Figure 1.
Ai: The oscillators with voltage trajectories and
and phase difference
determine the membrane potential at the ends of a cable with electrotonic length
. Aii: The interaction function
gives the phase shift of oscillator A as a function of
. This interaction function is shifted along the
-axis by the parameters
and
, which capture the oscillator and cable properties, respectively. Aiii: The stable phase-locked solution is determined by
and
and is either at
(e.g. for the solid curve) or at
(e.g. for the dash-dotted curve). Aiv: The stable phase-locked solution as a function of
. The value of
uniquely determines where the in-phase (black solid line) or the anti-phase solution (red dotted line) is stable, given a fixed value of
. B:
as a function of the electrotonic distance
between the oscillators,
ms and
ms (dotted line in panel D). For illustrative purposes we chose
so that the stable in-phase and anti-phase solutions are given by the white and gray areas, respectively. C:
as a function of the membrane resistance
for cable diameter
m, distance between the oscillators 1000
m, membrane capacitance
F/cm
, intracellular resistivity
k
cm and oscillator period
ms. D:
as a function of the oscillator frequency
. The distance between the oscillators is
(dotted line in B),
ms.
Figure 2.
A: Parameter as a function of the electrotonic distance
between the oscillators when the cable is passive (black) or with a regenerative (green) or a restorative (red) active current. The oscillator frequency is 8 Hz (dotted line in panel B). The membrane time constant of the connecting dendrite is
ms. The parameters for the active currents were determined for
(restorative) and
(regenerative) which are described in the Methods (see Equation 29). The current parameters when linearized around
mV are
,
and
ms for the regenerative current, and
,
and
ms for the restorative current, using the conductance densities given in the Methods. B:
as a function of the frequency of the oscillator (in Hz). The oscillators are separated by a cable with electrotonic length
(dotted line in panel A) for the same three conditions as in panel A.
Figure 3.
skewness controls phase-locking regimes and transitions.
The three panels A-B-C show triangular functions with different skewness with their peaks at
where
is a phase shift that results from the cable coupling. The oscillators are identical so that
. A: Right-skewed
with
(solid black line) plotted from left to right for three values of
together with the corresponding
(dashed blue line). Below each graph
is plotted (green lines) with the stable (black dots) and unstable (red dots) phase-locked solutions. The lower right panel shows the bifurcation diagram with the stable (solid black line) and unstable (dotted red line) phase-locked solutions. The right-skewed
yields gradual transitions between the in-phase and anti-phase solutions. B: Symmetrical
with
yields abrupt transitions between in-phase and anti-phase solutions. C: Left-skewed
with
yields bistable regions where both the in-phase and the anti-phase solution are stable.
Figure 4.
Phase-locking of two Morris-Lecar type II oscillators.
The oscillators (described in Methods) are coupled via a passive cable of electrotonic length ,
ms. A: Voltage trajectory (blue) and phase response function (black) of the Morris-Lecar type II oscillator, period
ms. B: Shape of
for
(solid curve),
(dashed curve) and
(dash-dotted curve). The functions have been rescaled and aligned in order to show the different degrees of skewness. C: Bifurcation diagram showing the stable (solid black line) and unstable (dashed red line) phase-locked solutions as a function of
. Cross marks give the stable phase difference determined with numerical simulations using
S cm
with
ms, and
mV. D: The middle two panels show simulations of the phase difference dynamics (red curves) for
(top) and
(bottom) with
S cm
. Space-time plots of the membrane potential along the dendritic cable cable are plotted for the first 200 ms (left) and for the final 200 ms (right) of the two simulations.
Figure 5.
Phase-locking behavior of subthreshold oscillators.
The oscillations are generated by interactions between and
(see Methods). A: Voltage trajectory (blue) and phase response function (black) of the oscillator. B: Corresponding bifurcation diagrams showing the stable (solid black lines) and unstable (dashed red lines) phase-locked solutions as a function of
. The bifurcation diagram is shown for a passive cable (top), a cable with a regenerative current (middle), and a cable with a restorative current (bottom). The restorative current
and regenerative current
(described in Methods) are inserted in the cable with relative densities of
and
, respectively. Linearizing these currents around
mV gives the parameters
,
and
ms for the regenerative current, and
,
and
ms for the restorative current. The membrane time constant of the connecting dendrite is
ms. Cross marks in the bifurcation diagrams give the stable phase difference determined with numerical simulations using
S cm
,
ms, and
is
mV,
mV and
mV, respectively for the three panels, so that the cable's resting potential is
mV. Note that the numerical simulations use the original (i.e. not the linearized) active currents in the connecting cable.
Figure 6.
Phase difference dynamics of three oscillators in a chain or a branched configuration.
The Morris-Lecar type II oscillators are separated by a passive cable, ms. Panels A and B show from left to right: a scheme of the model with below it the membrane potential of the oscillators at the start of the simulation; the dynamics of the phase difference
between the oscillators for
(top) and
(bottom); and the membrane potential of the oscillators at the end of the simulation. The properties of the Morris-Lecar oscillators and the dendritic cable are as in figure 4.
Figure 7.
Changing the phase-locked solution of dendritic oscillators with external input and its detection with an excitable soma.
A: Schematic drawing showing the configuration of two dendritic Morris-Lecar type II oscillators and a spike-generating soma (see Methods). All are separated by a passive cable with electrotonic length and
ms, with
S cm
. B: From top to bottom are shown the inputs to the two dendritic oscillators, the phase difference dynamics (red) and somatic firing rate (black), and the somatic membrane potential
(blue) with the spike threshold (dotted black line). Note that the spikes have been cut off in order to show the subthreshold membrane potential. C–D: Bifurcation diagrams describing the phase-locked solutions up to
seconds (C, see also figure 4C) and after
seconds (D) with dotted line at
giving the electrotonic distance between the dendritic oscillators.