Fig 1.
Active learning framework for network inference.
Recordings of spiking activity of a neuron population and the presented visual stimuli are fed into a GLM. The GLM and Variable Selection blocks work in tandem to decide which connections are relevant for explaining the system’s behaviour (the data) and building the directed connectivity graph (network). The active learning component analyzes the data obtained so far to optimize the visual stimuli to be presented for the next step of data acquisition, this is done to reduce graph uncertainty. This process is iteratively repeated. The bottom row shows how the network is gradually reconstructed as a function of acquired samples. Gray edges represent yet undiscovered edges present in the network, while red and blue edges represent discovered excitatory and inhibitory edges respectively.
Fig 2.
Adjacency matrices for networks SW1CL and SW3CL.
Red entries in the adjacency matrices denote an excitatory relation between the regressor and the child neuron, while blue entries denote inhibitory connections. The block diagonal structure present in the W matrix for network SW3CL evidences the three cluster structure of the network. These connectivity matrices were computed to generate the required spiking rate change for model parameter κ = 10. On both networks, we can observe the large number of stimuli that have no effect on the network.
Fig 3.
Comparison of elastic-forward BIC selection versus oracle lasso.
Whisker plot of performance indicators as a function of number of samples; elastic-forward BIC selection is shown in red, oracle lasso in blue. The whisker plot is obtained from 10 independent trials. External stimuli are drawn randomly from a uniform distribution; network SW1CL has a total of 24 non-zero parameters out 864 potential regressors, while network SW3CL has a total of 58 non-zero parameters out of 4536 potential regressors. The elastic-forward BIC selection outperforms the oracle lasso for larger sample sizes. This performance improvement is more noticeable in the SW3CL network, where edge weights are more diverse.
Fig 4.
Comparison of performance between the proposed active learning method versus uniformly sampling from all stimuli.
The experiment consisted of 500 sample interventions, with an initial 500 sample observation. Whisker plots are obtained from 10 independent trials. Left column show F1, precision, and recall performance on network SW1CL and right column on network SW3CL.
Fig 5.
Comparison of performance between the proposed active learning method versus uniformly sampling from all stimuli only over the direct stimuli response matrices H.
The experiment consisted of 500 sample interventions, with an initial 500 sample observation. Whisker plots are obtained from 10 independent trials. Left column shows F1, precision, and recall performance on network SW1CL and right on network SW3CL. On average, computing the active learning intervention step took 20s ± 9s on network SW1CL, and 98s ± 50s on network SW3CL.
Fig 6.
Comparison of elastic-forward BIC selection versus oracle lasso under active learning sampling.
Whisker plot of performance indicators as a function of number of samples; elastic-forward BIC selection is shown in red, oracle lasso in blue. The whisker plot is obtained from 10 independent trials. External stimuli are drawn from the recommended Active Learning distribution; network SW1CL has a total of 24 non-zero parameters out 864 potential regressors, while network SW3CL has a total of 58 non-zero parameters out of 4536 potential regressors. The elastic-forward BIC selection still outperforms the oracle lasso for larger sample sizes. Both methods show a performance gain on the samples drawn from the Active Learning distribution when compared to their uniform sampling counterparts.
Fig 7.
Comparison of the edge discovering process for network SW1CL using active learning versus random stimulation.
Rows show inferred connections over a simulated cluster as a function of samples. Red and blue edges show correctly detected excitatory and inhibitory connections respectively, while grey edges show connections as not yet detected. Left column shows detected edges as a function of time when the active learning stimulation policy is used, right column shows the same cluster with a completely random stimulation policy. Rightmost cluster shows the ground truth. This example shows a clear advantage in edge recovery when using active learning compared to random stimulation.
Fig 8.
Comparison of the edge discovering process for networks SW1CL and SW3CL using active learning versus random stimulation.
Rows show misclassified edges in the adjacency matrices W and H as a function of samples. Rows 1 and 3 show misclassified edges as a function of time when the active learning stimulation policy is used, while rows 2 and 4 show the same network probed with a random stimulation policy. The misclassified edge matrix under active learning quickly becomes sparse as the number of misclassifications goes to zero, random stimulation produces the same results but in a longer time frame.
Fig 9.
Recovered adjacency matrices for datasets 1 and 2.
Top and bottom rows show the recovered adjacency matrices for datasets 1 and 2 respectively. Columns from left to right show the full model and the AR model respectively. Inhibitory connections are shown in blue, and excitatory connections are shown in red. We can observe that self regression coefficients are always added to the model. We also observe that the overall sparsity of the recovered network is consistent across datasets, and that the first few rows of the H matrices show a heavy concentration of excitatory connections. These rows correspond to low spatial frequency (r) values in the Hartley basis functions (Eq (26)).
Fig 10.
Log likelihood difference of full model and AR model over the test samples.
The graph shows that, on average, spiking rate predictions are better on the full model than on the auto-regressive model. This grounds the idea of neuron to neuron interaction as being predictive of neuron behaviour.
Fig 11.
Difference in log likelihood of forecasted sequences.
Real stimulation sequence versus randomized stimulation sequences. Error bars represent one standard deviation over 20 trials.
Fig 12.
Comparison of performance between the proposed active learning method versus uniformly sampling from all stimuli.
The experiment consisted of 500 sample interventions, with an initial 1,000 sample observation. Whisker plot is obtained from 10 independent trials. Left column shows F1, precision, and recall performance on network recovered from dataset 1 and right column shows F1, precision, and recall performance on network dataset 2. On average, computing the Active learning intervention step took 108s ± 60s on dataset 1, and 86s ± 46s on dataset 2.
Fig 13.
Distribution of recommended stimuli (Pl+1) across interventions.
Initially, the distribution is uniform (top). The experiment consisted of 500 sample interventions, with an initial 1,000 sample observation. Left column shows the stimuli probability distribution history for dataset 1, while right column shows the distribution for dataset 2.
Fig 14.
Percentage of neurons that have an excitatory or inhibitory response to each possible visual stimuli.
Visual stimuli are represented in matrix form, where rows represent spatial frequency r and columns spatial orientation ϕ. The value for each entry in the matrix is the percentage of neurons in the datasets that show a response to that visual stimuli. This visualization shows that both datasets show a large number of directly responding neurons for low spatial frequency visual stimuli.
Fig 15.
Example of recovered cliques and corresponding spike trains.
Figures a), c), e), and g) show the excitatory (red) and inhibitory (blue) edges detected for neuron cliques in datasets 2 (a) and c)) and 3 (e) and g)). Figures b), d), f), and h) show the spike time series of the neurons in the clique. The nodes are numbered according to the corresponding neuron index. We can visually see that spike trains from neurons in the selected cliques show similar spiking behaviour throughout the experiment.
Fig 16.
Motif triplet counts in recovered adjacency matrix.
We count the occurrences of motif triplets for both datasets (we ignore edge weight and sign) by enumerating all neuron triplet combinations in the recovered networks and checking for graph isomorphism against all 5 motif triplet types. Top and bottom rows show results for datasets 1 and 2 respectively. We compare the obtained motif counts against a base model of small-world network topology and show the obtained p-values. These p-values are obtained by computation of the mean and standard deviation of each motif type in a small-world network with the same node count and edge density. The relatively large p-values obtained show that the small-world model is a good fit for the recovered network topology.
Fig 17.
Edge density distribution of recovered inter neuron connectivity.
We compare the edge density distribution of the recovered inter neuron connectivity matrices in both datasets. Edge counts shown from left to right are all edges (number of neurons connected to node, either as parent or child), outbound edges (number of child nodes), and inbound nodes (number of parent nodes). The edge counts are compared against a base model of small-world network topology, error bars denote two standard deviations obtained from simulation of small-world networks with the same number of nodes and connectivity degree.
Fig 18.
Percentage of cells in multiple cliques.
We count the percentage of cells participating in multiple cliques. The counts are compared against a base model of small-world network topology, error bars denote two standard deviations obtained from simulation of small-world networks with the same number of nodes and connectivity degree.
Fig 19.
Single sample (M = 1) Z-score approximation.
We show the single sample (M = 1) Z-score approximation according to Eq (46) for several κ parameters as a function of the ratio and
(E(yc|Ri = 0)). The
values were chosen to be representative of the observed rates in our real datasets.
Fig 20.
Comparison of elastic-forward BIC selection as a function of parameter ν.
Whisker plot of performance indicators as a function of number of samples; elastic-forward BIC selection is compared using several ν parameters against both oracle Lasso and the special case where the whole dataset is used at once, without splitting into random subsets (no-subset). The plots show relatively little difference between the various ν parameters, but the use of the ν parameter is better performing overall to both oracle lasso and no-subset model selection.
Fig 21.
Performance comparison between active learning and uniform sampling over LIF network.
Comparison of performance between the proposed active learning method versus uniformly sampling from all stimuli. Spiking trains were simulated using a Leaky Integrate and Fire model, but the recovered networks were done using the proposed Poisson GLM. The experiment consisted of 500 sample interventions, with an initial 500 sample observation. Whisker plots are obtained from 10 independent trials.
Table 1.
Variable reference list.