Fig 1.
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.
Fig 2.
Transient dynamics of the spiking network and IFA.
(A) Example simulation showing a transient ripple with IFA. From bottom to top: SPW-like drive (Eq (17)); histogram of membrane potentials (normalized), horizontal dotted lines: threshold and reset potential; raster plot showing spike times of 10 example units; population rate exhibiting transient ripple oscillation; wavelet spectrogram indicating instantaneous power (blue-yellow colorbar) for a frequency range of 0–400 Hz. Solid curve: continuous estimate of instantaneous frequency based on wavelet spectrogram with gray scale indicating maximal instantaneous power. Dotted line: cutoff frequency for peak detection fmin = 70Hz. Red lines in scalebars: power threshold (see Methods). White dots: discrete estimate of instantaneous frequency based on peak-to-peak distance in population rate. Network size N = 10, 000. (B) Quantification of IFA. Top: Grey dots: discrete instantaneous frequency estimates from 50 repetitions of the simulation shown in (A) with different noise realizations. Grey line: linear regression line with negative slope (χIFA = −3.04 Hz/ms) indicating IFA (see Methods, Eq (18)). Black line: asymptotic frequencies (cf. Fig 1B, top). Bottom: The same SPW-like drive was applied in all 50 simulations. Network size N = 10, 000. (C) Dependency of IFA slope χIFA on the slope of the external drive for different network sizes (color coded). (D) Instantaneous (dots) vs asymptotic (black lines) network frequencies (top) for piecewise linear drives (bottom) of decreasing slopes (left to right). Color/linestyle indicates network size N. Thin, colored linear regression lines illustrate decreasing strength of IFA for shallower drive (regression slopes summarized in C). Asymptotic network frequencies are derived via interpolation of the constant-drive results shown in Fig 1B, top. The asymptotic frequencies for N = 103 and N = 104 are nearly identical (dash-dotted and solid lines). Note that the drive is identical in panels A, B, and in the left panel of D.
Fig 3.
Illustration of the Gaussian-drift approximation.
Comparison of oscillation dynamics in the spiking network simulation (A) and the Gaussian-drift approximation (B) at constant drive (IE = 4.24). (Ai) Spiking network simulation. Empirical population rate rN (top), and density of membrane potentials (bottom, dimensionless voltage), exhibiting coherent stochastic oscillations with (weak) finite size fluctuations. (Aii) The average cycle of the oscillation dynamics in Ai (computed for 21 bins of 0.24 ms each). Top: population rate; middle: density of membrane potentials, orange marker: local minimum of mean membrane potential; bottom: standard deviation of membrane potential distribution, dashed line: theoretical asymptote in the absence of boundary conditions (Eq (26)). (Aiii) Snapshots of the membrane potential density over the course of the average cycle shown in (Aii). (Bi) Gaussian-drift approximation (numerical integration of DDE (8) with reset condition). Top: population rate, bottom: mean membrane potential μ(t) (black line) with Gaussian density of membrane potentials p(V, t) painted on in the background for better comparison with Ai. (Bii) Zoom into one oscillation cycle (black bar in Bi). Top: population rate r, middle: density of membrane potentials p(V, t) with mean μ(t), bottom: constant standard deviation
. The mean membrane potential μ(t) (black line) starts in each cycle at μmin (orange) and rises up until μmax (cyan) at time toff, at which point the population spike ends. In a phenomenological account for the single unit reset, μ is reset instantaneously to μreset (yellow). From there μ declines back towards μmin. (Biii) Snapshots of the membrane potential density p(V, t) over the course of one cycle (Bii). Colors mark t = 0 ∼ T (orange), and t = toff (cyan/yellow). Dotted lines in all voltage panels mark spike threshold VT = 1 and reset potential VR = 0. Note that in the theoretical approximation the spike threshold VT = 1 is no longer an absorbing boundary.
Fig 4.
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).
Fig 5.
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.
Fig 6.
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 < 0 and initial
. (B) Difference between instantaneous and asymptotic frequency for a range of reference drives
and initial mean membrane potentials 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.
Fig 7.
Transient dynamics for SPW-like drive with slope m = ±0.4/ms (A, cf. Fig 6), m = ±0.2/ms (B), or m = ±0.1/ms (C). Top panels (i): difference between instantaneous and asymptotic network frequency for the possible combinations of external reference drive and initial mean membrane potential μmin, shown separately for positive (top) and negative (bottom) slope of the drive. Trajectories shown in (ii) are overlaid. Bottom panels (ii): Example trajectories through the space shown in (i) for consecutive cycle under SPW-like drive with slope ±m. Top: instantaneous (colored markers) vs asymptotic (black solid line) network frequency as predicted by the theory. Grey dots indicate instantaneous frequencies in spiking network simulations (cf. Fig 2D, N = 10, 000). Black dashed line illustrates asymptotic network frequencies from spiking network simulations (cf. Fig 1B, N = 10, 000). See Table 1 for a quantitative comparison of simulation and theory. Middle: population rate as predicted by the Gaussian-drift approximation. Bottom: instantaneous vs asymptotic μmin (orange) and μmax (cyan). Gray line: trajectory of mean membrane potential μ(t). Green lines show SPW-like external drive (Eq (17)), green dots mark reference drive
of each cycle. The difference between instantaneous and asymptotic network frequencies (IFA) becomes less pronounced for smaller slope (C vs A, see also Table 1).
Table 1.
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).
Table 2.
Default parameters for the spiking network.
Fig 8.
Dynamical states of the DDE system.
Numerical integration of the DDE system (without phenomenological reset) demonstrating 4 distinct dynamical regimes for increasing external drive IE (bottom axis). Blue: Stable fixed point with zero rate. Red: Pathological, fast oscillations. Yellow: Period-2 oscillations. Green: Regular period-1 oscillations. Top: Numerically integrated trajectories of population rate r and mean membrane potential μ over time. Note the changes in scale for the population rate. Bottom: Phase space showing the trajectory (μ(t), r(t)) and the fixed point (IE, 0), which is only stable in the first case (black circle) and unstable otherwise (empty circle). The horizontal colored bar shows the full range of (relevant) drives from 0 to the point of full synchrony . The boundaries between the four dynamical regimes were approximated numerically. Black triangles mark the levels, for which the above example dynamics are shown.Extra ticks indicate: critical drive
, for which the spiking network undergoes a Hopf bifurcation (see section Linear Stability Analysis); lower bound
introduced for our theory to ensure that we only consider non-pathological dynamics (see paragraph Range of applicability).
Fig 9.
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).
Fig 10.
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.