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Contributed reagents/materials/analysis tools: PL. Wrote the paper: LK JR. Performed the research: LK. Designed the research: PL JR.

The authors have declared that no competing interests exist.

The concept of coding efficiency holds that sensory neurons are adapted, through both evolutionary and developmental processes, to the statistical characteristics of their natural stimulus. Encouraged by the successful invocation of this principle to predict how neurons encode natural auditory and visual stimuli, we attempted its application to olfactory neurons. The pheromone receptor neuron of the male moth

Efficient coding is an overarching principle, well tested in visual and auditory neurobiology, which states that sensory neurons are adapted to the statistical characteristics of their natural stimulus - in brief, neurons best process those stimuli that occur most frequently. To assess its validity in olfaction, we examine the pheromone communication of moths, in which males locate their female mates by the pheromone they release. We determine the characteristics of the pheromone plume which are best detected by the male reception system. We show that they are in agreement with plume measurements in the field, so providing quantitative evidence that this system also obeys the efficient coding principle. Exploring the quantitative relationship between the properties of biological sensory systems and their natural environment should lead not only to a better understanding of neural functions and evolutionary processes, but also to improvements in the design of artificial sensory systems.

According to the ‘efficient-coding hypothesis’

The efficient coding hypothesis has been much studied in the visual system

With a nonlinear stimulus-response function, the neuron encodes differently an equal change in stimulus intensity depending on the actual concentration (

(A) Stimulus-response function. The amount of transferred information is limited by the finite range of possible response states. Due to the non-linearity of the stimulus-response function, each response state encodes different relative changes in stimulus intensity. (B) Corresponding probability density function (pdf). Maximum information is transferred if all response states are used equally, i.e., if the area under the stimulus pdf is equal for each response state, as shown. In the limit of vanishingly small response states, the optimal stimulus CDF corresponds to the (normalized) stimulus-response function (adapted from

In this paper, we paralleled Laughlin's approach

Flying male moths rely on the detection of pheromone molecules released by immobile conspecific females for mating. The atmospheric turbulence causes strong mixing of the air and creates a wide spectrum of spatio-temporal variations in the pheromonal signal (

The figure is extracted and adapted from a digitized image of a smoke plume filmed in a wind tunnel 1 m across and 2 m long with source on the left side

The goal of this paper is to present arguments specifying in which sense the perireception and reception processes occuring in pheromone olfactory receptor neurons (ORNs) can be considered as optimally adapted to their natural stimulus. Although, in the light of previous studies on similar sensory neurons, the ORN may be considered

Pheromone components are detected by specialized ORNs located in the male antenna. We considered a specific ORN type of the moth _{air}) to the hair lumen (L); (2) the reversible binding of L to receptor R and the reversible change of the complex R_{L} to an activated state R^{*} (output signal); (3) the reversible binding of L to a deactivating enzyme N and its deactivation to product P which is no longer able to interact with the receptor.

The concentrations of individual components in the network 1–3 are denoted by square brackets and the concentration values are functions of time. For simplicity we omit here the explicit dependence on the time variable _{air} = [L_{air}](_{L} = [R_{L}](^{*} = [R^{*}](_{L} = [N_{L}](_{air} is fully described by five first order ordinary differential Equations 4–8 and two conservation Equations 9 and 10:_{tot} = _{L}+^{*}, as well as the total concentration of the deactivating enzyme, _{tot} = _{L}, do not change over time. We assume that at _{L}, ^{*}, _{L} and

Parameter | Value | Unit | Parameter | Value | Unit |

_{3} = | 0.209 | s^{−1}µM^{−1} | _{−3} = | 7.9 | s^{−1} |

_{4} = | 16.8 | s^{−1} | _{−4} = | 98 | s^{−1} |

_{5} = | 4 | s^{−1}µM^{−1} | _{−5} = | 98.9 | s^{−1} |

_{6} = | 29.7 | s^{−1} | _{I} | 29,000 | s^{−1} |

_{tot} = | 1.64 | µM | _{tot} = | 1 | µM |

From

The efficiency of information transfer in the system 1–3 depends critically on its stimulus-response relationship under single and repeated stimulus pulses. For transferring as much information as possible the response states must be optimally utilized. The actual amount of information transferred is limited by biological constraints. In the system studied, information transfer from _{air} (stimulus) to ^{*} (response) presents three main limitations.

First, it is limited by the finite number of receptor molecules per neuron which places an upper bound on the range of responses. Whatever the pheromone concentration (height of the step) the concentration of activated receptors cannot exceed

Second, temporal details in the stimulus course shorter than a certain lower limit Δ^{*} becomes too small to be effective. Therefore we set Δ

Third, information transfer in time is also limited by the response duration, which depends on the deactivation rate of the activated receptors. The time course of ^{*} in response to stimulations of different heights _{air} and limited duration (0.4 s) is shown in the inset of _{Δ}^{*} at the end of the stimulus pulse, i.e., _{Δ}^{*} = ^{*}(_{Δ}^{*} as the “response” of the system and for the sake of simplicity in the following, we omit index Δ. The duration of the falling phase (receptor deactivation) gets progressively longer for higher pheromone concentrations. This deactivation takes typically much longer than the time resolution parameter Δ^{*}), which is the time required for ^{*}(^{*} to ^{*}/2. The relationship between ^{*} and ^{*}) is shown in ^{*} corresponds to each value _{air}

(A) Temporal properties. Inset: concentration of activated receptors, ^{*}(_{air} (1, 5, 10 and 20 nM). The maximum of ^{*}(^{*} at the end of stimulation. (B) Stimulus-response function ^{*}(_{air}) for single pulses of the same duration as in (A). This curve depends on the temporal resolution and the choice of the response intensity. (C) Optimal cumulative distribution function of the responses, _{R}^{*}), determined by maximizing the information transfer per average half-time (see ^{*}(_{air}) and _{R}^{*}) were used for calculating the optimal stimulus probability distribution (shown in

In the simplest scenario (with no other constraints on the response range and stimulus-independent additive noise), the inputs should be encoded so that all responses are used with the same frequency ^{*} from 0 to 0.24 µM cannot be considered as equally “usable” (the long falling phases decrease the efficacy of the information transfer). Therefore, the longer the half-fall time of a given response ^{*} (i.e. the greater concentration ^{*} is) the less frequent it must be. The particular form of the optimal response cumulative probability distribution function (CDF), _{R}^{*}), which was determined by maximizing the information transferred and minimizing the average half-fall time (see

Examples of predicted temporal fluctuations in pheromone concentration are shown in

10 s (A), 50 s (B) and 350 s (C). Temporal positions of pulses in experiments and simulations do not need to coincide. Quantitative comparisons are done in

Characteristics | Predicted Values | Experimental Values |

Concentration CDF ( | Exponential | Exponential |

Spectra ( | Approx. flat to 0.2 Hz, | Approx. flat to 0.1 Hz or 0.5 Hz |

Close to −2/3 slope after | −2/3 slope to 1 Hz | |

Intermittency | 20% | 10–40% |

10–20% | ||

Total mean _{air} | 1.0×10^{−4} µM | – |

Total std. dev. of _{air} | 3.0×10^{−4} µM | – |

Peak value of _{air} | 3.8×10^{−3} µM | – |

Peak/mean ratio | 37 | >20 |

30–150 | ||

Peak/std.dev. ratio | 13 | >3 |

The mean concentration, standard deviation and their ratios are calculated from the complete stimulus course, including parts of zero concentration (see

Based on a simulated sample 4000 s long.

Concerning temporal aspects, the bursts of non-zero signal do not occur at periodic intervals but appear randomly. An important descriptor of the temporal structure is the intermittency

Concerning pulse height, the overall character of the predicted stimulus course is that pulses of high concentration are much rarer than those of low concentration. This feature of the predicted stimulus can be best quantified by the CDF, _{air}), of the stimulus. The shape of the CDF is one of the most important properties for comparing theoretical predictions to experimental measurements because it describes the relative distribution of odorant concentrations throughout the plume. In fact, because measuring pheromone concentration in the field is not presently feasible _{air} to actual measurements.

Given the definition of the optimal stimulus, function _{air}) can be directly computed (see _{air} are predicted to be less frequent than in the exponential model. The exponential CDF is in agreement with experimental CDF (

(A) Experimental CDF (solid) as measured at 75 m from a propylene (passive tracer) source and its best exponential fit (dashed) plotted on a logarithmic scale (taken from _{air} where they are less frequent than expected. Since this deviation is apparent only for events occurring with probability

Other predicted relative quantities (peak-to-mean ratios, dimensionless concentrations _{air}/〈_{air}〉) were compared with their experimental counterparts. The results, summarized in

Spectral density functions of the concentration time course, which analyze the contribution of various frequencies to the overall stimulus course, characterize other properties of the plume which are independent on the nature of the odorant (pheromone or ion source)

Several spectral density functions, shown in

Several spectral density functions were calculated from the predicted optimal pheromone stimulations (such as shown in

The goals of this study were to determine to which extent early olfactory transduction in olfactory receptor neurons can be considered adapted (in the evolutionary sense) to odorant plumes and to specify the plume characteristics to which it is adapted. The formulation and resolution of this problem benefited from successful studies of efficient sensory coding undertaken in the field of vision and audition. However, transposition from these sensory modalities to olfaction is not straigthforward, which may explain in part why it has not been attempted earlier. Specificities of olfaction concern both the odorant plume and the sensory system.

In theory and in practice, the quantitative description of odor plumes and their spatiotemporal distribution is less straightforward than that of visual or auditory scenes. Contrary to light and sound, for which the physical description is essentially complete, the turbulent phenomena which underlie the plume characteristics are still an incompletely mastered domain of physics

In Laughlin's classical experiment in vision a single time-independent variable, the contrast level, was measured

Moreover, there are no easy-to-use instruments to measure odor plumes in the field, comparable to luxmeters and microphones. For example, the absolute pheromone concentration cannot be easily known in field experiments _{air}/〈_{air}〉 and intermittency values). Other limitations of plume measurements are discussed below.

The essentially multidimensional and stochastic nature of the odor stimulus has a profound influence on the analysis of olfactory transduction system in its natural context, as undertaken here. Indeed to investigate the problems at hand, the kinetic responses of the system to a very large number of stimuli, varying in intensity, duration and temporal sequence must be known in order to simulate the diversity of stimuli encountered in a natural plume. This task is difficult, if not impossible, to manage in a purely experimental approach. However, this difficulty can be overcome with an exact dynamic model of the system because its response to the diverse conditions mentioned can be computed, provided it includes all initial steps from molecules in the air to the early neural response. This is the case of the perireception and reception stages of the moth pheromonal ORN and the reason why it was chosen in the present study. This choice brings about two questions, one about the validity of the model, the other on its position within a larger context.

The computational model employed has been thoroughly researched and improved over the last three decades ^{*} is always well above that corresponding to one activated receptor molecule per neuron (approximately 10^{−6.2} µM) because we do not investigate the effect of extremely small pheromone doses. Then, the response of the system can be considered as deterministic, in accordance with the efficient coding hypothesis

The system studied here constitutes only a small part of the whole pheromonal system, although its role is absolutely essential and all other parts depend on it. First, in ORNs, post-receptor mechanisms modify the receptor signal, primarily by a large amplification factor and by sensory adaptation. Second, the ORN population includes cell types with different properties, e.g. the ORN type sensitive to the minor pheromone components can follow periodic pulses up to 10 Hz

Different response states of the pheromone reception system have different efficacies from the coding point of view: the “high” states, with large concentrations of activated receptors, take much more time to deactivate than the “low” states, so that for some time after its exposition to a large concentration of pheromone the system is “dazzled”. It means that in the optimal stimulus the low pheromone concentrations must be more frequent than the high ones. This is a difference with respect to the classical problem where the efficacy of all response states at transferring information is considered the same, as in the vision of contrasts for example. The problem to solve is to find the right balance between two conflicting demands: to use all response states (including the high ones) and to react rapidly (the short transient responses must be as frequent as possible), i.e. to maximimize the information transferred

The solution to this optimization problem is provided by information theory as detailed in the

The main achievement of the present investigation was to predict the characteristics of the stimulus optimally processed by the receptor system based on its biochemical characteristics and an information theoretic approach. The predicted optimal plume was shown to be close to the actual plumes for a series of characteristics, namely intermittency, peak/mean ratio and peak/standard deviation ratio of pheromone pulses, probability distribution of dimensionless pheromone concentration and spectral density function of pheromone concentration (

These differences in precision of the predictions may be interpreted by taking into account technical factors. Increasing the noise rejection threshold leads to a decrease of the measured intermittency

Even if one considers that the pheromone olfactory system must be

As mentioned in the Results section, information transfer in the pheromone reception system is limited by the finite response range, (^{*}. This deactivation rate is described by the half-fall time ^{*}). The optimal performance of the system is thus reached by a trade-off between two conflicting demands: to employ full response range (maximum information) vs. to employ only the “fastest” responses (minimum average half-fall time). In other words we need to maximize the information transferred per average half-fall time. In the following we provide the mathematical framework that enabled us to find the probability distribution function over the response states ^{*} that realizes this trade-off.

The information transferred by the pheromone reception system in a selected time window (_{air}, and the corresponding response values, ^{*}. This relation is explicitly quantified by the mutual information, _{air}; ^{*}) (see ^{*}) is the entropy of the response probability distribution function and the conditional entropy ^{*}|_{air}) measures the uncertainty in the output given the input, or equivalently, the amount of noise in the information transduction ^{*}|_{air}) = 0. Thus maximizing the mutual information corresponds to maximizing the response entropy ^{*}). (Note that in the usual setting of signal independent and additive noise the term ^{*}|_{air}) is constant and then maximization of _{air};^{*}) again corresponds to maximization of ^{*}).)

The available response range, (^{*}) is (^{*}) is the probability of having ^{*} (expressed as a number of molecules). (In the following we use the base of logarithm 2 only to express all information-related quantities in the usual units of “bit”).

The value of ^{*} corresponding to one activated receptor molecule per neuron is approximately Δ^{*} = 10^{−6.2} µM ^{*}), defined as (^{*}) is the response probability density function. An approximative relation between ^{*}) and ^{*}) is given in (^{*} is very small compared to the whole response range (^{*}). The advantage of employing differential entropy is that it lends itself to an elegant approach for entropy maximization in terms of integrals.

We adopt the standard procedure for maximizing the differential entropy of a continuous probability distribution constrained by a known function ^{*}). “Constraining” means that the average value 〈^{*}) is under our control (see _{R}^{*}), which (i) maximizes the value of ^{*}) (Equation 13) and (ii) is such that the average 〈_{R}^{*}) is equal to the value we set. The well known solution to this problem (see ^{*}) = _{R}(^{*}) in Equation 16, so that the following equation between 〈

In order to simplify practical calculations we substitute ^{*}) = _{R}^{*}) into the definition of differential entropy 13 so that Equation 15 reduces to_{air};^{*}) from Equation 21 (shown in ^{*}) is a monotonously increasing function of ^{*}, the optimal probability density function _{R}^{*}) (Equation 17) is either monotonously increasing (_{R}^{*}) puts more weight on the “slow” response states (_{R}^{*}), which maximizes the information transfer per average half-time. The corresponding CDF _{R}^{*}), shown in

(A) Mutual information for the pheromone reception model in dependence on

The maximum of information transferred per average half-time (

The optimal stimulus course in time was calculated as follows. First, at time _{0} = 0 a random value _{0} is drawn randomly from a uniform probability distribution function over the range [0,1]. The concentration _{0} is obtained by solving the equation_{R}^{*}) is the optimal CDF given by formula 22 (_{air,0} for a pheromone pulse of duration Δ^{*}(_{air}) is the stimulus-response function (_{air,0} is plotted at _{0} (_{air,1} and time of appearance _{1} of the next pulse are determined. Time _{1} follows from the falling phase of activated receptors: optimality requires that no pheromone pulse appears before ^{*} returns to its resting level. In practice it is considered that the resting level is reached when ^{*} falls below 0.01 µM (less than 5% of the coding range). The concentration _{air,1} of the pulse at _{1} is determined in the same way as for the pulse at _{0} by drawing a new random number _{1} from the uniform probability distribution function over [0,1]. The same process can be repeated as many times as needed to create an optimal pheromone pulse train of arbitrary length.

It is common in the literature on the statistical analysis of plumes _{air}〉, describes the “true” mean concentration obtained from the whole record of concentration fluctuations in time, i.e., including the parts where no signal was available. On the other hand, the conditional mean concentration, 〈_{air}〉_{cond}, describes the mean concentration inside the plume, i.e., with zero concentrations excluded. The intermittency,

By combining Equations 23 and 24 we may symbolically express the optimal CDF of the stimulus, _{air}), as_{air}) cannot be expressed in a closed form, it can be well approximated by the exponential CDF^{−4} µM is the estimated value of 〈_{air}〉_{cond} by least-squares fitting of _{exp}(_{air}) to _{air}).

In order to compare concentration probability distribution functions from different measurements meaningfully, authors _{air}/〈_{air}〉. (In the _{air} = 0) of zero concentration be_{air}) must be renormalized _{air}/〈_{air}〉, and the value of _{air} = 0) but not the overall shape of CDF

The optimal stimulus course is represented by pulses of different pheromone concentrations, _{air}, occurring in time intervals 0.4 s long. In order to calculate the spectral density function of such stimulation course we sample the time axis with step Δ_{air,j}}, _{k}_{air,j}} is defined for _{air,j}} with such a pulse of unit height in the time domain

The authors are grateful to Christine Young for linguistic corrections.