Integral feedback mechanisms in regulating firing rate homeostasis.

(Ai) Neural net-work activity is determined by external inputs in a static network. (Aii) Hebbian plasticity amplifies the network responses to external inputs via a positive feedback mechanism. (Aiii) Homeostatic plasticity restores setpoint activity via a negative feedback mechanism, which is key to firing rate homeostasis. (Bi) Hebbian functional plasticity strengthens recurrent connectivity by potentiating the weights of certain groups of synapses. (Bii) Homeostatic synaptic scaling proportionally down-scale or up-scale all synaptic strengths upon chronic excitation and inhibition. (Biii) Homeostatic structural plasticity presents homeo-static spine loss upon chronic excitation, while divergent changes in spine density upon chronic inhibition have been observed. (C) Synaptic scaling and structural plasticity use intracellular calcium concentration ([Ca2+]i, C(t)) to track neural activity (AP, action potential, S(t)). Calcium concentration updates each time with calcium influx (βCa) upon the arrival of an action potential and decays with a time constant τCa. Homeostatic synaptic scaling is implemented as a weight-dependent rule which updates the synaptic weight w(t) with a scaling factor ρ. The discrepancy from the setpoint ɛ determines the direction of weight scaling. Structural plasticity is also calcium-dependent and governs the growth and retraction of axonal boutons and dendritic spines by the setpoint value. Two examples of the structural plasticity rule are presented here. They are either linearly or non-linearly dependent on the intracellular calcium concentration.

Heterogeneous modification of spine numbers and sizes upon three-day synaptic inhibition via NBQX.

(A) Example CA1 pyramidal neuron recorded in the entorhinal-hippocampal tissue culture for probing the effects of different NBQX concentrations. Scale bar: 500 µm. (Bi-Bii) Frequency and averaged amplitudes of sEPSC events for each recorded neuron in three groups (N = 18 for the control group, N = 18 for the 200 nM NBQX-treated group, N = 12 for the 50 µM NBQX-treated group). (C) Example Thy1-eGPF culture and the example dendritic segments from the radiatum layer (rad.) before and after three-day treatment. Scale bar: 200 µm and 5 µm. (D) Spine density at baseline and after the three-day treatment. All values were normalized by the corresponding baseline values. Lines with light shades are raw data (solid and dashed lines represent increased or reduced spine density, respectively). Dark-shaded lines with error bars are each group’s means and the standard error of the means (s.e.m.s). (N = 19 for the control group, N = 24 for the 200 nM-treated group, N = 33 for the 50 µM-treated group) (E) Cumulative distribution function of spine sizes before and after the three-day treatment (N = 489 for the control group, N = 736 for the 200 nM-treated group, N = 675 for 50 µM-treated group). Inset shows the corresponding averages of normalized spine sizes. (F) Normalized changes in spine sizes grouped by their initial spine sizes. Values on the x-axis are the upper limits of each group. (G) Each segment’s average change in spine sizes against its initial spine density. The marker size labels the net change in spine density over a three-day course. (H) Each segment’s average changes in spine sizes against its change in spine density. (I) The table summarises the experimental data, while the graph displays the extrapolated relationship between neural activity and spine densities or sizes.

Growing a neural network with three distinct structural plasticity rules.

(A) The point neuron model was used to study structural plasticity, where dendritic morphology is reduced. Dendritic spines are represented by pink sticks on the soma; axonal boutons are represented by empty or solid half circles. An empty circle with a dashed line labels an axon during retraction. Calcium concentration is linearly correlated with neural firing rate in our implementation, so neural activity would be used in the rest of the manuscript to reflect the hidden calcium dynamics. (Bi-Biii) Three activity-dependent growth rules of structural plasticity regulate the change of synaptic element numbers. (Bi) Linear rule with one setpoint (ɛ = 7.9). (Bii) Gaussian rule with two setpoints that one is zero (ɛ = 7.9 and η = 0). (Biii) Gaussian rule with two non-zero setpoints (ɛ = 7.9 and η = 0.7). Three shades indicate 100%, 50%, or 10% of the original growth rate (ν). Positive and negative values indicate, respectively, the speed of outgrowth and retraction of synaptic elements. (C) The neural network architecture of the Brunel network. 10 000 excitatory (blue) and 2 500 inhibitory neurons (red) are stimulated by external Poissonian inputs. All I-I, I-E, and E-I synapses are hard-wired with 10% probability. E-E synapses are subject to structural plasticity rules. (Di-Diii) Temporal dynamics of neural activity and network connectivity (Γ) during growth, respectively, guided by three rules. If not otherwise stated, the curve and the shaded area in activity plots represent the mean and standard deviation of the neural activity for the inhibitory population (I) and excitatory population (E). The network developed to an equilibrium state (Γ = 10%) in Di and Dii but not in Diii. The firing rates distribution and network connectivity matrices of the chosen time points, indicated by solid triangles, are included in Supplementary Figure 3 for Di-Dii, and in Figure 4 for Diii.

Silent neurons remained isolated in the network regulated by the biphasic Gaussian rule.

(A) Histogram of firing rates of excitatory and inhibitory neurons sampled at the time point indicated in Figure 3Diii. Almost half excitatory population was silent. The blue vertical line labels the mean firing rate of non-silent neurons. The orange dashed vertical line indicates the target firing rate (ɛ = 7.9 Hz). (Bi-Bii) Network connectivity matrix and the distribution of synapse numbers that individual excitatory neurons have. (C) Correlation heatmap between neural activity and synapse number of individual excitatory neurons. Neurons that were silent did not form synapses either. Neurons that fired around the target rate formed around 1 000 synapses from other active excitatory neurons. (D) Network architecture when facilitating current (Ifacilitating) was injected to boost the network development. (E) Temporal dynamics of neural activity and network connectivity when damping facilitating current were injected. The small inset shows the firing rate distributions of both excitatory and inhibitory neurons at a chosen time point (solid triangle). The facilitating current started at 750 pA and decayed linearly to zero at 4 000 s. (F) Different starting values of facilitating currents ended with different network connectivities. We used 750 pA throughout the manuscript.

Divergent regulation of network connectivity upon stimulation and deprivation by three structural plasticity rules.

(A) A subpopulation (10%) of excitatory neurons (S) was subject to activity perturbation. All E-E, S-S, and E-S synapses are subject to the biphasic Gaussian rule. (Bi) Activity perturbation protocol. Three different folds of the original intensity (FOI) of the Poisson generator were used as examples to represent stimulation (110% FOI), weak deprivation (95% FOI), and silencing (0% FOI). (Bii) Temporal dynamics of neural activity of the subpopulation (S), the rest of the excitatory neurons (E), and inhibitory neurons (I) under corresponding protocols. (Biii) Temporal evolution of the overall network connectivity and connectivity of different subgroups under corresponding protocols. Synaptic connection probability from E to S (E-S) is identical to that from S to E (S-E) here so only S-E traces are shown. (C) Network connectivity matrices at the end of three protocols. (D) Average incoming synapse numbers of S neurons under different FOIs. Empty green circles are data from networks under extreme stimulation or inhibition, where neural activity and network connectivity dynamics were unstable. (E) Examples of two neurons that received the same external inputs but have different calcium decay time constants (τCa). The upper panel shows the membrane potential; the middle panel shows the spike trains of the two neurons; the lower panel displays the integrated calcium concentration over time. (F) Connectivity traces of subnetwork upon silencing under three different conditions.

Activity perturbation and recurrent connectivity shaped the evolution of network connectivity.

(A) In the default network, we used the same growth rule for axonal boutons (A, light brown curve) and dendritic spines (D, pink curve). Alternatively, different η values could be used for axonal and dendritic elements. A silencing protocol was applied. (B) Neural activity of S, E, and I neurons under two conditions. (C) The network connectivity matrices at the end of the silencing protocol under two conditions. (D) Protocols used to examine the effects of recurrent connectivity and external stimulation. (E-F) Time courses of neural activity and connectivity upon silencing and external stimulation. Under Protocol 1, the growth rate in the left panel was ten times faster than that in the middle panel. In the right panel, external stimulation intensity was doubled.

Homeostatic synaptic scaling sculpted effective connectivity to interfere with struc-tural plasticity.

(A) Protocol of silencing and synaptic scaling enabling. Three different scaling strengths were applied, that ρ = 0 is for w/o scaling, ρ = 0.01 and ρ = 0.02 represent weak and strong scaling, respectively. (B-C) Time courses of network activity and connectivity. Γstruc. is synapse-number-based structural connectivity. Γeffec. denotes the effective connectivity which multiplies synapse numbers and synaptic weights. (Di-Dii) Structural and effective connectivity matrices of the whole network at t2. (E) Time courses of firing activity and average synaptic weights for an example active excitatory neuron (blue) and an example silent excitatory neuron (grey). In the weight plot, synapses from other silent neurons (solid grey line, S) and from other active excitatory neurons (dashed grey line, E) are specified. (E) Raster plots of 100 selected inhibitory neurons (red), excitatory neurons (blue), and silent neurons (grey) from the network at t1, t2, and t3 labeled in panel B.

Systematic study of the interaction between synaptic scaling and structural plas-ticity in response to activity silencing.

(A) We hypothesize that different combinations of synaptic scaling (SS) and homeostatic structural plasticity (HSP) may apply to different neuron types or dendritic segments within the same neuron, or the same type of neuron under different experimental conditions, such that the empirically observed structural plasticity was highly heterogeneous. (Sample neurons were reconstructed based on CA1 pyramidal neuron and dentate gyrus granule cells previously recorded in our lab.) (B) Simulation protocol. We systematically changed the values for the growth rate of the HSP rule (ν) and the scaling strength of the SS rule. (C-D) Example traces of neural activity and struc-tural plasticity of the deprived subpopulation (S neurons) under different parameter combinations. (E-F) Connectivity discrepancies and firing rate discrepancies when different growth rates and scaling strengths were combined. All the discrepancies were calculated by estimating the area between the actual time course and the equilibrium connection probability (10%) or the target rate (ɛ) from the time of silencing until the end of the simulation.