Chem. Senses 24: 351-372,
1999
© Oxford University Press 1999
A Neural Network Model of General Olfactory Coding in the Insect Antennal Lobe
Division of Insect Biology, ESPM, University of California at Berkeley, CA 94720-3112, USA
Correspondence to be sent to: Wayne M. Getz, Division of Insect Biology, Department ESPM, 201 Wellman Hall, University of California at Berkeley, CA 94720-3112, USA. e-mail:getz{at}nature.berkeley.edu
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Abstract |
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A central problem in olfaction is understanding how the quality of olfactory stimuli is encoded in the insect antennal lobe (or in the analogously structured vertebrate olfactory bulb) for perceptual processing in the mushroom bodies of the insect protocerebrum (or in the vertebrate olfactory cortex). In the study reported here, a relatively simple neural network model, inspired by our current knowledge of the insect antennal lobes, is used to investigate how each of several features and elements of the network, such as synapse strengths, feedback circuits and the steepness of neural activation functions, influences the formation of an olfactory code in neurons that project from the antennal lobes to the mushroom bodies (or from mitral cells to olfactory cortex). An optimal code in these projection neurons (PNs) should minimize potential errors by the mushroom bodies in misidentifying the quality of an odor across a range of concentrations while maximizing the ability of the mushroom bodies to resolve odors of different quality. Simulation studies demonstrate that the network is able to produce codes independent or virtually independent of concentration over a given range. The extent of this range is moderately dependent on a parameter that characterizes how long it takes for the voltage in an activated neuron to decay back to its resting potential, strongly dependent on the strength of excitatory feedback by the PNs onto antennal lobe intrinsic neurons (INs), and overwhelmingly dependent on the slope of the activation function that transforms the voltage of depolarized neurons into the rate at which spikes are produced. Although the code in the PNs is degraded by large variations in the concentration of odor stimuli, good performance levels are maintained when the complexity of stimuli, as measured by the number of component odorants, is doubled. When excitatory feedback from the PNs to the INs is strong, the activity in the PNs undergoes transitions from initial states to stimulus-specific equilibrium states that are maintained once the stimulus is removed. When this PNIN feedback is weak the PNs are more likely to relax back to a stimulus-independent equilibrium state, in which case the code is not maintained beyond the application of the stimulus. Thus, for the architecture simulated here, strong feedback from the PNs onto the INs, together with step-like neuronal activation functions, could well be important in producing easily discriminable odor quality codes that are invariant over several orders of magnitude in stimulus concentration.
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Introduction |
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TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
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Visual and auditory signals, as well as mechanosensory stimuli, can be precisely characterized by physical measurements of amplitude, direction and, in case of the first two, frequency (electromagnetic or air pressure waves). Chemosensory stimuli, on the other hand, are much more difficult to characterize using a few well-defined measures. A good measure of chemical signal amplitude does exist (i.e. chemical concentration) but, unlike vision (e.g. color) or hearing (e.g. pitch), not of signal quality. Further, chemosensory stimuli cannot easily code spatial information over great distances, primarily because the signals are molecules transported by turbulent air or water plumes that break the signal up into discrete packages with increasing distance from the source (Murlis, 1986
; Moore and Atema,
1991; Dittmer et al., 1995).
Thus olfactory signals are much noisier than other sensory modalities, which makes the olfactory
signal processing problem much different from that, say, of vision (Osorio et al., 1994) or audition.
Olfactory processing is primarily a signal classification problem that is complicated by
the fact that olfactory receptor neurons (ORNs) exhibit (i) phaso-tonic response profiles over
time; (ii) highly nonlinear stimulus–response relationships with respect to concentration
(Akers and Getz, 1992, 1993; Getz
and Akers, 1993, 1994; Lemon and Getz, 1997); and (iii) synergistic and inhibitory responses to blends compared with the response
to the individual components of blends (Atema et al., 1989;
Derby et al., 1989;
Fine-Levy and Derby, 1992; Getz and Akers 1995, 1997; Kang and Caprio, 1997). Olfactory
processing can also be a spatial information processing problem including transforming temporal
information related to the change in the frequency with which an organism is stimulated by
discrete packets of
molecules transported in turbulent flows into information on the direction and closeness of an
odor source (Moore and Atema, 1988; Murlis et al.,
1992).
The noise in odor signals and nonlinearities in the responses of sensory neurons at the
periphery challenges our ability to unravel olfactory coding in the animal brain. This challenge,
however, cannot be met by prevailing olfactory coding paradigms which have their origins in
pheromonal processing in insects (reviewed in Kaissling, 1987; Masson and Mustaparta, 1990).
In such pheromonal processing, each compartment of the multicompartment macroglomerular
complex (MGC) that is situated in the antennal lobes receives input from only one type of
extremely narrowly tuned receptor neuron (Hanson et al., 1992, 1994). The MGC
represents an architecture of dedicated pathways, where each pathway is specific for a particular
odorant component of the pheromonal blend. This architecture has led to the concept of a
`labeled-line' code in the insect brain—a code which has
proved inadequate in describing general olfactory processing in cockroaches (Sass,
1978; Selzer, 1984) and honey bees (Vareschi,
1971; Akers and Getz, 1992, 1993).
Unlike pheromonal coding,
general odor coding involves peripheral receptor neurons that are tuned to an array of odorants,
usually belonging to a related family of compounds (e.g. alcohols, esters, ketones: see Selzer, 1984; Fujimura et al., 1991; Akers and Getz, 1992; Getz and Akers, 1993).
The broader
response spectrum of general receptor neurons requires that we generalize the notion of dedicated
labeled lines: too few labeled lines exist to account for all the possible odorants that insects such
as cockroaches and honey bees are able to perceive. The obvious generalization is to consider the
code in terms of the across fiber patterns in a cable of neurons relaying information from one part
of the insect brain to another (Ache, 1991; Boeckh et al., 1990). Unfortunately, across
fiber patterns all too often are thought of in terms of average firing rates in each neuron among a
group of neurons thought to produce or transmit an olfactory code.
More recently, Laurent and co-workers (Laurent, 1996; Laurent et al., 1996)
proposed the concept of a combinatorial code in terms of the average firing rate over several
subintervals comprising the total interval of time over which coding occurs. This paradigm arose
because Laurent and colleagues (Laurent and Davidowitz, 1994; Laurent and Narghi, 1994;
Laurent, 1996; Laurent et al., 1996;
MacLeod and Laurent, 1996; Wehr and Laurent,
1996) found activity in projection neurons from the antennal lobe of insects
correlated with 20
Hz local field potential oscillations in the insect mushroom bodies where sensory modalities are
mixed and long-term memory is known to occur (Menzel et al., 1991). This type of
combinatorial paradigm, however, is essentially an extension of the across fiber pattern
paradigm: instead of having n numbers (or states) associated with a code across n neurons, one has n x m numbers if the coding interval of time is divided
into m subintervals.
A deeper understanding of the code can only be obtained if we move away from static or combinatorial paradigms to temporally varying (i.e. dynamical systems) paradigms. Temporally varying concepts of coding take cognizance of the dependence of the phasic activity of the coding neurons on initial conditions that, with time, become less important as the stimulus-specific neuronal states are approached. A combinatorial paradigm raises the critical question of how long an odor coding interval is (e.g. if the coding interval were 500 ms, then in the context of 20 Hz oscillations this would imply m = 10). Also, does the early part of the response of the coding neurons (i.e. the first or first few subintervals) reflect initial brain states rather than stimulus-specific brain states?
The only way to explore the properties of a temporally varying coding paradigm is through the analysis of appropriate dynamic neural network models. Here, we carry out such an analysis using a model that reflects the underlying architecture of the olfactory components of the insect antennal lobe. The function of each of these olfactory components, however, cannot be teased apart unless we start with a reasonably simple model and explore changes in the behavior of the model as complexity is added. Once this is accomplished, the model can be made more realistic using known anatomical and physiological features of the insect olfactory system.
More attention has been devoted to modeling vertebrate or generic olfactory systems
than insect olfactory systems (e.g. see Freeman and Skarda, 1985; Baird, 1986; Getz and
Chapman, 1987; Skarda and Freeman, 1987; Barnard, 1989; Li and Hopfield, 1989; Granger et al., 1990; Bower, 1991; Getz, 1991, 1994; Hopfield, 1991; Wang et al., 1991;
Bhalla and Bower, 1992; Meredith, 1992; Schild and Reidel, 1992; White et al., 1992;
Hendin et al., 1994; Aradi and Erdi, 1996; Linster and Hasselmo, 1997). Previous
analyses of mammalian (Freeman and Skarda, 1985; Baird,
1986; Skarda and Freeman, 1987;
Barnard, 1989; Bower, 1991), amphibian (White et al., 1992), and non-insect
invertebrate systems, such as the mollusc, Limax maximus (Delaney et
al., 1994; Kleinfield et al., 1994), are very
different from the analysis presented here
because they focus on relatively low frequency oscillations that are observed across sheets of
olfactory cortex or analogous invertebrate neuropil (see also Laurent and Davidowitz,
1994;
Laurent and Narghi, 1994; Laurent, 1996; Laurent et al., 1996; MacLeod and Laurent,
1996; Wehr and Laurent, 1996).
Olfactory processing in insects has been more widely studied in the context of
pheromone detection than general odor detection. General odor detection studies are gaining
ground (Fujimura et al., 1991; Akers and Getz, 1992, 1993; Getz and Akers, 1993, 1994, 1995, 1997; Laurent and Davidowitz, 1994; Laurent and Narghi, 1994;
Rospars and Fort,
1994; Laurent, 1996; Laurent et al.,
1996; MacLeod and Laurent, 1996; Wehr and
Laurent, 1996; Joerges et al., 1997; Lemon and Getz, 1997, 1999). The insect antennal
lobe, the part of the insect brain receiving input from olfactory receptor afferents, has a number
of underlying architectural features in common with the vertebrate olfactory bulb (Boeckh et al., 1990). Thus, some understanding of the function of the
architectural features of the
insect antennal lobe should provide insight into olfactory processing in all animals.
The architecture of our model is based on physiological and anatomical data that have
been obtained primarily from the American cockroach, worker honey bees, several species of
moths (e.g. Rospars, 1983, 1988; Boeckh and Ernst, 1987; Christensen and Hildebrand, 1987;
Flanagan and Mercer, 1989a, b; Gascuel and Masson, 1991; Malun, 1991a, b; Hansson et al., 1994) and a few other insects (as
reviewed in Smith and Getz, 1994). To date, modeling
studies of the insect antennal lobe have focused on reproducing the behavior of specific antennal
lobe neurons (Linster et al., 1993, 1994;
Linster and Masson, 1996), behavior (Masson
and Linster, 1996; Linster and Smith, 1997) or
self-organization properties based on a Hopfield
recurrent net architecture (Malaka et al., 1996). In some of these
models, important
architectural details pertaining to the insect antennal lobe have not been considered, although
others are more detailed (e.g. Linster and Smith, 1997).
The olfactory information processing questions we address here relate to the role
played by the architectural features and response properties of the intrinsic neurons (INs) and
projection neurons (PNs) of the insect antennal lobe in solving the `general olfactory
coding problem in insects' defined as follows (cf. Osorio et al., 1994):
In response to odor stimuli with naturally occurring levels of spatial and temporal variation, the general olfactory coding problem in insects is for the peripheral and antennal lobe neurons to produce responses in a subset of the antennal lobe projection neurons that, as an assembly, exhibit patterns of firing sufficiently similar across a range of concentrations of stimuli of the same quality and sufficiently different among stimuli of different quality to permit an appropriate classifier (either a higher level network such as the mushroom bodies of the protocerebrum or a measure in a linear space, as employed in this proposal) to identify odors of the same quality or to discriminate odors of different quality.
Several issues immediately arise from this definition relating to the temporal
properties of the code. The already mentioned relationship between period of oscillations in field
potentials (Laurent, 1996; Laurent et al., 1996; MacLeod and Laurent, 1996) and the
overall coding interval is one issue. Another is the influence the phaso-tonic response properties
of receptor neurons (Akers and Getz, 1992; Lemon and Getz,
1997) may have on the temporal
character of odor codes. A third issue is the role that nonlinear response characteristics of
receptor neurons [especially inhibitory responses to mixtures compared with pure
odorants—see Getz and Akers (1995)] may play in producing
olfactory codes. Beyond
these are the broader issues of how olfactory systems are able to separate distinct odor signals as
they mingle together in the environment or identify the quality of a particular odor from a
background of odors. These latter issues have been considered in several modeling studies by
Hopfield and collaborators (Li and Hopfield, 1989; Li, 1990; Hopfield, 1991; Hendin et al., 1994) and also in an earlier study by one of us (Getz, 1991). All of
these studies, however,
employ recurrent autoassociative networks that do not take cognizance of the glomerular neuropil
structures characteristic of the insect antennal lobe or vertebrate olfactory bulb.
The issues mentioned above raise questions that probe deeply into the temporal
properties of olfactory coding and we are not yet in a position to address all of them. In this
study, we shed some light on some of the basic questions raised for systems with constant input
stimuli and idealized peripheral ORNs. Studies involving inputs that have temporal, as well as
spatial, components can follow once we have obtained some understanding of the computational
properties of olfactory systems receiving spatially homogeneous on–off input of the type
considered here. We can only account for spatial heterogeniety if we model the input to each
glomerulus as coming from several different ORNs each located at a different site (sensillum) on
the antennae (cf. Malaka et al., 1996). In this case, the activation
of each ORN depends
on the structure of the odor plume with respect to the location of the ORNs on the antennae. Our
assumption of spatial homogeneity is functionally equivalent to assuming a single input into each
glomerulus with a firing rate that increases with concentration.
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Network approach |
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The model we present here to investigate olfactory processing in the insect brain can be viewed as a generalization of the subsystem architecture studied by Av-Ron and colleagues (Av-Ron and Rospars, 1995
; Av-Ron and Vibert, 1996). They consider several special cases of
two ORNs synapsing with two local INs which in turn synapse with one or two PNs, and they
model the activity of these neurons using four differential equations to describe the currents
producing the associated activation potentials (Av-Ron et al., 1991; Wang and Rinzel,
1992). Av-Ron and colleagues, however, do not include any feedback from the PNs
onto the INs.
As we will see, our results suggest that this PN–IN feedback may be an important
component of the olfactory coding processes in the insect antennal lobe.
In developing our model, we follow the dictum that `a good model should not
copy reality, it should help to explain it' (Segev, 1992). In
particular, we seek insights
into the type of code produced by a set of input ORNs synapsing on an underlying net of INs
from which a set of uniglomerular PNs project. To keep the model simple, however, we only use
one differential equation to characterize the activation potential in each neuron, so that the output
from our net is in terms of the probability that each of the PNs is spiking at a particular point in
time, rather than the details of the actual spike trains themselves. This approach leads to
formulating a set of n ordinary differential equations for a network containing n
neurons (Appendix A). The equations include parameters that
characterize the value of the
response threshold, the steepness of the neuronal activation function and the background firing
rate for each neuron, as well as the value of the connectance parameters (synapse strengths both
positive for excitatory connections and negative for inhibitory connections) among neurons.
Antennal lobe architecture
The olfactory neuropil in the antennal lobe of the insect deutocerebrum can be thought
of as an olfactory processing `black box' that receives input from peripheral ORNs
located in the antenna and produces output in relay neurons (i.e. PNs) that project from the
antennal lobe to higher processing centers in the protocerebrum (reviewed by Rospars,
1988;
Masson and Mustaparta, 1990; Hildebrand and Shepherd,
1997) (Figure 1).
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0 then we have IN feedback onto itself, which is not considered here); IN
inhibitory feedforward onto projection neurons (PNs) (Considerable progress has been made in characterizing the anatomy and physiology of the insect antennal lobe (Arnold et al., 1985
; Boeckh
and Ernst, 1987; Christensen and
Hildebrand, 1987; Malun, 1991a, b; Rospars, 1988; Flanagan and Mercer, 1989b; Gascuel and
Masson, 1991; Christensen et al., 1993; Hansson et al., 1994), although certain
critical neurological questions remain unanswered (e.g. whether certain neurons intrinsic to the
antennal lobe are uni- or bi-directional). Also, the general function of the antennal lobe is not
well
understood, although some progress has been made in terms of the MGC in the antennal lobes of
certain species of moth (Hansson et al., 1992, 1994). The inputs to the antennal lobes
are olfactory ORN axons that arborize in the antennal lobe glomeruli, which are grape-like
neuropil structures enclosed by a sheath of glial cell process (Homberg et al.,
1989;
Boeckh et al., 1990; Masson and Mustaparta, 1990; Gascuel and Masson, 1991;
Christensen et al., 1993). These arborizations synapse almost
exclusively with INs that
interconnect several glomeruli, but remain intrinsic to the antennal lobe. All INs identified thus
far appear to be inhibitory [they are GABA-ergic—but see Michelson and
Wong (1991),
for examples of excitatory GABA-ergic neurons]. The INs synapse with the dendrites of the PNs
which, in turn, appear to synapse back onto the INs before they exit a glomerulus (Malun,
1991a, b), thereby forming possible
excitatory–inhibitory feedback loops that parallel
granule–mitral cell feedback loops in the vertebrate olfactory bulb (e.g. Li and
Hopfield,
1989; Shepherd, 1994, Figure 7.17; Yokoi et
al., 1995; Erdi et al., 1997).
Besides providing the backbone to the glomerular network, the network of INs
informationally link all the glomeruli, although each individual IN may only synapse with
neurons in a small proportion of the total number of glomeruli. Even in species that have large
MGCs, such as the male sphinx and turnip moths, INs exist that arborize throughout the entire
antennal lobe, including the MGC (Christensen et al., 1993; Hansson et al.,
1994). At this time, it is not known if some or all INs have input regions in one
glomerulus and
output regions in several others, or as mentioned are bi-directional in terms of current flow.
Some of these INs appear to be symmetric in the sense of arborizing with the same intensity in all
the glomeruli they invade, while others have the asymmetry of arborizing intensely in only one of
the glomeruli and less intensely in others (Flanagan and Mercer, 1989a; Fonta et al.,
1991, 1993; Christensen et al., 1993). This apparent asymmetry, however, could be an
artifact of incomplete filling of neurons with dyes when using a pipette
technique.
In the honey bee and cockroaches, the ordinary glomeruli give rise to one
uniglomerular PN, but are also invaded by multiglomerular PNs. In all insects, however, the
MGC gives rise to several uniglomerular PNs. In the cockroach, only the male has an MGC
(Boeckh et al., 1990), which gives rise to ~15 PNs (Boeckh and Ernst, 1987). In the
honey bee, drones have an MGC consisting of four large glomeruli, queens have one enlarged
glomerulus and workers have only ordinary glomeruli [~166 of them—see review by
Masson and Mustaparta (1990)]. MGCs are implicated in pheromonal
processing (reviewed by
Rospars, 1988; Masson and Mustaparta, 1990; Hildebrand and Shepherd, 1997), while the
regular glomeruli are associated with general odor processing. Thus the firing patterns across the
uniglomerular PNs from the ordinary glomeruli provides a convenient representation of the
output of the antennal computation involving floral and food odor stimuli (Getz and
Chapman,
1989; Smith and Getz, 1994). Odor responsive
multiglomerular PNs also exist (Fonta et al., 1993), but they
appear to invade only the `core', not the
`periphery', of the glomeruli where they can feed information back to the INs
(Malun, 1991a, b). Thus, to begin, it makes sense
to concentrate on the set of uniglomerular PNs
as representing one level of ouput from the olfactory lobe. The additional complexity of
multiglomerular PNs can be considered once a deeper understanding of the uniglomerular
subsystem has been obtained.
Architecture of the model
The basic architectural features of the insect olfactory system outlined above are
schematized in Figure 1
. In this study, to keep the model simple, we
consider only one class of
INs, although more than one class appears to exist (e.g. seeSun et al., 1993), as
mentioned above. In particular, we assume that the jth IN receives excitatory input from
all the antennal (peripheral) ORNs that project into the jth glomerulus and also receives
feedback from the jth PN (the PN that ascends from the jth glomerulus). We do
not know whether this PN–IN feedback is excitatory, but assume it to be so in this study.
The model, however, can be used to explore the behavior of the network if we assume this
feedback to be inhibitory. We also assume that the jth IN inhibits all the other INs but
only inhibits the jth PN. Once the properties of this network are understood, the
potential function of other classes of INs can be investigated.
In summary, our architecture centers around the assumption that each glomerulus is
defined by a group of ORNs that excite a particular IN which we will henceforth refer to as
originating from that glomerulus. This IN then forms an inhibitory–excitatory feedback
loop with its associated PN. The INs each inhibit one another, although not themselves. Each PN
fires because self-excitation (or, equivalently, excitation from sources not explicitly identified in
the model) is disinhibited by inhibition of the IN synapsing with the PN in question [as suggested
from the empirical studies of Christensen et al. (1993); see also
Av-Ron and Rospars (1995)].
Time delays are included in our model so that the activity of an IN originating from a
particular glomerulus affects INs originating from neighboring glomeruli sooner than INs
originating from non-neighboring glomeruli. Topologically we assume that glomeruli are
arranged in a circle (i.e. each has two nearest neighbors, which for glomerulus 1 are glomerulus n and glomerulus 2) and that they are not all necessarily connected to one another. This
assumption of a circular arrangement appears to be supported by empirical observations
(Rospars, 1983; Arnold et al., 1985;
Rospar and Hildebrand, 1992). Finally, we assume
that ORNs can be categorized into one of n classes, and that all ORNs of the jth
class arborize in the jth glomerulus and synapse with the jth IN. In reality,
the categorization of ORNs is not as clean as originally thought (see Akers and Getz,
1992), but
the assumption that ORNs synapsing within the same glomerulus should at least be similar
appears to be the most reasonable assumption in the absence of evidence to the contrary (Rospars
and Fort, 1994).
Antennal lobe equations
The antennal lobe equations are obtained by restricting the synapse weights in the
general set of network equations (equation 3 in Appendix A) to be zero if
the neurons in question
do not connect with one another, positive if the connections are excitatory and negative if the
connections are inhibitory. We also treat the ORN spiking rates as given constant inputs. This
allows us to study the dynamic properties of the antennal lobe network without the confounding
influence of phaso-tonic (dynamic) ORN inputs. Once the computational properties of the
antennal lobe network are better understood, then the effects of more realistic ORN input can be
studied using dynamics models of ORN responses (e.g. see Av-Ron and Rospars,
1995; Av-Ron
and Vibert, 1996) to constant and temporally varying stimuli using available data
(Lemon and
Getz, 1997) as a basis for developing the ORN component of the model.
The model includes background firing rates for each neuron. These rates could be due
to self-excitation or unaccounted input from neurons not included in the architecture presented in
Figure 1 [e.g. certain ventral unpaired median neurons such as VUMmx1
have been implicated in
sending information from the subesophogeal ganglion to various parts of the insect brain
including the antennal lobes—e.g. see Hammer and Menzel (1995)], or related to
threshold effects (as discussed in Appendix A).
Receptor input
Olfactory ORNs in insects range from those specialized to respond to single critical
compounds (pheromone receptor neurons) to those responding to several classes of compounds
(for reviews see Masson and Mustaparta, 1990; Smith and
Getz, 1994; Hildebrand and Shepherd,
1997). Most, however, respond to large suites of compounds (Sass, 1978; Selzer, 1984; Fujimura et al., 1991; Akers and Getz, 1992, 1993; Getz and Akers, 1993, 1997) and exhibit
nonlinear responses with respect to both concentration gradients and pure compounds versus
mixtures of compounds (Getz and Akers, 1995), as well as phaso-tonic
profiles over stimulus
intervals of varying lengths (Lemon and Getz, 1997). Recently, Malaka et al. (1995) constructed response surfaces for pure compounds
and mixtures of these compounds, using
honey bee olfactory receptor data obtained by Akers and Getz (1993),
and used these surfaces as
input to neural networks performing olfactory computations (Malaka et al.,
1996).
The tuning and temporal properties of ORNs have been studied, even modeled (Linster
and Dreyfus, 1996; Linster and Masson, 1996), in some
detail. In our analysis we reiterate that
we do not include a dynamic model of the ORNs to capture their phaso-tonic response to
stimulation (Akers and Getz, 1992; Lemon and Getz, 1997), nor do we include details of the
nonlinearities [inhibitions and synergisms (see Getz and Akers, 1995)]
associated with the
response of olfactory ORNs to mixtures compared with their response to pure odorants. Rather,
because our focus is on how certain antennal lobe architectural features may determine antennal
lobe olfactory processing qualities, we model the response of the ORNs to constant olfactory
stimuli of fixed duration in terms of constant spiking rates.
Network output
The output from the network is the responses of the PNs that will vary depending on
the particular input stimulus, as well as the values of the network parameters. These values
include the dimension of the model itself, activation, response decay, transmission delay and
synapse weighting parameters, as well background firing rates. As already mentioned, we will
not consider stimuli with spatial heterogeneity, although this can be incorporated into the model
as has been done in the context of simpler network representations of antennal lobe computations
(Malaka et al., 1996). Even though our stimuli have no temporal
component other than
being switched off at the end of the stimulus interval, however, we still need to deal with
temporal structure in the output.
If a uniform stimulus is applied across the antenna for, say, 200 ms, then in the honey
bee and cockroach we know that the general olfactory receptor neurons reach maximum firing
rate within 50 ms and begin to adapt after ~150 ms (Akers and Getz, 1992; Lemon and Getz,
1997). The receptor neurons rapidly return to background firing rates after the
stimulus is
removed. Thus a 100 ms square pulse represents a good approximation to the critical processing
period for odors of moderate concentration (Getz and Akers, 1997). The
PNs, however, continue
to respond for several hundred milliseconds after the stimulus has been removed (Laurent et al., 1996; Lemon and Getz, 1999). Thus,
we will consider the response of the PNs for the
duration of a 100 ms input pulse in the ORNs and for 300 ms beyond the offset of this input
pulse.
Simple ways to compare two 400 ms output response profiles is to split them into bins
(say 50 ms intervals), calculate average firing rates over each of these eight subintervals in
question and treat the eight resulting numbers as an eight-dimensional vector (Figure
2). The
similarity of the quality of the odors coded by two output patterns can then be measured using the
inner product between the vectors (of average firing rates over eight 50 ms bins) associated with
the patterns in question. By interpreting the direction of the vectors in our eight-dimensional
space as coding for odor quality (Figure 2), and the magnitude of the
vectors as coding for
concentration (see Getz and Chapman, 1987), we can train the network
(i.e. progressively update
the values of the network parameters using a genetic algorithm—see Appendix
B) to
optimize the network's ability to recognize odor quality independent of concentration and
discriminate among odors of different quality (see Appendix B for more
details).
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In more sophisticated approaches, we can feed the PN output into a `classifier network' of some kind (i.e. the output becomes input into another network). From a biological perspective, this classifier network represents a higher order memory center, such as the mushroom bodies of the insect protocerebrum (reviewed by Menzel et al., 1991
;
also see Laurent et al., 1996). In this study, however, we define
two vector-based
measures of performance with respect to a given set of stimuli representing several different
odors each of which can be presented at one of three different concentrations (see Appendix B for details). The first performance index, Q (defined by equation 7 in Appendix B), is a measure of the average distance between stimuli of the same odor quality, and can thus be regarded as a quality voracity performance index. From the definition of Q it follows that the voracity of coding for quality increases or improves as the measure Q decreases in value. The second performance index, D (defined by equation 8 in Appendix B), is a measure of the average distance between stimuli of different odor qualities (Figure 2). It can thus be regarded as a discrimination performance index. From the definition of D it follows that the ability of the network to discriminate among odors improves as the measure D increases in value.
From these two measures a third measureR =D/Q can be defined (see equation 9 in Appendix B) that permits the performance of the network to be simultaneously measured with respect to both veracity and discriminability of quality coding. From the definition of R it follows that the overall performance of the network generally improves as the value of the measureR increases. Note, from the definition of these measures in Appendix B, the network cannot begin to distinguish between odors of different qualities at the same concentration and odors of the same quality at different concentrations unless R > 1, and cannot perform very well unless R >> 1.
Three and six odorant simulations
For all the simulations reported in this paper, the dimension of the net (i.e. the number
oglomeruli and output neurons) is n = 6 and the number of types of receptor neurons is
also 6 (cf. Figure 1), with response values defined by equations (6) (Appendix B). In our first set
of simulations, we confined our attention to three-dimensional stimuli of the form S = (C1,C2,C3), where Ci is a logarithmically transformed value of the actual concentration of
odorant i. Because empirical data indicate that the average rate of response of many
ORNs to odorants is proportional to the logarithm of concentration between threshold and
saturation levels (e.g. see Fujimura et al., 1991), the odors Ci can be scaled in future studies using real data. To keep things simple,
we assume that the
ORNs switch on and off instantaneously and respond tonically to constant stimuli. Further, in our
first set of simulations we consider two classes of ORNs: `linear specialist', which
respond to only one odor (namely, the ith ORN of this type responds to odorant i at a firing rate Ci), and `linear
differencers', which respond only to the presence of two odorants (namely, the ijth ORN of this type responds to odorants i and j at a firing rate |Ci – Cj|—see Appendix C). Thus the
second class of ORNs exhibit inhibition to mixtures containing the two odorants in question, a
phenomenon that is widespread in insects (Getz and Akers, 1995) and
also crustaceans (Derby et al., 1989).
The analyses of the performance of the network involved training the network using a
genetic algorithm (Appendix B) to classify and discriminate among
stimuli by setting the
concentrations of the odorants Ci, i = 1, 2, 3, in the
stimulus S = (C1,C2,C3)
to be Ci = C0 – 
C, C0 or C0 + C. Thus the
stimuli are all essentially characterized in terms of log concentration rate parameters C0 and C that translate either directly (linear specialist ORNs) or
indirectly (linear differencer ORNs) into ORN response rates (see equations (11a, 11b, 11c) in Appendix C
for more details). Thus the stimuli (C0 – C, C0 – C,0) and (C0 + C, C0 + C,0) represent the same odor (an
equicomponent mixture of odorants 1 and 2) at a high and low concentration, while the stimuli (C0 – C, C0 +C,0) and (C0 + C, C0 –
C,0) represent blends of the same two odorants but in different proportions and
therefore of different quality.
A more challenging problem for the network than classifying and discrimination
among stimuli consisting of three components is to one involving stimuli consisting of six
components [i.e. stimuli of the form S = (C1,C2, C3,C4,C5,C6)—see Appendix D]. As in the three odorant case, we
defined pure odorant
stimuli at three different concentrations in terms of response rate parameters C0 and DC and blends of these stimuli using these same two concentration
parameters (Appendix D). For comparative purposes, we retained six
classes of ORNs but went
beyond the elementary case of defining six specialist neurons (one for each odorant) by rather
defining three `binary specialists' (each responding now to two rather than one
odorant) and three `binary differencers' (each responding to the difference between
two pairs of binary odors). In this case, we no longer have receptor neurons that specialize on any
one single odorant (Appendix D).
Parameters and notational issues
The symbols and meaning of the various parameters used in the model and the values
used in the analysis are listed in Table 1. We used a genetic algorithm to
find the three sets of
IN–IN synapse parameters that optimize (at least in some local sense) the quality
performance index, Q, the discrimination performance index, D,
or the ratio index R = D/Q when the remaining
parameters in the network have the values indicated in Table 1. To avoid
confusion in discussing
our results, we use the following notation to describe the solutions to these three separate
optimization problems. First, for the given set of stimuli, we use Q(P), D(P) and
R = R(P) to denote the values of the measures
Q, D and R, (equations 7, 8, 9 respectively in
Appendix B) with respect to a given set P of parameter values.
Second, we
include the time variable t explicitly when we want to emphasize that the solution is the
best one found by our genetic (i.e. evolutionary) algorithm up to generation time t: for
example, R (PR(t)) is the best
solution obtained to the problem of optimizating the ratio measure after the genetic algorithm has
run for t generations, while R (PD(t)) is
the actual value of the performance measure R for the parameter set
that is the best solution to this problem of optimizing the discrimination measure D after
t generations. Third, we use the notation PQ*, PD* and PR* to
denote the candidate optimal parameter sets respectively as solutions to optimizing the measures
Q, D and R. Note our solutions are only
candidates to solving the optimization problem because they may correspond to local rather than
global optima. Also, they are only numerical approximations at that. Fourth, we use the notation
Q*, D* and R* to denote the candidate optimal
values of these respective measures; that is, R (PR(t)) 
R* as t 
, provided the
solution does not get trapped by a local optimum. More loosely, however, we use this notation to
denote the best solution and values we find within the time constraints we impose upon the
genetic algorithm. Fifth, the triplets (Q*, D (PQ*), R
(PQ*)), (Q (PD*), D*, R (PD*)) and (Q (PR*), D (PR*), R*)
(where
Q* = Q (PQ*), etc.) are the
values of the three measures in question when the criteria Q, D and R
respectively are optimized. Finally, at the risk of laboring the point, we use
the triplet (Q*,Q PD*),Q (PR*)) to denote all
the values of the
performance measure Q when evaluated respectively for parameter sets PQ*, PD* and PR*;
and, similarly, for the triplets (D (PQ*),D*,D (PR*)) and (R (PQ*), R (PD*),R*).
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Note that our optimization solutions are all in the context of the set of stimuli used to generate the performance measure Q, D and R (as defined in expressions 7 and9 in Appendix B). This dependence is consonant both with learning over intragenerational time scales and with natural selection over intergenerational time scales, because organisms should develop or evolve to maximize their ability to discriminate among stimuli that they encounter during their life times. Also note that each case (i.e. selection of parameters that are a priori fixed) requires the genetic algorithm to be rerun to obtain the optimal solution for the case in question.
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Results |
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TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
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Evolution over time
To address the question of how well the genetic algorithm converges on an optimal
solution, we ran several simulations for many tens of thousands of generations. In each of these
simulations, we used the set of parameter values listed in Table 1, but the
100 individuals for the
initial population in each simulation were picked at random (i.e. for each individual, the weights wij were randomly assigned one of their four possible
values). We found that the system converged much more rapidly for high (wj
= 1.2, j = 1, . . ., 6) compared with low (wj =
0.4, j = 1, . . . , 6) PN–IN feedback synapse values (Table 2). Because 5000
generations usually provided enough time for our genetic algorithm to achieve close to 90% of
the value obtained after 100 000 generations, to save computational time we restricted
our
simulations to 5000 generations (Table 2).
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Also evident from Table 2
is the fact that improvements in ratio R
(PR(t)) over time were not due to a steady improvement in both
quality verity Q (t) and stimulus discrimination D (t), but
sometimes corresponded to decreased performance in one of these
measures that was more than compensated for by an increase in performance of the other
measure. Also, in the low PN–IN feedback case (wj = 0.4, j
=
1, . . ., 6) improvements in the ratio R (PR(t))
were due more to improvements in stimulus
discrimination D (t) than to improvements in quality verity Q (t), while for the
high PN–IN feedback case (wj = 1.2, j = 1, . . . , 6) most of
the improvements in the ratio R (PR(t)) were due a dramatic improvement in quality
verity Q (t) over time
(Table 2). Thus, compared with the low PN–IN feedback case,
the high PN–IN feedback case produced a quality code that was more than three times as
good (2.5 is more than three times smaller than 8.35), but a discrimination code that was less
than twice as good (12.66 is less than twice as large as 6.9). Combining these two results, the
ratio R* for the high PN–IN feedback case is more than six times larger than the
ratio
R* for the low PN–IN feedback case. One of the reasons for this difference
(as we shall see in Figure 4 below once we present the dynamic responses
of the PNs to the three
odorant stimuli) is that in the low PN–IN feedback case the PNs relax to a
stimulus-independent state after the stimulus is removed, while in the high PN–IN
feedback case after the stimulus is removed the PNs remain in a state that is close to the state
they are driven to in the presence of the stimulus.
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In our simulations we found that some of the solutions appeared to get temporarily trapped by local minimum solutions for varying lengths of time. To deal with this problem, while still retaining a sense of the effects that local minima might have on a single net in an adaptive learning situation, we report in the analyses below the average values of our various performance measures for trajectories starting from five different randomly selected initial conditions. These five sets of 100 initial network configurations were used in all related simulations which, as mentioned above, were each run for only 5000 generations.
Effects of concentration
In the context of trying to understand the role of the different parameters on the
performance of the network, we began by examining the extent to which a relatively large value
of C (recall the highest and lowest concentrations used as stimuli are C+ = C0 + C and C- = C0 - C respectively) degrades
the performance of the network (Figure 3A) by setting C
= 1 and allowing the
value of C0 to take on integer values from 2 to 6 (cf. the horizontal axes in
Figure 3A–D). As expected, the performance measure R*
improves
steadily from ~23 to 199 as C0 increases from 2 to 6 and dominates the
values of the ratios R (PQ*) and R (PD*) respectively
obtained by optimizing only with
respect to the quality performance measure, Q, and then
only with respect to the discrimination performance measure, D. We also
generated values of Q*, D* and R* with respect to C0 for the cases C = 0.5, 1, 2 and 3 (Figure
3B–D). In general the discrimination performance measure, D* (Figure 3C), was indifferent to the value of C0 and
C, except for
low values of C0 (C0 = 2). As expected, however,
the quality performance measure was degraded (i.e. the value increased; Figure 3B), and thus the
overall performance measure, R*, was also degraded (i.e. the value decreased;
Figure 3D) with the value of the ratio C/C0.
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C = 0.5, 1, 2 and 3.
(Note that for the larger values of Activation and memory parameters
Next we examined the effect that the values of the activation function steepness
parameters, ß1 and ß2 (i.e. how rapidly neurons switch
from their off to on state as the level of membrane depolarization increases—see equation
2 in Appendix A and Table 1), have
on the optimal values Q*, D*
and R* (Figure 4A–C) for different values of C0
C = 1). Specifically, we compared the increasingly nonlinear activation
functions ß1 = ß2 = 0.3, 0.5 and 0.8. All three measures,
Q*, D* and R*, improve with increasing
nonlinearity in that activation function, and identification of quality becomes almost perfect (Q*
close to zero; Figure 3A) for the case ß1 = ß2 = 0.8 when
the ratio C/C0 is one-quarter or less (i.e. C0
= 4). Since the discrimination performance measure, D*, is insensitive to
this ratio (except for the one slightly anomalous point ß1 = ß2 =
0.3 and C0 = 3; Figure 4B), the overall
performance measure,
R*, is very large (>14 000) for the case ß1 = ß2 = 0.8 (Figure 4C).
We also examined the effect of the values of the neuron decay (memory) parameters µj and 
j (i.e.
how rapidly activity in neurons
ceases once stimulation is removed—see equation 4 in Appendix A and Table 1) on the
measures (Q (PR*),D (PR*),R*) by
evaluating
these measures for the
cases µj = j = 0.1, 0.2, 0.3, 0.4 and 0.5, j = 1, . . . , 5 (Figure 4D–F). Discrimination, D*, improved
(increase in value of the performance measure) with increasing values of the decay parameters
(Figure 4E), while quality assessment only improved (decrease in value)
for the decay parameter
increasing toµj = j = 0.3, held steady
at µj = j = 0.4 and deteriorated for
µj = j
= 0.5 (Figure 4D). The optimal
value of the ratio R* also improved with increasing values of µj
and j until the value
µj =
j = 0.3 was obtained, but then
decreased for µj = j = 0.4 and increased again for to µj = j = 0. 5 (Figure 4F).
Projection neuron feedback parameters
Finally, we examined how that values of the PN–IN feedback synapse
parameters 
j, j = 1, .
. . , 6 (see equation 4 in Appendix
A and Table 1), influenced the values of performance
measures Q, D and R. In particular, we analyzed the cases j
= 0.4, 0.6, 0.8, 1.0, 1.2, 1.4 and 1.6 for j = 1, . . . , 6 (increasing values imply
strengthened feedback effects), using our standard set of parameter values (Table 1) and the
values (C0,C) = (4,1) (Figure 4G–I). The optimal
discrimination and ratio values, D* (Figure 4H) and R*
(Figure 4I)
respectively, improved (i.e. increased) with increasing feedback, while the optimal quality Q* (Figure 4G) only improved (i.e. decreased) over the higher values
of the
PN–IN feedback synapse weights j, j = 1, . . . , 6.
In Table 3 we list the final set of IN–IN synapse values wij, i,j = 1, . . . , 6 (see equation 4 in Appendix A), after optimizing
R* for the standard set of parameters (Table 1 with the
PN–IN feedback
coefficients having the values j = 1.2, j = 1, . . . , 6) and
odor input parameters (C0,C) = (4,1). Note the values wij, i, j = 1, . . . , 6, are not forced to be
symmetric (i.e. wij is not necessarily equal to wji), as is
the case for a standard Hopfield recurrent network (Haykin, 1994, p.
557):
there is no reason why these synapse values should be symmetric if allowed to evolve in the
absence of a constraint that enforces such symmetry (the symmetry constraint facilitates
theoretical investigations of the stability properties of the network but is not needed in context of
the numerical simulations reported here).
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We also provide examples of the output in response to various input stimuli for the same set of PN–IN feedback synapse parameters,
j = 1.2, j = 1, . . . , 6 (Figure 5A–C), and compare this to the case where the
PN–IN feedback coefficients are j = 0.4, j = 1, . .
. , 6 (Figure 5D–I). Comparing panels A–C with panels
D–F in Figure 5, it
is clear that for the set of network parameters considered here, the high feedback has the effect of
rapidly stabilizing the output compared with the low feedback. Further, with high feedback, PN
activity stabilizes at stimulus-dependent equilibria, while with low feedback it stabilizes at a
stimulus-independent equilibrium. Thus, in the high-feedback case, PN equilibrium values can
represent the code, while in the low-feedback case only the 0–150 ms PN transient can
represent the code. In the latter case, the PN transient has the same pattern at low, medium and
high concentrations (Figure 5G–I), although the amplitudes of the
PN transients are
slightly more extreme in the case of the high (5I) compared with the low (5G) concentration stimulus.
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As we noted earlier, and is evident from Figure 5A–I, each of the PNs has an initial spiking rate that is around the middle of its firing range scaled so the maximum corresponds to a value of 1 (this is obtained by setting the corresponding PN membrane activation potential to zk(0) = 0.5, k = 1, . . . , 6—see Appendix B for details). For the low PN–IN feedback case
j = 0.4, j = 1, . . . , 6, the system appears to have a single relaxed state
(i.e. stable equilibrium), and we would expect that the system would be in this state prior to
stimulation. As the network evolves, this unstimulated relaxed state will change from stimulus to
stimulus so that its value is not known a priori. In this case, this relaxed state cannot be
used as a universal initial condition to implement the genetic algorithm. To get around this
problem in our simulations, we selected the intermediate values zk(0) = 0.5, k = 1, . . . , 6 as initial conditions for all PNs throughout the course of the
genetic algorithm. Further, if the feedback has the relatively high value j = 1.2, j = 1, . . . , 6, the system relaxes after
removal of the stimulus to states
that are stimulus dependent (and appear to be relatively close to the states to which the system is
driven by the stimulus being applied for the first 100 ms of the 400 ms evaluation interval). Thus,
if the system starts out in a relaxed state after application of a particular stimulus and a new
stimulus is applied for a sufficiently short interval of time, the system will not move far from its
initial state and relax back to that same state. This dynamic behavior implies that if codes are to
persist beyond the application of stimuli, each stimulus must be applied for a sufficiently long
period of time to enable the network to be driven to a new region of its state space. Then after
removal of the stimulus in question the network is able to relax to a state that is uniquely
determined by this new stimulus (i.e. independent of the initial condition). On the other hand, if
codes are contained in the initial transient, the network needs to be reset to the same initial state
prior to stimulation through a priming mechanism that is internally generated. One such
mechanism could be synchronizing oscillations that have been observed in the insect antennal
lobes (Laurent and Davidowitz, 1994; Laurent et al.,
1996). Since we do not have any
data on how the brain deals with this problem, as mentioned above, we have taken the neutral
stance of setting all the INs and PNs to have initial values that produce intermediate spiking rates
[i.e. y(0) = z(0) = 0.5 prior to the onset of a new stimulus—see
Appendix B and equation (4)]. Six odorant simulations
Turning now to the simulations carried out for the six odorant case (see Appendix
D),
the first question we asked is how well the network performs over 5000 generations compared
with the three odorant set of stimuli considered above. We carried out this comparison for the
measures Q, D and R (equations 7, 8, 9 in
Appendix B), using the standard set of parameters (Table 1) with C = 1, and for
C0 ranging over the integers 2 to 6 (Figure 6A–C). The ability of
the network to produce a stable quality code (small Q*; Figure 6A) was equally
poor in both the three and six odorant cases when C0 = 2 and comparably
good in both cases when C0 = 5 and 6. The quality coding, however, was
not as good for the six-odorant stimuli as it was for the three-odorant stimuli when C0 = 3 and 4. In fact, the network showed no improvement in quality coding for the
six-odorant stimuli when C/C0 fell from 1/2 to 1/3 and
then to 1/4, but improved dramatically when C/C0 fell to
1/5. Interestingly, the ability of the network to discriminate stimuli was better for the six-odorant
than for the three-odorant stimuli at C0 = 2, but the reverse was true for C0 = 3, 4, 5 and 6 (Figure 6B). Also, from C0 = 3
onwards, no improvement was evident in the ability of the network in either case to discriminate
among odors.
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The fact that the optimal quality performance measure, Q*, is much more sensitive than the optimal discrimination performance measure, D*, to the ratio
C/C0 is not surprising since a large ratio poses a
challenge to quality coding, but does not affect discrimination, unless improvements in quality
coding come at the expense of the networks discriminative abilities (as it does in the
three-odorant but not the six-odorant case for the stimuli and receptor neurons under
consideration). Note that the superior (larger) discriminative value D* in the
six-odorant case compared with the three-odorant case, when C0 = 2, is
not necessarily anomalous, since the receptor neurons defined by expressions (13) are not
generalizations of the receptor neurons defined by expressions (10). If we
set the odd or even
odorants to zero (i.e. C2 = C4 = C6 = 0 or C1 = C3
= C5 = 0) then the ORNs defined by
expressions (13) (Appendix D) exhibit excitatory
rather than the inhibitory binary interactions
that are evident in the ORNs defined by expressions (10) (Appendix C). Finally, the
discrimination-to-quality ratio, R*, improves steadily in both case as C0
increases from 2 to 6 (Figure 6C), but the network performs almost twice
as well in the
three-odorant compared with the six-odorant set of stimuli when C0 = 6.
We plotted selected outputs for the low (j = 0.4, j = 1, . . ., 6, Figure 7A,B) and high (j = 1.2, j = 1, . .
., 6, Figure 7C,D) PN–IN feedback cases (using the standard set of
network parameters
(Table 1) and stimulus parameters (C0,C) = (4,1). As
with the three-odorant case, high feedback produces extreme PN output (close to 0 or close to 1),
while low feedback allows the PNs to stabilize at intermediate values. Unlike the three-odorant
low feedback case, however, the equilibria are now different for different odors in both the high
and low feedback cases so that both the transient and equilibrium phases are able to contribute
towards coding odor quality.
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Discussion |
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TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
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With the constraints on the ranges of parameters investigated here, our results suggest that the performance of our network, as measured by the ratio R, (recall a good performance requires that R >> 1), is influenced at least threefold by the decay parameter µ ranging over 0.1–0.5 (Figure 4F), tenfold (an order of magnitude) by the PN–IN feedback synapse parameter ranging over 0.4–1.6 (Figure 4I) and an astonishing two orders of magnitude for the neuron activation function slope parameter ß (equation 2 in Appendix A) ranging over 0.3–0.8 (Figure 4C). The performance of the network, however, is severly degraded as the concentration variation ratio
C/C0 is increased from 1/4 (the value that
applies to all panels in Figure 4) towards 1 (see Figure 3). Thus the network, as it stands, has the potential to perform well for spatially
homogeneous stimuli containing many more component odorants than the three and six odorants
considered here (Figure 6), provided the value of the activation function slope parameter ß is
at least doubled from the value ß = 0.3 used to generate all the plots in Figure 7. Note that proportional changes in the input (i.e. in the logarithm of the concentration) do not produce proportional changes in the output (i.e. a linear response) because the neurons have nonlinear activation functions (threshold-like and saturation-like effects). Also the receptor neurons themselves are not all linear with respect to log-concentration of the components of the stimuli. The `linear differencers' ORNs, for example, are dependent on the absolute value of the difference and thus will have a V-shaped response when, for example, the concentration of one of the two odorants to which these ORNs respond increases from less than to more than a given concentration of the other odorant to which they respond. Also, in the six-odor simulation case, none of the receptor neurons specializes on any one odorant, further confounding the quality and concentration coding task of the network.
Further, numerically intensive investigations are required to assess how well the
system maintains its performance once the stimuli are given realistic temporal and spatial input
structures. Of course, the performance can always be improved, at additional computational cost,
by scaling up the dimensions of the network itself (i.e. increasing the number of glomeruli and
associated number of ORNs, INs and PNs). The real issue is not whether the network can
perform the computations required, but what the smallest network might be that will perform to
some specified error rate for a given set of stimuli with associated temporal, spatial and
concentration coefficients of variation. Further, the fact that the same type of ORNs are
distributed across the antenna in a manner that is more uniform than random (Getz
and Akers, 1994) also promotes the integration of temporal and spatial heterogeneity
in the stimulus, assuming as we have that similar ORNs are more likely than random to arborize
in the same glomerulus.
Perhaps one of the most interesting issues raised by our results relates to the possibility
that the coding in real systems could either be in the initial transient, as is necessarily the case
when the network equilibrium is independent of the stimuli (Figure 5D,E,F,H), or associated with the equilibria when the equilibria are stimulus dependent (Figure 5A,B,C). Of course, when the equilibrium is stimulus dependent,
the coding could be in both the transient and equilibrium phases of the PN responses. Further,
when the equilibrium is dependent only on the quality and not the concentration of the odor, as in
the case of the high PN–IN feedback values (j = 1.2, j = 1, . . . , 6) for the three odor stimuli (Figure 5A–C) and all the six odor stimuli (Figure 7A–D), then the system has a true quality code over the range of concentrations used to
train the network in optimizing the performance measure R (equation 9 in Appendix B). Even the transient phase for the low
PN–IN feedback values (j = 0.4, j = 1, . . . , 6) in the three-odor stimuli case, however, provides a
surprisingly robust quality code, despite the fact that once the stimulus is removed the system
returns to a stimulus-independent equilibrium state. For example, the only difference between the
transients for the odor stimulus S = (0,1,1) in the low (Figure 5G) and high (Figure 5I) concentration cases is the amplitudes of the
transients and not which PNs are firing at a high (>0.5), moderate (~0.4) or low (<0.3) rate.
For example, PNs firing at moderate rates are 2 and 3 when S = (1,0,0) (Figure 5D), 1 and 2 when S = (0,0,1) (Figure 5E),
and 5 and 6 when S = (1,1,1) (Figure 5F), while none of the PNs
fire at moderate rates when S = (0,1,1) (Figure 5G–I).
The temporal aspect of the coding is complicated by the fact that olfactory networks in
real insect systems exhibit rhythmic oscillations of ~20–30 Hz (Wehr and
Laurent, 1996). Thus, for example, the first one or two oscillations could be
transients with the remaining oscillations representing a periodic equilibrium that remains in
effect until the stimulus is removed or the ORNs become adapted to prolonged input. This idea
can only be tested, however, if sufficient data are collected over long enough time intervals to
distinguish initials transients from long-run cycles. Further, the meaning of transients can only be
gauged if the state of the antennal lobes (or olfactory bulb in vertebrates) is known prior to each
stimulation. Thus it is important to evaluate spike trains for sufficiently long periods of time
before, during and after stimulation to evaluate initial, transient and final lobe states associated
with particular stimuli. It is possible that the antennal lobe is reset to some initial condition by a
regulating signal originating in another part of the brain (e.g. a memory center in the mushroom
bodies), or that the lobe may revert to some kind of background noise state after stimulation and
be in this state prior to receiving ORN input in response to the next stimulus. If one stimulus
follows immediately after another, however, then the initial state for the second stimulus is the
final state for the first, so the situation becomes quite complex unless initial transients are not
part of the olfactory code.
Stopfer et al. (1997) recently demonstrated that odor
discrimination in honey bees remains functional, but at a degraded level of resolution, when
synchrony between antennal lobe and mushroom body oscillations is disrupted through the
application of picrotoxin to the brains of individual workers. A reason for this loss of resolution
might be that the mushroom body neuropil receiving PN input may depend on synchronized
antennal lobe/mushroom body oscillations for timing the beginning of a transient input signal. If
this were the case, then in the absence of the synchronizing oscillations the start of the transients
would no longer be read correctly (i.e. from a mathematical perspective, no way exists to mark
the initial state of the network and the appropriate initial condition is lost). Implicit in this
hypothesis is the assumption that similar odors are more likely to have similar transients that
would then be confused in the absence of a precise timing mechanism. On the other hand, very
different odors would have dissimilar transients that would still be discriminable even if the
timing mechanism were no longer precise, provided the stimuli were applied for a sufficiently
long period of time. This idea is supported by the transients in Figure 5D–F that, for very different stimuli, are quite distinct even though the initial and final
(equilibrium) PN values are the same for all stimuli.
The network constructed here does exhibit oscillations for some sets of parameters, and is more likely to oscillate as the delay parameters are increased in value. A thorough investigation of sets of parameters that produce oscillations of the type observed in vivo still needs to be undertaken, with attention being paid to the importance of inputs terms Ij and Ik in equations (4) (Appendix A), since these inputs would include feedback from other parts of the brain that may send entraining, cohering, reinforcing or timing signals. Further, it is clear from our analysis that we need empirical investigations on how the brain may reset the electrical activity of the antennal lobe prior to coding, because an internally consistent set of initial conditions is required for the production of a precise coding of odor quality.
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Conclusion |
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TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
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In the introduction, we raised several questions associated with the olfactory processing problem in general, as well as insects in particular. The results we report here shed some light on the specific questions and give us pause for thought on the general questions. In terms of the network we use here to model olfactory processing in the insect antennal lobe, our results demonstrate that, comparably speaking, signal attenuation (decay) has a moderate effect, PN–IN feedback has a large effect and a strong nonlinearity in the activation function (i.e. a steep slope) has an overwhelming effect on the network's ability to produce discriminable, concentration invariant, odor codes. Thus, a consistent theme that emerges from our results is that the stronger the nonlinearities in the network the more distinct the code appears to be. These nonlinearities reduce the likelihood that neurons maintain intermediate response values and cause neurons to either be `on' (close to 1) or `off' (close to 0). Further, high PN–IN feedback synapse values help maintain the code after the stimulus has been removed. Note that it takes only 30 neurons confined to binary `on–off' states to code the quality of over one billion (230) different odors. Finally, the network model presented here provides a unique tool for investigating the role of nonlinearities in the response of ORNs to mixtures versus pure odorants, although we have yet to conduct a systematic analysis of this question.
The process of running the model and trying to interpret the output raises a number of general issues that are critical for our understanding of olfactory coding in animals. The first issue relates to the initial state of the lobe or bulb (i.e. just prior to stimulation). The second issue is the amount of time it takes for the lobe or bulb to make the transition from the initial state to the particular state that represents the concentration-independent coding for the stimulus at hand. The third issue is the dynamic nature of the code itself (is the code an equilibrium state, a stable oscillation or a transitory pattern). The fourth issue is a function of the oscillations that occur in the lobe, their source and their relationship to the code itself.
In our simulations, in the absence of empirical information on the nature of the `resting' states of the insect antennal lobe, the initial conditions we used were half way between no activity and maximum activity for each of the neurons in the network. Clearly more comprehensive empirical studies are needed on resting versus coding states of all the neurons involved. Such studies would also provide insights into the kind of time constants associated with the transitions from resting to coding states. Further, the relationship between these time constants and observed frequencies of oscillations should help clarify the role of the oscillations in coding itself and whether the combinatorial hypothesis of Laurent and colleagues is a viable paradigm for olfactory coding.
The similarities between odor processing in all animals imply that investigation of the insect olfactory coding problem provides insight into olfactory processing in all animals. This study represents a first attempt at investigating how effectively the basic architecture of the insect olfactory lobe is able to form a stable concentration-independent code that solves the olfactory coding problem in insects. The network we model performs exceptionally well for the small homogeneous stimulus space considered here. In future studies, we need to account for the dynamic properties of ORNs, as well as present more challenging situations in terms of the dimensionality and heterogeneity (spatial and temporal) of the stimulus space. Such studies will provide us with a better understanding of the limitations of the network employed here in solving more complex problems. They will also provide insights into how the power of our network to form stable olfactory codes is affected by scaling (inclusion of more glomeruli, etc.). Finally, as is clear from the four issues discussed above, network models play a very useful role in identifying what aspects of the biology are most critical for obtaining a better understanding of olfactory processing in all animals.
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Appendix |
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TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
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A. Underlying equations
The simplest way of characterizing the state of the jth neuron in a net of n neurons (i.e. j = 1, . . . , n) is through the single variable vj, which can be regarded as the voltage or activation potential that is simultaneously the same value at all points on the neuron. Thus, from a conductance point of view, the neuron has no spatial structure or extent. Obviously this is a gross simplification, with spatial structure playing a crucial role in integrating input from highly arborized dendrites. At the most basic level, however, finite propagation speeds of spikes, of currents or of changes in voltage along neurons have a spatial dimension that can be incorporated into the model through explicit time delays. Such time delays also capture some of the delay characteristics associated with the operation of chemical synapses between the neurons themselves.
Having defined an activation potential vj, we assume that the output
Vj of the jth neuron (interpreted as the spiking rate for spiking
neurons) is related to the
activation potential through a nonlinear activation function 
h(vj) of type
h; that is,
 | (1) |
Note the subscript h allows for the fact that more than one type or class of
activation functions may exist. For spiking neurons, we assume that Vj is
scaled to take
values on the interval [0,1], where 0 represents no activity and 1 the
maximum firing rate. In this case h is a monotonically increasing
mapping of the
real line (-,+) onto the compact interval [0,1]. A commonly
used activation function that has this property is the logistic equation
 | (2) |
where ßh > 0 and 
h are constants
that
respectively determine how closely h approximates a step function
(h approaches a step function as ßh
) and where the
step occurs (the step occurs around vj = h).
If the jth neuron is efferent to m other neurons with spiking rates Vk, k = 1, . . . , m, and is also activated by some other source Ij
(this may also include a propensity to fire spontaneously, although in this context, Ij and
the parameter h in expression 2 are
functionally interchangeable). then the
standard electrical circuit equations, discretized with respect to time (setting and selecting the
units of t so that the maximum firing rate in each time interval is 1—i.e. the unit
of t is
~3–5 ms and will be regarded here as exactly 5 ms), are
 | (3) |
where wjk represent the strengths of the synapses of the m
neurons
afferent to the neuron in question and 
kj is a time delay associated
with how long
it takes to transmit information on the state of the kth neuron to the jth neuron
(it is assumed to
take on the value of 0 or integer values of time). Note that if the jth neuron has a
capacitance
Cj and resistance Rj, then, in terms of electrical circuit
theory [e.g. discretize
equation (14.14) in Haykin (1994)], the parameter µj in equation (3) is µ j = 1 - 1/( CjRj). The sign of each determines
whether the kth neuron inhibits or excites the jth neuron.
Equation (3) is generic and reflects no special architecture. It is easily
specialized to account for the antennal lobe architecture depicted in Figure 1 and discussed in the
text as follows. We use the symbols Xi, Yj and Zk to denote the
spiking rates of the ORNs (receptor neurons), INs (intrinsic neurons) and PNs (projection
neurons) respectively (Figure 1), and let xi, yj and zk denote the
corresponding activation potentials. As alluded to above, we do not include a dynamic model for
generating the ORN action potentials xi, but only model the dynamics of
the IN and PN
action potentials yj and zk respectively. We treat ORN
spiking rates, Xi, directly as given constant inputs. This allows us to study
the dynamic properties of the antennal
lobe network without the confounding influence of dynamic ORN inputs. Once the
computational properties of antennal lobe network are better understood, then the effects of more
realistic ORN input can be studied using dynamics models of ORN responses (e.g. see Av-Ron
and Rospars, 1995; Av-Ron and Vibert, 1996) to constant
and temporally varying stimuli using
available data (Lemon and Getz, 1997) as a basis for developing the ORN
component of the
model.
In the spirit of initially avoiding confounding complexities, we assume that all
INs have the same activation function 1 (i.e. expression 2 with h = 1) and all
PNs have the same activation function 2 (i.e. expression 2 with h = 2). Thus,
equation (3) for our specific architecture reduces to the two sets of
equations
 | (4) |
 |
 |
where the tildes, hats and bars are used to indicate the level at which the various
synapse, decay and delay parameters apply (Figure 1). For example, 
ij 
0
are the excitatory synapse weights between the ORNs and INs, wlj 
0
are
the inhibitory synapse weights among the INs,
0 are the inhibitory
synapses weights between the INs and their corresponding PNs, and 
ij 0 is the synapse weight for the excitatory
feedback of the PNs onto their corresponding
INs. The terms Ij and 
k are the background excitation of the INs and PNs
respectively. These background levels could be due to self-excitation, unaccounted input from
neurons or effects of placement parameters h, h = 1, 2, in
expression (2) (since we set h = 0 for all
neurons, as mentioned earlier we
can incorporate the effects of nonzero h by adding the real value of
h to the corresponding excitation term—cf. equations 2 and 3).
B. Network output and performance measures
The output from the network is the responses of the PNs which will vary
depending on the particular input stimulus, as well as the values of the network parameters.
These latter values are the dimension n of the model itself, the activation (ß1, ß2, 1, 2), decay
(µj, j, k
= 1, . . . , n), delay (lj, 
ij, 
lj,
j, k, l = 1, . . . , n), and weighting parameter (wlj, ij, j, wj, where j in our case
represents a sensory receptor of
type j and j, k, l = 1, . . . , n) parameters values, as well as on the
values of the
external/self-activating inputs ( Ij, k, j, k = 1, . . . , n). As mentioned above,
these inputs and and the activation threshold parameters 1 and 2
effectively enter the equations in the same way (i.e. the effects of changes in the value of the
activation thresholds 1 and 2 can also be accomplished by
making appropriate changes to the values of Ij and k—cf. equations 2 and
3). Thus, in our model, we set 1 and 2 to zero and
manipulate only the values of the inputs and to obtain a network `background'
state that facilitates coding, as described in more detail below. In reality, however, we should not
expect the activation position parameters 1 and 2 to be zero
and
need to take this into account when trying to obtain empirical measurements of the
external/self-activating inputs Ij and k. The total set of parameters can be
written as
 |
where the symbols are matrices (uppercase W) and vectors of the corresponding subscripted italicized symbols introduced above.
To permit the temporal structure in the response of the PNs to be taken into account over the 400 ms response interval motivated in the main text, we divide this response into eight 50 ms bins (Figure 2). Thus, if our iteration unit of time is 5 ms and the output from the jth PN in response to a 100 ms input stimulus Sq is given by (cf. equation 2)
 | (5) |
(i.e. t 
[0,400 ms], where
zj(t) is the solution to equation 4), then this
output can be aggregated into the
eight-dimensional vector (Figure 2)
 | (6) |
(Note, for notational simplicity, we have not indexed the fact that the network output variables zj and Zj also depend on stimulus Sq.) If we then concatenate the output across all n PNs (Figure 2), we have the input/output relationship that each stimulus, Sq, produces as an N-dimensional vector Pq = (p1q, . . . ., pnq), where, in this case, N = i x n = 48 (i = 8, n = 6).
Simple ways to compare two output vectors P1 and P2 is to take a vector norm (city block, square root of sum of squares, or maximum element, etc.) of the difference between these two vectors (Figure 2). Alternatively, we can interpret the directions of the vectors as coding odor quality and the magnitudes as coding for concentration and then train the network to optimize its performance in classifying input stimuli of the same quality as the same or discriminating between different odors irrespective of concentration.
The most appropriate way to measure the performance of a network, such as
the one represented by equations (1–3), is to evaluate how well the network is able to
code for quality independent of concentration and to discriminate among odors of different
quality. In many situations the measure of interest is the proportion of errors that are made in
misclassifying the quality of an odor or not discriminating between different odors (Getz and
Page, 1991). These measures, however, require that a priori we specify a
threshold (or
equivalently a level of tolerance for variations in odors of the `same' quality which
is also a level of resolution for discriminating among odors of `different' quality)
(e.g. see Getz and Akers, 1997). Because plotting the performance of the
system in terms of this
threshold variable greatly increases that computational work, as well as the graphical complexity
of the results, we will only present the results here in terms of average measures of verity of
quality and resolution of discrimination. This simplification in no way affects our ability to
address the issues at hand.
Suppose we idealize an environment to contain M odors Oq, q
= 1, . . . , M, each of which occurs at one of three concentrations
C1, C2 and C3, where the first is
relatively low, the second is medium
and the third is relatively high. Note that these odors may have different component odorants, or
may have the same components occurring in different concentrations (i.e. the ratio of
concentrations of components differs from one odorant to the next). Let Sqi
represent the 100 ms stimulus comprising odor q at, say, three concentrations i
= 1, 2, 3 (in
ascending strengths: low, medium, high), and let Pqi
represent the
corresponding 400 ms output from the network. Then, to ensure that the output is as invariant as
possible with respect to concentration, we are interested in minimizing the differences in the
directions of these two vectors (e.g. in minimizing differences between Pq1 and Pq3, assuming Pq2 falls somewhere between
Pq1 and Pq2
when concentration 2 is intermediate between
concentrations 1 and 3—see Figure 2and cf. Getz and
Chapman (1987).
Thus the problem of selecting a set of network parameters P that maintains a sense of
quality can be expressed as [any norm can be used, e.g. Euclidean (see Figure 2), but the actual norm we used in our calculations was the absolute value]:
 | (7) |
Note that the measure is scaled by the number of comparisons M, as well as the length N of the output vectors (recall N = 48, being six concatenated vectors each having eight time bins of spiking rates averaged over 50 ms intervals). By contrast, if we want to focus on maximizing the ability of the network to discriminate between pairs of odors all at, say, the medium concentration (i.e. concentration 2), then we can select the parameters that solve the following maximization problem:
 | (8) |
Finally, if we want to consider a criterion that simultaneously accounts for verity of quality across different concentrations of the same odor as well as discriminability among odors of different quality, we can select the parameters to solve the problem:
 | (9) |
Considering that Q is a measure of the average distance between stimuli of the same odor quality and D is the average distance between stimuli of different odor qualities, the network could not begin to distinguish between odors of the different qualities at the same concentration or odors of the same quality at different concentrations unless R > 1, and could not perform very well unless R > 1.
C. Three odorant simulations
The following equations characterize the response of six classes of sensory neurons, three linear specialist and three linear differencers, to the three odorant stimulus S = (C1,C2,C3):
 | (10) |
Thus, in response to a 100 ms input stimulus of the second odorant at
concentration C, the three input variables Xi, i = 1, 3 and
5, are zero over the whole 400 ms computational interval, while the three input variables Xi, i = 2, 4 and 6, are equal to C for 0 t 100 ms and 0
for the remaining 300 ms interval. In response to a 0.5:0.5 mixture of the first and third odorants
at total concentration C, however, inputs Xi, i = 2 and 5, are zero
over the whole
computational interval, while Xi, i = 1, 3, 4 and 6 are equal to C/2 for 0
t 100 ms and 0 for the remaining 300 ms interval.
The analyses involved training the network to the following predefined set of
stimuli, characterized in terms of two response rate parameters C0 and
C. The
stimuli were chosen to span a given range of concentrations in a way that would allow us to
control for the effect of concentration in a rigorous manner. They were also chosen to span a
range of odor qualities that included pure odorants, equicomponent binary mixtures and an
equicomponent blend of all three odorants. More particularly, we defined C- = C0 - C and C+ = C0 + C, and defined the
following 13 stimuli using a more informative superscript notation than that used in equations (7)
and (8) [the correspondence between the two notations is relatively clear:
superscripts 1, 2 and 3
in performance measures (7) and (8) are replaced
by
superscripts -, 0 and +, and the number of stimuli used to generate performance measure
(7) is M = 3 and performance measure (8) is M = 7]:
 | (11A) |
 | (11B) |
and
 | (11C) |
As discussed in the section above, each of these 13 stimuli that we used as input to equations (4) produced the corresponding output vectors Pqi, i = -, 0, +, q = 1, 2, 3 and Pq0, q = 4, . . . , 7. We used the output pairs (Pq-, Pq+), q = 1, 2, 3, to generate the quality measure Q defined in equation (7) and the outputs Pq0, q = 1, . . ., 7, (21 pairs) to generate the discrimination measure D defined in equation (8).
The challenging problem remained to solve for sets of parameters
PQ, PD and PR that respectively optimize
the
performance measures Q , D and R, as defined in expressions
(7, 8, 9), subject to
appropriate constraints (e.g. the parameters must be non-negative). Once
this was done, we could then compare these measures to see how well the system is able to
produce a stable quality code, discriminate among different odors and simultaneously perform
the two, somewhat conflicting, tasks. Since we anticipated that the network's ability to
maintain quality would be eroded as the ratio C/C increases from 0 to
1, we were motivated to compare the performance of the network for different
values of C and C.
D. Six odorant simulations
For this set of simulations the dimension of the odor space was increased from
three to six, with stimuli now of the form S = (C1,C2,C3,C4,C5,C6). As in the three-odorant case, we
defined pure odorant stimuli at three different concentrations in terms of response rate parameters
C0 and C. Specifically, we defined the 18
multiconcentration pure odorant
stimuli (cf. equation 11a) as
 | (12A) |
the fifteen single concentration equicomponent binary stimuli (cf. equation 11b) as
 | (12B) |
and the single equicomponent full blend (cf. equation 11c) as
 | (12C) |
As in the three-odorant case, we used the corresponding output pairs (Pq-,Pq+) q = 1, 2, . . ., 7, to generate the quality measure Q defined in equation (7) and the outputs Pq0, q = 1, . . ., 22 (231 pairs), to generate the discrimination measure D defined in equation (8). With six odorants, rather than the previous three, we needed to redefined the types of ORNs used to transduce chemical stimuli into electrical input (cf. equation 10).
The six classes of receptor neurons that we now used to produce the input into the network in terms of their response to the logarithmic concentrations Ci, i = 1, . . ., 6, of the six odorants in question were (cf. equations 10):
 | (13) |
 |
 |
Note that, unlike the three-odorant case, we no longer have receptor neurons that specialize on any one single odorant. Thus this six-odorant case is a step up in complexity in addressing the problem of how well our relatively simple neural network is able to code the stimuli defined by expressions (12a, b, c).
E. Fixed and optimized parameter values
A number of different numerical routines exist for solving optimization
problems (7, 8, 9) (Press et al., 1986). Here we use a genetic algorithm (Davis, 1990; Koza,
1992; Sou
ek, 1992; Bäck et
al., 1997) to solve these problems, since we
are not only interested in the solutions PQ, PD and PR that solve the three optimization problems (7, 8, 9) respectively, but how
optimal solutions are approached through time. [The algorithm we used was written by W.M.
Spear. We obtained the code in early August 1997 as freeware from the website
http://www.aic.nrl.navy.mil/~spears/freeware.html.
Spear describes the code as a standard
genetic algorithm, similar to Grefenstette's work, employing Baker's SUS
selection algorithm, with n-point cross-over maintained at 60%, a very low mutation
rate, and
selection based on proportional fitness.] This latter information may provide us with some
insights into how the antennal lobes might learn to perform well in terms of whether quality
identification or discrimination initially dominates. Further, we did not optimize over all
parameters simultaneously, but we fixed some of the parameters in P and optimized
over
the rest. Through this approach, we hope to learn how different components of the network
contribute to solving the olfactory processing problem in insects.
In the simulations discussed below, only the IN feedback parameters, wij, were
free to mutate in the genetic algorithm. All the other network parameters were fixed and
have the values list in Table 1. The initial conditions we used for the
simulations were yj(0) = zk(0) = 0.5, j, k = 1, .
. ., 6. We did not use the
relaxed (stable equilibrium) state as our initial condition because this state changes as the system
evolves, and for some sets of parameters several relaxed states exist. If the system has only one
relaxed state, we should expect the system to be in that state prior to stimulation. In reality, if the
system has several relaxed states, then either its state prior to stimulation will be a function of its
history or some mechanism may exist for setting the network to a particular initial state after each
stimulation. The latter mechanism would be required if the coding was dependent not only on the
stimulus involved but also on the initial state of the network.
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Acknowledgments |
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We thank Bill Lemon for helpful discussions, Benjamin Mattei for technical comments, Jean-Pierre Rospar and anonymous reviewers for extensive comments and suggestions on how to improve our presentation, and Barry Ache for patiently but insistently ensuring that our paper is readable to biologists.
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References |
|---|
|
TopAbstractIntroductionNetwork approachResultsDiscussionConclusionAppendixReferences |
|---|
Ache, B.W. (1991) Phylogeny of smell and taste. In Getchell, T.V., Doty, R.L., Bartoshuk, L.M. and Snow, J.B. (eds), Smell and Taste in Health and Disease. Raven Press, New York , pp. 3–18.
Akers, R.P. and Getz, W.M. (1992) A test of identified response classes among olfactory receptors in the honeybee worker. Chem. Senses, 17, 191–209.
Akers, R.P. and Getz, W.M. (1993) Response of olfactory sensory neurons in honey bees to odorants and their binary mixtures. J. Comp. Physiol., 173, 169–185.
Aradi, I. and Erdi, P. (1996) Multicompartmental Modeling of the Olfactory Bulb. Cybernetics and Systems, 27,605 –615.[Web of Science]
Arnold, G., Masson, C. and Budharugsa, S. (1985) Comparative study of the antennal lobes and their afferent pathway in the worker bee and the drone (Apis mellifera ). Cell Tissue Res., 242, 593–605.[Web of Science]
Atema, J., Borroni, P., Johnson, B., Voigt, R. and Handrich, L. (1989) Adaptation and mixture interaction in chemoreceptor cells: mechanisms for diversity and contrast enhancement. In Laing, D.G. (ed.), Perception of Complex Smells and Tastes. Academic Press, Marrickville, pp. 83–100.
Av-Ron, E. and Rospars, J.-P. (1995) Modeling insect olfactory neuron signaling by a network utilizing disinhibition. Biosystems, 36, 101–108.[Web of Science][Medline]
Av-Ron, E. and Vibert, J.-F. (1996) A model for temporal and intensity coding in insect olfaction by a network of inhibitory neurons. BioSystems, 39, 241–250.[Web of Science][Medline]
Av-Ron, E., Parnas, H. and Segel, L.A. (1991) A minimal biophysical model for an excitable and oscillatory neuron. Biol. Cybernet., 45, 487–500.
Bäck T., Fogel, D.B. and Michalewicz, Z. (eds) (1997) Handbook of Evolutionary Computation. Oxford University Press, Oxford.
Baird, B. (1986) Nonlinear dynamics of pattern formation and pattern recognition in the rabbit olfactory bulb. Physica D, 22,150 –175.[Web of Science]
Barnard, E. (1989) Analysis of the Lynch–Granger model for olfactory cortex. Biol. Cybernet., 62, 151–155.[Web of Science][Medline]
Bhalla, U.S. and Bower, J.M. (1993) Exploring
parameter space in detailed single neuron models: simulations of the mitral and granule cells of
the olfactory bulb. J. Neurophysiol., 69, 1948–1965.
Boeckh, J. and Ernst, K.D. (1987) Contribution of single unit analysis in insects to an understanding of olfactory function. J. Comp. Physiol. A, 161, 549–565.
Boeckh, J., Distler, P., Ernst, K.D., Hösl, M. and Malun, D. (1990) Olfactory bulb and antennal lobe. In Schild, D. (ed.), Chemosensory Information Processing. Springer-Verlag, Berlin, pp. 201–227.
Bower, J.M. (1991) Piriform cortex and olfactory object recognition. In Davis, J.L. and Eichenbaum, H. (eds), Olfaction: A Model System for Computational Neuroscience. The MIT Press, Cambridge, MA, pp. 265–285.
Christensen, T.A. and Hildebrand, J.G. (1987) Functions, organization, and physiology of the olfactory pathways in the lepidopteran brain. In Gupta, A.P. (ed.), Arthropod Brain: Its Evolution, Development, Structure, and Functions. Wiley, New York, pp. 457–484.
Christensen, T.A., Waldrop, B.R., Harrow, I.D. and Hildebrand, J.G. (1993) Local interneurons and information processing in the olfactory glomeruli of the moth Maduca sexta. J. Comp. Physiol. A, 173,385 –399.[Medline]
Davis, L. (ed.) (1990) Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York.
Delaney, K.R., Gelperin, A., Fee, M.S., Flores, J.A., Gervais, R., Tank,
D.W. and Kleinfeld, D. (1994) Waves and stimulus-modulated
dynamics
in an oscillating olfactory network. Proc. Natl Acad. Sci. USA, 91,669
–673.
Derby, C.D., Girardot, M., Daniel, P.C. and Fine-Levy, J.B. (1989) Olfactory discrimination of mixtures: behavioral, electrophysiological and theoretical studies using the spiny lobster Panulirus argus. In Laing, D.G., Cain, W.S., McBride, R.L. and Ache, B.W. (eds), Perception of Complex Smells and Taste. Academic Press, Orlando, FL, pp. 65–82.
Dittmer, K., Grasso, F. and Atema, J. (1995) Effects of varying plume turbulence on temporal concentration signals available to orienting lobsters. Biol. Bull., 189, 232–233.[Web of Science]
Érdi, P., Aradi, I. and Gröbler, T. (1997) Rhythmogenesis in single cells and population models: olfactory bulb and hippocampus.BioSystems , 40, 45–53.[Web of Science][Medline]
Fine-Levy, J.B. and Derby, C.D. (1992)
Behavioral discrimination of binary mixtures and their components: effects of mixture
interactions on coding of stimulus intensity and quality. Chem. Senses, 17, 307–323.
Flanagan, D. and Mercer, A.R. (1989a) An atlas 3-d reconstruction of the antennal lobes in the worker honey bee, Apis mellifera (Hymenoptera: Apidea). Int. J. Insect Morphol. Embryol., 18, 145–159.
Flanagan, D. and Mercer, A.R. (1989b) Morphology and response characterstics of neurones in the deutocerebrum of the brain in the honeybee Apis mellifera. J. Comp. Physiol. A, 164, 483–494.
Fonta, C., Sun, X.-J. and Masson, C. (1991) Cellular analysis of odour integration in the honey bee antennal lobe. In Goodman, L.J. and Fisher, R.C. (eds), The Behaviour and Physiology of Bees. CAB International, Wallingford, pp. 227–241.
Fonta, C., Sun, X.-J. and Masson, C. (1993)
Morphology and spatial distribution of bee antennal lobe interneurones responsive to odours.Chem. Senses
, 18, 101–119.
Freeman, W.J. and Skarda, C.A. (1985) Spatial EEG patterns, non-linear dynamics and perception: the neo-Sherringtonian view. Brain Res. Rev., 10, 147–175.
Fujimura, K., Yokohari, F. and Tateda, H. (1991) Classification of antennal olfactory receptors of the cockroach, Periplaneta americana L. Zool. Sci., 8, 243–255.
Gascuel, J. and Masson, C. (1991) A quantitative unltrastructural study of the honeybee antennal lobe. Tissue Cell, 23,341 –355.[Web of Science][Medline]
Getz, W.M. (1991) A neural network for processing olfactory-like stimuli. Bull. Math. Biol., 53, 805–823.[Web of Science][Medline]
Getz, W.M. (1994) Odor processing in the insect olfactory system. In World Congress on Neural Networks—San Diego. Lawrence Erlbaum Associates and INNS Press, Hillsdale, NJ, pp. 661–668.
Getz, W.M. and Akers, R.P. (1993) Olfactory response characteristics and tuning structure of placodes in the honey bee Apis mellifera.Apidologie , 24, 195–217.
Getz, W.M. and Akers, R.P. (1994) Honey bee olfactory sensilla behave as integrated processing units. Behav. Neural Biol., 61, 191–195.[Web of Science][Medline]
Getz, W.M. and Akers, R.P. (1995) Partitioning nonlinearities in the response of honey bee olfactory receptor neurons to binary odors.BioSystems , 34, 27–40.[Web of Science][Medline]
Getz, W.M. and Akers, R.P. (1997) Response of American cockroach (Periplaneta americana ) olfactory receptors to selected alcohol odorants and their binary combinations. J. Comp. Physiol. A, 180,701 –709.
Getz, W.M. and Chapman, R.F. (1987) An odor discrimination model with application to kin recognition in social insects. Int. J. Neurosci., 32, 963–967.
Getz, W.M. and Page, R. (1991) Chemosensory communication systems and kin recognition in honey bees. Ethology, 87, 298–315.[Web of Science]
Granger, R., Ambrose-Ingerson, J., Antón, P.S., Whitson, J. and Lynch, G. (1990) Computational action and interaction of brain networks. In Zornetzer, S.F., Davis, J.L. and Lau, C. (eds), An Introduction to Neural and Electronic Networks. Acadmic Press, San Diego, CA, pp. 25–41.
Hammer, M. and Menzel, R. (1995) Learning and memory in honeybees. J. Neurosci., 15, 1617–1630.[Abstract]
Hansson, B.S., Ljungberg, H., Hallberg, E. and Löfstedt, C.
(1992) Functional specialization of olfactory glomeruli in a moth. Science, 256, 1313–1315.
Hansson, B.S., Anton, S. and Christensen, T.A. (1994) Structure and function of antennal lobe neurons in the male turnip moth, Agrotis segetum (Lepidoptera: Notuidae). J. Comp. Physiol. A, 175,547 –562.[Web of Science]
Haykin, S. (1994) Neural Networks: A Comprehensive Foundation. Macmillan College Publishing, New York.
Hendin, O., Horn, D. and Hopfield, J.J. (1994) Decomposition of a mixture of signals in a model of the olfactory bulb. Proc. Natl Acad. Sci. USA, , 91, 5924–5946.
Hildebrand, J.G. and Shepherd, G.M. (1997) Mechanisms of olfactory discrimination: converging evidence for common principles across phyla. Annu. Rev. Neurosci., 20, 595–631.[Web of Science][Medline]
Homberg, U., Christiansen, T.A. and Hildebrand, J.G. (1989) Structure and function of the deutocerebrum in insects. Annu. Rev. Entomol., 34, 477–501.
Hopfield, J.J. (1991) Olfactory computation and object perception. Proc. Natl Acad. Sci. USA, 88, 36462–36466.
Joerges, J., Küttner, A., Galizia, C.G. and Menzel, R. (1997) Representations of odours and odour mixtures visualized in the honeybee brain. Nature, 387, 285–288.
Kaissling, K.E. (1987) R.H. Wright Lectures on Insect Olfaction. Edited by Konrad Colbow, Simon Fraser University, Burnaby, British Columbia, Canada.
Kang, J. and Caprio, J. (1997) In vivo responses of single olfactory receptor neurons of channel catfish to binary mixtures of amino acids. J. Neurophysiol. , 1, 1–8.
Kleinfeld, D., Delaney, K.R., Fee, M.S., Flores, J.A., Tank, D.W.
and Gelperin, A. (1994) Dynamics of propagating waves in the
olfactory
network of a terrestrial mollusk: an electrical and optical study. J. Neurophysiol., 72, 1402–1429.
Koza, J.R. (1992) Genetic Programming. MIT Press, Cambridge, MA.
Laurent, G. (1996) Dynamical representation of
odors by
oscillating and evolving neural assemblies. Science, 274,976
–979.
Laurent, G. and Davidowitz, H. (1994) Encoding
of olfactory information with oscillating neural assemblies. Science, 265, 1872–1875.
Laurent, G. and Narghi, M. (1994) Odorant-induced oscillations in the mushroom bodies of the locust. J. Neurosci., 14, 2993–3004.[Abstract]
Laurent, G., Wehr, M. and Davidowitz, H. (1996)
Temporal representations of odors in an olfactory network. J. Neurosci., 16, 3837–3847.
Lemon, W.C. and Getz, W.M. (1997) Temporal resolution of general odor pulses by olfactory sensory neurons in American cockroaches. J. Exp. Biol., 200, 1809–1819.[Abstract]
Lemon, W.C and Getz, W.M. (1999) Responses of cockroach antennal projection neurons to pulsatile olfactory stimuli. Ann. N.Y. Acad. Sci., 855, 517–520.[Web of Science][Medline]
Li, Z. (1990) A model of olfactory adaptation and sensitivity enhancement in the olfactory bulb. Biol. Cybernet., 62,349 –361.[Web of Science][Medline]
Li, Z. and Hopfield, J.J. (1989) Modeling the olfactory bulb and its neural oscillatory processings. Biol. Cybern., 61, 379–392.[Web of Science][Medline]
Linster, C. and G. Dreyfus (1996) A
model for
pheromone discrimination in the insect antennal lobe: investigation of the role of neuronal
response pattern complexity. Chem. Senses, 21, 19–27.
Linster, C. and Hasselmo, M. (1997) Modulation of inhibition in a model of olfactory bulb reduces overlap in the neural representation of olfactory stimuli. Behav. Brain Res., 84, 117–127.[Web of Science][Medline]
Linster, C. and Masson, C. (1996) A neural model of olfactory sensory memory in the honeybee's antennal lobe. Neural Computat., 8, 94–114.
Linster, C. and Smith, B.H. (1997) A computational model of the response of honey bee antennal lobe circuitry to odor mixtures: overshadowing, blocking and unblocking can arise from lateral inhibition. Behav. Brain Res., 87, 1–14.[Web of Science][Medline]
Linster, C., Kerszberg, M. and Masson, C. (1994) How neurons may compute: the case of insect sexual pheromone discrimination. J. Computat. Neurosci., 1, 231–238.
Linster, C., Masson, C., Kerszberg, M., Personnaz, L. and Dreyfus, G. (1993) Computational diversity in a formal model of the insect olfactory macroglomerulus. Neural Computat., 5, 228–241.[Web of Science]
MacLeod, K. and Laurent, G. (1996) Distinct mechanisms for synchronization and temporal encoding patterning of odor-encoding assemblies.Science , 274, 976–979.
Malaka, R., Ragg, T. and Hammer, M. (1995) Kinetic models of odor transduction implemented as artificial neural networks—simulations of complex response properties of honeybee olfactory neurons.Biol. Cybernet. , 73, 195–207.[Web of Science][Medline]
Malaka, R., Schmitz, S. and Getz, W.M. (1996) A self-organizing model of the antennal lobes. In Maes P., Mataric M.J., Meyer J.-A., Pollack J. and Wilson, S.W. (eds), From Animals to Animats 4: Proceedings of the Fourth International Conference on Simulation of Adaptive Behaviour. MIT Press, Cambridge, MA, pp. 45–54.
Malun, D. (1991a) Synaptic relationships between GABA-immunoreactive neurons and an identified uniglomerular projection neuron in the antennal lobe of Periplaneta americana : a double-labeling electron microscope study.Histochemistry , 96, 197–207.[Web of Science][Medline]
Malun, D. (1991b) Inventory and distribution of synapses of identified uniglomerular projection neurons in the antennal lobe of Periplaneta americana. J. Comp. Neurol., 305, 348–360.[Web of Science][Medline]
Masson, C. and Linster, C. (1995) Towards a cognitive understanding of odor discrimination: combining experimental and theoretical approaches. Behav. Process., 35, 63–82.
Masson, C. and Mustaparta, H. (1990) Chemical
information processing in the olfactory system of insects. Physiol. Rev., 70, 199–245.
Menzel, R., M.Hammer, G. Braun, J.Mauelshagen, M.Sugawa (1991) Neurobiology of learning and memory in honeybees. In Goodman, L.J. and Fisher, R.C. (eds), The Behaviour and Physiology of Bees. CAB International, Wallingford.
Meredith, M. (1992) Neural circuit computation: complex patterns in the olfactory bulb. Brain Rese. Bull., 29,111 –117.
Michelson, H.B. and Wong, R.K.S. (1991)
Excitatory synaptic response mediated by GABAA receptors in the
hippocampus.Science
, 253, 1420–1423.
Moore, P.A. and Atema, J. (1988) A
model of a
temporal filter in chemoreception to extract directional information from a turbulent odor plume.Biol. Bull.
, 174, 355–363.
Moore, P.A. and Atema, J. (1991) Spatial information in the threedimensional fine structure of an aquatic odor plume. Biol. Bull., 181, 408–418.[Abstract]
Murlis, J. (1986) The structure of odor plumes. In Payne, T.L., Birch, M.C. and Kennedy, C.E.J. (eds), Mechanisms in Insect Olfaction. Oxford University Press, Oxford, pp. 27–38.
Murlis, J., Elkington, J.S. and Cardé, R.T. (1992) Odor plumes and how insects use them. Annu. Rev. Entomol., 37, 505–532.[Web of Science]
Osorio D., Getz, W.M. and Rybak, J. (1994) Insect vision and olfaction: different neural architectures for different kinds of sensory signals? In Cliff, D., Husbands, P., Meyer J.-A. andWilson, S.W. (eds), From Animals to Animats 4: Proceedings of the Fourth International Conference on Simulation of Adaptive Behaviour. MIT Press, Cambridge, MA, pp. 74–80.
Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. (1986) Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, p. 818.
Rospars, J.P. (1983) Invariance and sex-specific variations of the glomerular organization in the antennal lobes of a month, mamestra brassicae, and a butterfly, Pieris brassicae. J. Comp. Neurol., 220,80 –96.[Web of Science][Medline]
Rospars, J.P. (1988) Structure and development of the insect antennodeutocerebral system. Int. J. Insect Morphol. Embryol., 17, 243–294.
Rospars, J.-P. and Fort, J.-C. (1994) Coding of odour quality: roles of convergence and inhibition. Computat. Neural Sys., 5, 121–145.
Rospars, J.P. and Hildebrand, J.G. (1992) Anatomical identification of glomeruli in the antennal lobes of the male sphinx moth Maduca sexta. Cell Tissue Res., 270, 205–227.[Web of Science][Medline]
Sass, H. (1978) Olfactory receptors on the antenna of Periplaneta: response constellations that encode food odors. J. Comp Physiol., 128, 227–233.
Schild, D. and Riedel, H. (1992) Significance of glomerular compartmentalization for olfactory coding. Biophys. J., 61, 704–715.[Web of Science][Medline]
Segev, I. (1992) Single neurone models: oversimple, complex and reduced. Trends Neurosci., 15, 414–415.[Web of Science][Medline]
Selzer, R. (1984) On the specificities of antennal
olfactory receptor cells of Periplaneta americana. Chemical Senses, 8, 375–395.
Shepherd, G. M. (1994) Neurobiology, 3rd edn. Oxford University Press, New York.
Skarda, C.A. and Freeman, W.J. (1987) How brains make chaos in order to make sense of the world. Behav. Brain Sci., 10, 161–195.
Smith, B.H. and Getz, W.M. (1994) Non-pheromomal olfactory processing in insects. Annu. Rev. Entomol., 39, 351–375.[Web of Science]
Souek, B. (ed.) 1992) Dynamic, Genetic, and Chaotic Programming.
John Wiley & Sons, New York.
Stopfer, M., Bhagavan, S., Smith, B.H. and Laurent, G. (1997) Impaired odour discrimination on desynchronization of odour-encoding neural assemblies. Nature, 390, 70–74.[Medline]
Sun, X.-J., Fonta, C. and Masson, C. (1993) Odour
quality processing by bee antennal lobe neurons. Chem. Senses, 18,355
–377.
Vareschi, E. (1971) Duftunterscheidung bei der Honigbiene—Einzelzell-Ableitungen und Verhaltensreaktionen. Z. Vergl. Physiol., 75, 143–173.
Wang, D., Buhmann, L. and von der Malsburg, C. (1991) Pattern segmentation in associative memory. In Davis, J.L. and Eichenbaum, H. (eds), Olfaction: A Model System for Computational Neuroscience. The MIT Press, Cambridge, MA, pp. 213–224.
Wang, X.J. and Rinzel, J. (1992) Alternating and synchronous rhythms in reciprocally inhibited model neurons. Neural Computat., 4, 84–97.
Wehr, M. and Laurent, G. (1996) Odour encoding by temporal sequences of firing in oscillating neural assemblies. Nature, 384, 162–166.[Medline]
White, J., Hamilton, K.A., Neff, S.R. and Kauer, J.S. (1992) Emergent properties of odor information coding in a representational model of the salamander olfactory bulb. J. Neurosci., 12, 1772–1780.[Abstract]
Yokoi, M., Mori, K. and Nakanishi, S. (1995)
Refinement of odor molecule tuning by dendrodendritic synaptic inhibition in the olfactory
bulb.Proc. Natl Acad. Sci. USA
, 92, 3371–3375.
Accepted February 9, 1999
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