Chem. Senses 27: 261-275,
2002
© Oxford University Press 2002
A Computational System for Simulating and Analyzing Arrays of Biological and Artificial Chemical Sensors
Department of Neuroscience, Tufts University School of Medicine, Boston, MA, USA
Correspondence to be sent to: John S. Kauer, Department of Neuroscience, Tufts University School of Medicine, Boston, MA 02111, USA. E-mail: john.kauer{at}tufts.edu
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Abstract |
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We have designed an approach for modeling olfactory pathways by which one can explore how the properties of individual receptors affect the information coding capacity of an entire system. The effect of receptor tuning breadth on system performance was explored explicitly. We presented model sensory arrays with sets of stimuli randomly and uniformly distributed in an `olfactory space'. Arrays of uniformly sized model receptors responding to 25-35% of the stimuli gave the best performance as measured by the ability to capture the most information about the stimulus set. Arrays of variably sized model receptors that were both more broadly and more narrowly tuned than this optimum could, however, perform better than uniform arrays. This method and the results obtained using it suggest a framework for considering the growing body of evidence on the functional properties of individual olfactory receptor and relay neurons from a systems coding perspective.
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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The peripheral olfactory system consists of a large, diverse array of sensory neurons and first-order relay neurons, each of which responds to many different stimuli (Gesteland et al., 1965
; Getchell,
1974; Duchamp et al.,
1974; Revial et al.,
1978,
1982;
Wellis et al., 1989;
Mori et al., 1992;
Sato et al., 1994;
Hirono et al., 1994;
Bozza and Kauer, 1998;
Malnic et al., 1999).
In the olfactory epithelium, single, mono- molecular stimuli elicit
time-varying changes in the activity levels of a population of olfactory
receptor neurons (ORNs) (Revial et al.,
1978,
1982;
Sicard, 1986;
Sato et al., 1994;
Youngentob et al.,
1995; Duchamp-Viret et
al., 1999; Vogler and
Schild, 1999). The axons of these primary sensory neurons project
to the olfactory bulb, where they segregate according to which olfactory
receptor (OR) protein they express
(Mombaerts et al.,
1996). In the olfactory bulb, ORNs synapse directly onto
mitral/tufted (M/T) cells, the output cells of the olfactory bulb, as well as
onto periglomerular interneurons. A substantial fraction of the output cells
also appear to be involved in representing each stimulus by modulating their
activity over time (Kauer, 1974b; Meredith
and Moulton, 1978; Harrison
and Scott, 1986; Hamilton and
Kauer, 1989; Wellis et
al., 1989; Wellis and
Scott, 1990; Imamura et
al., 1992; Mori et
al., 1992; Katoh et
al., 1993; Cinelli et
al., 1995). The representation of stimuli across such broad
arrays of neurons is an example of a distributed or population code (Kauer,
1987,
1991;
Schild, 1988;
Buck, 1996; Laurent,
1997,
1999;
Hildebrand and Shepherd, 1997;
Mori et al., 1999;
Christensen and White, 2001;
Kauer and White, 2001). The
fundamental property of this type of coding is that the ability of the entire
system to make fine discriminations depends not on any single component, but,
rather, how an animal perceives its chemical environment emerges from a
synthesis of information from many components at each level of the pathway.
This coding paradigm has the advantage of accounting for not only the
physiological data described above, but also for a number of other olfactory
system characteristics, including plasticity and tolerance of large lesions
(Lu and Slotnick, 1998). In
attempting to understand how a coherent odor perception emerges from this
representation, one is presented with the challenge of formally relating the
response properties of individual neurons to the global, system-wide
representation (Schild, 1988;
Rolls et al., 1996;
Alkasab et al.,
1999).
What methods can be used as the basis for describing such relationships?
One candidate approach is information theory (IT), which uses Bayesian
reasoning to quantitatively assess the contributions of individual elements to
overall system representations. IT was originally developed to quantify data
transmission over noisy electrical channels
(Shannon and Weaver, 1949),
but since it provides a methodology for measuring the statistical relationship
between two systems, the theory has been successfully used to study encoding
of sensory information in neural systems
(Bialek et al., 1991;
Theunissen and Miller, 1991;
Atick et al., 1992; Abbott et
al., 1996; Rolls et
al., 1996; Yamada et
al., 1996; Rieke et
al., 1997; Maynard et
al., 1999). IT techniques have been applied most often to
analysis of the fine temporal structure of sequences of action potentials [as
reviewed by Borst and Theunissen (Borst and
Theunissen, 1999)], but they have also been used to explore
distributed coding (Theunissen and Miller,
1991; Brunel and Nadal,
1998; Maynard et al.,
1999; Eurich and Wilke,
2000). Both its quantitative nature and its ability to
characterize relationships among numerous component elements recommend IT
analysis to the study of olfactory coding
(Alkasab et al.,
1999). Here, we examine how an IT approach may be used to measure
the degree to which receptors in biological systems or sensors in artificial
systems contribute to the overall performance of olfactory arrays.
In this paper, we establish a theoretical framework and modeling approach for simulating arrays of chemosensors, modeling their responses to stimuli and quantitatively assessing the coding capacity of the array using IT measures. Using these methods, we can explore how the properties of idealized individual sensors that model biological ORNs affect the distributed representation of an entire array. By systematically changing model sensors and quantifying array performance, we have begun to characterize some of the possible relationships between one sensor property—breadth of tuning—and the representation of odorant stimuli by the entire olfactory receptor array. This method also potentially offers an approach for quantifying how measured response profiles of individual biological ORNs represent fundamental properties of the chemical world.
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Methods |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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Models of odorant stimuli
The difficulty in describing the crucial physico-chemical properties that
define chemical stimuli as odors has long hampered our understanding of
olfactory systems. A particular difficulty has been discovering which
combinations of physical parameters of odorant molecules (e.g. molecular
weight, polarity, charge, chemical structure, or more complex properties based
on reactivity) determine ORN response. These parameters are expected to be
different for neurons expressing different olfactory receptor proteins. As a
simplified example, an ORN with responses that vary with changes in the
polarity of a stimulus molecule may be indifferent to the charge or molecular
weight of the stimulus, while the responses of another ORN may change with
another aspect of molecular structure. For biological ORNs, it seems most
likely that these parameters will correspond to complex combinations of
physico-chemical properties which determine the interaction between an odorant
and the OR protein. No comprehensive set of parameters has yet been found for
all odorant sets, although changes in responses with changing carbon chain
length appear to be relevant for some ORNs and second-order neurons
(Imamura et al.,
1992; Katoh et al.,
1993; Sato et al.,
1994; Yokoi et al.,
1995; Zhao et al.,
1998; Rubin and Katz,
1999; Uchida et al.,
2000).
We do not yet know even the number of parameters needed to characterize
general olfactory stimuli. However, we can estimate certain bounds on this
dimensionality for biological olfactory systems. Attempts to find stimulus
parameters that account for frog and zebrafish neuronal responses to limited
odorant sets have met with some success
(Revial et al., 1983;
Sicard and Holley, 1984;
Friedrich and Korsching, 1998). These studies, which have used variations of
principal component analysis applied to single unit responses, suggest that
there are at least three relevant parameters in the studied systems, although
these methods do not identify the physical parameters themselves. At the other
extreme, the number of different OR genes in rodents may be as large as 1000
(Ressler and Sullivan, 1993), although it is not yet known how many of them
are expressed at any one time. If each OR type corresponds to its own
independent parameter and if ORNs are simply active or inactive, an olfactory
system could describe 21000 different stimuli, which is more than
the estimated number of atoms in the universe
(Dyson, 1979). Thus, it is
unlikely that the responses of neurons expressing different receptors are
completely independent of one another. It is reasonable, therefore, to propose
that the dimensionality of `olfactory space' is greater than three, but much
less than 1000.
Given the uncertainty about the sizes, shapes and dimensionalities of olfactory space, we have made a number of assumptions for the present discussion. We created a generalized framework for modeling odorants and olfactory receptors, and have used IT analyses to examine some of the properties of their interactions. We begin by describing a set of stimuli in which each model odorant is represented by a point in a multi-dimensional experimental space (see Figure 1). This space is intended to model olfactory `odorant space', not perceptual `odor space'; that is, its dimensions are the set of physical parameters of stimuli which determine the response profiles of the sensors rather than behavioral dimensions (though in other experiments, analogous analyses might be performed in a perceptual space). Each stimulus in our model is specified by the values of these parameters, just as each odorant is defined according a set of physical parameters. In our simulations, we have used a three-dimensional experimental space, but the technique is not limited to this number: spaces of two, four or more dimensions are straightforward to implement (although high-dimensional spaces are demanding to compute and impossible to visualize). A three-dimensional experimental space means that the responses of all model receptors can be separated based on three independent stimulus parameters. Within this space, a cubic experimental volume is defined in which stimuli are uniformly randomly distributed; no stimuli occur outside this experimental volume. The arrangement of stimuli in this experimental volume is independent of the likelihood of each stimulus occurring, although the probability of occurrence is an important consideration in calculating IT measures.
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Models of receptors
In general, the physiological study of biological olfactory systems centers
around methodically observing response profiles and relating the qualitative
and quantitative attributes of odorants to changes in neuronal activity. In
the study of ORNs and M/T cells, a variety of techniques have been used to
measure neuronal responses as different odorant stimuli are applied under
controlled conditions. These include intracellular electrophysiological
recording (Getchell, 1977;
Wellis et al., 1989),
intracellular calcium measurements
(Restrepo et al.,
1993; Bozza and Kauer,
1998; Leinders-Zufall et
al., 1998), extracellular action potential measurement
(Kauer, 1974;
Meredith and Moulton, 1978;
Harrison and Scott, 1986;
Mori et al., 1992;
Duchamp-Viret et al.,
1999), 2-deoxyglucose uptake
(Stewart et al.,
1979; Johnson et al.,
1998), and in vivo imaging
(Cinelli et al.,
1995; Friedrich and Korsching,
1997; Joerges et al.,
1997; Rubin and Katz,
1999; Wachowiak and Cohen,
2001). Such responses have often been summarized as positive
(excitatory), negative (suppressive), or null (no change) responses based on
changes in action potential firing rate, membrane potential amplitudes or
other time-varying signals. In addition to temporally complex firing patterns,
olfactory neurons, like all CNS neurons, show variation in responses to
repeated applications of the same stimulus. This variability comes from
several sources: the inherent noise of the neuron
(Harrison and Scott, 1986;
Lowe and Gold, 1995), changes
in stimulus conditions, errors in the measurement process and changes in the
system as a result of prior odorant exposure
(Getchell and Shepherd, 1978;
Baylin and Moulton, 1979;
Kurahashi and Menini, 1997;
Friedrich and Laurent, 2001).
Upon different presentations, therefore, ostensibly identical odorant stimuli
can give rise to apparently different responses.
We have sought to create a framework which can capture these complexities without becoming unwieldy. In our model, a sensor—which represents an ORN—maps stimuli (points in the experimental space, as described in the previous section) to the probabilities of all possible responses (i.e. a probability density function). These responses could be binary (positive/null) responses, ternary (positive/negative/null) responses, labeled responses (`A', `B', `C', etc.) or complex time-varying signals. In the experiments described here, we have modeled the sensors as giving either positive or null responses. We have also made the receptive range (RR) of each sensor a cube in the experimental space: that is, a neuron gives a positive response (with probability 1) to all stimuli that fall within the cube, and a null response (also with probability 1) to all stimuli outside it (see Figure 1). The RR of each sensor is randomly placed within the experimental volume, and is required to be completely contained by the experimental volume to which stimuli are restricted. One consequence of this arrangement is that the center of the RRs of larger sensors are biased toward the center of the space (since they must remain completely inside the experimental volume). We can express the size of sensor RRs as a percentage of the experimental volume; this size is therefore also equivalent to the percentage of stimuli to which the sensor gives a positive response, since the stimuli are randomly distributed within the volume. Refinements of this scheme are possible (e.g. richer response sets, probabilistic responses, more complex RR shapes), but they do not alter the fundamental methodology: the input to a sensor is a stimulus and the output of the sensor is a probability density function of responses.
For computational purposes, an array of sensors can be treated as a single sensor. That is, if sensor A and sensor B each can give only positive and null responses, then the array of sensors A and B would have four possible response states: positive/positive, positive/null, null/positive and null/null. The response profile of a sensor array is then just the probability of each of these response combinations for each stimulus, and array responses to a stimulus set can be tabulated as for a single sensor.
Information theory measures
From these descriptions of stimulus sets and response profiles, the
performance of sensors and arrays in response to a set of stimuli can be
evaluated using IT measures. First, the entropy of the stimulus set is
calculated from the set of probabilities of occurrence of each stimulus.
Entropy quantifies the complexity of the stimulus set; it measures the average
a priori difficulty in randomly guessing which stimulus is present.
In comparing two stimulus sets, the set which is more complex has greater
entropy, and its state is more difficult to guess without other information. A
profile of sensor responses to a stimulus set also has an entropy; this is
calculated from the probability of each sensor response among the responses to
the entire stimulus set. The entropy of the response profile to a stimulus set
quantifies the maximum capacity of the sensor to carry information about the
state of the stimulus set. The mutual information (MI) (also known as
`transinformation') between sensor responses and the stimulus set measures the
extent to which complexity in a set of possible stimuli can be accounted for
by the complexity in the responses; that is, the correlated complexity. To
calculate the MI between a stimulus set and sensory responses, one needs to
know the response profile of the sensor; that is, the conditional probability
of the sensor giving each response to each of the stimuli. All of these
measures, including their mathematical definitions, are described in more
detail in a previous review (Alkasab et
al., 1999).
MI has a number of advantages as a measure of coding performance. First, MI
is an entropy measure—it directly quantifies the reduction in
uncertainty that is possible by utilizing a sensor. That is, if a stimulus set
has an a priori entropy of 10 bits and a sensor has a MI with that
stimulus set of 2 bits, then the uncertainty about the state of the stimulus
is reduced to 8 bits if one observes the output of that sensor. Secondly, it
measures how well a sensor or array makes discriminations averaged over
repeated stimulus applications. It therefore provides a link between
instantaneous response profiles and long-term performance. Finally, MI allows
one to account for the probability of occurrence of each stimulus; this
corresponds to the likelihood of encountering each odorant in the natural
world. Most chemical environments consist of some frequently encountered
stimuli and a larger set of detectable but rarely met odorants. Other neuronal
sensory systems have been observed to take advantage of these differences in
frequency of stimuli in the natural world. IT predicts that systems which fail
to account for the statistics of the natural world cannot describe the world
with optimal efficiency (Atick,
1992; Rieke et al.,
1995; Olshausen and Field,
1996). For these experiments, we have specified that each stimulus
has an equal likelihood of being presented to the system; distributions which
match what is known about natural chemical environments more closely are also
easily implemented within our system.
Experimental approach
We use these models to explore how the performance of a chemosensory array depends on the tuning breadths of its individual components. In this paper, because we are interested in the general properties of ensembles of sensors, we create many random arrangements of sensors and stimuli, and calculate the mean performance across these arrangements. This process is illustrated in Figure 2, which shows the details of the steps in generating a plot of performance against sensor size. First, the properties of the array and stimulus set are specified (step 1). In this example, 32 stimuli are randomly distributed according to a uniform distribution in a cubic experimental volume within a three-dimensional experimental space (step 2). We specify that each stimulus is equally likely, which means that the stimulus set has an entropy of 5 bits. The array consists of eight positive/null sensors; the RR of each sensor occupies 15% of the volume of the experimental volume. Since the stimuli are equally likely to be placed anywhere within the experimental volume, we expect each RR, on average, to contain 15% of the stimuli. The RRs of the sensors are also randomly placed within the experimental volume such that the entirety of each RR is contained within the boundaries of the volume. From this random placement of sensors and stimuli, the array response evoked by each stimulus is determined (step 3)—the stimulus—response map. This map is then reversed to list the realized sensor states and the stimuli which evoked them—the response—stimulus map (step 4). Note that although 256 responses are theoretically possible for an array of eight binary sensors, only 15 are actually observed for this array in response to this stimulus set. From the response—stimulus map, the MI is calculated to be 2.98 bits. The sensors and stimuli are then reshuffled to be placed randomly again within the experimental volume (step 5). The cycle of randomly rearranging sensors and stimuli and computing the performance of the resulting array is repeated hundreds or thousands of times, and the distribution of the results plotted in a histogram (step 6); in this case, the performance (i.e. the mean of the distribution) is 2.72 ± 0.44 (mean ± SD) bits for 500 repetitions. Next, the parameter of interest is incremented, and the process is repeated (step 7). In this case, the RR of each sensor is enlarged to a volume of 30% of the experimental volume. The stimuli and sensors of this new size are randomly placed, and the performance is evaluated for another 500 repetitions. This is repeated for arrays of sensors of RRs occupying 45, 60, 75 and 90% of the experimental volume. These results are then compiled into a plot (step 8) of array performance against sensor size for eight-sensor arrays discriminating 32 stimuli.
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Results |
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Varying sensor response breadth
To explicitly explore how RR size affects the performance of entire olfactory arrays of various sizes, we conducted a series of experiments using simple models in a three-dimensional experimental space. Sets of 1024 stimuli were randomly distributed according to a uniform distribution within a cubic experimental volume in this space. This design is intended to simulate the `generalist olfactory problem' of discriminating a large number of (possibly unknown) stimuli. For each array, the RRs of sensors were also randomly distributed such that the entirety of each RR was contained within the cube. The sensors in an array had uniform size; each responded one of two ways: either positive or null. Each of the set of 1024 stimuli was specified to be equally likely, which makes the entropy of the discrimination task 10 bits. For each array size and sensor breadth, we computed the performance of 1500 random placements of stimuli and sensors for arrays between 8 and 48 sensors and 300 repetitions for arrays of 64 and 128 sensors. The results of systematically varying sensor RR size from 5 to 95% of the experimental volume for arrays of 8, 16, 24, 32, 48, 64 and 128 sensors are shown in Figure 3A. For each array size, performance is poorest at the extremes of sensor RR size (5 and 95% of the experimental volume) and is greatest for arrays of sensors with RRs between 25% (for the 128-sensor array) and 35% (for the 8-sensor array). The peaks of the curves shift to the left as array size increases—that is, the optimal RR size is smaller for larger arrays. Figure 3B shows the distribution of the results of 300 trials with an array of 128 sensors of size 25%. For every array size (i.e. number of sensors), the difference in performance between each pair of different sensor sizes was significant. In addition, the performance differences between arrays with different numbers of the same-sized sensors were also significant. (For all comparisons, Welch's approximate t-test was used and P < 0.001.) Since changing the positions of receptors had a smaller effect on performance than changing RR size, we conclude that the breadth of response of sensors in an array is more important in determining the information capacity than the specific placement of sensors relative to the stimuli.
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Arrays containing different numbers of sensors were affected by this
changing sensor RR size in different ways. Increasing the number of sensors
increases performance (i.e. curves are higher for larger arrays), but MI
measures for larger arrays show diminishing increases in performance as more
sensors are added. The peak performance for arrays of each size is plotted in
Figure 3C. One of the
characteristics of a distributed code is that MI increases in proportion to
the logarithm of the number of sensors in the ensemble
(Abbott and Dayan, 1999). For
arrays of 8, 16, 24 and 32 sensors, this relationship holds; for these four
points, the data were fit by the relationship Imax = 1.56
log2(N) - 0.13(R2 = 0.998). However,
for larger arrays, maximum performance falls below this line. From this, we
conclude that as arrays get larger, new sensors contribute less new
information. Note that none of the arrays reach the maximum of 10 bits of
performance. This is a partly a result of the random distribution of sensor
RRs and stimuli; by chance, some stimuli will be so close to one another that
they cannot be distinguished, which pulls the performance below 10 bits. If
these arrays were tested against larger stimulus sets (with greater entropy),
this upper limit on performance would be higher, such that 128-sensor arrays
would achieve performances of >10 bits.
These results can be applied to the characterization of biological
olfactory codes in several ways. First, the RR breadth which results in the
maximum average MI (i.e. the sensors size which leads to the greatest
discrimination among the stimuli) for each array size is quite large, such
that each sensor gives a positive response to >250 stimuli (out of 1024).
In the context of biological olfactory systems, this result suggests that in
an olfactory system that must discriminate many disparate odorants (i.e. those
that span a broad range among the measures that determine their interactions
with ORNs), neurons which only respond to one or two stimuli would be
disadvantageous compared with more broadly tuned receptors. A system of
narrowly tuned sensors, in which each sensor responds to exactly one stimulus,
is represented at the left of Figure
3A. (Such arrays are limited to representing a number of odorants
equivalent to the number of different receptor types.) A system of receptors
with larger RRs, on the other hand, could encode many more odorants than the
number of receptor types. Finally, our results are predicated on a uniformly
random distribution of stimuli in both space and time. In physical terms, this
would correspond to a set of odorants which were distributed evenly across
physical parameters, and occur with equal likelihood. While this is not a good
model of natural chemical environments, the results are nonetheless consistent
with evidence from both amphibians and mammals, which suggests that olfactory
neurons do tend to be broadly tuned: olfactory neurons are generally found to
respond to more than one member of even small stimulus sets
(Gesteland et al.,
1965; Duchamp et al.,
1974; Getchell,
1974; Revial et al.,
1978,
1982;
Sicard and Holley, 1984;
Sato et al., 1994;
Bozza and Kauer, 1998;
Malnic et al., 1999;
Duchamp-Viret et al.,
1999,
2000).
As large as these optimal RRs seem, they are smaller than what would be predicted by considering the optimal breadth of an individual sensor. The MI of an individual binary sensor is greatest if it responds to exactly half of the stimuli. An array of independent, binary sensors would have a symmetric curve with an optimum RR size of 50%. The observed asymmetry of the curves in Figure 3 suggests that the relationships between sensors within the array play a major role in determining array performance (i.e. sensors are not independent). When a sensor is added to an ensemble, it makes a contribution to the total MI only if it conveys new information about the stimulus. The new sensor conveys no new information if its response can be predicted from the response of the rest of the array. The extent to which two sensors (or an array and a sensor) carry the same information can be termed their information redundancy. One way to quantify this redundancy in a noiseless case is to determine the MI between the additional sensor and the array. If it is high relative to the entropy of the new sensor, then the responses of the new sensor could be predicted from the existing array, and the new sensor has little new information. Conversely, if the MI between the array and the new sensor is low (i.e. if the response of the new sensor could not be predicted from the response of the array), then the sensor does convey new information and it can make a positive contribution to the performance of an array. Thus, we can use the MI between sensors to quantify the extent to which they carry redundant information.
This redundancy depends on the degree to which sensor RRs overlap. Clearly, sensors which substantially overlap will have redundant information; less obviously, sensors whose RRs do not overlap at all have greater information redundancy than sensors whose RRs partially overlap. This relationship between RR overlap and information redundancy explains some of the features of Figure 3A. Each curve represents a balance between optimizing the RRs of individual sensors and optimizing their RR overlap. Each individual sensor would be most informative if it were activated by half the stimuli. When we restrict sensor RRs to be completely within the experimental volume, however, every RR in an array has a high likelihood of containing the region at the center of the volume, resulting in many stimuli that activate all of the sensors (which therefore cannot be discriminated from one another). Smaller sensors allow central stimuli to activate different subsets of sensors, permitting them to be discriminated. However, if the sensor RRs become so small that they do not overlap at all, then many stimuli will activate no sensors, and therefore also not be discriminated.
We have tested this explanation explicitly by designing an experiment in which arrays of 32 sensors were measured for their ability to discriminate three stimulus sets: 1024 stimuli uniformly distributed throughout the experimental volume to which stimuli are restricted, 512 stimuli uniformly distributed in the inner 50% of this volume (i.e. a cube in the center of the experimental volume whose volume is half of the full volume), and 512 stimuli uniformly distributed in the outer 50% of the volume (i.e. the outer `shell' of the experimental volume). Sensors remain distributed throughout the entire experimental volume. The results of this experiment are shown in Figure 4. The central curve of Figure 4 (diamonds, solid line) is a replication of the results for a 32-sensor array from Figure 3A. When stimuli are restricted to the central half of the experimental volume, smaller sensors are at an advantage, resulting in a left-shifted curve. Conversely, for stimuli on the periphery of the experimental volume, larger sensors are advantageous, and the curve shifts to the right. From these data, one possible explanation for Figure 3A is that each curve starts at the far left with small sensors with little overlap, resulting in poor performance. As sensor RR increases, the overlap of sensors becomes sufficient for the array to discriminate stimuli and performance improves. Eventually, sensor size increases enough that there is a central region within which stimuli activate many sensors in the array; obviously, no stimuli within this region can be discriminated. As sensor size increases further and the size of this region increases, performance degrades. These considerations can also explain why the optimum RR size is larger for smaller arrays: the RRs of smaller arrays need to be large to overlap with those of their neighbors, and so the performance benefit of overlapping is realized at larger RR sizes. The relationship between RR overlap and information redundancy can also explain why increased array size does not give a constant performance improvement. If an array is already large, when a new sensor is added to it, it is unlikely to find an advantageous RR position, and it is therefore likely to contribute almost all redundant information.
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Considerations of RR overlap and information redundancy have direct
application to descriptions of distributed olfactory codes as well. Clearly,
an array of olfactory neurons will have the greatest coding capacity if each
neuron contributes unique information; this requires some overlap of response
profiles. IT analysis indicates that performance is enhanced when multiple
sensors/receptors are involved in the detection of any given stimulus, which
is what is generally observed in studies of biological olfaction: at each
level of an olfactory system, a substantial fraction of the neurons respond to
any one odorant (Adrian, 1950;
Gesteland et al.,
1965; Kauer, 1974;
Meredith and Moulton, 1978;
Stewart et al., 1979;
Hamilton and Kauer, 1985,
1989;
Harrison and Scott, 1986;
Wellis et al., 1989;
Wellis and Scott, 1990;
Imamura et al., 1992;
Mori et al., 1992;
Katoh et al., 1993;
Cinelli et al., 1995;
Johnson et al.,
1998). The IT analysis further indicates that when testing with
limited sets of stimuli (compared with the entire odorant universe), some
neurons expressing different ORs may appear to have similar response profiles
because their RRs overlap substantially. A small test stimulus set might not
include stimuli which would allow the distinct but overlapping RRs to be
discriminated, and the two receptors would appear to be functionally identical
(White et al.,
1999).
Sensor Heterogeneity
In investigations of responses from primary and secondary olfactory
neurons, the breadth of response profiles have been found to vary
(Duchamp et al.,
1974; Revial et al.,
1978; Imamura et al.,
1992; Mori et al.,
1992; Katoh et al.,
1993; Bozza and Kauer,
1998; Krautwurst et
al., 1998; Duchamp-Viret
et al., 1999; Malnic
et al., 1999), although the limited size of stimulus sets
makes it difficult to draw conclusions about the true RRs of these neurons.
Using the tools described here, we explored the effect that heterogeneity of
receptor response breadth may have on the ability of sensor arrays to encode
olfactory information. In this study, we created arrays of 24 and 36 sensors,
where each array contained a mix of sensors of two RR sizes: one class of
sensors which respond to many stimuli and another class of sensors which
respond to few. The different sizes were chosen such that a homogeneously
sized population of sensors of either size would yield approximately the same
performance (dashed vertical lines in
Figure 3A). For an array of 24
sensors, we chose RRs of size 15 and 50% because, from
Figure 3A, homogeneous arrays
of sensors of each size have approximately equal performance of 6.3 bits.
Homogeneous arrays of 36 sensors of either size 10 or 55% also have a MI of
6.3 bits. We constructed arrays with different numbers of small and large
sensors and observed the performance for various degrees of mixing. These
results are summarized in Figure
5. For arrays of 24 and 36 sensors, heterogeneous mixtures of both
large and small sensors outperform homogeneous arrays of either size. An array
of 12 large and 12 small sensors conveys 7.5 ± 0.2 bits of information
as compared with 6.3 bits for an array of 24 large sensors or 24 small
sensors. An array of 18 large and 18 small sensors conveys 7.9 ± 0.2
bits as compared with 6.3 bits for an array of 36 large or 36 small sensors.
These heterogeneous arrays also performed better than homogeneous arrays of
the optimum size: 7.1 ± 0.2 bits for 24 sensors (P < 0.001)
and 7.6 ± 0.2 bits for 36 sensors (P < 0.001) (dashed
horizontal lines in Figure
5).
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One possible explanation for this observation also relates to the consequences of RR overlap for array performance. As discussed above, arrays of small sensors are likely to have sub-optimal performance because they do not overlap sufficiently: too many stimuli do not activate any sensors. Larger sensors overlap too much: a significant proportion of the stimuli activate all (or nearly all) of the sensors. This is demonstrated by the results of Figure 4, which show that, on average, smaller sensors perform best in discriminating stimuli in the central region (where the sensor RRs overlap most), whereas large sensors tend to give the best performance in outer regions of the experimental volume (where they overlap least). In a heterogeneous array, the strengths of the small sensors compensate for the weaknesses of the large sensors, and vice versa. Small sensors are likely to contribute to making discriminations among centrally located stimuli with which large sensors might have difficulty. Reciprocally, the presence of large sensors means that discriminations can be made among stimuli in the periphery of the space which would be impossible with small sensors alone. This suggests the possibility that more broadly tuned receptors permit detection of stimuli in regions of the olfactory space where the inter-stimulus distance is large, while finely tuned receptors allow discrimination in regions of greater stimulus density. This could be tested by using different stimulus distributions.
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Discussion |
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TopAbstractIntroductionMethodsResultsDiscussionReferences |
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Recent studies of the response properties of neurons in which particular ORs have been identified (Zhao et al., 1998
; Krautwurst
et al., 1998; Malnic
et al., 1999; Murrell
and Hunter, 1999; Araneda
et al., 2000) and other studies in which the responses of
individual ORNs have been examined
(Duchamp et al.,
1974; Revial et al.,
1978,
1982;
Hirono et al., 1994;
Sato et al., 1994;
Bozza and Kauer, 1998;
Duchamp-Viret et al.,
1999) show that the receptor cells and the receptor proteins they
express generally respond to more than one compound. Ensembles of these
neurons encode the detailed information about the chemical environment on
which animals base their olfactory behaviors. In this paper, we have begun to
explore in a formal, systematic way how the discriminatory power of these
ensembles can emerge from the tuning properties of the individual
receptors. Our goal has been to create a framework in which to explore relationships between the response properties of individual olfactory neurons and coding properties of arrays. To do this, we created a system in which we could both define the response profiles of model olfactory sensors and evaluate statistically the performance of sensor arrays. Individual model sensors are defined as having continuous RRs in a multi-dimensional space. We have chosen such simple sensors because this assumption reduces large numbers of theoretically possible responses to a sparser subset. For example, an array of 32 binary sensors with positive/null responses has 232 possible states, and therefore could encode more than 4 x 109 different stimuli. However, this presumes that the response of each sensor is completely independent of the others; if the sensors are limited to having contiguous RRs in a space of dimensionality of <32, then the sensors cannot be independent, and many array states cannot be evoked by any stimulus. Specifically, if the RRs of a pair of sensors do not intersect, all of the states of the array in which both are activated cannot be elicited by any stimulus. Therefore, each pair of non-overlapping RRs reduces the number of available array states. The same argument applies to larger combinations of the sensors as well, which further reduces the number of realizable states. The ability of an array of receptors that is restricted in this way to discriminate among a defined stimulus set can be quantified using a MI measure. We have used this measure because it allows a comparison between the coding contributions of individual sensors and the overall performance of the array. Together, this combination of easily manipulated models and a statistical framework for evaluating overall performance comprise a powerful tool for exploring the properties of distributed codes.
This framework may be used to examine results describing the response
properties of biological olfactory receptor neurons
(Revial et al., 1978;
Krautwurst et al.,
1998; Zhao et al.,
1998; Duchamp-Viret et
al., 1999; Malnic et
al., 1999). In all of these studies, although relatively
small numbers of odorants were tested, receptor neurons were generally shown
to respond to more than one stimulus. This suggests that the neurons would, in
fact, respond to more odorants if additional ones could have been presented
(White et al., 1999).
Our results suggest that this property of broad tuning might be an
advantageous adaptation for a generalist chemosensory system: a system of
sensors which respond to a high proportion of possible stimuli (between 20 and
40%) provides substantially greater coding capacity than a system with more
selective sensors (between 1 and 10%). Another theoretical study of the
binding properties of arrays of biological receptors also predicts such
broadly tuned sensors (Lancet et
al., 1993). While such a generalist olfactory system might
not be able to capture all of the information about the chemical environment,
it can provide an efficient mechanism for covering broad regions of the
olfactory space. It seems that biological systems might use sensors with
relatively large RRs to generate the distributed code by which they describe
the environment (though sensors with small RRs may be involved for specialized
detection and discrimination).
In addition, considerations of sensor heterogeneity are relevant to the study of biological olfactory systems because they suggest that diversity of RR size may be advantageous. In experimental studies, differences in receptor breadth may be difficult to characterize completely because of the difficulty of obtaining response profiles to stimulus sets that comprise a significant fraction of the complete odorant universe for a given species. In spite of the difficulties of observing these differences, they may play a significant role in the distributed representation of olfactory information. Since it is unlikely in a system composed of hundreds of different receptor proteins that all the proteins would evolve with optimal tuning widths, it may be adaptive and biochemically efficient to generate a diversity of binding specificities. In addition, arrays of biological olfactory receptors might be intrinsically diverse. Olfactory receptors are in a constant state of turnover as a result of injury, apoptosis and renewal (Graziadei and Graziadei, 1979a,b); if the RR of a receptor neuron changes as it ages, this could have a profound impact on information coding in the olfactory system. It could even be that changes in response breadth over time that accompany the life cycle of ORNs provide advantages for the stimulus coding process.
Our study also suggests some bounds on the distribution of receptor tuning
breadths. Unfortunately, there are as yet only a few studies of the response
properties of ORNs expressing defined receptor proteins
(Malnic et al., 1999;
Araneda et al., 2000).
This is due to a variety of technical difficulties. First, the number of
possible stimuli which might be tested is large (some speculate on the order
of 104 for a macrosmatic animal) and needs to be defined for the
particular species being studied. Since there is no systematic description of
olfactory stimuli, meaningful sampling of this large set is not yet possible.
Secondly, because the time course of an olfactory response is generally longer
than for the other senses, relatively few stimuli with few repetitions can be
tested in each preparation. This combination makes it difficult to assess RR
size for olfactory neurons, since differences could arise as an artifact of
the selection of a particular stimulus set or measurement technique. However,
our results suggest that when sufficient data sets from populations of
odorants and sensors become available, they will show significant differences
in tuning breadths across populations of receptors—some olfactory
neurons will be found to respond to small sets of odorants and others to
larger sets (Kauer, 1980;
Hildebrand and Shepherd,
1997). In fact, a recent study of response profiles of olfactory
receptor neurons by Duchamp-Viret and colleagues
(Duchamp-Viret et al.,
2000) showed some evidence for bimodality in sensor breadth for
the rat, but not for the frog. A distribution of tuning widths follows
naturally from the large repertoire of receptor genes
(Sharon et al., 1998;
Fuchs et al., 2001):
the inevitable variation in receptor response properties results in a
population of receptors which have many tuning widths. Thus, biological
olfactory systems might exploit the advantages of distribution of receptor
breadths as a side effect of the expansion of the olfactory genome.
Since so little is known about general properties of olfactory stimuli, we made relatively arbitrary decisions in designing our model systems. Though our experiments illustrate some critical aspects of the relationship between sensors and array performance, the results are sensitive to the chosen distributions of stimuli. In particular, the fact that stimuli (and sensors) were constrained to a cubic experimental volume created an unintended difference between central and peripheral stimuli (Figure 4). In addition, the uniform distribution of stimuli and sensors resulted in a `ceiling effect' (Figure 3C) which affected high levels of performance. The results reiterate that knowledge of stimulus distributions and their relationships to sensor properties are critically important for understanding olfactory function. We intend to explore these relationships further by examining other stimulus distributions and the characteristics of sensor arrays that best discriminate them.
These initial studies suggest several additional avenues of further
research. First, the responses of the sensors modeled here are simpler than
those in biological olfactory systems in a number of ways. They give simple
positive or null responses, while olfactory neurons in general are clearly
capable of more complex output, including graded and time-varying events
(Kauer, 1974;
Kauer and Shepherd, 1977;
Harrison and Scott, 1986;
Meredith, 1986;
Hamilton and Kauer, 1989;
Wellis et al., 1989;
Laurent et al., 1996,
2001). Our model neurons have
a maximum information carrying capacity of 1 bit per stimulus application,
which means that the maximum information carrying capacity of an
N-sensor array would be N bits if there were no information
redundancy among the sensors. Sensors with more complex responses will lead to
systems with larger coding capacities. We are particularly interested in
modeling arrays of sensors with time-varying responses to explore the
evolution of information across an olfactory system over time
(Laurent et al.,
2001).
Secondly, olfactory receptors are subject to noise that arises at several
different levels, from the interaction between the odorant and receptors, to
the generation of the primary transduction currents and to spike generation.
For M/T cells, in addition to noise from the receptor neurons, there is noise
from the propagation of the receptor neuron spikes from the epithelium to the
bulb, noise of synaptic transmission from the receptor axon terminal, noise
from the generation of M/T cell spikes and noise from the global bulbar
neuronal network. All of these sources can reduce the capacity of the system
for carrying information about the stimuli. In addition, noise can have
effects on information capacity across an array—noise in common between
sensors (correlated or common-mode noise) has a different impact on the total
capacity of the array than non-correlated noise independent to each of the
sensors, since the effects of uncorrelated noise can be reduced by convergence
of multiple sensors of the same response type
(van Drongelen et al.,
1978). One might hypothesize that the large size of biological
arrays compensates for these sources of noise; models which explore this
balance can explicitly test that idea. The effect of other factors in
olfactory coding can also be tested using this technique: graded olfactory
neuron responses, multiple binding sites, temporally complex patterns of
activity and the effects of concentration all lend themselves to
representation in model systems.
In spite of the simplicity of the models used here, this approach of exploring the properties of networks from a formal statistical perspective is helpful for defining the limits of the initial representation of chemical stimuli in olfactory systems. Many data suggest that the representations of perceived odors tend to be distributed widely across the components of the olfactory pathway. Our approach provides a framework for analytically characterizing the growing understanding of the functional properties of individual neurons and for considering those properties in the context of system-wide coding. We anticipate using these techniques to approach a wide variety of questions, from the evolution of biological olfactory systems to the rational design of sensor arrays for artificial olfactory systems. These paths might lead us to a definitive description of how the pieces of the olfactory response can be put together to form the whole of a description of the chemical world.
|
Acknowledgments |
|---|
Supported by grants and contracts from the NIH-NIDCD (J.S.K.), Office of Naval Research (J.S.K.) and Defense Advanced Research Projects Agency (J.S.K. and J.W.). We thank Drs T.C. Bozza and K.M. Dorries for critically reading the manuscript.
|
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Accepted December 14, 2001
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 T. A. Cleland and C. Linster Computation in the Olfactory System Chem Senses, November 1, 2005; 30(9): 801 - 813. [Abstract] [Full Text] [PDF]
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