Microcircuit Synchronization and Heavy-Tailed Synaptic Weight Distribution Augment preBötzinger Complex Bursting Dynamics

Flowchart for model construction and testing. A, Rhythm-generating preBötC neurons project to pattern-generating preBötC excitatory neurons that in turn project to premotor neurons. Inhibitory neurons are not shown. B, Holographic photostimulation of 3–9 preBötC inspiratory neurons (of ∼1500 total excitatory neurons) in rhythmic neonatal mouse slices generate bursts at delays of ∼100–500 ms (Fig. 1F,G). C, Various network models tested to determine whether they capture the preBötC dynamics depicted in B, left: network topologies as graphs with edges connected through experimentally determined connection probability between putative rhythmogenic neurons (Rekling et al., 2000). The nodes were modeled as LIF neurons (right) with intrinsic and synaptic properties taken from experiments; right, synaptic activation from two (“laser”) stimulated model rhythmogenic neurons N1 and N2 (presynaptic) that project to neuron N3 (postsynaptic); their activation times represent the arrival of spikes form N1 and N2 at their respective synapses on N3 with weights indicated (left, black traces) consequently, changing the somatic potential of neuron N3 from resting potential. When the somatic potential increases above spike activation threshold V* = 12 mV (with V at −48 mV), N3 generates an action potential (*), followed by its potential dropping to V_rest = 0 mV (with V_rest at −60 mV) for a refractory period of 3 ms.

Network models. Various networks represented on a force-based (Fruchterman–Reingold algorithm in the Gephi software) layout. Linked nodes (neurons) are pulled together, and unrelated nodes are farther apart; most strongly linked nodes (through direct connections or common inputs) are at the center, and the least linked ones are at the periphery. Nodes (and their edges) are color coded based on the number of their projections (i.e., outward synapses with warmer shades representing more connections). In the hierarchical network (top left), the neurons of the two centers (as depicted in Fig. 2C) are pulled together into the center of the graph but can be distinguished based on their projections.

Simulation of holographic stimulation experiment. A, Model output when the same set of randomly selected 7 neurons (on the dashed lines) was stimulated to fire 7 spikes each at ∼25 Hz with 5 ms jitter (i.e., SD to model holographic uncaging of glutamate onto these neurons, as in Kam et al., 2013b). Each dot represents the time of spike (abscissa) from the corresponding neuron (ordinate). Note that once the network synchronized, it continues in the high-frequency firing (i.e., bursting mode) even after the stimulated spikes from the seven neurons and ended at ∼300 ms. B, Enlarged region from A (400–420 ms) exhibiting firing rate modulation because of the refractory period, of 3 ms, in the model neurons. C, Firing rate (FR) of stimulated neurons and their postsynaptic activated neurons in A in 5 ms bins. D, Average firing rate of network computed by averaging network activity in a moving window of 40 ms with 5 ms step increment.

ER graphs with lognormal synaptic weights reproduced the robustness of preBötC synchronization and trial to trial variability in the latency to synchronize seen in experiments. A, Model output when the same set of randomly selected 7 neurons (on the dashed lines) was stimulated to fire 7 spikes each at 25 Hz with 5 ms jitter (i.e., SD to model holographic uncaging of glutamate onto these neurons; as in the study by Kam et al., 2013b). Spike times for all neurons represented by circles color coded for 5 trials (green, blue, black, purple, and red). y-Axis represents arbitrary order of 1000 LIF preBötC neurons. In these trials, neurons 121, 126, 270, 491, 700, 749, and 943 were stimulated. In 4 of 5 trials (green, blue, black, and purple), the network synchronized, indicated by temporal alignment of spikes in all neurons, but at various latencies. In the fifth trial (red), the network did not synchronize (i.e., no vertical alignment of red dots). B, Firing rate (FR) of stimulated neurons and their postsynaptic activated neurons in A (color coded as in A) in 5 ms bins showing waxing and waning of their activity during and poststimulation. C, Average firing rate of network computed by averaging network activity in a moving window of 40 ms with 5 ms step increment. Dashed boxes in B and C represent unusual intervals where, despite relatively high synchronous activity, emerging from stimulated firing and their recruited neuronal firing, the network did not synchronize fully.

Randomly connected networks of excitatory neurons replicated experimentally observed preBötC synchronization. A, Degree, average path lengths (PLs) and average clustering coefficients (CCs) for localized networks used in this study. B, C, Synchronization probability (B) and mean latency to synchronize (C) for localized networks when (1–10) neurons were stimulated to replicate the experimental protocol described in Figure 1. Each colored trace represents a different network where synchronization probability and the mean latency to synchronize was computed over 10 trials; gray boxes span the parameter space that lie within the experimental range (170–370 ms; Kam et al., 2013b) for threshold number of stimulated neurons to induce preBötC bursts. D–F, Same as A–C, respectively, but for hierarchical 2-center networks (see Materials and Methods). G–I, same as A–C, respectively, but for random networks modeled on ER graphs. J–L, Same as A–C, respectively, but for networks with increased number of triplices (triangular directed edges, as described in Fig. 2) incorporated in the ER network, resulting in a small world network. Note that only ER networks replicate the experimental finding of minimum number of stimulated neurons required to induce a preBötC burst to be between 4 and 9 (H), with latency to synchronize exhibiting the widest range, closely representing the experimental results. For B, E, H, and K, across 10 different realizations of each network type, the minimum number of neurons required to synchronize the network with ≥80% reliability = 2–9, 3–4, 4–9, and 2–7, respectively. For C, F, I, and L range of mean latency to synchronize the network with the minimum number of stimulated neurons at ≥80% reliability = 55–145, 43–143, 88–187, and 27–121 ms, respectively. Error bars indicate SDs.

Computing efferent synaptic score from network topology. The efferent synaptic score (S) is neuron property defined by its network connectivity. A, Connectivity matrix of ER network with lognormal (LN) weight distribution. Presynaptic neurons (ordinate) project to other neurons labeled as postsynaptic (abscissa) with synaptic weights, randomly drawn from an LN distribution. Each dot represents synaptic weight (Syn W; color coded) of presynaptic #i (1–1000) to postsynaptic #j (1–1000). B, Top, Expanded section from a (100 × 100) showing the contribution of synaptic connections toward S. For example, the first-order S of neuron #80 (S₈₀₍₁₎) is the sum of all synaptic weights with which it connects to other neurons. In this example, neuron #20 is postsynaptic to neuron #80; thus, S₂₀₍₁₎ contributes to the second-order S of neuron #80 [S₈₀₍₂₎; second-order S (S₍₂₎) also accounts for the connectivity of the next neighbors of a given neuron; see text for the calculation of S₍₂₎); bottom, example of S calculation in a hypothetical network. For a given neuron, the first-order senderness, S_weight, is the sum of its efferent synaptic weights, and S_synout is the number of its efferent synaptic connections. C, Plot of the number of outward connections of neurons in the ER network with their synaptic weights color coded as in A and B. This network was used in Figure 5; S varies from neuron to neuron as their outward synaptic connections vary, per the ER graph, and as their output synaptic weights vary because of the LN weight distribution.

Network topology regulates spike transmission fidelity and network synchronization through input convergence. A, Illustration of interval selection for an example trial (Fig. 5, R1) of spike times after 7 randomly selected neurons were stimulated to fire up to 7 spikes at 40 ± 5 ms (mean ± SD) period. Spikes in these intervals were used to compute S and σ for plots in B–I, as described in the text. Intervals I₁–I₃ enclose the spike times of the stimulated neurons (demarked by dashed horizontal lines) as well spikes of their postsynaptic recruited neurons. The penultimate (preburst) interval (I_4A for R1) encloses the last set of spikes from stimulated and recruited neurons up to the last 1 ms bin, at which point the network activity always increased monotonically. Inset, Bottom, Expanded region from top, illustrating the selection of the end boundary for the preburst interval; shaded rectangles represent 1 ms bins, with darker color representing higher population firing frequency (kHz) denoted at the top of each bin. The final interval (I_4B for R1) represents the synchronized state of the network; see text for details of interval selection for various trials. B, C, S_weight (B) and S_synout (C) plotted against σ for each trial of Figure 5A, color coded for individual trials. Each circle corresponds to the results for 1 interval in a single trial, and lines connect evolution of network activity across successive intervals. SD of spike times in each interval (σ) is a measure of synchrony among the spiking neurons in that interval; intervals for two trials (R4 and R5) are indicated in color code for illustration. D, E, Total efferent synaptic weight Out_synW_recruited (D) and the number of efferent synapses Out_synN_recruited (E) of recruited neurons up to the preburst interval (i.e., the onset of network synchronization without any further external stimulation). Dashed line represents the mean S of the network. Note the failure of the purple trial to synchronize in interval I₄ (∼150 ms) even when it has high synchrony/low σ (in B, C). Here the S of recruited neurons is low, thus the network did not synchronize. F, G, S_weight/σ and S_synout/σ of neurons recruited by the activity of stimulated neurons in each interval (indicated at the top) across the five trials. Note that for the activity in the intervals that did not induce network synchronization in the network the S/σ was always lower than the ones that led to network synchronization. H, I, S_weight (H) and S_synout (I) versus σ, as in B and C, with additional simulations showing parameter space where the network activity bifurcates toward synchronization. For these simulations, 7–10 neurons were stimulated to fire 7 spikes at a 40–100 ms mean period with a 5–8 ms jitter (SD). Red trials did not synchronize. Green arrowheads represent a trial where a small change in the (S–σ) parameter space in the successive interval (with recruitment of only one additional neuron and Δσ = –0.5 ms) resulted in the bifurcation of network trajectory toward synchronization.

Lognormally distributed synaptic weights enhanced the fidelity of spike transmission that enables network synchronization with fewer inputs. A, B, Synchronization probability (A) and latency to synchronize (B) for 5 different ER networks with uniformly distributed synaptic weights that were equal to the mean weight used in Figures 5 and 6. C, Voltage of 5 randomly selected neurons from an ER network with uniform weights when a different set of randomly selected 7 neurons was stimulated like simulations in Figure 5A. D, Same as C when the same network connectivity was incorporated with lognormal weight distribution. The network synchronized at ∼78 ms and voltage traces reveal better coincidence detection in this network; vertical lines, ∼78 ms represent action potentials. E, F, Average firing frequency (E) and spike probability (F) of individual neurons when 10 randomly selected synaptic inputs (of 50) were activated at 6 different Poisson frequencies (10–200 Hz; indicated at the bottom of F), with either lognormal (LN) or uniform (Unif) synaptic weight distributions; these are composite results from 3 trials each of 10 different neurons at each stimulation frequency; p values from KS test. G, Histogram of LN weights used in E and F (red) compared with the distribution of weights of Figure 5; corresponding EPSP amplitudes for the weights are indicated in blue.