Fig 1.
A. A network of excitatory neurons (light blue triangles) fire stochastically, while their activity is driven by unstructured external input (red arrows) and modulated by a population of inhibitory neurons (yellow circles). Excitatory connections among the neurons can be weak (gray dashed arrows) or strong (black solid arrows), unidirectional or bidirectional. B. Cumulants of the spike trains (see Eq 1). The second-order cumulants Cij, Cii and Cjj are calculated based on the time difference between a pair of spikes (cross-covariance in green; auto-covariances in orange/red). The third-order cumulant Kij is calculated based on the time differences between three spikes (purple). The spike triplets can be two post- and one presynaptic spikes, or one post- and two presynaptic spikes. The time differences are: τ1 between a presynaptic spike and a postsynaptic spike, τ2 between different postsynaptic spikes and τ3 between different presynaptic spikes. C. STDP-induced plasticity by pairs and triplets of spikes. Left: An example of a classical pair-based STDP rule, with a learning window denoted by L2. Potentiation is triggered by a postsynaptic following a presynaptic spike (τ1 = tpost − tpre > 0), whereas if a presynaptic spike follows a postsynaptic spike (τ1 = tpost − tpre < 0), depression is induced. The total potentiation (depression) is given by the red (blue) area under the curve. Right: Examples of triplet STDP rules denoted by L3,y and L3,x. Potentiation (red) and depression (blue) are given by triplets of spikes: post-pre-post with a time difference , and pre-post-pre with a time difference
, respectively. D. Minimal triplet STDP rule where potentiation depends on triplets of spikes L3 and depression on pairs of spikes L2.
Fig 2.
Second-order cumulant contributions to plasticity.
A. The cross-covariance Cij between the presynaptic neuron j and the postsynaptic neuron i is obtained by summing over all the possible α- and β-paths from every possible source neuron k in the network. Each path is calculated via the corresponding weights in the connectivity matrix and the EPSC function (see Eq 3). B. Same as A but for the auto-covariance Cii of the postsynaptic neuron i (see Eq 4). In this case, γ is the second index to sum over the path from the source neuron k to the postsynaptic neuron i. It should be noted that the main difference between the α- and γ-path is given by the time dependence of the EPSC function, here represented in the Fourier domain for convenience.
Fig 3.
Structural motifs in the network under pair-based and triplet STDP.
A. Examples of structural motifs common for both the pair-based and triplet STDP framework. Here α and β constitute the path lengths of synapses from the source neuron to the postsynaptic neuron i and the presynaptic neuron j. α = 1, β = 0: Presynaptic neuron j projects to the postsynaptic neuron i. α = 0, β = 1: Postsynaptic neuron i projects to the presynaptic neuron j. α = 1, β = 1: Common input from source neuron k to presynaptic neuron j and postsynaptic neuron i. α = 2, β = 0: Presynaptic neuron j projects to the postsynaptic neuron i through another neuron k in the network. B. Illustration of the calculation of the common input motif with α = 1 and β = 1 framed in purple in A (there are also additional terms which are not illustrated). The motif coefficients Mα=1,β=1 (right) are calculated as the total area under the curve resulting from the product of the convolution of the EPSC function E (left) and the STDP functions (pair-based L2 and triplet L3, middle). C. Examples of structural motifs found only in the triplet STDP framework, where γ denotes the time-delayed path length from the source neuron to the postsynaptic neuron i. α = 1, γ = 1: Source neuron k projects twice to postsynaptic neuron i with a different time delay. α = 2, γ = 0: Feedback loop through another neuron k in the network (source and projecting neuron are the postsynaptic neuron i). α = 1, β = 1, γ = 1: Source neuron k projects to the presynaptic neuron j and postsynaptic neuron i via all the three possible paths. D. Illustration of the calculation of the motif with α = 1 and γ = 1 for the triplet STDP rule framed in purple in C, compare to B.
Fig 4.
Third-order cumulant contributions to plasticity can be broken down into four terms.
A. The first term contains all the α-, β- and γ-paths originating from the source neuron k to the spiking neurons i and j. B-D. The other terms take into account the possibility of an intermediate neuron l that acts as a new source neuron for two of the paths. These are referred to as ‘branched paths’, and the path length from the source neuron k to the intermediate neuron l is denoted with ζ. The branching describes the individual terms in Eq 5.
Fig 5.
Second- and third-order cumulants can be described in terms of structural motifs that contribute to weight change.
All motifs up to third order as they arise from the cross-covariance Cij (top row and Eq 3), the auto-covariance Cii (middle row and Eq 4) (both Cij and Cii together represent the second-order cumulant) or the third-order cumulant Kij (bottom row and Eq 5). The gray boxes indicate the ‘loop’ motifs. The novel motifs which follow from the triplet STDP rule are those that include the path γ (second and third row).
Fig 6.
Spontaneous emergence of assemblies via modulation of the triplet STDP rule.
A. The shape of the STDP function changes as a function of the modulation parameter η−, which preserves the overall level of depression by trading off the depression learning rate and the depression time constant. B. Motif coefficients as the modulation parameter η− increases. Points of interest given by the crossovers of the strength of particular motifs are indicated by a small arrow. Inset: Amplified scale around zero. Motif coefficients including γ paths are not illustrated, since they are always constant and positive in η− and do not provide a meaningful comparison to the other motifs. C. Top: Examples of connectivity matrices obtained with different values of η− at steady state. Unidirectional connections are shown in black, bidirectional connections in orange. Matrices are reordered using the k-means clustering algorithm (see Methods). Bottom: Mean fraction of unidirectional and bidirectional connections for 100 trials with different initial synaptic efficacies as a function of η−. Error bars represent the standard error of the mean. D. Graphs of the connectivity matrices in C. E. Averaged connectivity matrices over 100 trials at steady state. Note that the tighter clusters emerging near the edges of the matrices are the result of the clustering algorithm but do not affect the quantification of connectivity.
Table 1.
⋆ denotes that values are provided in the figures.
Fig 7.
Graph measures of the stable network configuration.
A. Mean clustering coefficient versus the modulation parameter η−. B. Mean global efficiency versus the modulation parameter η−. C. Mean modularity versus the modulation parameter η−. All results are calculated from 100 trials at steady state connectivity. Error bars represent the standard error of the mean.
Fig 8.
Spontaneous emergence of assemblies for four different motif combinations.
Considering only motifs related to the cross-covariance Cij (blue), from all cumulants (Cij, Cii and Kij) without the ‘loop’ terms (red), from the cross-covariance Cij plus the ‘loop’ terms (yellow) and from all cumulants (purple). A. Averaged connectivity matrices over 100 trials at steady state for four different motif combinations and modulation parameter η− = 13. Matrices are reordered using the k-means clustering algorithm (see Methods). B. Mean clustering coefficient versus the modulation parameter η−. C. Mean global efficiency versus the modulation parameter η−. D. Mean modularity versus the modulation parameter η−. All results are calculated from 100 trials at steady state connectivity. Error bars represent the standard error of the mean.
Fig 9.
Spontaneous emergence of assemblies due to the modulation of synaptic transmission.
A. Varying the time constant τι changes the shape of the EPSC function, shifting its peak by a few milliseconds. B. Relative value of the motif coefficients as a function of τι. While the common input motif M1,1 rapidly assumes dominance, the motif coefficient M1,2 crosses over in strength with the feedback motifs M0,1, M0,2 and M0,3. C. Averaged connectivity matrices over 100 trials at steady state and different values of the time constant τι. Matrices are reordered using the k-means clustering algorithm (see Methods). D. Mean clustering coefficient versus the time constant τι. E. Mean global efficiency versus the time constant τι. F. Mean modularity versus the the time constant τι. All results are calculated from 100 trials at steady state connectivity. Error bars represent the standard error of the mean.
Fig 10.
Emergence of assemblies in the presence of structured external input.
A. Mean clustering coefficient versus the pairwise correlation coefficient of the input pattern. The strength of the correlation was provided as ratios (0.01, 0.05, 0.125, 0.25, 0.375 and 0.5) of the possible maximum weight of each individual synaptic connection wmax. B. Mean global efficiency versus the pairwise correlation coefficient of the input pattern. C. Mean modularity versus the pairwise correlation coefficient of the input pattern. The rapid increase of the clustering coefficient and the modularity combined with a decrease of the global efficiency is a feature of robust assembly formation. Sufficiently strong correlations in the external signal generate tight assemblies. All results are calculated from 100 trials at steady state connectivity. Error bars represent the standard error of the mean.
Fig 11.
Impact of perturbations in the balance of potentiation and depression of the triplet STDP rule.
A. Averaged connectivity matrices over 100 trials at steady state for the four different cases of the perturbation parameter δ and modulation parameter η− = 13. B. Mean clustering coefficient versus the modulation parameter η−. C. Mean global efficiency versus the modulation parameter η−. We removed the cases δ = [0.01, 0.1] here since the global efficiency cannot be computed for weight matrices where all entries are identical. D. Mean modularity versus the modulation parameter η−. All results are calculated from 100 trials at steady state connectivity. Error bars represent the standard error of the mean.
Table 2.
Parameter values for supplementary figures.
⋆ denotes that values are provided in the figures.