Fig 1.
Working scheme of the smooth-index algorithm.
Starting from a ‘scrambled’ circuit, the algorithm treats the indices as smooth values along independent parameter axes and minimizes a given cost function, in this case all recurrent connections.
Fig 2.
Example performance of the smooth-index algorithm.
A Original matrix with a few recurrent connections. B Scrambled matrix, obtained by random index permutation. C Matrix resulting from smooth-index sorting. D Different cost functions as the algorithm performs the gradient descent. E Positions of all vertices zi during gradient descent. F Number of switches, calculated as the number of differences between two consecutive rank-sorted position vectors, during gradient descent.
Fig 3.
Performance of the smooth-index algorithm for matrices with different densities.
A Example matrices for N = 50 neurons, given a density of 50% in the lower triangle. B Performance of the algorithm as a function of the number of neurons. Plotted is the fraction of non-zero entries in the upper triangle of the matrix in case of the original connectivity matrix before scrambling (in black), after out-degree sorting (in red) and after smooth-index sorting (in blue). C CPU time needed for out-degree sorting (in red) and for smooth-index sorting (in blue) as a function of the number of neurons. Data in B and C represent the mean +- standard deviation (shaded area) obtained from 10 sorting runs. D-F Same as above, but for a density of 35% in the lower triangle. G-I Same as above, but for a density of 20% in the lower triangle.
Fig 4.
Comparison of the smooth-index (‘SI’) algorithm with feedback arc set (‘FAS’) sorting.
A Original connectivity matrix of 50 neurons with 50% density in the lower and 4% density in the upper triangle. B Same as A, but after scrambling. C Resulting matrix after removing recurrent connections identified by the FAS algorithm. Note that the ordering of the scrambled matrix is retained. D FAS reduced matrix, reordered applying the Schur decomposition. E Full, scrambled matrix reordered applying the permutation matrix as obtained from applying the Schur decomposition to the FAS-reduced matrix. F Scrambled matrix after SI-sorting. Note that it has fewer entries in the upper triangle than the one in E. G-I Performance of the SI and the FAS algorithm for matrices with three different densities in the lower triangle, from p = 50% to p = 20%. The density in the upper triangle was held constant at 4%. Plotted is the fraction of non-zero entries in the upper triangle of the matrix in case of the original connectivity matrix before scrambling (in gray), after FAS sorting (in red) and after smooth-index sorting (in blue). Data represent the mean +- standard deviation obtained from 10 sorting runs.
Fig 5.
Identification of recurrent synapses by the smooth-index algorithm.
A Original matrix. B Scrambled matrix. C Matrix holding the probability of each connection as being classified as recurrent from 1000 sorting runs. D Same as C, but after reversed scrambling. E Recurrence probability of all reciprocal connections. F Original connectivity matrix (in blue) with identified recurrent connections in red (probability threshold = 0.45).
Fig 6.
Application of the smooth-index algorithm to the single-column connectome of the Drosophila optic lobe.
A Connectivity matrix with synaptic weights, ordered according to the sequence of neuropils, from retina to lamina to medulla. B Total number of connections as a function of threshold (‘thrld’) for all connections (in black) as well as for excitatory (in blue) and inhibitory (in red) connections, separately. C Resulting 0,1-adjacency matrix. D Reordered 0,1-adjacency matrix according to the least number of upper triangle entries obtained in 1000 runs of the smooth-index algorithm. E Same as D, but shown with the weights of the original weighted matrix.
Table 1.
List of recurrent connections from the single column connectome of Drosophila.
Connections are sorted according to the probability by which a certain connection was classified (1st column, ‘p_recurrency’) as being recurrent in 1000 runs of the smooth-index algorithm. The values in the 2nd column (‘Synaptic Strength’) indicate the number of synapses at each connection, with positive values for excitatory, and inhibitory numbers for inhibitory connections. Highlighted are those connections which are not part of a reciprocally connected pair of neurons.
Fig 7.
Application of the smooth-index to bandwidth-limited matrices (A,B) and to block-diagonal matrices (C,D).
The smooth-index (‘SI’) algorithm is compared with the reverse Cuthill-McKee (‘RC’) algorithm. In A and C, one example sorting is shown for a 50 x 50 matrix. In B and D, the performance of both algorithms is quantified as the bandwidth of the reordered matrix relative to the original matrix. The performance is calculated as the average obtained from 10 runs +- the standard deviation (shaded area).
Fig 8.
Visualization and balancing the cost function with the Pauli term.
A Recurrency cost as as f(z2,z3). B Recurrency cost along the dashed line in A. C Pauli term as f(z2,z3). D Pauli term along the dashed line in C. E Recurrency plus Pauli cost as as f(z2,z3). F Recurrency plus Pauli cost along the dashed line in E. G Performance of recurrency minimization as a function of the Pauli term weight. H Performance of bandwidth minimization as a function of the Pauli term weight.