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Persaud KC, Marco S, Gutiérrez-Gálvez A, editors. Neuromorphic Olfaction. Boca Raton (FL): CRC Press/Taylor & Francis; 2013.
2.1. INTRODUCTION
The olfactory system has been optimized over evolutionary time to perform an exquisite function: analyze odorant molecules by their molecular features, and synthesize holistic representations of them when presented in complex mixtures. It has been estimated that the olfactory system is able to detect approximately 10,000 odors (Axel 1995) over a large range of concentrations. However, unlike the sense of hearing or vision, this modality has been elusive to psychophysical analysis because no simple set of physical properties, such as light wavelengths for sight or sound frequency for hearing, has been found. Rather, olfaction appears to be intrinsically multidimensional. Along with the multidimensional nature of olfaction, the striking similarity of different olfactory systems across phyla (Hildebrand and Shepherd 1997) suggests that its architecture has been optimized to reflect basic properties of olfactory stimuli.
The objective of this study is to analyze how odor intensity and odor quality information is encoded on the first stages of the olfactory pathway: the olfactory epithelium and the olfactory glomerular layer. To study the olfactory epithelium, we built computational models of olfactory receptor neuron (ORN) populations based on their experimental statistical distributions. These models are based on the detailed characterization of the odor concentration response of ORN populations reported by Rospars et al. (2003). To study the glomerular layer, we modeled the ORN axon projections and lateral inhibitory interactions occurring at the olfactory glomeruli. The odor intensity and odor quality information conveyed by these two stages of the olfactory system is evaluated using the information theory. We consider the amount of information transmitted as a measure of the efficiency of the coding strategy followed at each stage.
In this chapter we first present an introduction that contains background regarding the early stages of the olfactory system, the ORN models used in this work, and a description of the information theoretic measure used: the mutual information. Then, we present the three studies performed in this work. In Section 2.2, we study the coding of odor intensity at the olfactory epithelium. In the next section (2.3), we perform the odor intensity study at the following anatomical stage: the glomerular layer. Finally, in Section 2.4 the study of odor intensity is extended to odor quality as well and applied at the olfactory epithelium.
2.1.1. The Early Olfactory Pathway
We focus our study on the first two stages of the olfactory pathway: the olfactory epithelium and the glomerular layer within the olfactory bulb. In the olfactory epithelium, the molecular properties of the odorants are transduced into electrical signals through a collection of olfactory receptor neurons (ORNs). Mammals have tens of millions of ORNs (Hildebrand and Shepherd 1997; Doty 1991), which belong to as many as 1000 different types of receptors (Ma and Shepherd 2000). The prevailing hypothesis about olfactory primary reception is that ORNs do not respond to specific molecules, but rather to specific molecular features of an odorant molecule, commonly referred to as odotopes (Shepherd 1987, 1994), such as carbon-chain length, the presence of benzene rings, or different functional groups (e.g., ester, aldehydes). Considering that most odorants in the environment consist of mixtures of volatile molecules (e.g., roasted coffee has been estimated to contain the order of 600 volatile components), and that each molecule can contain several odotopes, an odorant is then detected as a large combination of specific odotopes.
ORNs project in a very orderly fashion into spherical regions of neuropil known as glomeruli. Each glomerulus receives axons from one type of ORN, and each ORN type projects into one or a few glomeruli (Vassar 1994; Ressler et al. 1994). Therefore, at the glomerular level, olfactory information can be thought of as being represented by an image of the molecular features of the stimulus. Two types of neurons can be found within the glomerular layer: projection neurons (mitral and tufted cells) and local interneurons (periglomerular and short axon cells) (Shepherd and Greer 2004; Aungst et al. 2003). Projection neurons receive inputs from their main dendrite, which is located at the glomeruli, and transmit this activation to higher areas of the olfactory system. The second type of cells, local interneurons, provides lateral connectivity between projection neurons at two different ranges of the glomerular layer. First, periglomerular cells provide inhibitory interconnection between projection neuron dendrites within glomeruli. Second, short axon cells interconnect tufted cells and periglomerular cells between glomeruli. Even though short axon cells are excitatory, they synapse to periglomerular cells, resulting in an inhibitory effect.
2.1.2. Olfactory Receptor Neuron Model
The olfactory receptor neuron response models used in this work were proposed by Rospars et al. (2003). These models capture the relationship between ORN output action potentials and odor concentration using three parameters: frequency, latency, and duration of action potentials (Figure 2.1). For each one of these parameters, they find a mathematical expression that best fits the variation of the parameter with odor concentration. This fitting is performed with the extracellular recordings of the spiking activity of receptor neurons in the frog olfactory epithelium when four odorants were presented at precisely controlled concentrations.
2.1.2.1. Response Model for Different Odorant Concentrations
The expressions proposed by Rospars et al. (2003) for the frequency, latency, and duration of the output action potentials are the following:
Frequency:
Latency:
Duration:
where FM is the maximum firing frequency, CT is the threshold concentration, and n controls the slope of the dose-response frequency curve. In Equation 2.2, Lm is the minimum latency, LM= La + Lm is the maximum frequency, and λ is the slope. In Equation 2.3 the maximum duration at the dose CM is DM. The concentration is defined as C = log10 M, and M is the molarity of the saturated vapor. Figure 2.2 shows the values of the spiking frequency (a), response latency (b), and response duration (c) extracted from Equations 2.1 to 2.3, respectively, as the odor concentration is increased from –7 to –3.
2.1.2.2. Response Model for Different Odorants and Odorant Concentrations
Rospars et al. (2003) also presented an anatomic electrical model that captured the frequency rate of the ORNs in terms of the odorant concentration and number of other structural parameters of the neuron. One of these parameters is the affinity of the ORN to specific ligands that characterizes its response to different odorants. The model accounts for the three main steps of chemotransduction: the change of conductance due to opening of ion channels, the potential of the axon at its initial segment, and the firing frequency. The following three equations correspond to each one of these steps:
Membrane conductance:
Receptor potential:
Firing frequency:
2.1.3. Mutual Information
Information theory (Shannon and Weaver 1949; Cover and Thomas 2006) provides a theoretical framework to quantifying the ability of a communication channel or coding scheme to convey information. This theory is suitable to evaluate the coding efficiency of a population of ORNs (Rieke et al. 1997). In this study, we are interested in measuring to what extent the amount of information at the input of the ORNs is transmitted to their output. To measure this quantity, information theory provides a magnitude called mutual information (MI), which is defined as
where yj and xi are the input and output states, p(yi) and p(xj) are their probability, and Ny and Nx are the number of possible input states, respectively.
2.2. STUDY OF ODOR INTENSITY CODING IN AN ORN POPULATION
In this section we estimate the information capacity of a synthetically generated population of olfactory receptor neurons (ORN) expressing the same olfactory receptor protein (ORP). This study is aimed at improving our understanding of odor intensity encoding at the olfactory epithelium. The ORNs of the olfactory epithelium belong to different types depending on the ORP they express (Buck and Axel 1991). Different ORPs allow binding to different odorant molecules or molecular features. Consequently, the quality of the odorants is captured at the olfactory epithelium as a population code across ORNs of different types. At the same time, the intensity of the odorants is thought to be mainly encoded by the firing frequency of the ORNs (Hildebrand and Shepherd 1997).
However, the apparent simplicity of this intensity coding scheme has been challenged by recent experimental results. Grosmaitre et al. (2006) have reported that ORNs expressing the same odorant receptor (MOR23) do not respond in the same way to the same amount of odorant. Even though all ORNs expressing MOR23 respond to the same odorant, each one of them has a different dose-response curve. The dose-response curve reflects the activity of each ORN in terms of the concentration of odorant used to elicit that activity. This cellular heterogeneity suggests that odor intensity is encoded as a population code across ORNs of the same type.
In this study we propose a computational approach to study the advantages of heterogeneous ORN populations, as opposed to homogeneous ones, to encode for odor intensity. We have generated a population of ORN mimicking the existent diversity on a population of ORN of the same type. The population has been generated following the statistical distributions experimentally found by Rospars et al. (2003) for the different parameters of the ORN: firing frequency, firing threshold, maximum frequency, response duration, and latency (Equations 2.1 to 2.3). To understand the role that each one of these parameters play on the odor intensity coding process, we analyze the information conveyed by the population as the statistical distribution of these parameters is varied.
We have performed this study for two types of ORN models: static and dynamic. Since the firing frequency is supposed to capture most of the odorant information, we built first a static model that considered only the firing frequency of the ORN and not time-dependent parameters. Second, we extended it by building a dynamic model of individual ORNs, which considers not only the firing frequency but also the latency, and the duration of the spike train. This will allow our study to evaluate the role of the time on odor intensity encoding.
2.2.1. Static ORN Populations
An interesting question that arises about the encoding ability of any neuron population is the following: What is the coding advantage of using a population with different neuron types (inhomogeneous population), as opposed to encoding using one with the same neuron type (homogeneous population)? With the aim of addressing this question, we have generated a series of ORN populations with increasing dissimilarity of the ORN within populations and compared its mutual information.
Before we analyze the information transmission, there are two issues that require special attention, namely, the stimuli distribution and the quantification level of the frequency axis. To the best of our knowledge, a statistical distribution for the input odor concentration to the ORNs has not been studied. Therefore, our position is to provide the simpler stimuli distribution to our populations: a uniform distribution. About the second issue, the discretization of the frequency axis, we are in a similar situation. Due to a lack of experimental studies, as far as we know, about the frequency resolution of ORNs, we have chosen to split the frequency axis in 10 bins. It is worth noticing that any other selection would only scale our results on mutual information.
To evaluate the mutual information of these populations, we generate random stimuli (concentration values) from a uniform distribution between –3 and –7 log units (dose C). The mutual information is computed over the accumulated response of all ORNs to all stimuli (Alkasab et al. 2002). To isolate the contribution of each one of the parameters Fm, n, and CT to the mutual information, we generated three series of ORN populations in which only one of the parameters is varied and the other two remain constant.
It is convenient to study the effect of each one of the parameters individually to determine its contribution to the information transmission. Figure 2.2a shows the relationship between spiking frequency and odor concentration as expressed by Equation 2.1. This relationship depends upon three parameters: threshold concentration (CT), n factor (n), and maximum frequency (FM). FM and CT are described in Figure 2.2a, and the n factor determines how fast the curve reaches the maximum frequency (FM). The different firing frequency characteristics of different ORNs in the olfactory epithelium can be described choosing different values of these three parameters. Therefore, we can generate a population of ORNs by assigning certain statistical distributions to each one of the parameters.
2.2.1.1. Variation of the n Factor
In this first series of populations, the ORN within the population will have the same Fm = 2.44 Hz and CT = –6 log unit, but the n parameter will be generated from a lognormal distribution with μ = 2.44 log units (these experimental values are extracted from Rospars et al. (2003)). The standard deviation (σ) of this distribution is used as a parameter to control the similarity of the ORNs. For σ = 0 log units, all the ORNs have the same parameters, so we have a homogeneous population. As σ increases, the variety of ORNs will increase accordingly. We have generated a series of populations for increasing values of σ. Each population comprises 10,000 neurons and is excited with 1000 stimuli. Figure 2.3a shows the mutual information of these populations in terms of their standard deviation. The mutual information increases initially until it reaches a maximum around σ = 1 log unit; after that, it decreases monotonically.
A first conclusion we can extract from this result is that a homogeneous population (σ = 0 log units) provides less efficient encoding than other populations with some variety of the n factor. This partially answers the question posed at the beginning of this section. A second conclusion that can be extracted is that there exists a σ value that provides an optimum encoding of the concentration information. This result can be further analyzed by observing the concentration-frequency curves for the different populations. Figure 2.3b shows these curves for several values of σ. For σ = 0 log units we have that all the ORNs share the same curve. As σ increases, the population provides a better coverage of the input-output space, making more homogeneous the output distribution, which in turn increases the mutual information. The population reaches a point (σ > 1) where the number of ORNs with very high or very low n is high enough to worsen the homogeneity of the output distribution.
2.2.1.2. Variation of the Threshold Concentration CT
In this second series of populations, the values of n and Fm are fixed to 0.5 and 2.44 Hz, respectively, and CT is generated from a uniform distribution. The range of the uniform distribution is in this case used to control the similarity of the ORNs. As in the previous subsection, the null range generates a homogeneous population, and increasing its value, the ORN similarity is decreased. Twenty-one populations were generated for range values between 0 and 4 log units. Each population comprises 1000 neurons and is excited with 1000 stimuli. Figure 2.4a shows the mutual information for populations with increasing width of CT. The mutual information has a behavior similar to that in the previous subsection, reaching a maximum for range = 1.6 log units.
Similarly to the previous subsection, two conclusions can be extracted from this result. First, having a nonhomogeneous population represents an advantage for the concentration information coding. Second, there exists an optimum width value for the concentration information coding. Figure 2.4b shows the frequency-dose curves for the different populations generated. We can see how as the range increases, the population covers more frequency-dose space and provides a more uniform distribution at the frequency space, which in turn increases the mutual information. However, as the new ORNs have increasing CT, they cover less dose range, and this produces a decrease of the homogeneity at the output space. These two factors compete with each other, producing the maximum of the mutual information.
2.2.1.3. Variation of the Maximum Frequency Fm
Finally, we generated a series of populations fixing the n = 0.5 and CT= –6 log units parameters and generating Fm from a lognormal distribution with μ = 2.44 Hz. The standard deviation of the distribution is again used to control the similarity of the ORN within a population. Sixteen populations are generated with standard deviation between 0 and 3. Each population comprises 1000 neurons and is excited with 1000 stimuli. Figure 2.5 shows the mutual information of these populations. In this case, the mutual information does not have a maximum and decreases monotonically. This result denotes that in terms of the maximum frequency parameter, a homogeneous population is more advantageous than a nonhomogeneous population of ORN.
Summing up, this partial theoretical study of the ORN populations predicts that nonhomogenous populations of ORNs encode more efficiently odor concentration information than homogeneous ones. Furthermore, this analysis provides theoretical evidence for the existence of an optimum spread of the ORN parameters to obtain an efficient odor information encoding. Interestingly enough, these results can serve to explain why the olfactory system has a nonhomogeneous population of ORNs to measure odor concentration.
2.2.2. Dynamic ORN Populations
To complete the previous study, we generated a dynamic ORN population based on an ORN model that considered also time-dependent parameters (latency and duration). It is worth noticing that this dynamic model is considerably different from the static one, and therefore required a different computation of the coding efficiency. The activity of these populations is expressed in timescale since each ORN has its spiking train perfectly defined in terms of latency, duration, and frequency. Figure 2.6 shows the spike trains generated by a population of 700 ORNs, where frequency, latency, and spike duration of each ORN are randomly generated from the distributions described in Rospars et al. (2003). Each row is the spike train generated by one ORN along the time where spikes are represented by blue dots.
To measure the odor concentration information encoded by the ORN populations, we have followed the subsequent procedure explained in Figure 2.7. We have generated populations of 5000 ORNs by assigning to each ORN a different combination of the three parameters mentioned before: spiking frequency, latency, and duration. The values of these parameters are randomly selected from their statistical distributions found by Rospars et al. (2003). To study these populations, it would be desirable to excite them with the distribution of concentrations that is presented to the biological ORN population. However, since there are no studies in this respect, to the best of our knowledge, we have used a random log-uniform distribution in the concentration range of [–7, –3] to excite the ORN populations. Then, the spike trains obtained from each ORN for each concentration are used to compute a histogram in the time axis. The histograms are computed using 10 ms as a bin width. We compute the mutual information of the population from the histogram obtained.
The generation of the ORN population has been driven by the objective of evaluating how efficiently the population encodes for odor concentration information. We would like to evaluate this coding efficiency as the population becomes more heterogeneous. To generate populations of increasing levels of heterogeneity, the parameters of all ORNs are fixed, except for one that is allowed to be different across ORNs. The range of values that this changing parameter is used to increase the heterogeneity of the ORN populations. We have generated a series of populations in which the first one has a null parameter range, meaning that all ORNs are equal. The subsequent populations are obtained by increasing the range of the parameters, therefore increasing their heterogeneity.
We generated two series of populations varying the parameters n and CT of Equations 2.1 to 2.3. Notice that the first parameter only affects the firing frequency (Equation 2.1), not latency and duration. We have chosen this parameter in order to compare the behavior of this complete model with that of the static model, which only considered firing frequency. The second parameter used to generate a series of populations (CT) allowed us to evaluate the behavior of the entire model since firing frequency, latency, and duration depend on CT.
Figure 2.8 shows the mutual information computed on the two series of populations as the range of n and CT is increased. In both cases the mutual information reaches a maximum for a certain value of the range. However, it is important to notice that the amount of bits increased from the null range to the maximum is higher when we vary CT. These results are consistent with those obtained with the static ORN model.
Several conclusions can be drawn from these results. The main of these conclusions is that the information transmitted by the ORN population is maximized when the population is heterogeneous (homogeneous population correspond to zero range value). This is theoretical evidence that supports the superior odor information coding efficiency of heterogeneous ORN populations. Second, there exists an optimal value of dissimilarity within ORN populations that maximizes information transmission. This dissimilarity level may be the one that is found in biological ORN. Finally, a last conclusion can be drawn comparing these results with those obtained with the static model. Using the static model, the increase of mutual information between the homogeneous populations and the optimum heterogeneous one was 0.2 bit, whereas in the current results the increase goes up to about 1.5 bits. This gives an idea of the larger amount of information that is accounted for in the dynamic model. Therefore, the time-dependent parameters that we have added to the ORN model are able to encode for additional information that was not captured by the static model.
2.3. STUDY OF ODOR INTENSITY CODING IN THE GLOMERULI
In this section we continue the coding efficiency study of the olfactory pathway extending the information theoretic analysis of the ORN population with the next stage of the olfactory pathway. This stage involves a projection of ORN axons to spherical regions of the neuropil called glomerulus. This projection is highly ordered since it has been observed that ORNs express the same olfactory receptor project to the same glomerulus. This allows studying odor intensity by focusing only on one glomerulus. Within these spherical structures we can find not only ORN axons, but also dendrites of mitral/tufted cells and periglomerular cells. The interaction of the existent types of cells defines a complex behavior that is not totally understood at this date. However, it is generally considered that there exists an inhibitory effect between ORN axons mediated by either periglomerular cells or microcircuits formed by ORN axons and mitral cell dendrites. These can be modeled as a lateral interaction between the incoming ORN axons before transmitting the odorant information (Figure 2.9). We used this simple interaction model to study the odor concentration information transmission, performing experiments first in a reduced model and second in a more comprehensive one.
2.3.1. Reduced ORN Population
To gain some insight on the effect of the lateral inhibition in the information encoding, we first studied a reduced population to facilitate the interpretation of the results. This toy population is formed by 13 ORNs characterized by their dose-response curves. Figure 2.10a shows the response curves that have been obtained from Rospars et al.’s (2003) model considering only the frequency response. The parameters used to generate the ORNs were the following: constant values of Fm = 10 Hz and CT = –7, and variable values of n from 0.5 to 3 with a step of 0.2.
The effect of the lateral inhibition is obtained by subtracting pairwise the dose-response curve of those ORNs that are connected. This is a first and simple approach to model these interactions, but consistent with the level of abstraction of our population model. The connectivity in our toy problem has been set to link all ORNs to that of a lower frequency response. Figure 2.10b shows the frequency response curve of the pairwise linked ORNs.
To compute the mutual information (MI) of the ORN population before and after the inhibitory stage, we present the population with a stimulus of random concentrations extracted from a uniform distribution in the range of [–7, –5]. The population response to these stimuli is shown in Figure 2.11 as the probability of occurrence of each frequency for the ORN population before (a) and after (b) inhibition. We can see how the effect of the inhibition distributes more evenly the probability of the frequencies. The probability distribution before inhibition (Figure 2.11a) is clearly peaked at 10 Hz because of the saturation of all dose-response curves at 10 Hz (Figure 2.10a), whereas the frequency probability after inhibition (Figure 2.11b) is more evenly distributed. In terms of information transmitted (MI), this translates into a larger amount of information transmitted after the lateral inhibitory layer. The MI values of the ORN population before and after inhibition are 4.35 and 5.46 bits, respectively. The information transmitted has increased by 1.1 bits due to the lateral inhibition.
2.3.2. Comprehensive ORN Population
To make a more thorough study of the effect of the lateral inhibition on the information transmission, we have built a population of ORNs that includes all the variability of ORNs experimentally observed. We have modeled a population of ORNs that project to a single glomerulus and analyzed the concentration information transmitted before and after the inhibitory interaction at the glomerulus. The ORN model used is that proposed by Rospars et al. (2003) for the firing frequency (Equations 2.4 to 2.6). In this mathematical model there are three parameters (Fm, CT, n) that determine the behavior of the individual ORNs. The statistical distributions of these parameters in the frog olfactory epithelium were experimentally found also in this paper. The maximum frequencies (Fm) are randomly extracted, for each ORN of the population, from a lognormal distribution with mean of 2.44 Hz and a standard deviation of 0.44 Hz. The CT parameter is randomly selected from a uniform distribution in the range of –7 to –3. Finally, n follows the inverse of a lognormal distribution centered at 0.2 and with a 1.16 standard deviation. We have used these statistical distributions to generate a population of 20,000 ORNs that project to the same glomerulus. Figure 2.12a shows the dose-response curve of 100 of these 20,000 ORNs. Before we explain the model of inhibition we have used, it is important to remark that our ORNs are characterized by its dose-response curve, which is the spiking frequency in terms of the concentration of the presented odor.
To model the effect of the inhibitory stage, we randomly connected ORN axons limiting the number of connected axons to one. Each axon will be inhibited by another axon, and only by that one. A difficult point was to model the effect of the inhibitory interactions in the dose-response curves. Dose-response curves assume constant frequency of the ORNs’ response for certain odor concentrations, and the interaction of two ORNs with different frequencies may lead to responses with nonconstant frequency responses. However, taking into consideration the time integrating effect of the neurons, we can consider that the result of an inhibitory incoming spike train is to reduce the spike frequency of the target neuron proportionally to the frequency of the inhibitory neuron. Therefore, the dose-response curve after inhibition can be obtained as a direct subtraction of the dose-response curves of the target and inhibitory ORNs. Figure 2.12b shows the dose-response curves of the ORNs after being inhibited by another ORN. In this figure only 100 of the 20,000 ORNs are shown to facilitate visualization.
At this point we can determine the odor concentration information transmitted after the lateral inhibitory stage computing the mutual information. To do so, we have presented the ORN population with 100 different odor concentrations randomly extracted from a uniform probability distribution in the range of [–3, –7]. After computing the dose-response curves of all ORNs of the population, we inhibit the ORN axons following the lateral connections, obtaining the final dose-response curves. The output frequency probability is obtained concatenating the response of all ORNs and computing the histogram from 0 to 40 Hz in 50 bins and normalizing afterwards.
We have studied the effect of the lateral inhibition using a parameter to regulate the intensity of the inhibitory interaction. This parameter will take values between 0 (no inhibition) and 1 (maximum inhibition). The inhibition is maximum when the dose-response curve of the inhibitory neuron is directly subtracted from the dose-response curve of the target neuron. We computed the response of our ORN population for 11 values of the inhibition intensity starting from 0, with an increasing step of 0.1, until 1 is reached. Figure 2.13 shows the output frequency distributions of the ORN populations for the increasing values of inhibition. The first subfigure (top left) represents the frequency distribution as if no inhibition was applied. As the inhibition intensity increases, the central peak of the figure decreases, vanishing almost totally for the maximum inhibition intensity. So the effect of the lateral inhibition is to reduce the importance that the frequencies around 10 Hz have at the output of the population and redistribute this activity among the other frequencies.
Computing the mutual information of the distributions of Figure 2.13 , we obtain the results of Figure 2.14. This figure shows the mutual information in terms of the intensity of the inhibitory interaction. A maximum of the mutual information is obtained for 0.3. Looking at Figure 2.13 , this value corresponds to a frequency distribution where the main peak is almost leveled, with the smaller peak forming a broader peak that distributes the output more evenly among the different frequencies. This shows that a certain amount of inhibition is able to redistribute the output frequencies more evenly, and therefore enhance the amount of odor concentration information that is transmitted. It is important to notice that the increase of mutual information found is not very high, and it does not seem to account completely for the positive effect that lateral inhibition is thought to have in the processing of the odorant information. However, the present results are evidence that lateral inhibition can increase the odor intensity information transmitted by the early stages of the olfactory system, and more significant increases of the mutual information may be obtained by using more accurate models of the inhibitory interaction. The validity of this conclusion is limited by the simplicity of the ORN and inhibitory model adopted in these simulations. A more detailed model of ORN and lateral inhibition may reveal some information transmission advantage not captured here.
2.4. STUDY OF ODOR INTENSITY/IDENTITY CODING IN AN ORN POPULATION
As mentioned in Section 2.1.1, odor identity is captured by the olfactory system using a combinatorial coding strategy. Individual ORNs are able to bind to a number of ligands with different affinities, and any ligand in turn is bound by a number of ORNs. Therefore, each odor (specific collection of ligands) elicits a particular pattern of ORN activation across the ORN population (Buck and Axel 1991). The objective of this section is to evaluate the advantages in information transmission of such a coding strategy to encode for odor identity and intensity. In the previous sections we have limited our studies to odor intensity by using an ORN model that considered the sensitivity of the ORNs to certain odors, but not the selectivity to different odors. In this section we will use an ORN model that takes into account both. This model was proposed by Rospars et al. (2003) and is described by Equations 2.4to 2.6.
2.4.1. ORN Affinity Distribution
To build ORN populations using the previous ORN model, we are still missing a piece of the puzzle. We need to know the distribution of binding affinities of the ORNs to certain ligands within an ORN population. Lancet et al. (1993) proposed to determine this distribution of affinities toward certain ligands using a combinatorial approach. They modeled receptors and ligands as having B binding subsites where each one could be in one of S possible configurations. They assumed a one-to-one correspondence between the B subsites of the receptor and those of the ligand. The binding was successful if at least L of the B subsites of ligand and receptor where in a complementary configuration. Assume that all receptors and ligands have the same B and S, and also that all subsite configurations are equiprobable. This model represents the receptor-ligand binding as composed by B independent and identical elementary experiments where the number of successful interactions (bindings) measures the affinity of receptor and ligand. Therefore, the probability distribution of ORNs for certain fixed ligands in terms of the number of subsites bound (L, affinity) follows a binomial distribution.
This affinity distribution allows modeling the interaction of a receptor with any ligand. In this way, we can characterize each ORN of a population with a vector of N affinities that describes the interaction between the ORN and N different ligands. To build an ORN population that is in accordance to this distribution, we generate groups of N values (affinity vectors) randomly extracted from a continuous binomial distribution. The affinity vector characterizes the frequency response of the ORNs, since the affinity parameter corresponds to the parameter Mg/2 in Equation 2.4. Figure 2.15b shows the dose-response curves of an ORN to the exposure of four ligands. These curves are characterized by an affinity vector of four values. This ORN has higher affinity to odor C than the rest. The following odors, in order of increasing affinities, are A, D, and finally B, which has the lowest affinity to this ORN. A ligand with higher affinity is able to elicit activity (firing frequency) of the ORN, starting from lower concentrations, and produces higher firing rates for the same amount of concentration than other ligands, as we can see for ligand C. To determine the response of the ORN of a mixture of odors, we assume hypoadditivity and consider that the response of the ORN is the highest response to any of the mixture components.
2.4.2. ORN Specificity Study
With this more complete population model, we study again the effect of the specificity of ORN in the coding efficiency. We try to answer the same question that is in previous sections: How does the specificity of receptors in an ORN population affect the quality and concentration of information transmitted by the population? To address this question, we generated a series of ORN populations with varying specificities of the ORNs within populations and compared their mutual information.
The regulation of the specificity of the ORNs is mediated by a modulation factor p that multiplies the ORN affinity vectors. This modulation factor is obtained with Equation 2.8:
where m is the ligand ranging from 1 to 10, and t is a parameter that modulates the specificity of the ORN. It ranges from –1 (high specificity) to 1 (no modulation). Figure 2.16a shows the shape of p for different values of t. When t= –1 the value of p is zero or has a low value for most of the ligands, except for ligand 10, which is 1. That means that this ORN will have a reduced response to any ligand except ligand 10, becoming very specific to detect it. As the value of t increases, so increases the number of ligands and the contribution of all ligands in p, resulting in a straight line for t = 0. This makes the ORN response less specific. When the value of t reaches 1, all ligands contribute almost equally, being equal or close to one value. This results in no modulation of the original ORN affinity vectors.
We generated populations of varying specificity composed of 100 ORNs able to respond to 10 ligands. The response of each ORN was determined by randomly generating 10-dimensional affinity vectors from a continuous binomial distribution (Figure 2.15a). Once these ORN affinity vectors are defined, we modulate them with the p factor (Equation 2.8) to regulate the specificity of the ORNs within each population. We built 11 different populations taking values of the t parameter from –1 (high specificity) to 1 (no modulation), separated by 0.2 each. We presented the population with 100,000 randomly generated odors (mixtures of ligands), where each odor is defined as a 10-dimensional vector with the concentration of each component. The concentrations of the ligands are randomly generated following a uniform distribution within the range of –3 to –7 log units.
To determine the efficiency of these populations when encoding for odor quality and odor intensity with these stimuli, we computed the mutual information between the input and output signals as described in Section 2.1.3. Figure 2.16b shows the results obtained for the 11 populations. The mutual information for the population with the more specific ORNs (t = –1) is the lowest. The mutual information increases then as the specificity of the ORNs decreased within the populations, and it saturates for a maximum value when t reaches 1. This result clearly shows that populations with specific ORNs encode less information than those with less specific ORNs. The saturation of the mutual information value for t approaching 1 is due to the absence of modulation from p. It is close to 1 for all ligands. So, the population exhibits the amount of specificity generated when the affinities are randomly chosen from a binomial distribution. This reflects that this distribution is optimal for the transmission of information.
2.4.3. Population Dimensionality Study
The second study we performed is motivated by the following question: How does the dimensionality of the ORN population affect the information transmission? To address this question, we generated a series of populations with an increasing number of ORNs, from 1 to 15. Each ORN was able to respond to 10 different ligands, and the affinity vectors characterizing this response were obtained with a continuous binomial distribution. In order to compute the input-output information transmission, we present the model with 10,000 input mixture odors. These odors are generated, as in the previous section, as a mixture of the 10 ligands that are randomly extracted from a uniform distribution within the range of –3 to –7 log units. Figure 2.17 shows the value of the input-output mutual information for the population for an increasing number of ORNs. The mutual information increases with the number of ORNs reaching saturation for 10 ORNs. This result shows that beyond 10 neurons, the information encoded by the population does not increase further.
From these results we can draw several conclusions. The first is in line with those obtained in the studies of odor intensity. It is much more efficient to encode the information of a mixture of odors with nonspecific receptors. This analysis provides theoretical evidence for the existence of an optimum spread of the ORN parameters to obtain an efficient odor information encoding. From the second study we can conclude that the dimensionality of the population determines the coding capacity of the population.
2.5. CONCLUSIONS
From the three information theoretic studies presented in this chapter, we can draw several conclusions. First, this work presents theoretical evidence that supports the advantages of nonspecific ORN populations for the encoding of odor intensity as well as for odor identity. Furthermore, it has been shown that there exists a value of the variability of the ORNs that maximizes the transmission of odor information. In the case of odor intensity, this has been shown for static and dynamic models of ORNs.
Second, the lateral inhibition at the glomerular level has been shown to improve the coding efficiency of odor intensity. The limited improvement observed in this work is probably explained by the simplicity of the inhibition models. More detailed models of lateral inhibition can lead to a more significant improvement on information coding efficiency.
Finally, it is worth noting that the studies conducted in this chapter did not consider an important aspect of neural activity: neural noise in the signals. Considering noise, it would be possible to provide a more realistic estimate of the amount of information transmitted by the early olfactory pathway. We can consider the present results as an upper bound of this information.
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