Figure 1.
Starburst amacrine cells obey Morris-Lecar dynamics with voltage term , refractory variable
, sAHP variable
and acetylcholine concentration
. A. Time course of individual SAC dynamics following current injections of indicated size, injected at
for 1.5 s (shown in grey). Model SACs' refractory period shows dependence on amount of current injected. B. Single cell dynamics showing sparse spontaneous depolarizations. Different colors represent different SACs. C. Distribution of inter-event intervals in stochastic simulation of isolated SAC. A threshold of −50 mV was employed. Bar plot shows mean and standard deviation.
Figure 2.
Model produces realistic cholinergic waves.
A. Network dynamics showing spatiotemporal patterns of retinal waves B. Distribution of wave sizes, speeds, durations and inter-wave intervals from 2500 s of simulation. Mean wave size is 0.017 mm(
0.059 mm
), mean wave speed is 0.11 mm/s (
0.022 mm/s), mean wave duration is 0.63 s (
0.90 s), and mean inter-wave interval is 49 s (
25 s). C. SACs exhibit variable participation in waves. Pearson correlation coefficient between a cell in the center of the domain and all other cells. The correlation coefficient for each variable is plotted as a function of euclidean distance between cells. Computed using one 2500 s simulation, with activity recorded every 0.1 s. Solid curve represents a loess moving average estimate of mean correlation as a function of distance. Shaded region highlights all points within one standard deviation of this mean curve.
Figure 3.
Construction of traveling wave-front.
A. Fast-slow dynamics in the canonical Fitzhugh-Nagumo model of action potential generation. Black curve represents a trajectory of an action potential through phase space, in which a fast transition occurs between the rest (blue dot) and excited state (green dot), followed by slow excited dynamics (green to purple dot), another fast transition between the excited and refractory state (purple to yellow), and slow dynamics while refractory (yellow to black). Red arrows represent flow lines, and the blue curve is the nullcline which defines the slow manifold (
nullcline not drawn for clarity). B. The fast system here is described by three dynamical variables (
,
, and
). Shown here is the trajectory connecting the rest (blue) and excited (green) fixed points, defining the wavefront. C. Temporal voltage dynamics of the wave front.
Figure 4.
Parameter regimes which produce propagating activity.
Numerical determination of retinal wave excitability threshold for different timescales and excitability threshold determined through singular perturbation analysis, both as functions of
,
. Each point on each curve indicates a point in parameter space in which the wavefront transitions from propagating to receding. Points in parameter space below each curve are therefore not excitable, while those above are excitable.
Figure 5.
Modeling biophysical manipulations.
A. Synaptic connection strength is varied. Sub plots from left to right: speed of wave front
at rest (when
) as a function of conductance
, velocity indicates maximum wave-front speed since
and
is monotonically decreasing, point at which
becomes zero represents excitability threshold; wave-front speed as function of refractory variable
for three different values of
; from 5000 s of simulation of model with indicated values of
interwave-inteval; wave speed distribution; and mean wave size. B. Sub plots from left to right: dynamics of refractory variable
of individual SAC following depolarization with different sAHP timescales
, black line indicates refractory value above which
and thus represents an absolute refractory time period in which SAC is not sufficiently excitable to participate in future wave activity; from 5000 s of simulation of model with indicated values of
inter-wave interval; wave speed distribution; and mean wave size.
Figure 6.
Power-law distributed wave-size retinal waves.
A. Parameter space in which avalanches are expected (gray, Equation 3) and three sample points B. Wave size distributions (points) following 5000 s of simulation on a 128128 domain for specified values of
and
. Solid lines represent log-linear least-squares lines of best fit, having slopes:
(
, green),
(
, red) and
(
, blue) C. Correlation in membrane potential between cells of a given distance apart.
Table 1.
Parameters for retinal waves model.
Table 2.
Dimensionless parameters for retinal waves model.