Effect of synaptic connectivity on long-range synchronization of fast cortical oscillations

Abstract

Cortical gamma oscillations in the 20- to 80-Hz range are associated with attentiveness and sensory perception and have strong connections to both cognitive processing and temporal binding of sensory stimuli. These gamma oscillations become synchronized within a few milliseconds over distances spanning a few millimeters in spite of synaptic delays. In this study using in vivo recordings and large-scale cortical network models, we reveal a critical role played by the network geometry in achieving precise long-range synchronization in the gamma frequency band. Our results indicate that the presence of many independent synaptic pathways in a two-dimensional network facilitate precise phase synchronization of fast gamma band oscillations with nearly zero phase delays between remote network sites. These findings predict a common mechanism of precise oscillatory synchronization in neuronal networks.

FIG. 1.

Model properties. A: steady-state response pattern of excitatory (PY) neuron for 3 different levels of the resting potential. Black, σ = 0.06; green, σ = 0.09; blue, σ = 0.17. B: inhibitory postsynaptic potential (IPSP) in the postsynaptic PY neuron (bottom) triggered by a spike in presynaptic interneuron (IN) (top).

FIG. 2.

Fast gamma oscillation in vivo. Patterns of activity onset and synchronization during active states in a neocortical slab 6 × 10 mm. A: the position of recording electrodes. Distance between electrodes ∼1.5 mm. B: the active state started from around electrode 5 (see left vertical line) and propagated to other electrodes. C, top: cross-correlation of the onset of active states (electrode 5 is the reference, analyzed fragments from −100 to +100 ms of electrode 5 half-amplitude). Bottom: cross-correlation during active states (electrode 5 is the reference, analyzed fragments from +200 to +400 ms of electrode 5 half-amplitude). Note the absence of correlation between activities in the slab (electrodes 3–8) and outside the slab (electrodes 1 and 2).

FIG. 3.

Effect of synaptic coupling on fast network model oscillations. A: structure of synaptic connectivity in a cortical network model including 512 PY neurons and 128 INs. AMPA, excitatory synapses; GABA, inhibitory synapses. B: field potential (top) and representative PY neurons (bottom) during gamma oscillations. C, top: frequency of field potential oscillations is plotted as a function of inhibitory feedback: excitatory PY → IN and inhibitory IN → PY coupling. Bottom: frequency of field potential oscillations is plotted as a function of excitatory PY → IN coupling and resting potential (σ): σ = 0.09 corresponds to the onset of spiking; σ = 0.17 corresponds to ∼20-Hz sustained firing rate. D: spectrograms representing the power spectrum of the field (top) and PY (bottom) oscillations as a function of PY → IN coupling for a fixed value of IN → PY coupling (g = 0.0007).

FIG. 4.

Synchrony of oscillations in 1- and 2-dimensional (2D) network models. Group of PY neurons [with indexes (107,406) for 1D or (107,406) × (107,406) for 2D] was entrained to gamma oscillations by DC input. A: 1D model: 512 PY neurons and 128 INs. Ai, top: connectivity structure. Bottom: spatiotemporal patterns of activity in a 1D network (g_IN-PY = 0.0007, g_PY-IN = 0.0015). Spikes are shown in red; dark blue indicates hyperpolarization. Aii: cross-correlation of the local field potential (averaged over 20 PY neurons) between the center of the network and remote sites. Aiii: time lags to the main peak of the cross-correlation function between local field potentials in different spatial locations (see methods) plotted for various strengths of PY-IN coupling. Color indicates number of time lags within 1 ms bins (dark blue indicates no lags). B: 2D model: 512 × 512 PY neurons and 256 × 256 INs. Bi, top: connectivity structure. Red and green dashed circuits illustrate radii of PY-IN and IN-PY connectivity for a single PY and IN neurons (shown with black circuit). Bottom: snapshots of activity in the IN population during global coherent gamma oscillation at times t₀, t₀+5 ms, t₀+8 ms. Bii: cross-correlation of the local field potential (averaged over 20 × 20 PY neurons) between the center of the network and remote sites along the x axis. Biii, top: the staircase structure of resonance modes (f_PY/f_FP ) is shown as a function of PY-IN coupling. Bottom: same as in Fig. 3Aiii but for a 2D network. Inset: field potential oscillations near the onset of DC input (↑).

FIG. 5.

Two-dimensional network dynamics at the transition between 2 resonance states. A: representative snapshots of network activity at 4 different times illustrated with IN population. Network size: 640 × 640 PY neurons and 320 × 320 INs. B: spatial distribution of the resonance modes computed as the ratio of the total number of IN spikes to the total number of PY spikes in each network site during a given time interval. Orange (green) area indicates f_PY/f_FP = 1/3 (f_PY/f_FP = 1/2) resonance mode, respectively. Note that the population of neurons in f_PY/f_FP = 1/3 mode (orange area) is broken into several disjointed sets. C: examples of oscillations in 2 network sites (1 and 3 in B) belonging to different resonance areas (black indicats field potential; green and red indicates PY neurons). D: cross-correlation of the field oscillations among 4 network sites indicated in B. The synchrony was high between sites 3 and 4 (C_3–4, the same population); moderate between sites 1 and 2 (C_1–2, disjoint populations of the same type); low between 1 and 3 or 1 and 4 (C_1–3, C_1–4, populations oscillating in different resonance modes).

FIG. 6.

Effect of network connectivity structure on gamma range synchronization. A: “cross-like” connectivity structure in a 2D network. B: oscillations in the network with cross-like connectivity: 512 × 512 PY neurons and 256 × 256 INs. Left: snapshots of spiking activity in the IN population. Right: cross-correlation of the field potential oscillations (averaged over 10 × 10 PY neurons) between the center of the network and remote sites along the x axis as a function of the distance between the sites. C: multiple independent pathways with different synaptic lengths connect any 2 sites in a 2D network. Circles illustrate the synaptic connectivity of INs (red dots). Dashed lines illustrate different pathways connecting 2 selected INs (black dots).

FIG. 7.

Effect of network dimensionality on gamma range synchronization. A: oscillations in the 16 × 512 PYs (8 × 256 INs) network model. Snapshots of IN activity as 3 different times (left) and cross-correlation of field potentials along x axis (right). B: oscillations in the 8 × 512 PYs (4 × 256 INs) network model. C: normalized probability density distribution of time lags to the main peak of the cross-correlation function between local field potentials at different spatial locations as a function of the network size along the y dimension. Logarithmic scale. Results are averaged across 15 independent trials with randomly selected initial conditions. Note disappearance of the isolated peak for the network of size <1 footprint (8 PY neurons). D: spectrogram of the local field potential [averaged across all neurons within (150,350) × (1, N) area, where N is number of PY neurons along y axis] as a function of the network's size along the y-dimension. Logarithmic scale (dB). E, left: cross-correlation of field potentials along x axis in 2D model with radius of the synaptic footprint 4 for PY-PY and PY-IN synapses (total 47 and 48 neurons, respectively) and 2 for IN-PY synapses (total 13 neurons). Right: cross-correlation of field potentials along the chain of neurons in 1D model with radius of the synaptic footprint 24 for PY-PY and PY-IN synapses (total 47 and 48 neurons, respectively) and 6 for IN-PY synapses (total 13 neurons). Not only is delay to the peak of cross-correlation function larger in 1D model but also the amplitude of the correlation function is significantly reduced in 1D model.

FIG. 8.

Effect of connectivity patterns on the network synchronization. A: connectivity structure for 2 different networks, 512 × 512 PY neurons and 256 × 256 INs. Left: circular footprint. The radius of the synaptic footprint: 18 for neurons with AMPA mediated PY-PY synapses, 18 for neurons with AMPA-mediated PY-IN synapses and 9 for neurons with GABA_A-mediated IN-PY synapses. Right: cross-like footprint with all-to-all connections along the x axis. B: snapshots of activity in the IN population. C: cross-correlation of the field potential oscillations between the center of the network and remote sites along the y axis as a function of the distance between the sites. D: power spectra of field potential oscillations. For both networks, the main peak was at ∼40 Hz with subharmonics at ∼80 Hz. Network with all-to-all connections along the x-axis also showed a 2nd peak at ∼50 Hz. Logarithmic scale. E: normalized probability density distribution of time lags to the peak of the cross-correlation function between local field potentials at different spatial locations. Logarithmic scale.