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Modeling the Architecture of the Olfactory System ∗ Modeling the Architecture of the Drosophila Ol-factory System The object of this paper is to present a model of the architecture of the olfactory systemof the Drosophila.
Our initial model of the fly olfactory system (see Figure 1) consists of four cascadedbuilding blocks (networks). Each block models a transduction, information representa-tion or processing stage of odor information.
The first block represents the network of receptors and the dendritic arbor (cilia) of the OSNs. The input to this network is provided by odor molecules binding to thereceptors. Each output of the receptor network is an analog waveform that models thedendritic current feeding the soma of exactly one of the olfactory sensory neurons.
The second block is the network of OSNs. The input to this network is identical to the output of the receptor network. Its output models the multidimensional spike traingenerated by the OSN assemblies.
The third block models the glomerular network. The glomerular network accepts as input the spike train generated by the OSNs and has as output the dendritic treecurrent feeding each of the somas of the PNs. Hence, the output of the glomerularnetwork is modeled as an analog waveform. In addition to the direct axon/dendritic ∗BNET Technical Report #2-05, Department of Electrical Engineering, Columbia University, New arbor connectivity, the circuit diagram of the glomerular network is characterized by theinterconnectivity between glomeruli due to local interneurons and (possibly by) feedback.
Feedback between the glomerular network and the network of PNs is shown by the redarrow in Figure 1.
The fourth building block models the network of projection neurons. The input to this network coincides with the output of the glomerular network whereas its output ismodeled as a multidimensional spike train.
olfactory system begins with themapping of an odor by the subset vates. This odor receptor map (im-plemented as a network) creates atime dependent receptor image (or OSN and the PN images (i.e., spiketrains) can be readily recorded us- Figure 1: The Architecture of the Olfactory System ing whole-cell recording, the recep- tor image and the glomerular im-age are difficult to measure exper- imentally. Elucidating the nature of odor representation in the initial stages of theolfactory system, however, calls for understanding the main structural characteristics ofall four images. We shall demonstrate in the next section how to recover the receptorand the glomerular image from the OSN and the PN image, respectively. This enablesus to identify the wiring diagram of the glomerular network from neuron spiking data.
An olfactory system must solve the problem of odor detection, recognition and seg- mentation [12]. Segmentation is necessary because the odor environment often containstwo or more odors. The system must be able to identify these objects separately andsignal their presence to higher brain areas. Odor recognition must be concentrationinvariant over a broad range of odor concentrations [3].
In the language of communication theory odor recognition calls for finding good par- titions of the multidimensional odor space, mapping odors into such partitions througha sequence of processing steps and identifying the partitions with the appropriate odors.
Accordingly, we shall model the overall operation of the olfactory system of the Drosophilaas a real-time odor detection system (in the sense of [15]). From a theoretical point ofview, the detector is viewed as an instantiation of a hypothesis testing system. In its simplest form, the null hypothesis or H0 is “lack of odor” (only spontaneous activity ispresent). The alternative or H1 is “odor plus spontaneous activity is present” (sponta-neous activity is usually interpreted as noise). This basic model can be easily generalizedto both the case of an arbitrary number of odors as well as the case of segmenting (orextracting) an odor from a mixture of odors.
What is a possible realization of the overall odor detection system beyond the four distinct networks depicted in Figure 1? Since the data available about the higher braincenters is scant, the following comments have only a suggestive value. Genetic experi-ments have determined that the precise spatial map in the antennal fly lobe is representedin the protocerebrum [18]. Therefore, the dimensionality of the higher order olfactorynetworks can be expected to be preserved, that is, it is the same as that of the net-works in Figure 1. The functionality of the glomerular network is interpreted here asa spatial (non-linear) filter whose dimensionality (or cardinality) is given by the maxi-mum number of taps (outputs corresponding to distinct glomeruli). For a single odor ofgiven concentration, a number of taps “light up”. These active taps give rise to a single“symbol” (odor representation in the antennal lobe). For a segmentation problem withtwo odors, a more complex pattern of activity will be discernible at these taps. Thisactivity pattern typically exhibits “intersymbol interference”. (The two patterns eachcorresponding to a single odor overlap, or interfere.) Up to the same number of tapscharacterize one or more olfactory networks residing in the higher brain centers. Therole of these networks is to simply remove, to the extent possible, the intersymbol inter-ference. Finally, the last decision step in the detection system is executed by a networkof cardinal neurons (also labeled as taps) in a “voting network” that maps symbols intounique odor outcomes.
Neural decision models are also employed in other areas of (systems) neuroscience [13] where they are often associated with behavioral experiments. Our model, however,is markedly different from other computational models proposed for the olfactory systemof a number of insects or mammals [5], [1], [2], [3]. The latter models use timing basedcomputation of synchrony of oscillatory waves observed in the olfactory system of theseorganisms. To the best of our knowledge there is no clear indication in the publishedliterature of oscillations in the olfactory system of the fly.
Relationship to Experimental Observations The task of the olfactory system is to separate different odor representations throughprocessing. Evolutionary tinkering [7], we believe, created an olfactory system in thefly that exploits through a small sequence of processing steps the distance betweenodors in the coding space and, thereby, makes accurate detection and segmentationdecisions. In what follows we shall discuss methods for (i) evaluating the odor codingspace and, (ii) extracting an input/output characterization of the antennal lobe of thefly. Referring to Figure 1, our model of the architecture of the olfactory system calls for a complete characterization of the space of receptor images and the transfer function ofthe glomerular network. A combination of experimental and theoretical tools is used forthis purpose.
By defining the odor coding space as the space spanned by receptor images, the most faithful representation of odors in the olfactory system is obtained. By further assumingthat the set of dendritic currents is bandlimited (i.e., these currents have a boundedrate of change), the construction of the coding space and its investigation with tools ofinformation theory becomes tractable.
In the experiemental literature, the input/output description of the olfactory lobe is typically given in terms of the activity patterns of OSNs and of PNs [18], [19], [17]. Byassuming that the OSN and PN spike trains are essentially equivalent representationsof their respective input image (i.e., dendritic tree currents), the input/output trans-fer function of the olfactory lobe is reduced to the transfer function of an equivalentglomerular network accepting the receptor image as input and the glomerular image asoutput.
The input to each glomerulus, however, is the sum of all activity patterns of the axons of the neurons expressing the same receptor. We postulate that this convergencecan be modeled as a beamforming operation. In beamforming, a widely used strategyin array processing [8], the summation of multiple observations of the same phenomenaleads to an increase in the signal-to-noise ratio. Operationally, the input to the equiva-lent glomerular network can be aggregated using an appropriate sum of receptor images.
By reducing the dimensionality, input aggregation greatly simplifies the evaluation ofthe transfer function of the equivalent glomerular network. Working with the (receptor,glomerular) image pair instead of the (OSN, PN) image pair also adds an additionaldegree of “hardware independence”. This is because spike trains generated by anatom-ically identical neurons might physiologically be substantially different. The receptorand glomerular images are largely insensitive to these differences.
In the next section we shall present our approach to modeling and characterization of the odor coding space and the input/output characterization of the glomerular network.
Functional Characterization of the Architecture What are the limits of the olfactory system in the Drosophila? Is there a number of odorsbeyond which the system is not capable of recognizing odors with high probability? Aback of the envelope calculation suggests that, for a given concentration, the numberof recognizable odors could be anywhere between n and 2n, where n is the numberof glomeruli. For computing this rather rough estimate, we assumed that the timeaverage of the neural activity is averaged on some small time interval. To investigatesuch questions, there is a need to go beyond empirical results and set up a formalmathematical model. The first step in the process calls for defining the odor coding space. Once established, the coding space can be investigated with tools of informationtheory [4].
One simple way to define the odor coding space is to consider the space spanned by the set of OSN images. This approach has the advantage that, for given odor stimuli,the OSN spike trains can be readily measured in different flies. The disadvantage ofthis methodology, however, is that spiking neurons both within the same organism andamong different sample organisms might vary. As a result the typical raster diagramsdepicting the spike trains of the OSNs while qualitatively similar, display visible vari-ations. What is the information the OSNs carry? Are variations in the timing andnumber of spikes among anatomically identical neurons in different sample organismsdue to the underlying “hardware”? A satisfactory answer to these questions is neededin order to tackle the nature of the processing taking place in the antennal lobe.
We shall model the receptor image as a continuous bandlimited function where s denotes the olfactory sensory neuron, o denotes the odor, r the receptor, c theconcentration and t the time variable. Thus, for a given odor receptor pair (o, r) andodor concentration c, a time function models the dendritic tree current feeding the somaof an OSN expressing the receptor r.
Each coordinate of the receptor image in Figure 1 is mapped by an OSN into a spike train. (ts ) denotes the sequence of spike times at the output of network of the olfactory sensory neurons. The set of these neurons is denoted by S. The coding space is definedby U = {us, s ∈ S|us = us(o, r, c, t), o ∈ O, r ∈ R, c ∈ C}, where O is the family ofodors, R is the set of receptors and C is the concentration range. For example, o = CO2is an odor in the family O and r =OR22a is a member of the receptor set R. For thefruit fly the number of detectable odors in not known. The set of receptors currentlystands at 60. Thus, the coding space at the input to the OSN network is parametrizedalong three dimensions. One of the dimensions is given by the set of odors, one bythe set of receptors and one by the concentration. Hence, the family U of continuoustime bandlimited functions u introduced above represents the set of the odor space/timecodes.
The above formalism can be used for reasoning about the nature of the odor code.
A purely combinatorial space code would aggregate (abstract) information of the timecomponent in u thus effectively providing information about the odor only throughthe activated receptors. Since receptors with the same identity uniquely map into thesame glomerulus, the activation of the glomeruli could be used for odor recognitionand segmentation. A purely time code on the other hand, would aggregate receptorinformation and map it into the time domain. Explicit knowledge about the activated receptor would not be made available to higher processing centers. Without capacityconstraints, a combined four dimensional (odor, receptor, concentration, time) codeprovides for the largest possible coding space. Information theory [4] teaches us thatthis space can not be enlarged by processing.
While taking measurements of the receptor image u appears to be at least for now a daunting task, algorithms for estimating the dendritic current based on informationprovided by the OSN spike trains have been developed. The OSN spike train can bereadily recorded for various values of the triplet (o, r, c).
The characteristics of the spike train including, the odor response spectrum, the spon-taneous firing rate, the signaling mode (excitatory or inhibitory) and the time constantof the response to stimuli (response termination) [6] must be reflected in the dendriticcurrent as well.
An algorithm for perfect recovery of the stimulus of an integrate-and-fire neuron from reading the spiking times at its output was derived in [10] and [11]. This algorithm canreadily be tailored for recovering the receptor image u = u(o, r, c, t), t ∈ R, based on theknowledge of the trigger (spike) times (tk), k ∈ Z. In order to simplify the notation inthis section we dropped the superscript s specifying the olfactory sensory neuron and, Rand Z denote the real numbers and the integers, respectively. The structure of a decoderimplementing the perfect recovery algorithm is highly intuitive. Spikes are generated attimes sk, sk = (tk+1 + tk)/2, with weight ck, k ∈ Z, and then passed through an ideal lowpass filter with unity gain for ω ∈ [−Ω, Ω] and zero otherwise, where Ω is the bandwidthof x. Thus, the output of the decoder is given by where g(t) = sin(Ωt)/πt for all t, t ∈ R. The ck’s, are the solution to a linear equationthat will be discussed below. For describing the adaptation of our main theoretical result, the following notation will be used: g = [g(t − s ) du] and finally, r = RC·ln[1− δ−y(t0) ]· Ω , where y(t0) is the resting potential, δ is the threshold voltage, R and C are the parameters ofthe (leaky) integrate-and-fire model and, a and b are constants (see below).
Theorem 1 (Perfect Recovery Algorithm) Let u = u(o, r, c, t), t ∈ R, be a boundedstimulus |u(o, r, c, t)| ≤ a < b bandlimited to [−Ω, Ω]. If r < 1, the stimulus u can beperfectly recovered from (tk)k∈ as where G+ denotes the pseudo-inverse of G.
We have shown [9] that this algorithm can be re-written in such a way as to become threshold insensitive. Therefore, the receptor image u does not depend on the value ofthe threshold voltage of the olfactory sensory neuron. This is rather obvious from anexperimental standpoint. However, since direct measurements of the receptor image arenot yet available, an estimate of the receptor image needs to be provided that does notdependent on the particular details of the cellular mechanism that generates spikes.
Beamforming: Characterizing the Aggregated Input to theGlomerular Network Our working assumption in this subsection is that OSNs expressing the same receptorconverge on the same glomerulus in order to increase the signal-to-noise ratio at theinput to the olfactory lobe. This is akin to beamforming in array processing whereobservations in vector form correspond to multiple sensing devices [8]. The dimension ofthe observation vector is given by the number of neurons expressing the same receptor.
Intuitively, beamforming of spiking neuron data would suggest that the activity patternof neurons converging on the same glomerulus can be generated by an “ideal” equivalentneuron with improved signal-to-noise ratio.
Input/Output Characterization of the Glomerular Network The interconnectivity between the glomeruli in the antennal lobe is determined by aset of local interneurons [14]. In order to understand how these interneurons affect thetransfer of odor information between the input and output, the transfer function of theantennal lobe can be identified following an established methodology in system theoryand spectral analysis [16]. The transfer function is simply the ratio between the Fourierspectrum of the glomerular and the receptor images. The transfer function provideskey information regarding the functionality of the circuit diagram of the glomerularnetwork. Note that the PN image and the OSN image can not be used for evaluatingthe transfer function because the neuron-induced non-linearities lead to unmanageablespectral components.
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