Fig 1.
Emergence of modular connectivity by learning two selective stimuli in spiking networks.
(A) Experimental protocol consists of the stimulation of two non-overlapping neuronal populations of QIF neurons with plastic synapses. Networks are made of 80% of excitatory and 20% of inhibitory neurons. Stimuli are presented in temporal alternation. Results of the performed numerical experiments are reported for a network with all anti-Hebbian inhibitory neurons (B); with all Hebbian inhibitory neurons (C); with 50% anti-Hebbian and 50% Hebbian inhibitory neurons (D). Raster plots display the firing times of excitatory (red dots) and inhibitory (blue dots) neurons during the simulations. Matrices represent the temporal evolution of the connection weights at different times: t = 0s (random initialization of the weights), t = 20s (middle of the learning phase) and t = 40s (end of the learning phase). The color denotes if the connection is excitatory (red), inhibitory (blue) or absent (white) and the color gradation the strength of the synaptic weight. The final configuration of the connection weights is shown schematically on the right in each case.
Fig 2.
Consolidation of imperfectly learned memories.
(A) Temporal evolution of the connectivity matrix during spontaneous activity in the absence of stimulation. Initial connectivity with two unfinished modules at t = 0 is reinforced over time. The excitatory (inhibitory) connections are marked as in Fig 1. (B) Evolution of the distribution of link weights (probability density functions, PDFs, in a linear-logarithmic scale) in the connectivity matrices at t = 0s (light green), t = 400s (cyan) and t = 4000s (magenta). (C-J) Evolution of the network activity and various metrics in absence of stimulation for a sample of 30 seconds. Some spontaneous recalls are highlighted, for population P1 (green shade) and P2 (orange shade). Pink shadow marks an epoch of asynchronous irregular firing activity without recalls. (C) Raster plot with excitatory (inhibitory) neurons marked in red (blue). (D) PDFs of the coefficient of variations for all the neurons during the entire simulation (grey) and for a homogeneous Poisson process (yellow). (E) Instantaneous Kuramoto order parameters R for the networks (gray) and for neurons in population P1 (green) and P2 (orange), and their corresponding PDFs over the entire simulation (F). (G) Temporal evolution of the mean firing rates for populations P1 and P2, and (H) their corresponding frequency distributions (in linear-logarithmic scale) showing a peak at 2 Hz and long time tail. (I) Instantaneous change rates of synaptic weights in both populations P1 and P2 over time and their PDF (in linear-logarithmic scale) (J) over the entire simulation time, showing the prevalence of positive weight changes (reinforcement) with respect to negative ones (depression).
Fig 3.
Recovery of damaged modular structure by spontaneous recalls.
A Recovery experiment starting from randomized excitatory (A), inhibitory (B), or inhibitory plus excitatory-inhibitory (C) synaptic connections. In all the three cases, we study the partial recovery of the original modular organization of the synaptic weights over time. This recovery is mediated in all the cases by the emergence of spontaneous recalls, —transient events of partial synchronization between neurons associated with one of the two originally stored memories, highlighted in green for cluster 1 and orange for cluster 2.
Fig 4.
(A) Training the network with M = 4 non-overlapping stimuli. The green, orange, pink and cyan brackets represent clusters 1, 2, 3 and 4. Results are qualitatively the same as for the example with two memory items. (B) Stability diagram of the model for a network of N = 100 neurons. The red line separating the stability (orange) and instability (purple) regions, represents an upper limit for the number of inhibitory neurons (
) needed to maintain M independent memory items. The number of excitatory neurons in each realization is
−
delimiting the non-accessible areas (white region) of the diagram. The star symbol marks the maximal memory capacity of the network, corresponding to
items with
inhibitory neurons (cyan lines, 33 anti-Hebbian and 33 Hebbian). Magenta and light-green lines highlight the ranges of number of inhibitory neurons (and therefore of the
ratios) that allow the stabilization of M = 20 and M = 10 memory items, respectively. (C) Table comparing the number of neurons and E ∕ I ratios of different parts of the human [62–74] and mouse [75–77] brain and their hypothetical memory capacity obtained by extrapolating the results of our model.
Fig 5.
Stability of a network of N = 1000 neurons trained with M = 10 stimuli.
(A) Post-learning raster plots of the neuronal activity at times: t=0h, t=2h and t=4h. In each plot, the spike count, normalized to the total number of neurons N and estimated over time bins of 0.2 sec, is displayed below the time axis. Selected spontaneous recalls are highlighted by colored shadows while a peak of global activity is highlighted by a grey shadow. Some of these events are zoomed on the right side of the panels. (B) Post-learning weights matrices at times: t=0h, t=2h and t=4h. (C) Post-learning evolution of mean intra- (solid lines) and inter- (dashed line) clusters weights for excitatory to excitatory (E-E dark red), excitatory to inhibitory (E-I red), anti-Hebbian inhibitory to excitatory (A-E dark blue), anti-Hebbian inhibitory to inhibitory (A-I cyan), Hebbian inhibitory to excitatory (H-E blue) and Hebbian inhibitory to inhibitory (H-I steel blue) connections.
Fig 6.
Training the network with overlapping stimuli.
(A) Experimental protocol for a network of N = 100 neurons trained with M = 2 stimuli that share 8 excitatory neurons. (B) Simulation and learning results. Connectivity matrices show the evolution of the synaptic weights leading to the emergence of two modules which overlap over 8 hub neurons. The raster plot shows the simulated neuronal activity of the network during the initial resting phase, the learning stage and the post-learning period. The activity during the post-learning phase is characterized by a variety of spontaneous recall events involving P1 neurons and the hubs (green shadow), P2 neurons without the hubs (orange shadow) and the hubs alone (pink shadow). (C) Schematic diagrams representing the formation of the synaptic connectivity with two stable populations of neurons and overlapping hubs. Each population is composed at least of a population of excitatory neurons Ex and a population of hub neurons H (in red), one population of Hebbian inhibitory neurons I and one population of anti-Hebbian inhibitory neurons I
(in blue) (x = 1 , 2). Dashed circles identify groups of neurons admitting synchronization events (memory recalls).
Fig 7.
Plasticity and soft bound functions.
Plasticity functions: (A) Hebbian asymmetric STDP (Eq. (7)); (B) Hebbian symmetric STDP
(Eq. (8)); (C) anti-Hebbian symmetric STDP
(Eq. (9)). Soft bound functions for excitatory (D) and inhibitory (E) neurons. The bounds for potentiation are shown in red and for depression in blue. The displayed functions refer to M = 2 encoded stimuli, i.e. to f = 0 . 1.
Table 1.
Parameters for the network of QIF neurons.