Fig 1.
Connectivity among KCs and PNs.
(A) Clusters of KCs and their connectivity. The connections shown here are the low-inhibition connections (blue) and high-inhibition (red) between specific KCs and the medium inhibition between members of a cluster (green). The high-inhibition all-to-all connections are not depicted. Each neuron is connected (as an example) to three other neurons outside of its cluster. A line with a single dot at the end means that the connection is one-way only. The neurons in one cluster connect to each other and to elements outside the cluster. Those neurons that receive input from more than one KC in a cluster form the cluster to be activated next. Each KC could connect with a neuron that is not part of the next cluster; these connections are labeled KCn, KCn+1, etc. and arise because KCs can belong to other clusters as well. (B) Conditions for the connectivity matrix in the proposed model and in the [11] model. In the proposed model, all components of the connectivity matrix ρ are negative (inhibition), whereas the [11] model uses both inhibition and excitation. Ck is the k-th cluster, i.e. i ϵ Ck for all neurons in cluster k. (C) Example of connections between KCs and PNs. KCs work as coincidence detectors, being activated only when both the PNs they are connected to are activated. Shown, a population of 6 KCs divided in two clusters. Four PNs are connected to the KCs so that each KC sees half of the PN population. When PNs 1, 2 and 3 are active, cluster 1 is activated. Cluster 2 is activated with PNs 2, 3 and 4.
Fig 2.
PN output of our model compared to the Laurent-Rabinovich model.
Firing rates of 100 KCs and 30 PNs in mushroom body and the antennal lobe, respectively, for the proposed model and an implementation of the Laurent-Rabinovich model. Each panel represents the activity of the KC and PN populations at times t1, t2, t3 and t4, in which the clusters of the sequence are at their maximum value (see Fig 5), and each square in a panel represents a neuron (10x10 for KCs, 6x5 for PNs). (A) An SHS model, as presented in [11], used for the KC population, plus a random 1 KC to 20 PNs projection. (B) Data generated by the proposed model, with clusters of three KCs each.
Fig 3.
Illustration of experimental findings.
(A) Raster plots for the full response (three phases) to a stimulus of PNs (left) and KCs (right) using a modified version of the generative model (see Methods for details; also see the S1 Text). This modified version was only used for the generation of this figure and no inversion was performed on it. Although the model consisted of 30 PNs and 100 KCs, most KCs were inactive during the simulations. Only the 27 active KCs are shown. The simulation was run for ten trials and the responses of different neurons are separated by thick black lines. For the PNs, periods of inhibition and excitation can be observed. During the steady state phase the population no longer evolved. KC responses are very sparse, with only a few spikes per KC throughout a trial. (B) Instantaneous firing rate for a sample PN. (C) The periods of excitation and inhibition (with respect to a baseline) during the dynamic phase for the PN in B. The baseline shown is calculated as half the minimum PN activity that the generative model can produce before it is zero.
Fig 4.
Diagram of the proposed model.
The generative model was used in two ways: first, to generate the PN data that was used as input to the Bayesian inference; Second, to describe the internally expected dynamics of the PN and KC populations. Top left: Specifically, the generative model uses the Lotka-Volterra equations to generate activity of the KC population. Bottom left: This activity was projected into the PN population via Eq 2 giving synthetic PN activity that was used as input to the Bayesian inference. Bottom right: Through the unscented Kalman filter, the PN activity observed was balanced with the expected dynamics of the KCs (prescribed by the generative model). Top right: This enabled the model to infer KC activity that is consistent with the PN input data.
Fig 5.
Example activity of the generative model’s PN-KC network.
Y axis represents instantaneous firing rate, X axis is time. (A) Activity of 20 sample PNs that all connect to the same KC, showing epochs of excitation and inhibition. The shaded area is the onset sequential phase; the rest is the steady state phase. (B): Activity of the KCs during the onset sequential phase where each colored line represents a cluster of three KCs. In total, 15 KCs are shown (five clusters), with maximum firing rates around times t1, t2, t3 and t4. As in Fig 3, the baseline shown is calculated as half the activity level of a PN when only one KC of those connected to it is active.
Fig 6.
Performance of the model in difficult tasks.
The connectivity matrix has two sequences which were similar to each other. (A-B) Each single line represents one trial (out of 100 trials) simulated from the model. The Y axis represents the recognition variable, which is calculated from the difference between the Euclidean distances from the observed KC activity to the correct one and from the observed KC activity to the incorrect one (see Eq 6 in Methods). If the KCs are displaying the correct representation (encoding the displayed odor), the corresponding line is near the top. If the KCs are displaying the incorrect representation (i.e. the other sequence stored in the system), the corresponding line is near the bottom. Lines near the middle are as close to the correct representation as to the incorrect one. The activity of the clusters in the expected (correct) sequence is in the plot background as a time reference to show that many single trials quickly jump to the correct representation after the second cluster in the sequence starts: The shaded areas (2 colors) represent two clusters in the sequence (as in Fig 5B). (A) Results for a task in which the PN data for the two stored sequences are very similar during the first cluster: Only three out of twenty PNs are different for the two sequences during the first cluster. Due to noise, the KC activity represents sometimes the incorrect odor for a brief period of time. (B) Results for a task in which the PN data for the two stored sequences are identical during the first cluster but dissimilar for the rest of the odor. During exposure to the first segment of the odor, the representations on the KCs are correct around 50% of the time, as expected, consistent with random chance. When the second cluster in the sequence begins (at ca. 170ms), the KC activity quickly jumps to the correct representation. (C) Average reaction time of 100 trials plotted against difficulty of the task (as defined with the number of PNs that belong to the representations of both odors; see Methods). The PN representations of the two stored sequences are similar during the first 170ms; afterwards, they diverge and become easily identifiable. The more similar the PN representations of two stored odors are during this initial period, the longer it takes the KCs to identify it. The maximum reaction time, around 200ms, corresponding to the case in B, is obtained when all 20 PNs are the same. The case in A corresponds to 17 identical PNs.
Fig 7.
PN and KC activity for the full response to a stimulus.
(A) Sample of four PNs’ activity throughout the full response. Left: experimental data as reported in [7]; Right: data generated by our model. The qualitative behavior of the PNs observed in experiments can be matched by our model by choosing an appropriate observation matrix. (B) Bayesian inversion of the model: Inferred KC activity as a response to the PN data generated from the full response shown in A. Each color represents a cluster of three different KCs. It can be seen that the inferred KC activity displays the three phases of the odor response, although the generative model given by Eq 1 includes only the sequential dynamics.
Fig 8.
Performance of the model under adverse conditions.
(A) Simulations to test robustness against noise in the PNs. Percentage of correct trials (i.e. trials in which the model displayed the correct KC representation) for low SNRs. For these simulations, we added white noise of different variances to all PN activity. The model is said to display the correct response if the Euclidean distance in the firing-rate-space between the inferred and expected (i.e. used to create the data) KC responses is smaller than a threshold (see main text). A decrease in performance can be observed only for SNRs smaller than 2.6. (B) Simulations to test robustness against KC states not at baseline at the onset of an odor. Histogram of reaction times (see Methods) for trials with initial KC states set to a baseline (light gray) or a random state (blue). These reaction times were calculated for two completely dissimilar sequences with high SNR. Therefore recognition is rapid. The reaction times for trials with random initial conditions are slightly more varied and the mean reaction time is 19ms, as compared to 17ms when the initial KC activity is at baseline.
Fig 9.
Time-jittered KC activity from the model.
Using one connectivity matrix, we generated PN data with no jitter and used that data to create time-jittered PN activity. The KC activity output by our model is shown: each row represents a sequence displayed by our model; each color represents a cluster in the sequence. The first sequence (top) has no time-jitter in the input. The following sequences are time-jittered, i.e. the time each cluster is active is different. It can be readily seen that the Bayesian inference has no trouble handling time-jitter that cannot be generated directly by the generative model.
Fig 10.
Performance of the model with noisy PN input.
The performance of our model was measured in terms of displaying the expected sequence in response to the input while a number of PNs that connect to a single KC display large amounts of noise. Three noise levels were used (different colors); with high noise (SNR of only 1), the activity of a PN is mostly noise-driven, because the fluctuations due to noise are comparable to activity of the noise-free PNs. With an SNR of 2 (called here low noise), the noise variations are about one third of the maximum noise-free PN activity. Medium noise had an SNR of 1.5. On the x-axis, we plotted the number of noisy PNs; on the y-axis, the percentage of trials in which the KCs displayed the correct representation (100 trials for each number of noisy PN/noise combination). Left: performance of an inverted Single-neuron SHS generative model. We used a modified version of our generative model, in which single KCs, as opposed to clusters of KCs, are activated sequentially. This is then projected to the PNs as described for our proposed model and inverted with the described unscented Kalman filter. Right: performance of proposed model with Cluster SHS. The large differences in the performance of the two models can be attributed to the within-cluster lateral input that the KCs receive (i.e. intelligent coincidence detector) in the Cluster SHS model: The performance of the single-neuron SHS model degrades for smaller number of noisy PNs, for all noise levels. In particular, for high noise rather small numbers of noisy PNs lead to poor performance.
Fig 11.
Dimension reduction of the PN response to a stimulus.
(A) Data as presented in [7] showing the trajectories in the first three components of principal component analysis for three different odors (different plots) and three stimulus durations for each odor (blue, red and green for 0.3s, 1s and 3s, respectively). The dynamic phase of activity lies between baseline activity (represented by B) and the fixed point (FP), where a number of turning points can be seen, between 2 and 4 for different cases. (B) First three elements of principal components analysis of the data obtained with the proposed model of the 30 PNs during the first phase of the full response, for different number of clusters in a sequence (4, 6 and 10 clusters). The number of elements equals the number of sharp turning points in the trajectory (big circles in the left-most plot). Around each turning point, the data points agglomerate, reflecting the period in which a cluster remains active. Since we used data from only the onset sequential phase, the trajectories we present are not closed as in experimental data where the data acquisition is done up until the system returns to baseline, a few seconds after odor offset. The points B and FB, as before, represent the baseline (at odor onset) and the fixed point (end of onset sequential phase).