Table 1.
Parameter descriptions and canonical values used throughout this study, unless otherwise stated.
Figure 1.
Linear analysis of the neuron models.
Linear analysis of the model neurons with fixed external conductances. A: Resting potential as a function of the background inhibition-to-excitation ratio k for the canonical value of (solid lines), and as a function of the background synaptic excitation
for the canonical value of
(dashed lines). Solid lines refer to the bottom axis; dashed lines to the top axis. Circles show the canonical values of the corresponding parameters. B: Effective membrane time constant as a function of k and
. Line styles, colors and symbols as in A. C: Effective intrinsic frequency as a function of k and
. Line styles, colors and symbols as in A. D: IPSPs in response to a single presynaptic pulse delivered at time t = 0, in the presence (dashed lines) or absence (solid lines) of fixed background conductances. Inset shows the membrane potential response to an instantaneous perturbation.
Figure 2.
Sample activity in the canonical GIF and IF networks.
A: Population rate (top), spike raster plot (middle), and v trajectories of selected neurons (bottom) in the GIF network for a representative parameter set. C: Single-cell ISI histogram for the GIF network. The red arrow indicates the period of network oscillations. The black arrow indicates the mean single-cell ISI. B, D: The same as A and C, for the IF network. Oscillations are more prominent in the GIF network.
Table 2.
Single-neuron and network statistics for the IF and GIF models considered.
Figure 3.
Power spectrum of population activity.
Power spectrum of the population rate for the four model networks considered. Population activity exhibits a peak at ∼100 Hz, with higher synchrony corresponding with slightly higher population frequency. Note the tall, non-Gaussian peaks in the GIF networks, corresponding to non-linear, fully developed oscillations. In contrast, power spectrum peaks in the IF networks are bell-shaped, a signature of sinusoidal oscillations.
Figure 4.
Correlation between synaptic inputs, membrane potential trajectories and RMPC among pairs of neurons as a function of distance.
A: Average Pearson correlation between incoming synaptic currents to neuron pairs as a function of their distance. Note the high correlation for nearby neurons, which decreases with distance. B: Average Pearson correlation between the membrane potential of neuron pairs as a function of their distance. The modulation with distance is negligible. C: between firing patterns of neuron pairs as a function of their distance.
decreases at short distances due to short-latency mutual inhibition. Line styles and colors as in Figure 3.
Figure 5.
Effects of membrane potential depolarization and coupling on network dynamics.
A: Membrane potential distributions of GIF and r-matched IF neurons in response to the background input alone, for different values of the background inhibition-to-excitation ratio k. In all cases, the voltage threshold for spike generation has been adjusted in order to match the rate response of the canonical GIF (73.7 Hz). Color and line style code as indicated. Inset shows the mean membrane potential as a function of k for the four model networks shown in Figure 3 (same line colors and styles). B, D: Synchrony (as assessed by
, B) and firing rates (D) as a function of coupling strength
. C: Enlarged view of B for low values of coupling strength. E: Ratio between corresponding curves in C. Solid lines show ratios between canonical models (
(GIF)/
(IF)), while dashed lines show ratios between models that have been adjusted in order to exhibit the same rate response to the background input (purple, blue and light blue,
(GIF)/
(r-matched IF); brown, red and orange,
(r-matched GIF)/
(IF)). Dots in B–E indicate simulated points, lines are drawn to guide the eye.
Figure 6.
Intrinsic and synaptic currents.
Population rate (A), along with the mean membrane potential v (B), the mean synaptic current
(C), and the mean intrinsic current
(D) across neurons for the GIF network in a short representative time window. Blue lines show simulation results, black lines are least-squares sinusoidal fits. The red vertical lines indicate the peak of the sinusoidal fit to the corresponding traces, the green vertical line indicates the trough of the sinusoidal fit to the mean membrane potential. Phase leads (
, in radians) with respect to the mean membrane potential oscillation are shown for the mean synaptic current
and the mean intrinsic current
.
Figure 7.
Distribution of intrinsic currents conditioned on the phase of the population rhythm.
Probability density functions of intrinsic currents in the GIF (blue) and IF (red) networks. Solid lines indicate unconditional probability densities, dashed (dotted) lines indicate probability densities conditioned on the peak (trough) of the mean
oscillation.
Figure 8.
Deviation from independence of intrinsic currents for adjacent neurons.
A: Deviation from independence of intrinsic currents
flowing through the membrane of pairs
of adjacent neurons in the GIF network. Shades of red (blue) indicate
values that occur more (less) often than what expected under the assumption of independence. C, E: The same as A, but the probability density functions
and
are conditioned on the peak (C) or trough (E) of the mean
oscillation. B, D and F: The same as A, C and E, for the IF network.
Figure 9.
Phase relationships of synaptic and intrinsic currents and their effect on synchrony
A: Phase of the sinusoidal fit to the mean synaptic current plotted against the local level of synchrony (as assessed by the amplitude of the sinusoidal fit to the mean membrane potential
). GIF: blue; IF: red. Crosses: canonical models; circles: r-matched models. B: As in A, for the mean intrinsic current
. Only GIF networks are shown, as
is always equal to
in IF networks. C: As in A, for
.
Table 3.
Circular-linear correlation analysis corresponding to the data plotted in Figure 9.
Figure 10.
Mean and standard deviation of the membrane potential across neurons.
A: Covariation of the mean membrane potential and the mean intrinsic current across neurons in the GIF (blue) and IF (red) networks. B, C: Bivariate probability density function of the mean and standard deviation of the membrane potential variable across neurons for the GIF (B) and IF (C) networks. Brighter colors indicate higher probabilities.
Figure 11.
Neuronal dynamics if inhibition is shunting, rather than hyperpolarizing.
A: Resting potential as a function of the background inhibition-to-excitation ratio k for the canonical value of (solid lines), and as a function of the background synaptic excitation
for the canonical value of
(dashed lines). Solid lines refer to the bottom axis; dashed lines to the top axis. Circles show the canonical values of the corresponding parameters. All parameters as in Table 1, except
= 4 mV. Compare with analogous results for hyperpolarizing inhibition shown in Figure 1A. B: Synchrony (as assessed by
) as a function of the background inhibition-to-excitation ratio
for different values of the coupling strength
. Voltage threshold for spike generation
has been increased to 15 mV in the canonical models, in order to compensate for the depolarization of the resting potential and to keep the models in the fluctuation-driven regime. Voltage thresholds for r-matched models have been scaled accordingly for each value of
. GIF: purple, weak coupling (
); blue, medium coupling (
); light blue, strong coupling (
). IF: brown, weak coupling; red, medium coupling; orange, strong coupling. All other parameters as in Table 1, except
= 4 mV,
= 4 mV. Dots indicate simulated points, lines are drawn to guide the eye.