Effects of plasticity on E/I connectivity motifs in a two-neuron circuit.

A. Schematics of the two-neuron circuit model. Two time-varying coupled Poisson units, one excitatory and one inhibitory, are connected by a fixed excitatory weight wexc and a plastic inhibitory weight winh. The instantaneous rates are computed as the convolution between each incoming spike and an exponential synaptic kernel that either increases (exc) or decreases (inh) the instantaneous firing probability. The input to the excitatory neuron hexc is fixed, whereas the inhibitory input changes dynamically, keeping the inhibitory neuron at a fixed rate rinh. Only the inhibitory synapse (dashed line) is plastic. The circuit is simulated in two configurations: unidirectional (U), with wexc = 0 and mutual (M), with wexc = 0.5. B,C,D. Pairwise interaction kernels of the three iSTDP rules tested (Eq. 1). We used the following numerical parameters. B. B = 1, αpre = −25, αpost = 0, τ+ = 50 ms. C. B = 0, αpre = −0.12, αpost = 1.45, τ+ = 50 ms, τ− = 500 ms. D. B = 0, αpre = −0.14, αpost = 7.95, τ+ = 30 ms, τ− = 30 ms. E,F,G. Change in inhibitory synaptic weight over time for the M (brown) and U (light orange) circuit configurations, under the rules above each column. Dashed lines are analytic predictions for the fixed point (Methods, Eq. 11). H,I,J. Change in excitatory firing rate over time for the M (dark blue) and U (dark green) configurations, under the rules above. Dashed horizontal lines are the analytic predictions for the fixed-point (Methods, Eq. 11).

Parametric study of different shapes of iSTDP kernel in the reduced 2D model.

A. Representation of four possible shapes of iSTDP kernel, labeled by their effects in the covariance-dominated regime (i.e. with near-zero rate-dependent terms). B. Parametric study based on the analytic solution for each of the four iSTDP kernels, varying αpost and the net area under the iSTDP kernel, B (see Eq. 1). The color map represents the level of inhibition, where 1 indicates that the excitatory neuron is completely silenced, and 0 indicates no inhibition, so that the excitatory firing rate matches the input current C,D,E. Final inhibitory weight as a function of the incoming excitatory weight for three different iSTDP rules. Dots are numerical simulations, gray lines are analytic results (Methods, Eq. 11), color shades indicate different input levels. C. Results for the symmetric, self-stabilizing rule. The green dots represent the same conditions shown in Fig. 1F. D. Results for the antisymmetric, self-stabilizing rule. The green dots represent the same conditions shown in Fig. 1G. E. Results for the symmetric rule in a rate-dominated regime. F,G,H. Corresponding plots showing the rate of the excitatory neuron after convergence.

Numerical parameters used for Fig. 1. The parameters in Fig. 2 match those of Fig. 1 unless otherwise indicated in the figure caption.

Numerical parameters for the random RNN in Fig. 3.

RNNs of conductance-based LIF neurons self-organize into specific E/I motifs under different iSTDP rules.

A left. Representation of the random network with (Nexc = 900, Ninh = 100) spiking neurons. The inh-to-exc weights form all-to-all connections and are plastic, all other weights are fixed. Exc-to-exc and exc-to-inh connections are sparse. A middle. Kernel of the symmetric, covariance-dominated iSTDP rule used in network 1 (panels B-F). A right. Kernel of the asymmetric, covariance-dominated iSTDP rule used in network 2 (panels G-K). Parameters in Table 2. B. Evolution of population mean firing rates over time, with final distribution of rates over the full populations. C. Evolution of inh-to-exc synaptic weights over time and final distribution. The weights are split into a “mutual” (brown) and a “unidirectional” (light orange) group. Only few weights reach the saturating value, set to 80. D. Example of incoming (blue) and outgoing (red) weights from a single inhibitory neuron, with correlation between them, and distribution of correlations for the full inhibitory populations across 10 random network realizations. E. Random portion of the fixed and sparse exc-to-inh weights (the full matrix, of size 900×100 is shown in SI, Fig. S4). F. Portion of the all-to-all and plastic inh-to-exc connections after learning, with indices matching panel E. For clarity the matrix has been binarized, with threshold-value wsmall defined as 0.1 times the 0.9-quantile of the full weight distribution. See SI, Fig. S4, for a continuous representation of synaptic weights. G-K. Equivalent plots in a second network with identical network dynamics and initial conditions, but using the asymmetric iSTDP (see also SI, Fig. S6)

Numerical parameters for the one-dimensional ring model. The parameters that regulate neural dynamics are the same as in Table 2.

Inhibitory self-organization in a spiking RNN topologically organized as a one-dimensional ring.

A. Representation of the model. 800 excitatory neurons are arranged in a one-dimensional ring topology, connected to each other and to their inhibitory counterparts with weights that depend on their distance in tuning space (Methods, Eq. 15). 100 “PV” inhibitory neurons connect all-to-all to the excitatory neurons with plasticity following a symmetric iSTDP rule, 100 “SST” neurons connect all-to-all to excitatory neurons with plasticity following an antisymmetric iSTDP. For both rules B = 0, αpre = −0.1, αpost = 0. B. Full weight matrix after weight convergence. The excitatory weights (blue) are fixed, the inh-to-exc weights (red, lower right) are determined by plasticity. See SI, Fig. S4 for full weight matrix. C. Profile of the average outgoing weights as a function of tuning distance. For convenience, the profiles are normalized so that the maximum is either 1 or −1. D. Average effective interaction profile between the excitatory neurons, as a function of tuning (see Methods). E. Evolution of mean firing rate over time. F. Pearson correlation during spontaneous activity between an excitatory neuron and its neighbors, ordered by tuning distance and averaged over the entire population.

Response properties of the one-dimensional ring network.

A. Representation of the stimuli (black) and of the network’s response (green), as a function of the neuron’s location relative to the stimulus center, averaged over 100 trials. B. Response to the center neuron only as a function of stimulus size. C. Schematics of the iso-contra stimulation protocol. The iso direction corresponds to excitatory inputs to excitatory neurons with matching receptive field. The contra direction generates instead an inhibitory input on the SST population. D. Profiles of the center, full iso and contra stimuli to the network. E. Mean response of the center neuron, averaged over 100 trials, for the three stimuli.

Numerical parameters for the spiking RNN with intrinsically-generated fluctuating activity (Eq. S55)

Schematics of the reduced circuit model and of interactions between covariance density and iSTDP kernel.

A. Representation of the model used for Fig. 1 in main text. Two time-varying coupled Poisson units, one excitatory and one inhibitory, are connected by a non-plastic excitatory weight wexc and a plastic inhibitory weight winh. The instantaneous rates, here shown as traces, are computed as the convolution between each incoming spike and an exponential synaptic kernel that either increases (exc) or decreases (inh) the instantaneous firing probability. The input to the excitatory neuron hexc is fixed, whereas the inhibitory input changes dynamically, keeping the inhibitory neuron at a fixed rate rinh. Only the inhibitory synapse (dashed line) is plastic. B. Unidirectional configuration. Here wexc = 0 and the covariance density between the pre (inh) and post (exc) neuron can only be negative in the pre-post (Δt > 0) section of the curve, due to the inhibitory interaction given by winh. C. Mutual configuration, wexc = 0.5. Here the covariance density is positive in the post-pre (Δt < 0) section of the curve due to wexc, and negative for (Δt > 0) for the effect of winh. D. Effect of a symmetric, covariance-dominated iSTDP kernel on a circuit in the M configuration: the positive left branch of the covariance density interacts with the kernel, resulting in a positive Q factor in the weight update. (Eq. 1). E. Effect of an antisymmetric iSTDP kernel. Here the positive covariance density interacts with the negative branch of the kernel, resulting into synaptic depression. Hence, the Q term is negative, and mutual connections are discouraged.

Analytic fixed point solutions for different iSTDP interaction kernels (complements Fig. 2 in main text).

A. Representation of the model (see also Fig. 1A). B-E. iSTDP rules, represented by their kernel and their additional parameters (left panels) and study of the associated weight dynamics as a one-dimensional differential equation (right panels), black dots represent fixed points. B Rate-dominated iSTDP rule. C-E. covariance dominated rules presented in the same order as in Fig. 2A, main text. Note that the configurations in E and F produce unstable fixed points.

Effects of plasticity for E/I motifs in a reduced model with unconstrained rinh.

See Section S1.3 for model description and details on the analytic calculation. A. Model schematics. Unlike the model in Fig. 1A, the inhibitory rate rinh is unconstrained and determined by the net inhibitory input hinh. B-J. Corresponding to the respective panels in Fig. 1, for the model with unconstrained rinh. K-L. Value of rinh during the simulation. Note that in the U configuration (dark green) rinh is stationary and corresponds to hinh.

Full weight matrix for the random sparse RNN simulation with symmetric iSTDP (see Main Text, Fig. 3B-F).

Randomly connected spiking RNN with intrinsically-generated fluctuating asynchronous irregular activity.

See Table S1 for numerical parameters. A-B. Spiking activity of a subset 100 excitatory neurons (left) and distribution of Fano factors of excitatory neurons (right) before plasticity. C-D. Spiking activity and Fano factor distribution after weight convergence, using either a symmetric (C) or an antisymmetric (D) iSTDP kernel. E-F. Change of mutual and unidirectional inhibitory-to-excitatory weights during learning in the circuit. Equivalent to Fig. 3C and H in Main Text.

Full weight matrix for the random sparse RNN simulation with antisymmetric iSTDP (see Main Text, Fig. 3G-K).

Full weight matrix for the recurrent network with 1D ring connectivity structure after weight convergence with two inhibitory populations.

A. Full matrix of Pearson correlations between excitatory neurons in the one-dimensional ring model during spontaneous activity, after convergence of plastic weights. B Spike raster showing the response of the 1D ring model to localized stimuli (see Fig. 4 in main text).

Spontaneous and evoked activity in a 1D ring structure with rate-dominated iSTDP.

A. Model schematics, compare with Fig. 4A. The excitatory component is identical to the previous case, but the inhibitory population follows a single, rate-dominated iSTDP. B. Full weight matrix after weight convergence, compare with Fig. 4B. C. Mean responses of the excitatory population to stimuli of different sizes, averaged over 100 trials (compare withFig. 5A). D. Response of the center neuron to stimuli of different sizes (compare withFig. 5B). E. Pearson correlation during spontaneous activity between an excitatory neuron and its neighbors, ordered by tuning distance and averaged over the entire population (compare withFig. 4F). F. Pearson correlation of spontaneous activity for every pair of excitatory neurons (compare withFig. S8A).

One-dimensional ring structure with only one inhibitory population (either PV or SST).

A-E. The RNN has 800 excitatory neurons, 200 “PV” neurons and no “SST” neurons. The network evolves under a PV-like covariance-dominated iSTDP rule with B = 0, αpre = −1.0, αpost = 1.0. A. Evolution of population mean firing rates over time. B. Evolution of inh-to-exc synaptic weights over time. C. Profile of the average outgoing weights as a function of tuning distance. For convenience, the profiles are normalized so that the maximum is either 1 or −1. D. Inhibitory to excitatory weight matrix after weight convergence. E. Raster plot of excitatory neurons. F-J. Same but for a RNN with 800 excitatory neurons, 200 “SST” neurons and no “PV” neurons. The network evolves under a SST-like covariance-dominated iSTDP rule with B = 0, αpre = −1.0, αpost = 1.5.

Effects of plasticity for different levels of mean inhibitory input.

A. Symmetric kernel. B. Evolution of population mean firing rates over time. The green vertical lines represent sequential increases of the mean input by a factor of 1.5, 2.0, 2.5, 3. C. Evolution of inh-to-exc synaptic weights over time. D. Final distribution of weights, split into a “mutual” (brown) and a “unidirectional” (light orange) group. E-H. Same as above but for asymmetric iSTDP.