3D wiring explain why the volume fraction of wiring
in the brain (40 to 60%; see below) is lower than in chips (up to 90%)? In
chips, the components are arranged to reduce the total length of wiring. This
same design principle has been established in the nematode worm
Caenorhabditis elegans, which has 302 neurons arranged in 11 clusters called
ganglia. An exhaustive search of alternative ganglion placements shows that
the layout of ganglia minimizes wire length (4).

Cortical projections in the early sensory processing areas are
topographically organized. This is a hallmark of the six-layer neocortex, in
contrast to the more diffuse projections in older three-layer structures such
as the olfactory cortex and the hippocampus. In the primary visual cortex,
for example, neighboring regions of the visual field are represented by
neighboring neurons in the cortex. Connectivity is much higher between
neurons separated by less than 1 mm than between neurons farther apart (see
below), reflecting the need for rapid, local processing within a cortical
column.an arrangement that minimizes wire length. Because cortical neurons
have elaborately branched dendritic trees (which serve as input regions) and
axonal trees (which project the output to other neurons), it is also possible
to predict the optimal geometric patterns of connectivity (5.7), including
the optimal ratios of axonal to dendritic arbor volumes (8). These
conclusions were anticipated nearly 100 years ago by the great neuroanatomist
Ramon y Cajal: "After the many shapes assumed by neurons, we are now in a
position to ask whether this diversity... has been left to chance and is
insignificant, or whether it is tightly regulated and provides an advantage
to the organism.... We realized that all of the various conformations of the
neuron and its various components are simply morphological adaptations
governed by laws of conservation for time, space, and material" [(9), p.
116].

The conservation of time is nicely illustrated in the gray matter of the
cerebral cortex. Gray matter contains the synapses, dendrites, cell bodies,
and local axons of neurons, and these structures form the neural circuits
that process information. About 60% of the gray matter is composed of axons
and dendrites, reflecting a high degree of local connectivity analogous to a
local area network. An ingenious analysis of resource allocation suggests
that this wiring fraction of 60% minimizes local delays (10). This fraction
strikes the optimum balance between two opposing tendencies: transmission
speed and component density. Unlike the wires in chips, reducing the diameter
of neural wires reduces the speed at which signals travel, prolonging delays.
But it also reduces axon volume, and this allows neurons to be packed closer
together, thus shortening delays.


Global Organization of the Communication Network

Long-range connections between cortical areas constitute the white matter and
occupy 44% of the cortical volume in humans. The thickness of gray matter,
just a few millimeters, is nearly constant in species that range in brain
volume over five orders of magnitude. The volume of the white matter scales
approximately as the 4/3 power of the volume of the gray matter, which can be
explained by the need to maintain a fixed bandwidth of long-distance
communication capacity per unit area of the cortex (11) (Fig. 1). The layout
of cortical areas minimizes the total lengths of the axons needed to join
them (12). The prominent folds of the human cortex allow the large cortical
area to be packed in the skull but also allow cortical areas around the
convolutions to minimize wire length; the location of the folds may even
arise from elastic forces in the white matter during development (13).

Fig. 1. Cortical white and gray matter volumes of 59 mammalian species are
related by a power law that spans five to six orders of magnitude. The line
is the least squares fit, with a slope around 1.23 ± 0.01 (mean ± SD) and
correlation of 0.998. The number of white matter fibers is proportional to
the gray matter volume; their average length is the cubic root of that
volume. If the fiber cross section is constant, then the white matter volume
should scale approximately as the 4/3 power of the gray matter volume. An
additional factor arises from the cortical thickness, which scales as the
0.10 power of the gray matter volume. [Adapted from (11)] [View Larger
Version of this Image (44K GIF file)]

The global connectivity in the cortex is very sparse, and this too reduces
the volume occupied by long-range connections: The probability of any two
cortical neurons having a direct connection is around one in a hundred for
neurons in a vertical column 1 mm in diameter, but only one in a million for
distant neurons. The distribution of wire lengths on chips follows an inverse
power law, so that shorter wires also dominate (14). If we created a matrix
with 1010 rows and columns to represent the connections between every pair of
cortical neurons, it would have a relatively dense set of entries around the
diagonal but would have only sparse entries outside the diagonal, connecting
blocks of neurons corresponding to cortical areas.

The sparse long-range connectivity of the cortex may offer some of the
advantages of small-world connectivity (15). Thus, only a small fraction of
the computation that occurs locally can be reported to other areas, through a
small fraction of the cells that connect distant cortical areas; but this may
be enough to achieve activity that is coordinated in distant parts the brain,
as reflected in the synchronous firing of action potentials in these areas,
supported by massive feedback projections between cortical areas and
reciprocal interactions with the thalamus (16, 17).

Despite the sparseness of the cortical connection matrix, the potential
bandwidth of all of the neurons in the human cortex is around a terabit per
second (assuming a maximum rate of 100 bits/s over each axon in the white
matter), comparable to the total world backbone capacity of the Internet in
2002 (18). However, this capacity is never achieved in practice because only
a fraction of cortical neurons have a high rate of firing at any given time
(see below). Recent work suggests that another physical constraint.the
provision of energy.limits the brain's ability to harness its potential
bandwidth.


Energy Usage Constrains Neural Communication

As the processor speeds of computers increase, the energy dissipation
increases, so that cooling technology becomes critically important. Energy
consumption also constrains neural processing. Nervous systems consume
metabolic energy continuously at relatively high rates per gram, comparable
to those of heart muscle (19). Consequently, powering a brain is a major
drain on an animal's energy budget, typically 2 to 10% of resting energy
consumption. In humans this proportion is 20% for adults and 60% for infants
(20), which suggests that the brain's energy demands limit its size (21).

Energy supply limits signal traffic in the brain (Fig. 2). Deep anesthesia
blocks neural signaling and halves the brain's energy consumption, which
suggests that about 50% of the brain's energy is used to drive signals along
axons and across synapses. The remainder supports the maintenance of resting
potentials and the vegetative function of neurons and glia. Cortical gray
matter uses a higher proportion of total energy consumption for signaling,
more than 75% (Fig. 2), because it is so richly interconnected with axons and
synapses (21). From the amounts of energy used when neurons signal, one can
calculate the volume of signal traffic that can be supported by the brain's
metabolic rate. For cerebral cortex, the permissible traffic is 5 action
potentials per neuron per second in rat (Fig. 2) (22) and <1 per second in
human (23). Given that the brain responds quickly, the permissible level of
traffic is remarkably low, and this metabolic limit must influence the way in
which information is processed. Recent work suggests that brains have
countered this severe metabolic constraint by adopting energy-efficient
designs. These designs involve the miniaturization of components, the
elimination of superfluous signals, and the representation of information
with energy-efficient codes.

Fig. 2. Power consumption limits neural signaling rate in the gray matter of
rat cerebral cortex. Baseline consumption is set by the energy required to
maintain the resting potentials of neurons and associated supportive tissue
(r.p.) and to satisfy their vegetative needs (nonsignaling). Signaling
consumption rises linearly with the average signaling rate (the rate at which
neurons transmit action potentials). The measured rates of power consumption
in rat gray matter vary across cortical areas and limit average signaling
rates to 3 to 5.5 Hz. Values are from (19), converted from rates of
hydrolysis of adenosine triphosphate (ATP) to W/kg using a free energy of
hydrolysis for a molecule of ATP under cellular conditions of 10.19 J. [View
Larger Version of this Image (20K GIF file)]


Miniaturization, Energy, and Noise

The observation that 1 mm3 of mouse cortex contains 105 neurons, 108
synapses, and 4 km of axon (24) suggests that, as in chip design, the brain
reduces energy consumption by reducing the size and active area of
components. Even though axon diameter is only 0.3 µm (on average), sending
action potentials along these "wires" consumes more than one-third of the
energy supplied to cortical gray matter (22). Thus, as with computer chips,
an efficient layout (discussed above) and a high component density are
essential for energy efficiency. but, as is also true for chips,
miniaturization raises problems about noise.

When a neuron's membrane area is reduced, the number of molecular pores (ion
channels) carrying electrical current falls, leading to a decline in the
signal-to-noise ratio (SNR) (25.27). The noise produced by ion channels, and
by other molecular signaling mechanisms such as synaptic vesicles, is
potentially damaging to performance. However, the effects of noise are often
difficult to determine because they depend on interactions between signaling
molecules in signaling systems. These interactions can be highly nonlinear
(e.g., the voltage-dependent interactions between sodium and potassium ion
channels that produce action potentials) and can involve complicated spatial
effects (e.g., the diffusion of chemical messengers between neurons and the
transmission of electrical signals within neurons). A new generation of
stochastic simulators is being developed to handle these complexities and
determine the role played by molecular noise and diffusion in neural
signaling (26, 28, 29). With respect to miniaturization, stochastic
simulations (25) show that channel noise places a realistic ceiling on the
wiring density of the brain by setting a lower limit of about 0.1 µm on axon
diameter.

The buildup of noise from stage to stage may be a fundamental limitation on
the logical depth to which brains can compute (30). The analysis of the
relationships among signal, noise, and bandwidth and their dependence on
energy consumption will play a central role in understanding the design of
neural circuits. The cortex has many of the hallmarks of an energy-efficient
hybrid device (28). In hybrid electronic devices, compact analog modules
operate on signals to process information, and the results are converted to
digital data for transmission through the network and then reconverted to
analog data for further processing. These hybrids offer the ability of analog
devices to perform basic arithmetic functions such as division directly and
economically, combined with the ability of digital devices to resist noise.
In the energy-efficient silicon cochlea, for example, the optimal mix of
analog and digital data (that is, the size and number of operations performed
in analog modules) is determined by a resource analysis that quantifies
trade-offs among energy consumption, bandwidth for information transmission,
and precision in analog and digital components. The obvious similarities
between hybrid devices and neurons strongly suggest that hybrid processing
makes a substantial contribution to the energy efficiency of the brain (31).
However, the extent to which the brain is configured as an energy-efficient
hybrid device must be established by a detailed resource analysis that is
based on biophysical relationships among energy consumption, precision, and
bandwidth in neurons.

Some research strongly suggests that noise makes it uneconomical to transfer
information down single neurons at high rates (29, 31). Given that a neuron
is a noise-limited device of restricted bandwidth, the information rate is
improved with the SNR, which increases as the square root of the number of
ion channels, making improvements expensive (25). Thus, doubling the SNR
means quadrupling the number of channels, the current flow, and hence the
energy cost. Given this relationship between noise and energy cost, an
energy-efficient nervous system will divide information among a larger number
of relatively noisy neurons of lower information capacity, as observed in the
splitting of retinal signals into ON and OFF pathways (32). Perhaps the
unreliability of individual neurons is telling us that the brain has evolved
to be energy efficient (31).


Saving on Traffic

Energy efficiency is improved when one reduces the number of signals in the
network without losing information. In the nervous system, this amounts to an
economy of impulses (33) that has the additional advantage of increasing
salience by laying out information concisely. Economy is achieved by
eliminating redundancy. This important design principle is well established
in sensory processing (34). Redundancy reduction is a goal of algorithms that
compress files to reduce network traffic.

In the brain, efficiency is improved by distributing signals appropriately in
time and space. Individual neurons adopt distributions of firing rate (35,
36) that maximize the ratio between information coded and energy expended.
Networks of neurons achieve efficiency by distributing signals sparsely in
space and time. Although it was already recognized that sparse coding
improves energy efficiency (37), it was Levy and Baxter's detailed analysis
of this problem (38) that initiated theoretical studies of energy-efficient
coding in nervous systems. They compared the representational capacity of
signals distributed across a population of neurons with the costs involved.
Sparse coding schemes, in which a small proportion of cells signal at any one
time, use little energy for signaling but have a high representational
capacity, because there are many different ways in which a small number of
signals can be distributed among a large number of neurons. However, a large
population of neurons could be expensive to maintain, and if these neurons
rarely signal, they are redundant. The optimum proportion of active cells
depends on the ratio between the cost of maintaining a neuron at rest and the
extra cost of sending a signal. When signals are relatively expensive, it is
best to distribute a few of them among a large number of cells. When cells
are expensive, it is more efficient to use few of them and to get all of them
signaling. Estimates of the ratio between the energy demands of signaling and
maintenance suggest that, for maximum efficiency, between 1% and 16% of
neurons should be active at any one time (22, 23, 38). However, it is
difficult to compare these predictions with experimental data; a major
problem confronting systems neuroscience is the development of techniques for
deciphering sparse codes.

There is an intriguing possibility that the energy efficiency of the brain is
improved by regulating signal traffic at the level of the individual synaptic
connections between neurons. A typical cortical neuron receives on the order
of 10,000 synapses, but the probability that a synapse fails to release
neurotransmitter in response to an incoming signal is remarkably high,
between 0.5 and 0.9. Synaptic failures halve the energy consumption of gray
matter (22), but because there are so many synapses, the failures do not
necessarily lose information (39, 40). The minimum number of synapses
required for adequate function is not known. Does the energy-efficient
cortical neuron, like the wise Internet user, select signals from sites that
are most informative? This question draws energy efficiency into one of the
most active and important areas of neuroscience: synaptic plasticity.


Reconfiguring the Network

Long-distance communication in the brain occurs through all-or-none action
potentials, which are transmitted down axons and converted to analog chemical
and electrical signals at synapses. The initiation of action potentials in
the cortex can occur with millisecond precision (41) but, as we have just
discussed, the communication at cortical synapses is probabilistic. On a
short time scale of milliseconds to seconds, presynaptic mechanisms briefly
increase or decrease the probability of transmission at cortical synapses
over a wide range, depending on the previous patterns of activity (42). On
longer time scales, persistent correlated firing between the presynaptic and
postsynaptic neurons can produce long-term depression or potentiation of the
synaptic efficacy, depending on the relative timing of the spikes in the two
neurons (43).

A new view of the cortical network is emerging from these discoveries. Rather
than being a vast, fixed network whose connection strengths change slowly,
the effective cortical connectivity is highly dynamic, changing on fast as
well as slow time scales. This allows the cortex to be rapidly reconfigured
to meet changing computational and communications needs (44). Unfortunately,
we do not yet have techniques for eavesdropping on a large enough number of
neurons to determine how global reconfiguration is achieved. Local field
potentials (LFPs), extracellular electric fields that reflect the summed
activity from local synaptic currents and other ion channels on neurons and
glial cells, may provide hints of how the flow of information in cortical
circuits is regulated (16). Oscillations in the 20- to 80-Hz range occur in
the LFPs, and the coherence between spikes and these oscillations has been
found to be influenced by attention and working memory (45, 46).


Conclusions

The more we learn about the structure and function of brains, the more we
come to appreciate the great precision of their construction and the high
efficiency of their operations. Neurons, circuits, and neural codes are
designed to conserve space, materials, time, and energy. These designs are
exhibited in the geometry of the branches of dendritic trees, in the precise
determination of wiring fractions, in the laying out of maps in the brain, in
the processing of signals, and in neural codes. It is less obvious, but
highly likely, that the unreliability of single neurons is also a mark of
efficiency, because noise in molecular signaling mechanisms places a high
price on precision. To an extent yet to be determined, the noise and
variability observed among neurons is compensated by plasticity.the ability
of neurons to modify their signaling properties. Neural plasticity also has
the potential to direct the brain's scarce resources to where they will be of
greatest benefit.


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