Transient bimodality in the membrane potential distribution can affect instantaneous ripple frequency dynamics in the spiking network.

(A) Same layout as in Fig 2A: Spiking network response to an isolated downwards ramp stimulus with the same slope as in Fig 2D, middle, after time t > 10 ms (N = 10, 000). Note that units that participate in the third population spike tend not to spike in the fourth population spike (red lines in raster plot), which is an indication of a residual bimodality in the membrane potential distribution from one cycle to the next (only faintly visible in voltage plot, see red square). (B) Same layout as Fig 2B: instantaneous network frequencies pooled together from 50 such ramp-down-only simulations. What appears as a continuous non-monotonic “wiggle” in the instantaneous frequencies of Fig 2D, middle (gray dots) is now clearly identifiable as a single outlier cycle (marked in red). (TIF)</p

Illustration of analytical approximation of DDE dynamics.

(A) One cycle of the oscillatory solution of the DDE for IE = 3.6. Dotted lines for rate (top) and mean membrane potential (bottom) are the result of a numerical integration of the DDE (Eq (31)). All other lines illustrate our analytical considerations. Top: population rate r(t) (black). Bottom: mean membrane potential μ(t) (full grey line: Eq (A2) during upstroke, Eq (39) during downstroke, gray area: indicating the width of the Gaussian density p(V, t)). Constant external drive IE (green line), total input I(t) = IE − II(t) (dashed line, Eq (42)). Local extrema of the mean membrane potential occur at the intersections of μ and I: Cyan: local maximum μmax (Eq (38)). Orange: approximate local minimum μmin (Eq (43)). Vertical dotted lines mark end of population spike toff as well as intervals toff + kΔ, k ∈ {−2, −1, 1}. Arrows illustrate simplifying assumption (A1). The beginning of the cycle (ton = 0) is determined by μmin. Horizontal gray bars mark the length of one cycle (here T = toff + Δ = 3.44 ms (Eq (47)), corresponding to a network frequency of fnet = 290.7 Hz, (Eq (32))). Inset: magnification highlighting the differences between numerical solution (dotted) and analytical approximation (full line). Due to assumption (A2) μmax is slightly overestimated. Note that the second intersection of μ and I occurs shortly before time toff + Δ. Hence μmin is slightly larger than the true local minimum. (B) Same as A, but with an account for the reset on the population level. At the end of the population spike, μ is reset instantaneously from μmax to μreset (Eq (49)) (yellow marker). This leads to a lower μmin (Eq (50)) and hence a slightly longer period (T = 4.24 ms), i.e. lower network frequency (fnet = 235.8 Hz). (C) Illustration of phenomenological reset. Cyan: density of membrane potentials p(V, toff) at the end of the population spike, centered at μmax (before reset). Cyan hatched area: fraction of active units (saturation, Eq (48)). Grey hatched area: resetting the active portion of p. Yellow: p(V, toff) after reset, centered at μreset. The reset value μreset is calculated as the average of the density that results from summing the grey-dashed (active units) and cyan-non-hatched (silent units) density portions (Eq (49)). Default parameters (see Table 2).</p

Analytical approximation of the oscillation dynamics for constant drive.

Comparison of dynamics in theory (full lines) and spiking network simulation (dashed lines, N = 10, 000). Top: Network frequency (black triangles) and mean unit frequency (blue circles). Red markers: Hopf bifurcation. Vertical lines indicate the range for which the theory applies (see Methods, Eq (52)). Middle: Saturation s increases monotonically with the drive (Eq (48)). Bottom: characterization of the underlying mean membrane potential dynamics via local maximum μmax (cyan, Eq (38)), local minimum μmin (orange, Eq (50)) and population reset μreset (yellow, Eq (49)). Default parameters (see Methods).</p

Analytical vs numerical evaluation of oscillatory solutions in the Gaussian-drift approximation.

Network frequency (top) and dynamics of the mean membrane potential (bottom) quantified in terms of its periodic local minimum μmin (orange) and local maximum μmax (cyan) for different levels of external drive IE. The analytical approximations (solid lines: with reset, dashed lines: without reset) are very close to the results of numerical integration of the DDE Eq (31) (round markers: with reset, square markers: without reset). Including the reset does not affect μmax but leads to a decrease in μmin (Eqs (50) vs (43)) and thus a decrease in network frequency. Results are shown in the relevant range of external drives (vertical dotted lines). For parameters see Table 2.</p

IFA in theory and simulation.

Quantification of the IFA slope χIFA in the spiking network simulations and the theoretical approximations shown in Fig 7Aii–7Cii for different slopes m of the external SPW-like drive. The error ϵ (Eq (67)) quantifies the average relative deviation of the theoretically predicted instantaneous network frequencies (colored markers in Fig 7) from the simulation results (grey dots in Fig 7).</p

Serial correlation coefficient at lag 1 as a function of the time scale separation.

<p><b>A</b> Serial correlation coefficient in the case of channels (corresponding to ) for stochastic adaptation (circles, Eq. (3)) and for deterministic adaptation (squares, Eq. (2)). Theoretical curves for stochastic adaptation, Eq. (72), and deterministic adaptation, Eq. (9), are displayed by a dashed line and a solid line, respectively. The zero baseline is indicated by a dotted line. <b>B</b> shows the corresponding plot for channels (corresponding to ) and . The gray-shaded region marks the relevant range for spike-frequency adaptation. was varied by changing at fixed (), all other parameters as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1001026#pcbi-1001026-g002" target="_blank">Fig. 2</a>.</p

The influence of network architecture and the shape of the external drive on the asymptotic and instantaneous ripple oscillation dynamics.

A covariation of network architecture and stimulus profile demonstrates that IFA is modulated by, but occurs largely independent of, the shape of the asymptotic network frequency as a function of the external drive (Fig A). Furthermore, we illustrate that a simple square pulse cannot elicit IFA in the feedback-based inhibition-first model. (PDF)</p

Transient dynamics and IFA for piecewise constant external drive.

(Ai, Aii) Dynamics under constant drive depending on the initial mean membrane potential. Top: Population rate. Bottom: Constant external drive (green) and mean membrane potential (gray) with initial value μmin (orange marker). Dotted horizontal lines mark spike threshold VT = 1 and reset potential VR = 0. (Aiii) Direct comparison of the first oscillation cycles in Ai/Aii (gray shaded area) with the asymptotic cycle dynamics. Orange and cyan horizontal lines mark the asymptotic values for and , respectively. Left: shorter cycle for . Middle: asymptotic period for . Right: longer cycle for . The color of the population rate curve (left, right) expresses the difference in cycle length as a difference in instantaneous frequency (colorbar in B). (B) Difference between the instantaneous frequency of a cycle with constant drive IE and initial condition μmin, and the asymptotic frequency for a range of external drives IE and initial mean membrane potentials μmin. Black line: asymptotic (cf. Fig 4, bottom, orange line). Markers indicate example cycles shown in C. Arrows indicate convergence to the asymptotic dynamics after one cycle. If the drive changes after each cycle (dotted lines), the seven examples lead to the trajectory shown in C. Cycles 2 and 6 are also shown in Aiii (left, right), together with their common asymptotic reference dynamics. ȈFA for piecewise constant drive with symmetric step heights. Shaded areas mark oscillation cycles. Bottom: The external drive is increased step-wise, up to the point of full synchrony (green staircase). As in A, lines in all panels indicate the asymptotic dynamics associated to the external drive of the respective cycle. Circular markers indicate transient behavior. Cyan: μmax. Orange: μmin. Reset not marked to enhance readability. Gray line: trajectory of the mean membrane potential. Middle: Population rate. Top: the instantaneous network frequency (markers) is first above and then below the respective asymptotic network frequencies (black line). Same colorbar as B. All quantities are derived analytically from the Gaussian-drift approximation. Vertical axis labeled “voltage, drive” in panels A and C applies to membrane potential and external drive.</p

Constant-drive dynamics of the spiking network.

(A) Four dynamical regimes depending on the external drive: asynchronous irregular state, sparse synchrony, full synchrony, multiple spikes (left to right, N = 10, 000). Top: population rate rN, middle: raster plot showing spikes of 30 example units, and histogram of membrane potentials v (normalized as density, threshold and reset marked by horizontal dashed lines), bottom: power spectral density of the population rate. (B) Top: network frequency fnet (black) and mean unit firing rate funit (blue) for a range of constant external drives Iext. Grey band marks approximate ripple frequency range (140–220 Hz). Red markers indicate the critical input level (Hopf bifurcation) and the associated network and unit frequency, as resulting from linear stability analysis, see section Linear Stability Analysis in Methods, Eq (70), and [34]. Linestyle indicates network size (N ∈ [102, 103, 104]). Middle: saturation s = funit/fnet. Bottom: coefficient of variation of interspike intervals, averaged across units.</p

Transient dynamics and IFA for piecewise linear external drive.

(A) Exemplary transient dynamics during rising vs falling phase of the external drive (“up” vs “down”), given fixed (green dot). Bottom: external drive (green lines), and trajectories of mean membrane potential (gray lines) depending on initial conditions (orange dots). Dotted horizontal lines mark reference drive , spike threshold VT = 1, and reset potential VR = 0. Top: Population rate. Color (colorbar in B) indicates the difference in cycle length (shaded area) compared to the asymptotic reference (middle panel). Left: shorter cycle for m > 0 and initial (orange dot vs orange line). Middle: asymptotic period for constant drive () and initial . Right: longer cycle for m μmin. Left: linearly increasing drive with slope m = +0.4/ms. Right: linearly decreasing drive with slope m = −0.4/ms. Black line: asymptotic for constant drive. White line: initial membrane potential μmin for which . Stars mark the examples shown in A for . Round markers and arrows indicate the trajectory shown in C for piecewise linear drive, numbered by cycle. White space where either: no asymptotic oscillations occur (), or (bottom left): no transient solution exists (see Methods, Eq (64)). (C) IFA for symmetric, piecewise linear (SPW-like) drive. Shaded areas mark oscillation cycles. Bottom: The external drive is increased up to the point of full synchrony (green line). Colored lines indicate asymptotic dynamics. Gray line: mean membrane potential trajectory μ(t). Markers quantify transient behavior. Cyan: μmax. Orange: μmin. Reset not marked to enhance readability. Middle: Population rate. Top: the instantaneous network frequency (markers) is first above, then below the resp. asymptotic network frequencies (black line). Same colorbar as B. Dashed lines: plateau phase of variable length with , during which the network settles into the asymptotic dynamics. The IFA slope χIFA was derived for an assumed plateau length of 20 ms (as in Fig 2). All quantities are derived analytically. Vertical axis labeled “voltage, drive” in panels A and C applies to membrane potential and external drive.</p