Cortical Dynamics in Presence of Assemblies of Densely Connected Weight-Hub Neurons

Abstract

Experimental measurements of pairwise connection probability of pyramidal neurons together with the distribution of synaptic weights have been used to construct randomly connected model networks. However, several experimental studies suggest that both wiring and synaptic weight structure between neurons show statistics that differ from random networks. Here we study a network containing a subset of neurons which we call weight-hub neurons, that are characterized by strong inward synapses. We propose a connectivity structure for excitatory neurons that contain assemblies of densely connected weight-hub neurons, while the pairwise connection probability and synaptic weight distribution remain consistent with experimental data. Simulations of such a network with generalized integrate-and-fire neurons display regular and irregular slow oscillations akin to experimentally observed up/down state transitions in the activity of cortical neurons with a broad distribution of pairwise spike correlations. Moreover, stimulation of a model network in the presence or absence of assembly structure exhibits responses similar to light-evoked responses of cortical layers in optogenetically modified animals. We conclude that a high connection probability into and within assemblies of excitatory weight-hub neurons, as it likely is present in some but not all cortical layers, changes the dynamics of a layer of cortical microcircuitry significantly.

Keywords: connectivity; hub neuron; neural assembly; spike frequency adaptation; up-state/down-state oscillation.

Figure 1

Networks with weight-hub neurons. (A) Histogram of the sum of inward weights for a random (solid line, network without weight-hubs) and inward correlated (filled, network with weight-hubs) network topology. While the random topology (without weight-hubs) shows an approximately normal distribution, the inward-correlated topology has a broader, lognormal-like distribution. Weight-hub neurons form the tail of this distribution. Both networks, without and with weight-hubs, have the same lognormal distribution of individual weights shown in (B) (red line). Inset: In a heterogeneous network with inward correlations, most neurons receive many weak (thin arrows) connections (top) whereas weight-hub neurons (bottom) receive many strong connections. (B) Fitting the experimental distribution (red line) of synaptic weights (EPSP amplitudes) by a two-element “homogeneous” distribution (dashed areas). The lognormal distribution (solid line) was fitted to experimental data (Lefort et al., 2009) and is used to find the values of weak and strong weights, w_nh and w_h, respectively. Inset: Splitting the excitatory population into two subpopulations. Weight-hub neurons receive strong synaptic weights (w_h) and non-hub neurons receive weak synaptic weights (wnh). All connection probabilities are low (p_nh; nh: non-hub) except for the hub-to-hub connections (p_h).

Figure 2

Irregular up- and down-state transitions in a network with three assemblies of densely connected weight-hubs. (A-Top) Membrane potential of sample non-hub (black, labeled 1–12) and weight-hub (labeled 13–15) and inhibitory neurons (red, without label). Inside each group neurons have been sorted by their firing rate. Brown bars indicate time intervals which are considered as up-states for bottommost inhibitory neuron. (A-Bottom) Raster plot of several neurons of each population (same color). (B) Rate distribution of excitatory neurons. Numbered labels indicate the firing rate of neurons whose membrane potential traces are shown in (A). Inset: The distribution of firing rates is close to a normal distribution (red curve) on a (semi-) logarithmic scale. (C) Histograms of the up-state duration for each group of excitatory neurons. The coefficients of variation for the assemblies are 0.06, 0.10, and 0.16, which signifies regular durations. The non-hub neurons (filled histogram) exhibit a broad distribution of up-state durations with coefficient of variation of 0.42. Inset: The Excitatory population contains three assemblies of weight-hubs and a large population of non-hubs. (D,E) Distribution of pairwise Pearson correlation coefficients of transition times from down- to up-state (D) and from up- to down-state (E) inside each subpopulation (solid lines) and over all 145,530 pairs of neurons (dashed lines). Transitions of two neurons are counted as coincident if they happen in the same time bin of 20 ms. (F) Averaged Pearson correlation coefficients of transitions from down- to up-state (upper triangle) and up- to down-state (lower triangle).

Figure 3

(A) Network of 454 excitatory and 90 inhibitory neurons with identical neuron parameters, organized into homogenous subpopulations with dense connectivity within each assembly and non-hubs. Membrane potential traces of a non-hub neuron (A1), neurons from each of the three weight-hub assemblies (A2, A3, A4) and an inhibitory neuron (Inh) (A5). (A6) Raster plot of several neurons of each population (same colors). Oscillations of the assemblies are different in terms of the up-state and down-state durations. Non-hub and inhibitory neurons receive input from the three oscillating assemblies and exhibit irregular oscillations. Note that there are 359 non-hub neurons in the network, which is the majority of cells. (B) Heterogeneous network as in Figure 2, but sparse connectivity (p = 20%) inside assemblies. Membrane potential of non-hub (B1), three weight-hub neurons (B2–B4) and inhibitory neurons (B5) and the raster plot of several neurons of each population (B6). The Up-state/down-state oscillation vanishes. Weight-hubs (B2–B4) occasionally emit spikes since they receive stronger synapses from Poisson neurons, while non-hub neurons (B1) do not spike at all. (C) Oscillations in a network with a single assembly of densely connected weight-hubs are more regular than in Figure 2. Membrane potential of a non-hub (C1), a weight-hub (C2) and an inhibitory neuron (C3). (C4) Raster plot of randomly selected neurons of each population. While weight-hub neurons (green ticks) exhibit a high firing rate in the up-state, non-hub neurons (black ticks) show only a small number of spikes. Inhibitory neurons (Inh) are shown in red.

Figure 4

Simulated response to light-evoked stimulation of non-stimulated excitatory neurons in cortical L2/3 (A) and in L5 (B). Approximately 15% of all neurons (weight-hubs, non-hubs and inhibitory) are stimulated in each layer for a time period of 300 ms (blue bar). Membrane potentials (lines) and spikes (ticks) of weight-hubs (green) and non-hubs (black). L2/3 neurons (A) show little depolarization due to sparse connectivity between weight-hubs, while L5 neurons (B) display a long-lasting depolarization and a significant number of spikes. This effect is due to the dense connectivity between weight-hubs in the L5 network model, but not in the L2/3 model. (C) Simulation of L5 in case of a modified assembly model that only has strong internal synaptic weights but is not innervated as strongly from other neurons as the weight-hub assembly in (B), see main text for details. In the absence of weight-hub neurons, L5 does not generate long-lasting depolarization in response to the stimulation. (D) Simulation of L5 in case of having sparse connectivity (p = 20%) inside assemblies. The network is unable to produce long-lasting depolarization.

Figure 5

Cross-correlations of neuronal activity. (A) Two pairs of subthreshold membrane potentials (spikes have been removed) with low (A1) and high (A2) correlation and a pair of spike trains (A3). Spikes are counted as coincident if they fall within the same bin of 10 ms. (B,C) Distribution of Pearson correlation coefficients of subthreshold membrane potentials (B) and spike trains (C) of pairs of neurons inside each subpopulation (solid lines) and over all 145,530 pairs of neurons (dashed lines). (D–F) Averaged Pearson correlation coefficients between the membrane potentials (upper triangle) and the spike trains (lower triangle). Correlations are computed for pairs of neurons in the respective subpopulations of (A–D) the heterogeneous network of Figure 2, (E) the homogeneous network of Figure 3A and (F) the sparsely connected weight-hubs network of Figure 3B. Because inhibitory neurons do not fire any spikes in the sparsely connected weight-hubs network (F), the spike train correlations of them are not defined (white area).

Figure 6

Transition from up-down-state oscillations to “active” state. (A1–A6) Network of Figure 2 receiving external Poisson process stimulus from t = 3 s to t = 5 s (blue bar). Neurons show up-down state oscillation before and after the stimulus, while they exhibit higher firing rate (20.9 Hz for inhibitory neurons and 12.4 Hz for excitatory neurons, split into 37.4, 33.6, 33.0, 6.4 Hz for assemblies 1, 2, 3 and non-hubs neurons respectively) during the stimulation period. (B) Distribution of firing rates across neurons in the network during stimulation interval (blue bar in A). (C) Distribution of Pearson pairwise correlation coefficients (bin size = 10 ms) of spike trains of pairs of neurons inside each subpopulation (solid lines) and over all pairs of neurons (dashed lines) during stimulation interval.

Figure 7

Mean-field analysis. (A) The network feedback (C_fb, Equation 17) affects the quasi-stationary dynamics of the system. The red curve is the noisy gain function g of the GIF neuron model (mean spike count in a group of 50 independent neurons over 10 ms, divided by 50 × 10 ms, shaded area marks ±3 SEM) measured during the initial 10ms after switching on a synaptic current of mean 〈I_syn〉 (see Section Materials and Methods, Equation 14). The green lines (solid, dashed, and dash-dotted) show the relation of firing rate and synaptic current caused by network feedback (see Section Materials and Methods, Equation 16) for increasing C_fb. The slope of the green lines has an inverse relation with the effective coefficient C_fb of the population. Intersections of the red curve with one of the green lines indicate potential stationary states (fixed points) of a network of non-adapting neurons. Populations with a high C_fb (dashed and dash-dotted green lines) have three fixed points, stable low point, high point and unstable switch point. If the population described by a network feedback given by the dashed lines is driven by a mean current higher than I_s, it rapidly converges to the high point. On the other hand, a population with a low C_fb (solid green line) has only one intersection at the low point. Inset: Increasing C_fb causes an increase in the high firing rate r_h (magenta curve, left vertical scale) and a decrease of the switch current I_s (blue curve, right vertical scale). (B) The noisy gain function of adaptive neurons is different during the first 10 ms after stimulus onset (solid red curve) than later (dashed red curves) (C) The duration of up- and down-states as a function of the time constant of the excitatory neuron firing threshold kernel γ(s). Only the time constant of γ₂(s) (the exponential with the longer time constant) was manipulated, while γ₁(s) remained as reported in Table 1. The black stars indicate the experimentally extracted value of the time constant, which was used in the other figures. (D) Same as C, except that here we manipulated the amplitude of γ₁(s) (the exponential with the larger amplitude). The error bars show the standard deviation of up/down state durations over 10 simulation trials of 10 s duration each.

Figure 8

K-means clustering identifies weight-hub neuron assemblies. Each dot represents one neuron and its color denotes the corresponding subpopulation in the simulation shown in Figure 2. (A) Clustering of all neurons into two clusters. The first stage of the classification algorithm successfully identifies weight-hubs and non-hubs, but does not distinguish between different assemblies of weight-hubs. The red circles show the center of the clusters and the dashed line displays the classification boundary. (B) Clustering of weight-hub neurons (identified in A) into different assemblies. The K-means algorithm with three clusters identifies the assembly of each weight-hub neuron with 93.7% accuracy.