Spiking network model of pDp.

A. Schematic of pDpsim. OB, Olfactory bulb; E, excitatory; I, inhibitory neurons. B. Spike raster of a random subset of 50 mitral cells in the OB representing 2 odors. During odor stimulation, firing rates of 10% of mitral cells were increased and firing rates of 5% of mitral cells were decreased (baseline rate, 6 Hz). C. Spike raster of random subsets of 50 E and I neurons in response to 2 odors. D. Representative membrane potential trace (top) and excitatory (EPSC, black) and inhibitory (IPSC, red) currents (bottom) in one excitatory neuron in response to 2 odors. Purple trace shows net current (EPSC + IPSC). E. Odor-evoked inhibitory (red) and excitatory (black and blue) currents as measured in a hypothetical voltage clamp experiment (conductance multiplied by 70 mV, the absolute difference between holding potential and reversal potential; Rupprecht et al., 2018). Representative example of 1 network, averaged across neurons and odors. F-H. Measured values of the observables used to match pDpsim to experimental data. Each dot represents one network (average over 10 odors); n = 20 networks. Pink shading shows the experimentally observed range of values. F. Baseline and odor-evoked population firing rate. G. Left: gOE is the synaptic conductance in E neurons contributed by afferents from the OB during odor stimulation. Middle: gsyn is the total odor-evoked synaptic conductance. Right: % recurrent input quantifies the percentage of E input contributed by recurrent connections during odor stimulation. H. Correlation coefficient between odor-evoked activity patterns in Dp. The dotted line indicates the mean correlation between odor patterns in the OB.

Neuronal assemblies (memories).

A. Schematic of assemblies. Each assembly contains the 100 E neurons that are most densely connected to the mitral cells activated by a given odor. Connection probability between these E neurons is increased by a factor α. In Scaled I networks, weights of all I-to-E synapses are increased by a factor χ. In Tuned networks, the 25 I neurons that are most densely connected to the 100 E neurons are included in each assembly. In Tuned I networks, the probability of I-to-E connections within the assembly is increased by a factor β. In Tuned I+E networks, probabilities of I-to-E and E-to-I connectivity within the assembly are increased by factors β and γ, respectively. n = 20 networks with 15 assemblies each were simulated for each group. B. Firing rates averaged over all E or I neurons (full lines) and over all assembly neurons (dashed lines) as a function of α (mean ± SD. across 20 networks). C. Mean E neurons firing rates of Scaled (left) and Tuned (right) networks in response to learned odors as a function of connection strength and probability, respectively. Squares depict parameters used in following figures unless stated otherwise. D. Spike raster plots showing responses of 50 E neurons to two odors in a Scaled I and the corresponding Tuned E+I network (same neurons and odors in the corresponding rand network are shown in Fig. 1C). E. Top: Mean firing rates in response to learned odors as a function of time, averaged across assembly or non-assembly neurons. Bottom: Correlation between activity patterns evoked by different trials of the same learned odor as a function of time. The pink bar indicates odor presentation. F. Mean firing rate in response to learned odors or novel odors. Each data point represents one network-odor pair (n=20 networks, 10 odors). G. Amplification within and outside (non-A.) assemblies, calculated as the ratio between mean firing rates in response to learned odors averaged across the same populations of neurons in a given structured network (Scaled I, Tuned I, or Tuned E+I) and the corresponding rand network. H. Quantification of co-tuning by the correlation between time-averaged E and I conductances in response to different odors, average across neurons (n = 20 networks). I. Quantification of co-tuning by the ratio of dispersion of joint conductances along balanced and counter-balanced axes (Methods). Each data point corresponds to one network (n = 20). Mean +/- SD.

Changes of output activity to gradual modifications of inputs.

A. Morphing of a novel odor N into a learned odor L. Intermediate mixtures were generated by gradually decreasing/increasing the fractions of active mitral cells defining odors N/L. B. Pearson correlation between activity patterns evoked by the learned (full line) or novel (dotted line) odor and the intermediate odors in pDp as a function of the corresponding correlations in the OB. C. Firing rates in response to intermediate odors averaged across assembly neurons (learned odor, full line) or pseudo-assembly neurons (novel odor, dotted line) as a function of correlations between the OB activity patterns representing the intermediate odors and the OB activity pattern representing the learned odor. B, C: averages over 8 networks.

Geometry of odor representations in pDpsim.

A. Odor subspace delineated by 2 learned (L, M) and 2 novel (N, O) pure odors at the vertices of a square. Pixels within the square represent odor mixtures. B. Left: Pearson correlations between OB activity patterns defining pure odors. Right: Correlation between one pure odor (L; top left vertex) and all other odors. The odor from one vertex gradually morphs into the odor from another vertex. C. Projection of activity patterns in the OB onto the first 2 principal components (PCs). Colors represent patterns in the odor subspace shown in A. D. Projection of activity patterns in pDp in response to the odor subspace onto the first 2 PCs. Representative examples of different pDp networks. E. Density plot showing distribution of data points and demonstrating clustering at distinct locations in PC space for Scaled I networks. F. Quantification of dimensionality of neural activity by the participation ratio: activity evoked by novel odors and related mixtures (left), activity evoked by learned odors and related mixtures (center), and activity evoked by all stimuli (right). Each data point represents one network; dotted line represents the participation ratio of OB activity. G. Variance along the first 40 PCs extracted from activity patterns in Rand and Tuned E+I networks. Insets: variance along PCs 200 - 400. H. Angles between edges connecting a pure odor response and related versions thereof (see inset). The analysis was performed using the first 400 PCs, which explained >75% of the variance in all networks. n = 21 angles per pure odor in each of 8 networks (Methods). Similar results were obtained in the full-dimensional space.

Distance relationships and classification of odor representations.

A. Activity patterns used as class distributions and vectors (B) or training and test sets (C). Same odor subspace as in Figure 4. B. Left: schematic illustration of Mahalanobis distance dM. Right: dM between one activity vector (v) and reference classes (Q) in rand and Tuned E+I networks. dM was computed based on activity across subsets of 80 E neurons drawn from the four (pseudo-) assemblies with equal probability (top) or from the whole population (bottom). Note that dM between patterns related to a learned odor and non-matching reference classes was higher in Tuned E+I networks, particularly when E neurons were drawn from assemblies. C. Pattern classification probability quantified by QDA. PTarget quantifies the probability that an activity pattern from the test set (odor mixtures, see A) is assigned to a target class from the training set (pure or closely related odor; see A). Left: classification probability as a function of the similarity (Pearson correlation) between the test and target odors in the OB (input patterns). Note enhanced classification probability for patterns evoked by odors similar to learned odors in Tuned E+I networks. Right: Classification probability for patterns similar to the training set (see A).

Representation of correlated patterns and resilience against additional memories.

A. Subspace delineated by four positively correlated odors. Top: Correlations between pure odors. Bottom: Projection of OB activity patterns onto the first two PCs. B. Firing rates (top) and PC projection of output activity of a Tuned E+I network with 15 E/I assemblies that were not aligned to any of the four pure odors of the subspace. C. Firing rates (top) and PC projection of output activity after creation of two additional assemblies representing two of the pure odors (Y and Z). Left: Tuned E+I network. Right: Scaled I network. Note that in the Scaled I network, but not in the Tuned E+I network, firing rates evoked by newly learned odors were increased and patterns evoked by these odors were not well separated in PC space.

Schematic of geometric transformations.

A. Randomly connected networks tend to preserve the geometry of coding space. Such networks can support neuronal computations, e.g., by projecting activity patterns in a higher-dimensional coding space for pattern classification. B. We found that balanced networks with E/I assemblies transform the geometry of representations by locally restricting activity onto manifolds. These networks stored information about learned inputs while preserving continuity of the coding space. Such a geometry may support fast classification, continual learning and cognitive computations. Note that the true manifold geometry cannot be visualized appropriately in 2D because activity was “focused” in different subsets of dimensions at different locations of coding space. As a consequence, the dimensionality of activity remained substantial. C. Neuronal assemblies without precise balance established discrete attractor states, as observed in memory networks that store information as discrete items. Networks establishing locally defined activity manifolds (B) may thus be considered as intermediates between networks generating continuous representations without memories (A) and classical memory networks with discrete attractor dynamics (C).

Values of the neuronal parameters.

The superscripts indicate the reference where the experimental measurements can be found. 1 Rupprecht and Friedrich, 2018 2 Blumhagen et al., 2011

Values of the connectivity parameters of different networks {probability pYX and synaptic strength wYX in pS}

Percentage of cells selected from the 150 activated mitral cells defining one pure odor (vertex at top left) for uncorrelated pure odors.

Percentage of activated cells available for selection for uncorrelated pure odors.

C is obtained by multiplying the values by 1.5.

Values of α, β, γ, χ used in the simulations. α: increase in E-E connection probability within assemblies.

β: increase in I to E connection probability. γ: increase in E to I connection probability.

Structured networks reproduce key features of Dp.

A. Connection probability between classes of E and I neurons in a rand, a Tuned I, and a Tuned E+I network. B. EPSCs and IPSCs in a Tuned I network as observed in a hypothetical voltage clamp recording, averaged across neurons and odors. An equivalent plot for a rand network is shown in Figure 1E. C,D. Values of observables for rand networks (same as Figure 1G-H) and different structured networks (Scaled I, Tuned I, Tuned E+I). E. Network with increased connectivity between the E assembly neurons (α=5) and the 25 I neurons that are most densely connected to the mitral cells activated by a given odor. F. Mean firing rate of the network in E in response to learned odors as a function of connection probability. Selecting I neurons based on their afferent connectivity could not stabilize activity efficiently.

Structured networks: additional results.

A. Raster plots showing responses of assembly (A) or non-assembly neurons to a learned odor. B. Coefficients of variation of the inter-spike interval (ISI) in assembly neurons. C. Correlation between activity patterns across E neurons evoked by the same novel odor in different trials as a function of time. Pink bar indicates odor presentation. Note that correlations in response to novel odors are similar across networks and different from responses to learned odors in Scaled and Tuned networks (Fig. 2E) D. Population firing rate in response to learned odors for networks with different probability of E-E connectivity within assemblies (α). E. Co-tuning (correlation between E and I currents in individual neurons) as a function of α.

Pattern completion: additional results.

A. Artificial reactivation of E assemblies. During 6 Hz baseline activity of the olfactory bulb, a subset of the assembly neurons was artificially reactivated by current injection (500 ms, 28 nA). Mean firing rates were quantified in the injected assembly neurons (i), in the remaining, non-injected assembly or pseudo-assembly neurons (ii), and in the non-assembly neurons (iii) as a function of time. The orange bar indicates duration of current injection. B. Correlation between activity patterns across E neurons (output patterns) evoked by a series of input patterns representing a morph of one learned odor into another learned odor. Correlations between output patterns are plotted as a function of the correlation between the corresponding OB patterns.

Transformations and dimensionality of activity patterns: additional results.

A. Projection of activity patterns representing the odor subspace (Fig. 4A) onto the first 2 PCs of the corresponding rand networks (representative examples of one network each). B. Scree plot for the PCA results shown in Figure 4D. C. Error in the reconstruction of odor-evoked activity patterns as a function of the number of PCs. Euclidean distances between all pairs of activity patterns were calculated in the full-dimensional state space (Df) and in reduced-dimensional embedding spaces (Dl). The reconstruction error was defined as 1 – (correlation between Df and Dl). D. Participation ratio of the neural activity sampled from different numbers of neurons (50 iterations). E. Loadings of neurons on the first two PCs of a rand and a Tuned E+I network (Figure 4D). Each line represents one neuron. Neurons that are part of the assemblies representing the two learned odors are color-coded in magenta. F. For each network and PC, the 100 E neurons with the highest absolute loadings were selected and grouped into 3 categories: neurons part of the two assemblies representing the learned odors, neurons part of the two “pseudo-assemblies” representing the two novel odors, and the remaining neurons (non-assembly).

Further analyses of pattern distances.

Differences in dM between rand and Tuned networks may involve differences in the distance between class centers and/or differences in intra-class variability. To dissect the contributions of these effects we compared three distance measures:

  1. The Euclidean distance between class centers:

  2. The mean Euclidean distance between patterns of one class and the center of another class (dÊ):

  3. The Mahalanobis distance dM:

Here, μV and μQ are the centers (average patterns) of classes V and Q, respectively; v is a vector from V = [v1,v2,…,vn], and SAB is the inverse of the covariance matrix of the neuronal population within reference class Q. Note that dÊ corresponds to dM without normalization by variability (covariance).

A. Analysis of dE. Left: dE in rand and Tuned E+I networks based on 80 E neurons drawn from (pseudo-) assemblies. Right: same based on 80 E neurons drawn from the entire population. Note that dE between learned and other odors was increased in Tuned E+I networks as compared to rand networks, particularly when neurons were drawn from assemblies. B. Equivalent plots for dÊ. Note that distances were increased nearly symmetrically, similar to dE. C. Equivalent plots for dM (same plots as in Fig. 5B). Note that dM was increased asymmetrically. These observations show that the changes in dM relative to rand networks involved an increase in the distance between class centers (dE) and a non-isotropic change in intra-class variability (comparison between dÊ and dM). These effects were prominent when E neurons were drawn from assemblies. An important contribution to the increase in dM in the direction from learned odors to reference classes representing novel odors was made by the increased distance between class centers. In the other direction, dM was smaller, implying that variability in the reference class was higher. Nonetheless, variability in the relevant direction did not fully counteract the increased distance between class centers in Tuned E+I networks. As a consequence, dM was still increased slightly relative to the corresponding rand networks. Most of these effects were still observed, albeit weakly, when E neurons were drawn from the whole population.