A simple cartoon of variance coding.

As a simple cartoon which illustrates this hypothesis, we have simulated a single conductance-based integrate-and-fire PN, driven by a set of ORNs, each endowed with a simple model of synaptic-depression. This simple model exhibits the following dynamical features: (i) the relationship exhibits high gain and saturation, and (ii) for different values of on the plateau of the relationship, the variance in PN activity decreases as increases, even though remains roughly constant.

Within this simple model, we describe each ORN as a Poisson process with fixed rate (). The coupling strength between the ORNs and the PN is modulated by a term (), which is intended to model vesicle-depletion at the ORN synapses. As each ORN fires, this term will give rise to synaptic-depression between the ORNs and the PN. If , the synapses between the ORNs and the PN are exhausted. If , the synapses between the ORNs and the PN are completely refreshed. The model details are given in a section entitled “An idealized model used to illustrate variance coding” in Methods.

With this simple model, it can be seen that the PN firing-rate is a nonlinear function of the ORN firing-rate , and that saturates (plateaus) at values of (See Fig. 2A). The time-averaged mean total excitatory conductance of the PN enjoys a similar nonlinear relationship (Fig. 2B). Notably, for values of , the time-averaged mean vesicle-depletion parameter increases as a function of , and the standard deviation in the total PN conductance decreases as a function of (Fig. 2C and Fig. 2D). This decrease in standard deviation is associated with a decrease in coefficient-of-variation for the total PN conductance. Qualitatively speaking, the PN is more ‘mean driven’ when , and the PN is more ‘fluctuation driven’ when , even though the firing-rate of the PN is similar in both cases (Fig. 2E and Fig. 2F). This can be quantified by measuring, for example, the autocorrelation of the PN. In the case , the PN autocorrelation shows several significant peaks, the first of which is at , indicating periodic-firing at (Fig. 2E). In the case , the PN autocorrelation does not indicate a strong periodicity to the PN firing-patterns (Fig. 2E).

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Figure 2. A simple illustration of variance coding.

Here we presume the simple model described in the section entitled “An idealized model used to illustrate variance coding”. [A] There is a nonlinear relationship between the ORN firing-rate and the PN firing-rate. [B] There is also a nonlinear relationship between the ORN firing-rate and the time-averaged conductance of the PN. [C] As the ORN firing-rate increases, the time-averaged vesicle-depletion parameter increases and saturates. [D] Since the average vesicle-depletion parameter increases as the ORN firing-rate increases, the variance in the PN conductance is a decreasing function of ORN firing-rate, for sufficiently high ORN firing-rates. Two different points along this curve are indicated, corresponding to two different PN dynamical regimes with similar PN firing-rates. The ‘’ and ‘’ symbols indicate, respectively, an irregularly firing-regime and a regularly firing-regime. [E] As a result of the fact that the PN conductance has a low variance when the ORN firing-rates are high, the PN activity is very regular when the ORN firing-rate is high. In contrast, the PN activity is less regular when the ORN firing-rate is not as high. This is reflected in the normalized PN autocorrelation, which shows several significant peaks when the variance in the PN conductance is low (‘’-regime, left). In contrast, when the variance in the PN conductance is high the autocorrelation does not show significant peaks (‘’-regime, right). [F] The regularity in the PN spiking activity is seen in PN voltage trace, as shown for the ‘’-regime (top) and ‘’-regime (bottom). [G] The variance in the PN conductance is seen in PN conductance trace, as shown for the ‘’-regime (top) and ‘’-regime (bottom). [H] In this panel we show the voltage-trace of a putative Kenyon cell, a conductance-based integrate-and-fire-neuron, driven by either the PN from the -regime (top) or the PN from the -regime (bottom). Thick vertical lines indicate firing-events for this putative KC. When driven by the regular activity of the -PN, the KC mainains an elevated subthreshold voltage, but does not fire often. On the other hand, when driven by the irregular activity of the -PN, the KC does not maintain an elevated subthreshold voltage but fires after each burst in -PN-activity. This provides a simple illustration of one possible way in a variance-code could be ‘read-out’ by downstream neurons.

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The simple cartoon described above only considers synaptic-depression resulting from vesicle-depletion. The real AL displays evidence of presynaptic-inhibition as well. Nevertheless, the same general principle still holds regardless of the source of synaptic-depression at the ORN synapses, as long as the PNs become more mean driven as ORN firing-rates increase. In fact, it is possible to show analytically that similar results hold across a wide range of parameters for an idealized system similar to this one (see the section entitled “A simple analyzable cartoon of variance coding” in Methods).

If this picture is accurate in the real AL, then the PN dynamics within any given glomerulus in the AL will change as a function of ORN input to that glomerulus, even when the mean PN firing-rates have saturated for that glomerulus. These dynamical changes will only be observable through measurements of statistics that are ‘higher-order’ than mean firing-rate. We note that synaptic-depression of the ORN synapses is not the only mechanism via which the PNs may become more mean-driven as ORN firing-rates increase — other mechanisms, such as spike-frequency adaptation, could also contribute to this effect. As long as the postsynaptic influence of each ORN spike decreases as increases, the PN activity will become more mean-driven as increases. As the PN activity becomes more mean-driven, we expect the firing-sequences produced by that PN to become more regular [22].

An illustration of variance coding within a large-scale model.

We also observe this phenomenon within our large-scale model (described in Methods), which contains both presynaptic-inhibition and vesicle-depletion. To illustrate this phenomenon at work, we created a panel of 16 odors, all of which saturated the PN firing-rates (i.e., produced average PN firing-rates at the ‘plateau’ of the curve for the model). We presented each of these odors to the model network times.

For each of the trials of each stimulus we measured the -component vector of PN firing-counts collected over the following odor onset. Each component of this vector represents the number of spikes fired by one of the PNs during this time. We then used this vector to perform each possible -way and -way stimulus discrimination task (see the section entitled “Odor Discrimination” in the Methods). Each of these -way and -way discrimination tasks results in a discriminability rate (i.e., the fraction of correctly categorized trials – note that chance performance for a -way task is , and chance performance for a -way task is ).

We construct a histogram of the discriminability rates for the -way discrimination tasks, and as expected (see Fig. 3A), the typical discriminability rate for the system is not particularly high (recall that each odor saturated the PN firing-rates). Similarly, the -way discrimination tasks performed using PN firing-rate vectors also do not yield high discriminability rates (Fig. 3B). However, if instead of merely using PN firing-rate information we also use information regarding PN-PN correlations within the system, then the typical discriminability rates for the -way and -way tasks increase (see Fig. 3C,D). To produce the discriminability rates shown in Fig. 3C,D, we measured not only the -component vector of PN firing-counts for each odor trial, but also the -component vector of PN-PN correlations (with correlation time ). As expected, these higher-order statistics contain enough information to discriminate odors significantly more reliably than mere firing-rates.

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Figure 3. A manifestation of variance coding within the large-scale model.

The large scale model (described in Methods) exhibits a phenomenon similar to the variance coding shown in Fig. 2. We constructed a panel of 16 odors, all of which only directly stimulated the same glomeruli (although to differing degrees). Moreover, we chose every odor within this panel such that the ORN firing-rates of the directly stimulated glomeruli were sufficient to saturate the firing-rates of the associated PNs (i.e., the directly stimulated ORN firing-rates were 12 Hz, see Fig. 10). Given this panel of odors, we presented each odor multiple times, and used the collection of -component PN firing-rate vectors (measured over the period immediately following odor onset) to perform a variety of odor discrimination tasks (see Results for details). [A] The histogram of discriminability rates associated with -way discrimination tasks when only firing-rate data is used. Note that is chance level for these tasks (chance level is also shown in panels B,C,D). [B] The histogram of discriminability rates associated with the -way discrimination tasks when only firing-rate data is used (note that is chance level for these tasks). [C] The histogram of discriminability rates associated with -way discrimination tasks when firing-rate data and -point correlations (correlation time ) are used. [D] The histogram of discriminability rates associated with -way discrimination tasks when firing-rate data and -point correlations (correlation time ) are used. Note that the typical discriminability rate is higher when correlations are used. [E] Here we plot the difference in mean discriminability for the -way discrimination task between the cases (i) when firing-rate data and -point correlations are used, and (ii) only firing-rate data is used. We plot this difference as a function of the parameters and used in our large-scale model. The vesicle-depletion parameter ranges from to across the vertical axis, and the presynaptic-inhibition parameter ranges from to across the horizontal axis. The data shown in panels A–D is taken from the simulation indicated by the dashed square. Note that, as the total amount of synaptic-depression decreases, the discriminability computed using only firing-rates is closer to the discriminability computed using both firing-rates and -point correlations. [F] Similar to panel-E, except for the -way discrimination task, rather than the -way discrimination task.

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The difference between the performance of these low-order and high-order readouts is more noticeable when the synaptic-depression in the system is strong. Conversely, in a network with no vesicle-depletion and reduced presynaptic-inhibition, the low- and high-order readouts yield more similar discriminability-rates (see Fig. 3E,F). Thus, the presence of strong synaptic-depression within our system is one factor which allows the network's dynamics to encode input-specific information within the PN-PN correlations.

For the example shown in Fig. 3, the difference between the typical 2-way discriminability rates observed when using high-order versus low-order readouts is maximized when the synaptic-depression is strongest; the effect of variance coding is seen quite clearly. However, for the 3-way discriminability rates, the difference between the high- and low-order readouts is greatest when the presynaptic-inhibition is not too strong. A natural question is: why does the performance for the 3-way discrimination task not parallel that for the 2-way task? Why is the difference in performance between high- and low-order readouts not maximized when both presynaptic-inhibition and vesicle-depletion are at their strongest?

This effect arises in part because the 3-way task is quite difficult and the observation time over which the task is carried out is rather short — in this case. As we will argue below, one consequence of strong presynaptic-inhibition is that the network's ability to perform fine discrimination will be compromised when is small. In order to perform very well on fine discrimination tasks when is small, the network should have only moderate amounts of presynaptic-inhibition (consistent with Fig. 3F).

An illustration of the tradeoff between reliability and sensitivity within a large-scale model.

In this subsection we will show how the hypothesis introduced above manifests within our large scale model. First we will discuss some features of this model which are pertinent to this hypothesis, then we will discuss our hypothesis in more detail.

We used simulations to investigate and benchmark our large-scale model (see the sections regardin benchmarking in the Methods). By analyzing these simulations we determined that, even after benchmarking, there were still a handful of free parameters that were left unconstrained. Two parameters in particular were not fully constrained by our benchmarking: (i) the strength of vesicle-depletion as characterized by , and (ii) the strength of presynaptic-inhibition as characterized by . Within our large-scale model the combination of these two parameters produced synaptic-depression of the ORN synapses. While the total amount of synaptic-depression was constrained by our benchmarking, the relative strengths of versus were not constrained.

As an example of this lack of constraint, consider the following benchmark: assume that we expect the average PN firing-rate within the AL to saturate at a certain level when stimulated sufficiently by ORN input. What we found was that there is a spectrum of possible AL architectures which could produce this desired firing-rate : (A) on one end of the spectrum is an AL in which there is hardly any vesicle-depletion of the ORN synapses, but for which the LNIs give rise to substantial presynaptic-inhibition at these synapses. This type-A AL would be characterized by a large value of and a small value of . (B) on the other end of the spectrum is an AL in which vesicle-depletion is primarily responsible for synaptic-depression, and the presynaptic-inhibition of the ORN synapses due to LNIs is negligible. For this type-B AL would be small and would be large.

An example of this spectrum is given in Fig. 4. Given a fixed value for the saturated firing-rates of PNs in a strongly driven glomerulus, there exists a -parameter family of values which corresponds to networks exhibiting saturated firing-rates equal to . This -parameter family of values ranges from networks with high and low (i.e., type-A networks) to networks with high and low (i.e., type-B networks).

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Figure 4. A tradeoff between reliability and sensitivity within our large-scale model.

We performed a systematic scan of our large-scale network model, varying and (see the section entitled “An illustration of the tradeoff between reliability and sensitivity within a large-scale model” in the main text for details). For each point in this parameter array we measured various features of the network dynamics (such as mean PN spike-counts and reliability), as well as the performance of each of these networks on a -way odor discrimination task. [A] Shown is the mean PN spike-count of PNs in the first glomerulus, for each pair of parameter-values , . Overlaid on top of the mean spike-counts are contour lines for the spike-count. Four of these contours are highlighted in magenta, and will be referenced later. [B] Indications of the type-A and type-B network regimes. [C] Shown are the standard deviation in PN spike-counts of PNs in the first glomerulus (see colorbar on far left). [D] Reproduction of panel-C, along with the contours of panel-A. [E–H] Shown are contour plots associated with for various values of . These panels use the colorbar shown to the far left. [I] Here we plot the standard-deviation in spike-count (taken from panel-D) as a function of the distance along each of the contours indicated in panel-D, with values bi-linearly interpolated as necessary. [J] Here we plot the discriminability values indicated in panel-E as a function of the distance along each of the contours shown in panel-D. The contours are indicated using the colorcode from panel-I. [K–M] Similar to panel-J, except for , , and respectively.

https://doi.org/10.1371/journal.pcbi.1002622.g004

Shown in Fig. 4A are the mean stimulus-driven PN spike-counts for several networks with varying values of and . To construct this example we performed a systematic scan of parameter space for our large-scale network model. We selected a -dimensional array of parameter values for , ranging from and from . For each fixed within this array, we ran a large-scale simulation using a panel of odors, and we ran trials per odor. The first eight of the odors used stimulated three glomerular channels — the first glomerular channel was stimulated strongly, and an odor-specific subset of two other glomerular channels was stimulated weakly. The ninth odor only stimulated the first glomerular channel strongly. We remark that the simulations used to construct this array differ only in their values of and . The architecture and connectivity of the rest of the model network were fixed.

In Fig. 4A we show the mean spike-count of PNs in the first glomerulus, for each pair of parameter-values , . The mean spike-count is calculated as the mean of the number of spikes/ time-bin averaged across all trials, and further averaged over the period following odor onset, and further averaged across all odors. Overlaid on top of the mean spike-counts are contour lines for the spike-count. Each of these contours represents a -parameter family of networks with a different constant mean stimulus-induced spike-count. Note that, as indicated in Fig. 4B, these contours extend from regions of high and low to regions of low and high . In this example Type-A networks correspond to the lower-left corner of the array, and Type-B networks correspond to the upper-right corner of the array. Thus, in Fig. 4A it can be seen that is constant along contours extending from type-A networks (lower left) to type-B networks (upper right).

We observed two important systematic differences between the candidate networks along these -parameter families. First, type-B networks are more reliable than type-A networks. This can be understood as follows. First consider the ORN inputs to PNs in a type-B network (for which synaptic-depression is dominated by vesicle-depletion). A typical odor stimulates many ORNs to fire at a high rate. Each of the ORN synapses likely has a high quantal release rate [11], implying that the fraction of active vesicles remaining after several rapid ORN spikes is likely to be small. Moreover, there are such ORNs which converge onto each PN within their target glomerulus [23]. Thus each PN within a strongly stimulated glomerulus receives a large number of input spikes from a large number of presynaptic ORNs, each firing with a high rate, each synapse of which is likely to experience profound vesicle-depletion. Moreover, the vesicle-depletion experienced by the ORN synapses is only dependent on the ORN activity, and is independent of the activity of the AL. Thus, we expect the ‘feed-forward’ synaptic-depression observed within a type-B network to always exhibit very similar dynamic transients from trial to trial, with the only differences due to the variation in ORN spike-sequences induced by the trial-to-trial variability of the Poisson input to the ORNs [19]. Now, on the other hand, let us consider the ORN inputs to PNs within a type-A network. In such a network, synaptic-depression is primarily governed by ‘feedback’ from the AL in the form of presynaptic-inhibition. ORNs in a type-A network rely on the odor-specific firing patterns of LNIs in order to exhibit synaptic-depression, and therefore may receive different amounts of presynaptic-inhibition from trial to trial (or over disjoint time-windows within a single trial). Moreover, there are only a few LNIs per glomerulus, and a given stimulus may not cause all these LNIs to fire at high rates. A few extra LNI spikes induced on any one trial may substantially change the footprint of synaptic-depression across the ORN synapses, thus leading to even more extra LNI spikes later on, and so forth. This ‘feed-back’ mechanism allows the synaptic-depression observed within type-A networks to exhibit quite different dynamic transients from trial to trial. Put another way, the ‘feed-back’ structure within type-A networks allows the trial-to-trial variability in LNI activity to affect and magnify the trial-to-trial variability in ORN input to the AL. In conclusion, we expect that ORN inputs to PNs in type-A networks will be less reliable than the corresponding ORN input to PNs for type-B networks when measured either (a) over multiple trials, or (b) over different time-windows within a single trial.

The second systematic difference between networks along such a -parameter family is that type-A network-dynamics is more sensitive than type-B network-dynamics to subtle changes in ORN input. To see why this might be true, let's revisit the argument used above. Consider a subtle change in ORN input which is only large enough at first to shift PN and LN firing rates slightly. This subtle change in ORN input will not create a large shift in the PN input for type-B networks, yet the same subtle change in ORN input may give rise to a few different LNI firing-events in the type-A network, which may then presynaptically inhibit different ORNs, giving rise to even more different type-A-network-activity, and so forth. In other words, due to the feedback between the type-A LNIs and the type-A ORNs, we expect the type-A system's dynamics to be more sensitive than the type-B network's dynamics to certain perturbations in input.

These systematic differences (i.e., type-A networks are less reliable, but more sensitive to perturbations in input than type-B networks) manifest within our large-scale model.

To quantify reliability for each network along such a -parameter family, we measured the trial-to-trial standard deviation in PN spike-counts of PNs in the first glomerulus, for each pair of parameter values , . The standard-deviation is calculated as the standard-deviation of the number of spikes/ time-bin across all trials, averaged over the period following odor onset, then further averaged across all odors. The coefficient-of-variation in spike-counts is equal to the standard-deviation in spike-count divided by the mean. Thus, along contours of (where the mean is constant) the coefficient-of-variation in spike-count will be proportional to the standard-deviation in spike-count. Shown in Fig. 4D are the standard deviation in PN spike-counts along with the 4 contours highlighted in Fig. 4A. These four contours (labelled ,,,) each correspond to a -parameter family of networks exhibiting a fixed mean spike-count, and are each associated with a different color (black, red, green, cyan, respectively) on the colorbar to the far left. In Fig. 4I we plot the standard-deviation evaluated along these contours. Note that, since contour is longer than contour , the graphs shown in Fig. 4I are not directly comparable. However, there is a clear trend amongst all these graphs: As one moves along the -parameter family of networks with constant mean stimulus-induced spike-count from type-A networks to type-B networks the standard-deviation in spike-count decreases as long as the mean spike-count is sufficiently high (i.e., contours ). This is equivalent to the statement that, along contours , type-B networks are more reliable than type-A networks.

Recall that, for each network (i.e., for each fixed value of ), we ran trials for each of different odors. Using this data, we can quantify the sensitivity of each of these networks to input perturbations. For each odor trial we measure the -component PN firing-rate vector averaged over the time-window including and immediately following a odor presentation. We use these time-averaged firing-rate vectors to perform each of the -way odor discrimination tasks, and thus we obtained a distribution of discriminability rates for each -way odor task (see the section entitled “Odor Discrimination” in the Methods). For each network we then record the -percentile of the distribution of discriminability rates (across odors), denoted by . We chose to display , as this -percentile discriminability rate illustrates our conclusions most clearly. However, our main results do not change if we choose another percentile in the range . Higher percentiles, such as the -percentile, are usually all near correct-classification, since the set of odors used contain several rather distinct odors. Note that will depend on and . Shown in Fig. 4H are the contour plots associated with for . In Fig. 4M we plot these discriminability rates along the contours shown in panel-A. For each of these contours the maximum discriminability (when ) occurs at the type-A end of the spectrum. This indicates that the discriminability of type-A networks (using firing-rates measured over long observation times) is superior to that of the type-B networks. This is a reflection of the fact that type-A networks are more sensitive than type-B networks to subtle changes in input.

A combination of vesicle-depletion and presynaptic-inhibition is required to optimize discriminability over short observation-times within a large-scale model.

Within our model network we have observed a further functional consequence associated with the tradeoff between reliability and sensitivity described above. Type-A networks are indeed more sensitive than type-B networks to shifts in input, and this sensitivity is reflected in the long time (or trial averaged) PN firing-rate vector associated with any given input. As a result, type-A networks outperform type-B networks in odor-discrimination tasks when the discriminability rate is calculated using a long time observation (such as , as shown in Fig. 4H,M). However, type-A networks are less reliable than type-B networks, and thus, if the observation-time of any given odor stimulus is sufficiently short, the higher variability associated with the single-trial short-time responses of type-A networks will pollute the performance of any discrimination task which uses only these short observations. On the other hand, since type-B networks are rather reliable, shortening the observation-time associated with a discrimination task will not affect the discriminability rate associated with that task for a type-B network as much. As demonstrated in our model network, if the observation-time of any given odor trial is shortened from (as shown in Fig. 4H,M) to merely after odor onset, the decreased reliability associated with type-A networks will drastically lower the discriminability rate of the odor-discrimination tasks which use only these short observations (see Fig. 4E,J). Moreover, since the type-B networks are more reliable than type-A networks, the decrease in discriminability associated with reducing the observation-time of the discriminability task is lower for type-B networks than it is for type-A networks (compare Fig. 4J,M). Most intriguingly, there is a midpoint in the spectrum — a balance between vesicle-depletion and presynaptic-inhibition — which gives rise to the maximum discriminability rates using only short-time observations. This optimal point depends on the length of the observation-time associated with the discrimination-task. With long observation-times type-A networks are optimal. With very short observation-times type-B networks are optimal.

This feature is shown in more detail for our large scale model in Fig. 4E,F,G,J,K,L, which illustrate the discriminability capabilities of our model for a variety of observation times . Note that, for any particular contour ,,,, The point of maximum performance occurs closer to the type-B extreme when is small, and this maximum occurs closer to the type-A extreme when is large. In other words, when is low, type-B networks outperform type-A networks, whereas when is large type-A networks outperform type-B networks.

In conclusion, we have demonstrated that for a particular set of discrimination tasks the network which performs optimally lies in between the type-A and type-B extremes. Moreover, as the observation-time associated with this task increases (or decreases) the optimal point shifts towards the type-A (or, respectively, type-B) end of the spectrum. Although the details of Fig. 4 only pertain to a particular discrimination task, we mention now that this systematic dependence of the optimal point on observation-time is actually a natural consequence of the fact that type-A networks are more sensitive, and type-B networks are more reliable. Indeed, as we will argue below (in a section entitled “A simple cartoon of optimizing discriminability over short observation-times”), this feature is to be expected for a rather general class of discrimination tasks in which estimates of the mean firing-rates (sampled over an observation-time) are used to classify the input.

A simple analyzable cartoon of the tradeoff between reliability and sensitivity.

In this section we introduce a simple deterministic -neuron model network which will allow us to discuss various aspects of hypothesis-2. This simple network has the property that the sequence of neuronal firing-events is a sensitive function of the network's initial conditions as well as the input to the network and the source of synaptic-depression within the network. The model itself consists of LNIs, each driven by a single ORN. Each LNI (labelled and ) is modeled by a simple phase-oscillator (similar to a current-based integrate-and-fire neuron), and each ORN is modeled by a fixed input-current (i.e., and ). This input current indicates the rate at which each neuron would fire if there were no presynpatic-inhibition or vesicle-depletion.

In this system the strength of presynaptic-inhibition is modeled by a constant parameter . As increases, the firing-events of each neuron have a greater inhibitory effect on the input to the other neuron. Specificially, whenever LNI fires, the ORN input to LNI is shut off for -time. Similarly, whenever LNI fires, the ORN input to LNI is shut off for -time. To ensure that both neurons fire, we assume . As the amplitudes of the ORN processes are constant, the vesicle-depletion in this system is assumed to attain a steady state, and is modeled via a single constant parameter , which reduces the ORN input to both and .

In keeping with the description above, the membrane potentials for LNI and obey the differential equations(1)and whenever the potential reaches , we say that LNI fires, and reset to . The spiketime of neuron is recorded as . Similarly, whenever reaches , we say that LNI fires, and record the spiketime of neuron as . The term is equal to , unless neuron has fired within of the current time, in which case . Similarly, the term is equal to , unless has fired within of the current time, in which case . Note that, since , the terms and are each either or at each time.

This simple network is easy to analyze, and the firing-rates of each LNI in the network, as well as the interspike-interval-distributions (, and ) can be directly calculated in terms of the inputs to the network and the sources of synaptic-depression , and . See the section entitled “A simple model illustrating the tradeoff between reliability and sensitivity” in the Methods for more details.

An example of such calculations is shown in Fig. 5. If we fix , then the firing-rates of the two neurons is a decreasing function of for small (see Fig. 5A). As expected, this decrease in firing-rate corresponds to the two neurons interfering with and slowing down one another. However, this interference causes the firing-events of each neuron to occur at irregular intervals, and hence the variance in the ISI-distributions of these neurons is a monotonic increasing function of for small (see Fig. 5B). Thus, as increases from , the neurons fire less, and have a lower trial-to-trial reliability.

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Figure 5. A simple analyzable cartoon of the tradeoff between reliability and sensitivity.

In this example , and . In panels A and B the vesicle-depletion parameter . In panels C,D,E and F, the vesicle-depletion parameter , such that the mean firing rate is held constant. [A] Graphs of (solid), (dashed), (gray), and (gray dashed), as functions of , for the case . [B] Graphs of var (solid) and var (dashed) as functions of , for the case . [C] Graph of as a function of , subject to the constraint that remain constant. The constant value of chosen (essentially arbitrarily) in this case is the value of shown in panel A for . Other choices of yield similar results. Note that this graph is monotonically decreasing, implying the existence of a 1-parameter family of networks possessing the same — ranging from type-A networks with low and high , to type-B networks with high and low . [D] Graphs of (solid), (dashed), (gray), and (gray dashed), for the case . [E] Graphs of var (solid) and var (dashed) as functions of , for the case . [F] Graph of the optimal choice of (implying a vesicle-depletion parameter of ) for which discriminability is maximized, as a function of the sample number . The notion of discriminability is described in the section entitled “A simple cartoon of optimizing discriminability over short observation-times”. In this case the observation error is fixed at . Note that for low , discriminability is maximized for a type-B network. However, as increases, discriminability is maximized by type-A networks. The graph shown plots for , as for this particular simple example the derivative of reaches a vertical asymptote at .

https://doi.org/10.1371/journal.pcbi.1002622.g005

If we fix , then the firing-rates of the two neurons are decreasing functions of . Thus, there is clearly a -dimensional family of synaptic-depression parameters which gives rise to networks exhibiting the same mean firing-rates for any fixed set of inputs. This -parameter family ranges from type-A networks (with high and low ) to type-B networks (with low and high ) – see Fig. 5C. We can index networks along this -parameter family using , assuming that is chosen so that the average firing-rate is maintained (see Fig. 5D).

Intuitively, one expects that for an extreme type-B network (i.e., ) the activity should be perfectly regular: each neuron fires independently of the other neuron. On the other hand, for an extreme type-A network (i.e., ), each neuron fires in spurts, constantly disrupting the periodicity of the other neuron's activity. This interplay between the neurons (resulting from presynaptic-inhibition) gives rise to a greater variability in the ISI-distributions of the neurons within the type-A networks. This increased variability implies that the neurons in the type-A networks have a lower trial-to-trial reliability than the analogous neurons within the type-B networks (assuming that different trials have different initial conditions).

This same intuition can be extended to see that, in the type-A network, there is a ‘rich-get-richer’ phenomenon: the neuron which gets more input will slow the other neuron down more than it is slowed down by the other neuron. Thus, the presynaptic-inhibition magnifies the sensitivity of the type-A networks, increasing the difference in firing-rates between the two neurons when the input to these neurons is similar. In other words, the firing-rates produced by the type-A network should be more sensitive than those produced by the type-B network to small differences in inputs.

This intuition is borne out by analysis. When constraining so that the mean firing-rate is constant, we see that as increases from the network becomes more sensitive (i.e., the difference in firing-rates increases) and less reliable (i.e., the variance of the ISI-distributions of the two neurons increases). These functions are plotted in Fig. 5D,E. Thus, for this simple system, we can show analytically that hypothesis-2 holds: type-A networks are more sensitive, and type-B networks are more reliable. See the section entitled “A simple model illustrating the tradeoff between reliability and sensitivity” in Methods for more details.

A simple cartoon of optimizing discriminability over short observation-times.

As postulated above, and illustrated for both a large-scale and idealized network architecture, we expect there to be a -parameter family of networks with the same mean firing-rate for any fixed set of inputs. This -parameter family ranges from type-A networks (with significant presynaptic-inhibition, lower reliability and higher sensitivity) to type-B networks (with significant vesicle-depletion, higher reliability and lower sensitivity). It turns out that, under rather general conditions, the networks which perform best on discriminability tasks with finite observation-times are the networks in the middle of this spectrum (i.e., networks with a combination of presynaptic-inhibition and vesicle-depletion).

To illustrate this principle, we will use the network architecture discussed in the section entitled “A simple analyzable cartoon of the tradeoff between reliability and sensitivity”. It should be noted however, that the argument we will present here is not specific to the 2-neuron architecture discussed above, and a modified version of this argument will hold for any -parameter family of networks ranging from the type-A to the type-B extremes discussed above.

To begin, let us consider the following discriminability task. Assume that a simple 2-neuron network (of the type described in the section entitled “A simple analyzable cartoon of the tradeoff between reliability and sensitivity”) is driven by one of two inputs — either () neuron is driven at rate and neuron is driven at rate , or () is driven at and is driven at . The steady-state ISI distributions and will depend on the unknown input , as well as the known system parameters . By performing a measurement of the system, is it possible to tell which input (i.e., either or ) is driving the system? Let us assume that our measurement process consists of steps. First, we estimate the mean of by drawing samples from . For example, we can either measure within a single system for a long time, or we can measure multiple systems from an ensemble. The longer a single system is measured for, the larger the effective number of samples is, assuming that measurements of the system that are sufficiently well-separated in time are effectively independent. We use the notation rather than to draw analogy with the observation-time discussed in the section entitled “A combination of vesicle-depletion and presynaptic-inhibition is required to optimize discriminability over short observation-times within a large-scale model”. Let us denote by this estimate for the mean of . Second, we assume that our measurement process includes some external noise modeled by a random variable . For the sake of presentation, let's assume that is drawn from (i.e., a Gaussian distribution with mean and variance ). Thus, our final measurement of the mean of is some estimate .

Our goal is to determine from this measurement whether the input to the system is or . By analyzing the signal to noise ratio of this measurement process (see the section entitled “Analysis of signal-to-noise ratio in a general discrimination task” in Methhods), we can show that the discrimination error associated with the best linear-classifier for this problem is well-approximated by:(2)where is the difference in the means of and , and is the average variance of and . Because is monotonic increasing, is minimized when the ratio is maximized. Recall the structure of the simple 2-neuron networks described above — both and are monotonically increasing functions of , and can be defined implicitly through (by fixing ) as a monotonically decreasing function of . For these simple networks, when , this discriminability error is minimized when is as large as possible, and the maximum is achieved when correspond to a type-A network. Conversely, when , then the discriminability error is minimized for corresponding to a network in between the type-A and type-B extremes. In this case, one can show that if are sufficiently small, then the optimal (for which the error is minimized) increases as increases. (see Fig. 5F, which displays for the case , , ).

As we mentioned earlier, The argument given in this section is quite general, and similar reasoning can be applied whenever any measurement is made by sampling from a distribution and adding an observation error . Given a -parameter family of networks indexed by , and a measurement of any dynamical feature , with sensitivity described by and reliability described by , the error associated with the best linear-classifier can be approximated by an equation similar to Eq. 2, where increases as the observation-time of the measurement increases.

A population-dynamics approach towards verifying Hypothesis 2 within more general networks.

In the preceding sections we have described in detail a specific -neuron network which exhibits the phenomena associated with hypothesis-2. However, the reasoning used in these sections cannot readily be applied to more complicated heterogeneous networks composed of more realistic model neurons. Indeed, while there exist networks for which hypothesis-2 holds (e.g., the large-scale networks described earlier on), there also exist networks for which hypothesis-2 does not hold. A natural question is: given a specific network architecture, what dynamic phenomena will that network exhibit? In the remainder of this section we will apply a rather general method [24], [25] which can be used to assess the equilibrium dynamics of pulse-coupled networks, and which can be used to determine which network architectures exhibit phenomena associated with hypothesis-2 (e.g., the tradeoff between reliability and sensitivity discussed above). With this analysis we will able to see that hypothesis-2 holds for a rather large class of networks, and in particular holds for a class of sparse randomly connected networks, provided that the network size is sufficiently large.

For the purposes of illustration, let us consider a network of discrete-state glomeruli (LNIs), each driven by a different ORN. We will model each ORN-LNI pair as a discrete-state discrete-time Markov process which is as simple as possible, while still retaining the following features: (i) each LNI generates spikes, (ii) each ORN input spike contributes to the vesicle-depletion of that ORNLNI synapse, and (iii), each LNI spike gives rise to presynaptic-inhibition of some subset of ORNLNI synapses. This model does not take into account excitatory interactions; PNs and LNEs are not included. While these excitatory interactions certainly contribute to hypothesis-2, they do not substantially change the following analysis, and we delay discussion of their effects until the end of this section.

Within this simple network model we will model the ORN-LNI pair using the state-variables , and which represent LNI membrane-potential, ORN vesicle-depletion and ORN presynaptic-inhibition, respectively. The architecture of the model is determined by the connectivity matrix which encodes the presynaptic-inhibitory coupling between ORN-LNI pairs. The other parameters of the model include the feedforward input rates to each ORN-LNI pair, as well as the overall strength of vesicle-depletion and the overall strength of presynaptic-inhibition . The details of the model are given in a section entitled “A discrete state model used to analyze hypothesis-2 within general networks with arbitrary architecture” in Methods (see Eq. 16).

We are interested in how the dynamics of such a network depends on the connectivity matrix , and also on other parameters such as the inputs , the vesicle-depletion strength and the presynaptic-inhibition strength . For reference, let us specify precisely what we mean by ‘hypothesis-2’. Let us say that the LNI satisfies ‘hypothesis 2.0’ if there exists a -parameter family of small variations in which maintain the firing rate of the LNI (denoted by ), such that this -parameter family ranges from high and low (i.e., type-A) to low and high (i.e., type-B). Let us say that the LNI satisfies ‘hypothesis-2.1’ if, given a small increase in along this -parameter family, the reliability of the LNI decreases (i.e., var increases as the network parameters are shifted towards a type-A network). Finally, let us say that the LNI satisfies ‘hypothesis-2.2’ if, given a small increase in along this -parameter family, the sensitivity of the LNI to its own input increases (i.e., increases as the network parameters are shifted towards a type-A network).

Let us assume that we have some large network in which (i.e., the input to each LNI is the same), and that is given (but otherwise arbitrary). One can readily show that hypothesis-2.0 holds — namely, for sufficiently small , each ORN-LNI pair with at least one presynaptic-inhibitory input has the property that there exists a -parameter family of parameters (ranging from high and low to low and high ) for which the firing rate remains fixed. This is simply because, as either or increases, the firing rate decreases (i.e., and are both negative) as long as is sufficiently small.

In the rest of this section, we will analyze reliability (i.e., hypothesis-2.1). Let us concentrate on a single ORN-LNI pair (say, the such pair) embedded within this larger network, and assume for the moment that the ORN is presynaptically-inhibited by the LNI (with ). If we were to increase the strength of the presynaptic-inhibitory connection between LNI and ORN (i.e., if we were to increase ) without decreasing the strength of vesicle-depletion , then the firing rate would drop. If, instead, we were to increase while decreasing simultaneously so as to maintain (as required by hypothesis-2.1), then the ISI distribution of the LNI would change (but the firing rate would remain constant by construction). In thi case the differential shift in the ISI distribution of the LNI associated with increasing the strength of connection (while appropriately decreasing ) gives rise to an increase in var. Thus, if coupling strengths are sufficiently weak, then the derivative of var with respect to increasing (while appropriately decreasing ) is positive. Using population-dynamics techniques from [24], [25], this reasoning can be systematically extended to consider every connection in the network, not merely the connection .

Formally speaking, this analysis is nothing more than a Taylor-expansion of var in terms of the coupling strengths of the network. Namely, var depends on many parameters (e.g., , , , ), and if we assume that is implicitly dependent on in such a way that is constant, then we can Taylor-expand var in terms of and the components of . If we retain all terms up to second order, such an expansion has the form(3)

where is a -order contribution, each term corresponds to a -order contribution, and each term corresponds to a -order contribution. Each and term is a correction to var associated with a particular subnetwork containing the LNI (see Fig. 6 for an example). For example, the term corresponds to the subnetwork in which the LNI (see Fig. 6 for an example). For example, the term corresponds to the subnetwork in which the LNI presynaptically-inhibits its own ORN (the ORN) — is the differential correction to var associated with increasing the connection strength , while appropriately decreasing . This correction is actually negative (i.e., if one were to increase , then the LNI would become more reliable). As another example, the term (with distinct) corresponds to the subnetwork in which both the LNI and the LNI presynaptically-inhibit the ORN. If both and were to increase, then the change in var would be well-approximated by the change in the terms (in this manner, the -order correction captures the change to var which is not accounted for by and ). Indeed, given any specific network containing the ORN-LNI pair, one can determine the effect of increasing (or, equivalently, increasing all of the components of simultaneously) by dissecting the specific network and determining the contributions made by the various comprised subnetworks to var. See Fig. 7 for an illustration of this technique.

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Figure 6. An example of subnetworks which come into play when considering the sensitivity or reliability of the LNI.

On the left in panel-A we show a particular network, with various ORN-LNI pairs (shown as ovals and circles respectively) connected via presynaptic-inhibitory connections. We will adopt the convention that the ORN-LNI pair is fixed (highlighted in dark gray), whereas the indices are not fixed, but are considered distinct from and from each other. Several dynamic features associated with the LNI can be determined by considering an expansion of the dynamics of this full network in terms of subnetworks. Shown on the right in panels-B,C,D are -order, -order and -order subnetworks of the full network which are relevant for determining the sensitivity and reliability of the LNI. The -order subnetwork consists of the ORN-LNI pair alone. The two -order subnetworks shown are those incorporating a single presynaptic-inhibitory connection — namely (top) and (bottom). The full network has embedded within it three -order subnetworks of the form , and one -order subnetwork of the form . The five -order subnetworks shown are those incorporating two presynaptic-inhibitory connections. Listed in reading order, these subnetworks are denoted by , , , , and . The full network has embedded within it , , , and of these subnetworks, respectively.

https://doi.org/10.1371/journal.pcbi.1002622.g006

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Figure 7. An example of subnetworks which influence reliability.

In this example we assume a network of the form explained in the section entitled “A population-dynamics approach towards verifying Hypothesis 2 within more general networks”. The input to each LNI is constant, and the strength of vesicle-depletion . Note however, that we do not assume that the connectivity is fixed. We adopt the convention that are distinct indices. [A] Here we illustrate the shift in the ISI-distribution of the LNI (i.e., ) that would occur (up to -order) if the connectivity were increased while decreasing so as to maintain the firing-rate of the LNI (denoted by ). The ISI-distribution of the LNI when uncoupled from the rest of the network is shown with a dotted-line for reference. The rate at which changes with respect to an infinitesimal increase in the coupling strength is shown with a dashed-line. This rate is magnified by a factor of for visibility. The sum of this rate and the uncoupled is shown with a solid-line for a qualitative representation of the new that would occur if the connectivity were increased by . The inset shows this same data (dotted and solid lines) with time plotted on a logarithmic scale for ease of view. For this particular term in the subnetwork-expansion, as increases (and the dotted shifts to more closely resemble the solid ) the var increases. The rate at which var increases as is increased is approximately for this system (as indicated by the legend ‘var(ISI)+E-3’). A separate calculation can be performed which shows that the rate at which the sensitivity of the LNI (i.e., ) changes as is increased is approximately (as indicted by the legend ‘snstvty+E-2’). Thus, by strengthening the presynaptic-inhibitory connections from several other LNIs onto the ORN-LNI pair (while simultaneously reducing so as to maintain ), we can readily show that, to -order, these shifts collectively increase both var and the sensitivity . [B–G] In these panels we show similar plots illustrating the influence of various other subnetworks on the reliability of the LNI. These plots use axes identical to those shown on the inset in panel-A. Listed in reading order, these subnetworks are denoted by , , , , , and . Note that the contribution of the autapse actually decreases var, and the contributions of the -edge subnetworks all decrease the sensitivity (albeit with magnitudes that are dwarfed by the contribution of the -edge subnetwork ).

https://doi.org/10.1371/journal.pcbi.1002622.g007

By analyzing the various terms in this expansion, one can determine that by increasing the strength of certain elements of , it is possible to actually lower var, and make the LNI more reliable. For example, by exclusively strengthening without increasing the other , the reliability of the LNI would increase, in seeming contradiction to hypothesis-2.1. However, in a typical random network (containing many ORN-LNI pairs, and many presynaptic-inhibitory connections), the subnetworks which increase var dominate those that lower var, and thus a uniform increase in will increase var. By analyzing the magnitudes of the various terms in Eq. 3, one can quantify this statement for any particular class of networks. For example, consider a random network of neurons for which each , and each element of is independently chosen to be either or with probability and respectively (i.e., a Erdos-Renyi random graph with sparsity coefficient ). For any fixed LNI , which does not presynaptically-inhibit its own ORN, there will be approximately subnetworks of the form , subnetworks of the form , subnetworks of the form , subnetworks of the form , and subnetworks of the form (where we assume are distinct). If and are modified for such a network so as to maintain , then as is increased var will increase as well (since the reduction in var caused by the subnetworks of the form and is more than cancelled out by the subnetworks of the form ). If, on the other hand, the LNI presynaptically-inhibits its own ORN, then in addition to the various subnetworks mentioned in the previous case, there will be a single subnetwork of the form , and approximately subnetworks of the form . If , are not sufficiently large, then the contribution to var will be dominated by the subnetwork, and increasing (while decreasing to maintain ) will actually reduce var. An example of the critical below which hypothesis-2.1 fails is shown in Fig. 8, which illustrates that hypothesis-2.1 holds with high likelihood for all LNIs within Erdos-Renyi random networks obeying the dynamics specified in Eq. 16 (assuming that each ), so long as is sufficiently large.

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Figure 8. An analysis of sparsely-coupled Erdos-Renyi random networks, using a -order subnetwork-expansion.

Given a network of the form described in the section entitled “A population-dynamics approach towards verifying Hypothesis 2 within more general networks”, with and fixed, one may ask if, for the LNI, the reliability of this LNI would decrease if the presynaptic-inhibitory strength were to be increased (while simultaneously decreasing so as to maintain the firing-rate ). Let us denote this condition by ‘hypothesis-2.1’. By analyzing the terms in the -order subnetwork-expansion, one can readily conclude that hypothesis-2.1 holds if the LNI does not presynaptically-inhibit its own ORN, and there is at least one other LNI which does presynaptically-inhibit the ORN. However, if the LNI presynaptically-inhibits its own ORN, then hypothesis-2.1 holds only if the size of the network is sufficiently large. This critical network size (above which hypothesis-2.1 holds with high probability) is a function of the background firing-rate of the ORNs , the strength of vesicle-depletion , and the sparsity-coefficient of the random network. In panel-A we plot , where we have calculated such that, for values of , a randomly selected LNI within an E-R random network generated with sparsity-coefficient is highly likely (probability 75%) to obey hypothesis-2.1, given that the LNI in question presynaptically-inhibits its own ORN. Values of are displayed according to the colorscale shown on the right. In the remaining panels B–F we plot for different values of . Note that, unless is small and is large, it is highly likely that hypothesis-2.1 holds (even for LNIs which presynaptically-inhibit their own ORNs) for all LNIs within an E-R random network of size .

https://doi.org/10.1371/journal.pcbi.1002622.g008

This type of approach can also be used to analyze hypothesis-2.2. Using similar techniques as above, one can readily show that hypothesis-2.2 holds with high likelihood for all LNIs in an E-R random network of any size regardless of whether or not those LNIs presynaptically-inhibit their own ORNs, as long as their ORNs are presynaptically-inhibited by at least one other LNI.

Through the application of the above analysis, one can show that all sufficiently large E-R random networks obeying the dynamics of Eq. 16 have a high likelihood of satisfying the three properties associated with hypothesis-2 — namely, (0) there exists a -parameter family of network parameters ranging from type-A to type-B for which the firing rate of any particular LNI is maintained, (1) as the network parameters are shifted along this -parameter family in the type-A direction, that LNI's reliability decreases, and (2) as the network parameters are shifted along this -parameter family in the type-A direction, that LNI's sensitivity to its dedicated ORN channel increases. For the class of networks considered in Fig. 8 the critical above which hypothesis-2 holds is roughly . Although this critical value obtained from our idealized model should not be compared quantitatively with the number of LNIs in the real fly AL, we expect that a similar qualitative result should hold for more realistic models — namely that hypothesis-2 should hold for any model as long as the presynaptic-inhibitory network includes sufficiently many inter-glomerular connections and the number of LNIs is sufficiently large.

This type of subnetwork analysis can also be used to probe the relationship between connectivity and dynamics that exists in more complicated heterogeneous networks, including scale-free and small-world networks, as well as networks in which the different neurons are governed by different equations. In each case, one can use the distribution of subnetworks within the larger network to determine dynamical features associated with neurons inside that network. For example, one can easily show that, for a scale-free network obeying the dynamics specified in Eq. 16, if the ORN-LNI pair has low incoming-degree and is presynaptically-inhibited by ORN-LNI pairs with high incoming degree, then the LNI is likely to violate hypothesis-2.1. This is simply because there will be an overwhelming number of subnetworks of the form , which each contribute to the decrease of var when is scaled up (assuming as before that is appropriately decreased to maintain the firing-rate ).

Finally, before we conclude, we address the effect of excitatory interconnectivity in the AL, which was not taken into account in the simple model analyzed above. Let's assume for a moment that, in the above model, we had added excitatory cells (say, LNEs) which were also affected by the ORN input channels. When analyzing the effect of modifying on the LNIs, the excitatory interactions associated with the LNEs only come into play at the -order. For example, if indices correspond to LNIs and index corresponds to an LNE, the simplest coupling terms by which LNE- can affect LNI- are of the form . Similar to the -order contributions shown in Fig. 7, The contributions of this term are dwarfed by the contributions of the LNI-LNI terms and . When analyzing the effects of modifying on the LNEs themselves, the situation is similar, with excitatory interactions playing a secondary role to the dominant contribution of first-order presynaptic-inhibitory connections. Thus, as long as the number of LNIs which affect each AL-neuron is not much less than the number of LNEs which affect that neuron (and as long as excitation and inhibition have effects of comparable magnitude), we expect the analytical results of this section to hold for networks which include both excitatory and inhibitory neurons.

Model synaptic currents.

ORN, PN and LNE excitatory synapses and LNI GABAergic synapses are modeled by fast-activating synaptic currents [34], [42]. Excitatory and GABAergic transmission are both modeled via stereotyped instantaneous neurotransmitter release in response to a presynaptic action potential, with ORN synapses experiencing vesicle-depletion. Although there is evidence for a slowly activating inhibitory current in the honeybee [43], we are not aware of any similar evidence for such slowly activating inhibitory current in the fly AL. There is, however, evidence for both long-timescale GABA-B type inhibitory currents, as well as short-timescale GABA-A type inhibitory currents in the fly AL [11]. To account for these two timescales, both a long- and short-timescale inhibitory current are incorporated into this model — the relevant synaptic currents include fast excitation (nAch-type, timescale 5–10 ms), fast inhibition (gabaA-type, timescale 10–15 ms) and slow inhibition (gabaB-type, timescale 100–400 ms).

Model connectivity.

Experimental observations indicate that all the ORNs that express the same odorant receptor gene have similar odor responses and project to the same glomerulus in the brain [18], [44]. Each PN receives direct ORN input from a single glomerulus [6], and thus all the ORNs and PNs corresponding to a given glomerulus constitute a discrete processing channel. We have designed the architecture of our model to reflect these experimental observations — our model ORNs and PNs only project to PNs and LNs associated with their own glomerulus.

Experimental observations also indicate that glomeruli are interconnected by a network of local interneurons. There have been several experimental results implying that the lateral connectivity between glomeruli in the AL has a strong inhibitory component, and that this inhibitory component is partly due to presynaptic-inhibition (i.e., LNIs synapsing on the ORN axons presynaptically) [12], [13], [41]. It has also been revealed that ORN-PN synapses are quite strong, and likely experience substantial synaptic-depression through vesicle-depletion [11]. Our computational model reflects these observations, and model LNEs and LNIs project to neurons both within and outside their own glomerulus. In addition, the model LNIs only affect the ORNs presynaptically. That is, deposition of neurotransmitter from LNIs onto ORNs only suppresses the efficacy of the synapse at the ORN axon (without suppressing the membrane potential at the ORN soma). In order to model vesicle-depletion, the efficacy of the synapse associated with each ORN is depleted each time that ORN fires [22].

There is debate as to whether or not there is a functionally relevant large-scale structure to the lateral connections within the AL, such as center-surround excitation/inhibition, or chemotopy [7], [45]. The lateral connectivity in our model network is structured so that the interconnections between ORNs, PNs, LNEs and LNIs are sparse and randomly determined. Thus, the results of our model may be expected to generalize to a variety of networks with a degree-distribution similar to that of an Erdos-Renyi random-graph. The connectivity of the network is encoded by a matrix , with labelling the existence of a connection between cell and cell . Each entry in the connectivity matrix is randomly chosen to be either or independently, with connection probabilities specific to the cell types and glomerular channel assignments of neurons and . Both the connection probabilities, as well as the coupling strengths are chosen in a manner consistent with the literature (details given in the sections regarding benchmarking below).

Model odor input.

The model ORNs themselves are each stimulated by Poisson input. In background (i.e., in the absence of an odor stimulus) the Poisson input rate is 350 Hz, and the input strength is low enough that the ORN background firing-rate is , and the corresponding PN background firing-rate is (consistent with experimentally observed firing-rates — [6], [41]). Odor presentation within this network is modeled by stimulating the ORNs corresponding to a subset of glomerular channels (typically around half of the glomerular channels) with additional high-rate Poisson input, in addition to the background Poisson input (consistent with experiments which indicate that an odorant typically activates multiple ORN types and triggers activity in to of the glomeruli — [18], [20], [46], [47]). In this manner, a model odor is represented in a combinatorial fashion — an odor is defined by the degree to which that odor drives the various ORN input channels. To simulate odors of different chemical composition, we stimulate different subsets of ORN input channels, whereas to simulate odors of the same chemical composition (but of differing concentration) we stimulate the same input channels to differing degrees, as motivated by the observation that varying the concentration of a given odor tends to modulate the firing-rates of responding ORNs in vivo [45], [48]–[50]. The odor-dependent noisy input signal is sufficient to drive individual ORNs strongly stimulated by the odor to firing-rates of . We choose a time-course of the odor-specific ORN current stimulus that is comparable to the time-course observed experimentally [18]. Specifically, at the time of ‘odor onset’ we increase the input-specific drive to the ORNs slowly (over ), and at odor offset we decrease the input-specific ORN drive even more slowly (over ). While there is some evidence of more complicated odor-specific temporal structure to ORN odor response, we will model only the temporally simplistic ORN response detailed above, so as to focus on emergent dynamics within the AL which manifest solely as a result of AL interconnectivity.

Stimulus driven dynamic transients reflect PN reliability and saturation.

We tuned the ORNPN connection probabilities and connection strengths in our model so as to produce stimulus-triggered dynamic transients which are qualitatively similar to experiment. This stimulated set of ORNs directly activates a corresponding subset of glomeruli. As the glomeruli respond to the ORN input, the glomerular activity pattern shifts and spreads to include other glomeruli not directly stimulated by the odor.

As shown in Fig. 9, the stimulus triggered firing-sequence of a typical model PN is more reliable than the corresponding sequence for a typical ORN associated with the same glomerulus, and the PN PSTH typically peaks after but before the ORN PSTH peaks (consistent with [20]). The sensitivity and reliability of PN activity within our model is critically dependent on three features: (i) the convergence ratio of ORNs to PNs must be sufficiently high that any given PN receives strong reliable input (summed over ORNs) immediately after odor onset, (ii) there must be some synaptic-depression at the ORN synapses, otherwise the PN activity does not peak before the ORN activity peaks (rather, the PN activity saturates and remains constant), and (iii) the combination of ORN activity and synaptic-depression at the ORN synapses must give rise to essentially ‘mean-driven’ (i.e., low-variance super-threshold) input to the PNs during odor presentation. These three network mechanisms seem to be necessary and sufficient to give rise to high-firing-rate PN activity which peaks prior to the peak in ORN activity, and which is more reliable than any single ORN's activity. We note that the source of synaptic-depression is not constrained by this particular phenomena — either vesicle-depletion or presynaptic-inhibition (generated by the recruitment of LNIs in other glomeruli) can give rise to mean-driven ORNPN input, and hence to reliable PN firing sequences. As shown in Fig. 10, the PN firing-rate is a nonlinear function of ORN firing-rate when averaged across odors and trials and PN/ORN pairs within any given glomeruli, as is consistent with experiment [20].

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Figure 9. PNs are more reliable than their individual ORN inputs.

Shown are averaged response curves for a typical model PN (magenta, solid) and model ORN (green, dashed) associated with the same glomerulus in our model. The grey overlay indicates the odor presentation period. Spikes were counted in bins. The mean spike-count per bin (averaged over trials) is shown on the left. The standard-deviation in spike-count per bin is shown in the center, and the coefficient of variation (standard deviationmean) is shown on the right. Note that, qualitatively similar to experiment [20], the model PN activates more quickly, has higher firing-rates, and is more reliable than the ORN.

https://doi.org/10.1371/journal.pcbi.1002622.g009

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Figure 10. The relationship between ORN firing-rates and PN firing-rates is nonlinear.

Shown is a scatterplot of model PN and model ORN firing-rates associated with a typical glomerulus in our model. Spike rates were measured during the epoch during which PN firing-rates peak following odor presentation. Note that, qualitatively similar to experiment [20], the model PN firing-rates saturate for relatively small values of ORN firing-rates.

https://doi.org/10.1371/journal.pcbi.1002622.g010

PNs are more broadly responsive to odors than their ORN class would indicate.

Experimentally, it has been observed that most PNs respond to multiple odors [45], [51], and are more broadly responsive than their input ORNs [9], [20] — there are many different PNs which respond to odors that do not stimulate their respective ORNs, and there are PNs which do not respond very strongly even when their respective ORNs are strongly stimulated. Thus, due to the lateral connectivity within the AL, the total set of activated glomeruli corresponding to a particular odor is generally not in one-to-one correspondence with the set of activated ORN olfactory receptors stimulated by that odor. Many believe that the inter-glomerular crosstalk is critically important for redistributing the glomerular activity within the AL ([41], but also see [7], [45]). It has been postulated that this re-expression of the odor at the glomerular level is advantageous for the fly, as the combinatorial code linking different odors to different glomerular subsets serves to separate similar odors more efficiently than the corresponding combinatorial code at the olfactory receptor level [20].

We tuned the lateral connection probabilities and connection strengths within our model to produce a dynamic regime capable of rich glomerular activation patterns which are qualitatively similar to experiment. As shown in Fig. 11, the lateral connectivity in our model is sparse enough that oscillations do not develop during the initial odor-response, and yet strong enough that the activity of PNs and LNs within each glomerulus is not in direct correspondence with the activity of their respective ORN inputs. This lack of correlation between PN response and ORN response across odors can be quantified by measuring the PN-PN and PN-ORN rank-correlation, which is in good qualitative agreement with experiment [20]. In our model, as in experiment, LN activity is similar to PN activity (data not shown). For example, LN firing-rate can get quite high, and typically peaks shortly after odor onset [13].

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Figure 11. PNs exhibit broader odor responses than their associated ORNs.

[A] Shown are trial-averaged firing-rate curves for various model PNs (magenta, solid) and associated model ORNs (green, dashed) in response to various model odors. Note that, qualitatively similar to experiment [12], the activity of the model PNs does not necessarily reflect the activity of the associated model ORNs. [B] Shown are the PN-ORN (green) and PN-PN (red) Spearman rank-correlation histograms for the model PNs and associated model ORNs (averaged over all PN and ORN pairs associated with each given glomerulus, and then further averaged over glomeruli — see [20] for the statistical methods used). Note that, qualitatively similar to experiment, the mean of the PN-ORN histogram is closer to than the mean of the PN-PN histogram, indicating that, while PNs associated with a given glomerulus tend to respond to the same odors, they do not necessarily respond to the same set of odors which stimulate their associated ORNs.

https://doi.org/10.1371/journal.pcbi.1002622.g011

ORNPN induced EPSCs attenuate as frequency of ORN synapse activation increases.

Our model of vesicle-depletion and presynaptic-inhibition is phenomenological, and is intended to allow us to qualitatively reproduce and investigate the functional role of synaptic-depression at the ORN synapses. We have chosen a parsimonious model for vesicle-depletion, involving only one timescale of , and one parameter . Similarly, our model of presynaptic-inhibition involves only the timescales of synaptic inhibition and a coupling strength (see the section regarding vesicle-depletion and presynaptic-inhibition below). In order to ensure that our phenomenological model of synaptic-depression at the ORN synapses (a combination of vesicle-depletion and presynaptic-inhibition) was qualitatively accurate, we followed the experimental paradigm of [11]. We constructed a numerical experiment to measure the attenuation timescale of ORNPN input (see Fig. 12). We first forced the ORN synapses to activate periodically at (to simulate ORN background firing-rates), and then once the system equilibrated to this periodic input, we increased the frequency of ORN stimulation (to ) and measured the input current to each PN as a function of time. For a larger given frequency , the attenuation of the PN EPSC will occur more quickly. We tuned the coupling strengths so that the attenuation time scale (as a function of ) qualitatively matched experimental observations [11]. Our modeling work indicates that the attenuation time-scale match experiment as long as and are sufficiently high (the exact ratio of to is not strongly constrained by this particular experiment).

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Figure 12. Synaptic-depression at the ORN synapses.

Shown are current traces associated with a model PN in response to direct current stimulation of the model ORNs associated with that PN. Analogous to experiment [11], the model ORNs associated with the model PN have been stimulated by periodic input current prior to the epoch shown in the figure. At the start of the epoch shown in this figure, the ORN stimulation is increased to , , , or . The trial-averaged model PN EPSCs in response these different stimulations are plotted (over a time interval of ). Above each EPSC curve, we show the envelope of the response in gray. This envelope is calculated by fitting a piecewise linear function to the maxima of the EPSC response sampled at the rate of stimulation. Note that, similar to experiment, the envelope of the PN EPSC attenuates more quickly when stimulated at than when stimulated at .

https://doi.org/10.1371/journal.pcbi.1002622.g012

A further constraint on our model of synaptic-depression can be obtained by considering the correlation between total ORN activity, and reduction in subthreshold voltage observed at any given PN [12]. To obtain a roughly linear relationship between ORN activity and PN inhibition (as shown in Fig. 13), our model requires .

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Figure 13. Presynaptic-inhibition is partly responsible for ORN PN synaptic-depression.

Shown is a scatterplot displaying the correlation between total ORN activity across all glomeruli in response to various odors, and the suppression of spontaneous EPSPs associated with a particular PN associated with a glomerulus which has been ‘shielded’ (i.e., the odor stimulus chosen does not affect the input drive to that glomerulus). In analogy with [12]. The PN suppression is measured as the difference in integrated PN membrane potential between (i) the scenario in which the PN receives spontaneous spikes from its associated ORNs in the absence of any odor, and (ii) the scenario in which the glomerulus associated with that PN is shielded and an odor is presented, in which case the activity generated within the other glomeruli reduce the effect of the spontaneous spikes impingent on the PN, and the spontaneous EPSPs are absent or greatly diminished. Note that, due to presynaptic-inhibition within the model, the correlation between PN EPSP magnitude and total ORN activity is qualitatively similar to experiment [12].

https://doi.org/10.1371/journal.pcbi.1002622.g013

Intrinsic currents.

The intrinsic currents for each neuron consist of fast sodium and potassium currents and . These currents obey equations of the following form:The maximal conductances are , and . The reversal potentials are , .

The gating variables take values between and and obey equations of the following form: and are described in [52]:

Synaptic currents.

Given two connected neurons in the network, the synaptic conductances of the postsynaptic neuron are increased whenever the presynaptic neuron's membrane potential rises above a threshold of (i.e., when the intrinsic currents of the presynaptic neuron generate an action potential). The excitatory-synaptic current associated with the neuron in the network is governed by an equation of the following form:(4)(5)with the excitatory reversal potential . The efficacy of the excitatory-type synapse associated with the neuron obeys the equation:(6)where , the times are the spiketimes of the neuron, , and (adapted from [33], [34], [53], [54]). The term represents synaptic-depression at the ORN synapses, and is used to model both vesicle-depletion and presynaptic-inhibition at these synapses. This term is identically if corresponds to a PN, LNE or LNI. The dynamics of when corresponds to an ORN will be discussed later. The differential equation Eq. 6 is structured so that for all time, and in the absence of firing . The synaptic GABA-A current for the PNs, LNEs and LNIs obeys equations analogous to Eqs. 4,6, with . The efficacy of the GABA-A-type synapse associated with the neuron obey the equation(7)with . The synaptic GABA-B current for the PNs, LNEs and LNIs obeys an equation of the following form:with inhibitory reversal potential . The quantity for the neuron obeys the equation:with . The efficacy of the synapse for the neuron obeys an equation analogous to Eq. 7, with . Thus, the slow GABA-B type synaptic current has a rise and decay time-scale, whereas the fast excitatory and GABA-A type synaptic currents only have decay time-scales (again adapted from [33], [34], [53], [54]).

The synaptic coupling strengths , , and depend only on the cell types of neurons and , and are chosen so that only presynaptic ORNs, PNs and LNEs give rise to excitatory-type currents, and only LNIs give rise to GABA-A and GABA-B-type currents. The strengths are given by the following arrays:(8)

The parameter varies between (no presynaptic-inhibition) and (strong presynaptic-inhibition). Note that the PNPN and LNIPN coupling strengths are all negligible, to account for the experimental observations that PNs may not be targeted by other PNs, or by local inhibitory interneurons. However, we do allow for PNs to connect to LNEs (and LNIs), as observed in [41]. If these strengths are set to we can retune the remaining connectivity strengths so that our major conclusions still hold (data not shown).

Network connectivity.

The intra-glomerular connection probabilities are given by the array:(9)and the inter-glomerular connection probabilities are given by the array:(10)

Modeling vesicle-depletion and presynaptic-inhibition at the ORNPN and ORNLN synapses.

The ORNs in this model do not directly experience synaptic conductances from either the PNs or the LNEs. The excitatory-type from PNs and LNEs onto ORNs is identically (see the array shown in (8) above). The GABA-A and GABA-B-type conductances associated with an ORN (say, with index ) alter the term in Eq. 6, thus affecting the efficacy of synapses from the ORN onto the AL. The evolution of in our model is governed by two parameters: (described above), and (described below). These two parameters will allow us to consider a -parameter family of model networks in which the strength of vesicle-depletion and presynaptic-inhibition can be altered independently (see Fig. 4). The equations governing the evolution of arewith the vesicle-depletion parameter obeying the differential equation:with . The parameter varies from (no vesicle-depletion, ) to (complete vesicle-depletion with each firing-event). As the vesicle release rate per synaptic event is likely quite high within the real fly AL [11], values of are most reasonable from a physiological standpoint. Note that the vesicle-depletion parameter is bounded between and .

Odor simulation.

The stimulus current to the ORN is governed by the equationwhere is a response-function such that if , and if , with . The spiketimes are drawn from a Poisson-process with rate . The strength of this background input is . The spiketimes are drawn from a Poisson-process with rate which depends on the time since odor onset, the odor being presented, as well as the ORN under consideration. Typical values for range from (when the odor does not directly stimulate the ORN) to (when the ORN is being strongly stimulated by the odor). The strength of this odor-specific input is .

The time-dependence of the odor-specific input-rate is governed by the factorwith and representing the onset and offset times of the odor stimulus (respectively), and with , . A typical odor (stimulation of the ORNs within of the glomerular channels) activates of the PNs, which fire at about . Not all of the PNs exhibit increased activity upon odor stimulation — a given odor will typically cause a few PNs which are not directly stimulated to actually decrease in activity (as a result of inter-glomerular inhibition). On presentation of a typical odor, the typical PN PSTH rises very quickly, and peaks after , before the ORN PSTH peaks (at by construction). The typical PN PSTH decays more quickly than the ORN PSTH ( vs ms). Due to lateral connectivity, PNs respond to a broader selection of odors than their ORN inputs and most PNs have a ‘rank order’ of odors that is different from the rank order of their ORN inputs. Note that PNs exhibit a rise in firing-rate after odor offset only very rarely. This is a consequence of the fact that the current model only incorporates odors which increase the firing-rate of the stimulated ORNs. It has been observed that some odors actually decrease the firing-rate of certain ORN types, with perhaps a resurgence of ORN firing-rates after odor offset (as shown in DL1 response to cis-3-hexen-1-ol, cyclohexane and ethyl acetate — [20]). These types of inhibitory OdorORN responses could potentially give rise to richer PN dynamics (potentially triggered by PNs/LNs which fire greatly after odor offset), and these phenomena will be studied in more detail in future work.

Odor discrimination.

In order to estimate the model network's ability to discriminate different odors, we measure the odor-dependent probability distribution of each PN's firing-rate, and use standard methods from classification theory [55]. For completeness we describe our procedure applied to firing-rate vectors.

Assume that we are estimating the network's ability to discriminate between different odors. For each , we perform multiple trials of odor , and estimate the probability distributionOnce the are sufficiently well estimated, we perform and classify individual odor trials. A single trial of odor (randomly chosen to be either or with probability) will give rise to a vector such that is the number of firing-events produced by the PN during that trial. By looking at a fixed and comparing and , we can use the PN to identify a possible candidate stimulus (either odor if , or odor if ). This process can be performed for each , and in this way each PN ‘votes’ for a candidate stimulus. We tally these votes, weighting each one by the log of the information ratio associated with each PN. We use the weightingwhere is the ‘hit-rate’ associated with the PN:and is the ‘error-rate’. This particular weighting is chosen so that votes for stimulus with error-rate have the same combined weight as a single vote for stimulus with a far smaller error-rate of . The sum of the weighted votes is compared to determine the candidate stimulus underlying this particular trial. If the candidate stimulus matches the true stimulus, the trial is classified correctly. If not, the trial has been classified incorrectly. By going through this process with multiple trials, we can generate a probability that any given trial will be classified correctly. To perform -way discriminability tasks, we go through an analogous procedure, performing all pairwise discriminability tasks for each sample observation, and ultimately selecting the candidate stimulus corresponding to the majority (with ties automatically counted as incorrect).

We have chosen this particular procedure because it allows us to take advantage of components of the firing-rate vector which carry substantial information (as measured by the information ratio), without requiring an estimate of the joint distribution of firing-rates (across PNs) for any particular odor. Thus, this measure of discriminability is more sensitive than typical linear discriminators which use the Euclidian distance between firing-rate vectors (see [20]), but does not succumb to the curse of dimensionality associated with the large number of distinct PNs.