Functional roles for synaptic-depression within a model of the fly antennal lobe

Abstract

Several experiments indicate that there exists substantial synaptic-depression at the synapses between olfactory receptor neurons (ORNs) and neurons within the drosophila antenna lobe (AL). This synaptic-depression may be partly caused by vesicle-depletion, and partly caused by presynaptic-inhibition due to the activity of inhibitory local neurons within the AL. While it has been proposed that this synaptic-depression contributes to the nonlinear relationship between ORN and projection neuron (PN) firing-rates, the precise functional role of synaptic-depression at the ORN synapses is not yet fully understood. In this paper we propose two hypotheses linking the information-coding properties of the fly AL with the network mechanisms responsible for ORN-->AL synaptic-depression. Our first hypothesis is related to variance coding of ORN firing-rate information--once stimulation to the ORNs is sufficiently high to saturate glomerular responses, further stimulation of the ORNs increases the regularity of PN spiking activity while maintaining PN firing-rates. The second hypothesis proposes a tradeoff between spike-time reliability and coding-capacity governed by the relative contribution of vesicle-depletion and presynaptic-inhibition to ORN-->AL synaptic-depression. Synaptic-depression caused primarily by vesicle-depletion will give rise to a very reliable system, whereas an equivalent amount of synaptic-depression caused primarily by presynaptic-inhibition will give rise to a less reliable system that is more sensitive to small shifts in odor stimulation. These two hypotheses are substantiated by several small analyzable toy models of the fly AL, as well as a more physiologically realistic large-scale computational model of the fly AL involving 5 glomerular channels.

Figure 1. A schematic of the large-scale network model.

[Left]: The network consists of 5 glomerular channels, each incorporating 60 olfactory receptor neurons (ORNs in green) which stimulate a ‘glomerulus’ consisting of 6 projection neurons (PNs in red), 6 excitatory local neurons (LNEs in magenta) and 6 inhibitory local neurons (LNIs in blue). The PNs, LNEs and LNIs are connected to one another randomly within each glomerulus, and the LNEs and LNIs also affect the neurons in other glomeruli. The LNIs affect the ORNAL synapses via presynaptic-inhibition. [Right]: The non-negligible connection strengths are listed on top, with the slow-inhibitory connection strengths listed separately from the fast-inhibition strengths. The relevant connection probabilities are listed on the bottom. The parameter refers to , which characterizes the overall strength of presynaptic-inhibition. See Methods for full details.

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.

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.

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.

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 .

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.

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 ).

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 .

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 , the model PN activates more quickly, has higher firing-rates, and is more reliable than the ORN.

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 , the model PN firing-rates saturate for relatively small values of ORN firing-rates.

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 , 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 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.

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 , 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 .

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 . 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 .

Figure 14. A simple analyzable cartoon of the tradeoff between reliability and sensitivity.

[A] Shown is a schematic of the simple network, consisting of 2 ORNLNI pairs, each of which presynaptically inhibits the other. [B] Shown on top are sample voltage-traces for the two LNIs (represented by and ) for the case . Shown on the bottom are sample voltage-traces for the two LNIs in the case that is nonzero. Note that after LNI A fires, is constant for -time. Similarly, after LNI B fires is constant for -time. A pair of voltages for LNI B are circled. This pair of voltages corresponds to a point on the graph of the return map , namely . For this point on the graph of , , and . [C] Shown on the top and bottom are return maps for the values , and , respectively. [D] Shown on the top and bottom are return maps for the values , and , respectively.