Figure 1.
Luminance contrast and input-drive dependent gamma oscillation frequency.
(A) V1 LFP mean relative power spectra (20–60Hz, line thickness represents ±1 SEM) during presentation of square wave gratings of 8 different luminance contrasts (line color), mean data from Monkey S [8].(B) Gamma band peak frequency (top) and power (bottom) as a function of contrast (only the 6 highest contrast conditions). (C) Schematic architecture of the pyramidal (red)—interneuron (blue) gamma network (PING). (D) Example time period of population spike histogram (2ms bins) during steady excitatory drive input (0.06 mS/cm2). Spikes of the excitatory (red) neurons occurred earlier than from inhibitory (blue) neurons within a gamma cycle. (E) The absolute power spectra for different input excitation levels (mimicking contrast). (F) Quantification of (upper panel) gamma frequency (black), I-cell spike rate (blue), E-cell spike rate (red) and (lower panel) gamma power as a function of excitatory input drive.
Figure 2.
Impact of oscillation frequency on the interaction between two PING gamma networks.
(A) Illustration of the Arnold tongue. The potential for two oscillators to synchronize (grey area) is positively correlated with coupling strength and negatively correlated with the difference in intrinsic frequency. (B) The main structure of the two coupled PING gamma networks. Excitatory drive difference (detuning, [Δω) and the coupling strength C between the two networks were modulated. (C) Example simulation output from the networks, upper) smoothed LFP signal, arbitrary scaling, lower) time-frequency representation. (D–F) Example simulations with different network parameters: Left to Right: Arnold tongue: Parameters Δω and Coupling Strength are indicated as black dot in Arnold tongue diagram. Power spectra, shown in black line for simulated LFP for network 1 and in dashed grey line for network 2. Population raster plots shown for simulation window [2.2sec to 2.55sec]: network 1 = neuron 1–100, network 2 = neurons 101–200. E-cells spikes are indicated by red dots, I-cells spikes by blue dots. Polar plot of phase difference are shown to the right. Bar length indicates percentage of time (from 5s trial) within each phase bin. G-I) Reconstruction of Arnold Tongue when manipulating coupling strength and detuning (detuning = input-drivenetwork1—input-drivenetwork2). Arnold Tongue corresponds to a region with strong phase locking (G, bright colors), common emergent frequency (H, green color), and systematic phase difference (I, color-coded) between coupled networks, Network 1 lagged in phase compared to network 2 (red in I) when network 1 received a lower drive. Conversely, network 1 had a leading phase relationship with network 2 (blue in I) when network 1 received the higher excitatory drive Δ phase is only shown for conditions of substantial phase locking (>∼0.3) (J) Overlaid representation of emergent frequency difference (black line), intrinsic frequency difference (dashed line) and phase difference (red line) for simulations with inter-network excitatory connections of 0.6 mS/cm2.
Figure 3.
Assembly formation and complementary rate/phase code.
(A) The overall network ring structure of the PING model with nearest-neighbor connections (B) Example of a simulation population spike raster output (top I-cells, bottom E-cells), variation in E-cell excitatory drive is indicated to the right.(C to E) Detailed results for three example E-cells a, b and c. (C) Location of each example E-cell along the ring structure is indicated by the level of input-drive (black) as well as the squared derivative of input (red). (D) Phase-locking values between each example E-cell and all other E-cells (estimated by cross-correlation peak). (E) Phase difference between example E-cell and all other E-cells with phase-locking threshold >0.25 (for illustration, see Methods). (F) Matrix showing phase-locking between all possible pairs of E-cell pairs, location in the ring is indicated by the level of input-drive, as in C. (G) Phase difference between all possible E-cell pairs with same phase-locking threshold as above. Blue indicates that the X-axis neuron leads the Y-axis neuron, red indicates the reverse.
Table 1.
The synaptic parameter values for AMPA and GABA-A.
Table 2.
Connectivity number and strength of synaptic connections in the Hodgkin-Huxley networks of Fig. 1 and Fig. 2.
Table 3.
Connectivity number and strength of synaptic connections in the ring-PING network (Fig. 3).
Figure 4.
Reconstruction of stimulus input based on phase and frequency coding.
See Methods for derivations of the coding schemes. A) The stimulus input Sorig to be reconstructed B) Reconstruction based on frequency Sest(ω) (here E-ell rate) alone C) based phase-differences among E-cells Sest(θ) D) based on a combined frequency and phase code Sest(ω,θ). E) The reconstruction performance, measured by mutual information (MI), was from lowest to highest MI = 0.18 for Sest(θ), MI = 0.65 for frequency code Sest(ω) and MI = 0.92 for combined code Sest(ω,θ).
Figure 5.
Comparison of Hodgkin-Huxley networks (HH) and phase-oscillator model.
A) The voltage membrane of an E-cell and I-cell is shown as modeled by the HH-dynamical equations. The generative mechanism of PING gamma oscillation is the rhythmic interaction of E- and I-cells. In the middle, the LPA (local population average of E-cell spikes, see Methods) is shown. The fluctuations in the LPA represent the synchronous rhythmic interactions among the local population of E- and I-cells. Using the Hilbert transform, one can easily derive the instantaneous phase of the LPA as shown below. B) The instantaneous phase of a phase-oscillator is shown. In the phase-oscillator model the phase-variable is modeled directly by one simple dynamical equation (see Methods) mainly governed by the intrinsic frequency and interactions by other oscillators. Our assumption in this study is that the instantaneous phase derived from local LPA in the HH-PING network can be approximated by phase-oscillators.
Figure 6.
Reproduction of Hodgkin-Huxley results of Figs. 3 and 4 by a phase oscillator model.
(A) Replication of Fig. 3 Hodgkin-Huxley network architecture, here with 160 phase-oscillators with the same connectivity structure. B) Example extract of simulation output. Color represents the current phase of individual oscillators. The location of each oscillator in the ring architecture is indicated by the relative input level (intrinsic frequency) C) Matrix of phase-locking values between all possible oscillator pairs, equivalent to Fig. 3F. D) Phase relation between all possible oscillator pairs. Pairs with phase-locking < ∼0.3 (see Method) are masked for illustrative reasons. Blue indicates that the X-axis oscillator leads the Y-axis oscillator, red indicates the reverse. E) Stimulus reconstruction of Sorig (intrinsic frequency) based on the frequency code Sest(ω), the phase code Sest(θ) and the combined frequency and phase code Sest(ω,θ) F) The reconstruction performance, measured by mutual information (MI), from lowest to highest MI = 0.17 for phase code Sest(θ), MI = 0.75 for frequency code Sest(ω) and MI = 0.95 for combined code Sest(ω,θ).
Figure 7.
Phase-oscillator model with natural image input.
A) The general approach: The natural image was compressed to 100×100 pixels and transformed from a luminance image into a contrast image. Contrast values were used to define the intrinsic gamma-frequencies of the 100×100 phase-oscillator lattice on a one-to-one pixel to oscillator basis. B) Example synchrony fields (color) of two reference oscillators to all other oscillators (color) overlaid onto border segmentation of the corresponding image. One example (black dot) was located outside of the main object (top row) and the other within the object (bottom row). Phase relation maps of example oscillators are shown to the left that represent the phase relation to all oscillators with phase-locking >0.3 (see Methods). Blue (red) indicates that the oscillator leads (lags) compared to the reference oscillator. C) Segmentation-border triggered analysis. From the online image database [71] segmentation borders as indicated by 30 human subjects are available. We used these segmentation borders to analyze spatial synchronization around them (see Methods). Borders are thought to be associated with high contrast variation (and hence detuning) [112]. Top plot shows the mean absolute contrast spatial derivative (averaged over the 80 images) confirming that segmentation borders are indeed associated with higher contrast/detuning. Below the mean synchronization profile for reference oscillators a and c located 3 pixels on each side of the window’s center (i.e., the boundary location) and reference oscillator b located at the boundary location. Spatial synchronization is reduced over the segmentation border in line with the higher detuning at the segmentation borders (see S4 Fig. for more details).
Figure 8.
Arnold reconstruction and information content of the frequency, phase and combined codes for natural image input.
A) Arnold tongue reconstruction from the natural images (n = 80) by using detuning and coupling between all possible pairs of oscillators in the lattice network. We used lattice distance here as approximation for coupling (see Methods). The resulting Arnold tongue in terms of (I) phase-locking (II) phase-relation and (III) (emergent) frequency difference is shown. In (IV) a cross-section of the Arnold-tongue with overlaid representation of phase-locking (black), phase-relation (red) and intrinsic frequency (dashed) for an oscillator-pair of a lattice distance of 3 pixels (direct coupling strength = 0.62). B) Stimulus reconstruction of the natural image contrast. Intrinsic frequency of all phase oscillators determined by local contrasts of example image Sorig [71] were reconstructed based on the frequency code Sest(ω), the phase code Sest(θ) and the combined frequency-phase code Sest(ω,θ) C) The reconstruction performance, measured by mutual information (MI), was from lowest to highest MI = 0.28 for Sest(θ), MI = 0.46 for frequency code Sest(ω) and MI = 0.67 for combined code Sest(ω,θ). Error bars give ±3 SEM (n = 80).