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.

FIGURE 2.1

Three parameters of the ORN output action potentials characterized by Rospars et al.’s (2003) model: latency (L), duration (D), and frequency (F).

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 F_M is the maximum firing frequency, C_T is the threshold concentration, and n controls the slope of the dose-response frequency curve. In Equation 2.2, L_m is the minimum latency, L_M= L_a + L_m is the maximum frequency, and λ is the slope. In Equation 2.3 the maximum duration at the dose C_M is D_M. 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.

FIGURE 2.2

ORN model response in terms of concentration: (a) spiking frequency, (b) response latency, and (c) response duration.

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 y_j and x_i are the input and output states, p(y_i) and p(x_j) are their probability, and N_y and N_x are the number of possible input states, respectively.

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 F_m, n, and C_T 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 (C_T), n factor (n), and maximum frequency (F_M). F_M and C_T are described in Figure 2.2a, and the n factor determines how fast the curve reaches the maximum frequency (F_M). 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 F_m = 2.44 Hz and C_T = –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.

FIGURE 2.3

(a) Mutual information of the static populations with different values of the variance of the ORN n parameter. (b) Frequency-dose curves for populations with increasing value of σ.

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 C_T

In this second series of populations, the values of n and F_m are fixed to 0.5 and 2.44 Hz, respectively, and C_T 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 C_T. The mutual information has a behavior similar to that in the previous subsection, reaching a maximum for range = 1.6 log units.

FIGURE 2.4

(a) Mutual information of the static populations with different values of the C_T distribution range. (b) Frequency-dose curves for populations with increasing value of the C_T range.

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 C_T, 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 F_m

Finally, we generated a series of populations fixing the n = 0.5 and C_T= –6 log units parameters and generating F_m 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.

FIGURE 2.5

Mutual information of static population with increasing values of the standard deviation of F_M.

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.

FIGURE 2.6

Raster plot of the spike trains produced by a population of 700 ORNs. Each of the rows represents different ORNs, and spikes are represented by 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.

FIGURE 2.7

Computation of the mutual information of an ORN population. From the reduced ORN model we built ORN populations and obtained their time responses. The mutual information is then computed from the histogram of the ORN population response.

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 C_T 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 (C_T) allowed us to evaluate the behavior of the entire model since firing frequency, latency, and duration depend on C_T.

Figure 2.8 shows the mutual information computed on the two series of populations as the range of n and C_T 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 C_T. These results are consistent with those obtained with the static ORN model.

FIGURE 2.8

Mutual information of two series of dynamic ORN populations as the dissimilarity of the populations increases.

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.

2.3.1. 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 F_m = 10 Hz and C_T = –7, and variable values of n from 0.5 to 3 with a step of 0.2.

FIGURE 2.10

Dose-response curves of toy population before (a) and after (b) lateral inhibition.

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.

FIGURE 2.11

Frequency probability of the toy population output before (a) and after (b) inhibition for a uniformly randomly distributed [–7, –3] input.

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 (F_m, C_T, 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 (F_m) 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 C_T 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.

FIGURE 2.12

Dose-response curves of complete population before (a) and after (b) lateral inhibition.

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.

FIGURE 2.13

Output frequency distributions for different lateral inhibition intensities.

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.

FIGURE 2.14

Mutual information of an ORN population in terms of the intensity of the lateral inhibition within the glomerulus.

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.

FIGURE 2.15

ORN affinity distribution. (a) Distribution of affinities given by a continuous binomial distribution. (b) Dose-response curves of an ORN characterized to respond to four odors.

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.

FIGURE 2.16

Specificity study. (a) Modulation factor (p) for six values of t. High specificity for t = –1 and low specificity for t = 1. (b) Mutual information of the populations as the ORN specificity decreases.

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.

FIGURE 2.17

Population dimensionality study. The mutual information to encode for 10 odors as the dimensionality of the population increases from 1 to 15.

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.