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Intra-ripple frequency accommodation in an inhibitory network model for hippocampal ripple oscillations
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.
doi: https://doi.org/10.1371/journal.pcbi.1011886.g005