Figure 1.
Simulated traces from the FHN (black) and AFHN (red) models for varying HF stimulus frequency
. Parameters: Radial frequency
is identified in each panel,
,
.
Figure 2.
Repetitive firing in the AFHN model.
(A) Simulated traces for
and different values for
. The dashed lines indicate
. (B) Phase-plane portrait for variable
and
. In each panel, the
-nullcline (green) is shown for 3 values of
. The
-nullcline (blue) is independent of
and
. (C)
-
parameter space, denoting regions of rest, repetitive firing, and block. The limit cycle lower and upper limits (
, Eq. 6 ) and rheobase (
, Eq. 9 ) as functions of
. (D) Frequency and amplitude of action potentials, as functions of
and
.
Figure 3.
Steady-state of the AFHN model.
The steady-state transmembrane potential and degree of refractoriness
are shown as functions of the HF stimulation parameter
. Critical values of
for repetitive firing
and for evoking a single action potential following a brief applied current pulse
are identified. See text for description of critical values.
Figure 4.
Excitability in the AFHN model.
(A) Simulated traces during brief
stimuli pulses of amplitude
for
, 1, and 1.5. In simulations that
exceeds the threshold
, an action potential is elicited. Inset shows an expanded time course. (B) Strength-duration curve ( Eq. 8 ) for several values of
. (C) Rheobase (
, Eq. 9 ) and chronaxie (
, Eq. 10 ) as functions of
.
Figure 5.
Simulated traces from the HH (black) and AHH (red) models for varying HF stimulus frequency
. Parameters:
rad/s (where
is the frequency identified in each panel),
= 30
,
.
Figure 6.
Repetitive firing in the AHH model.
(A) Simulated traces for
and different values for
. The dashed lines indicate
mV. (B)
-
parameter space, denoting regions of rest, repetitive firing, and block. The limit cycle lower and upper limits (
) and rheobase (
) as functions of
. (C) Action potential frequency and amplitude, as functions of
and
.
in units of
.
Figure 7.
Excitability in the AHH model.
(A) traces following brief (0.1 ms) cathodal and (B) anodal stimulus pulses, for different values of
. Threshold
indicated in each panel. (C) Cathodal and anodal strength-duration curves for different values of
. (D) Cathodal and anodal threshold
(for 0.1 ms stimuli), rheobase, and chronaxie, as functions of
. Current pulse amplitudes in (A): 64-66 (left); 633-635 (middle); 600, 800, 1000 (right); in (B): 198-200 (left); 397-399 (middle); 400, 600, 800 (right); in
.
Figure 8.
Steady-state of the AHH model.
(A) Steady-state gating variables ,
,
and time constants
,
,
as functions of
in the AHH neuron model for different
values. (B) Steady-state values for the transmembrane potential
and the gating variables (left), and gating variable time constants at
(right), as functions of
. Vertical dashed lines indicate
,
, and
(see text for description). In the top panels, the horizontal dashed line indicates
mV for
. Time constants in units of ms, and
in units of
.
Figure 9.
Electrical activity in a network of AHH model neurons.
(A) Rastergram of action potentials and the pseudo-electroencephalogram (pEEG). Arrows indicated quiescent neurons. Parameters: ,
. (B) pEEG (left, Eq. 32 ), the corresponding power spectrum (middle, value indicates the average neuron firing rate), and synchrony measure
(right, Eq. 33 ), as functions of
.
in units of
.
Figure 10.
HF stimulation of a network of AHH model neurons.
The critical value for repetitive activity in a AHH model neural network, as a function of the probability of excitatory synaptic connections
. The mean
(thick black) over
simulations,
1 (solid black) and 2 (thin black) SEM (standard error of the mean, standard deviation of
) are shown. Single cell critical values of
(
and
) are identified (see text for description).
in units of
.
Figure 11.
Period of a stable limit cycle in the AFHN model.
For , the period
of the stable limit cycle is approximately the sum of the time required to traverse the two stable branches of the
-nullcline,
, denoted by the points
and
, respectively. As
increases, the amplitude and period of the stable limit cycle, indicated by the second set of points labeled
, both decrease. See text for description of other variables in figure. Parameters:
.
Figure 12.
Sub- and super-threshold brief stimuli in the AFHN model.
(Left) For ,
indicates the threshold for evoking an action potential. Two trajectories starting near
are identified by arrows: (1, left arrow) when the initial condition
,
returns quickly to the resting potential
; (2, right arrow) when
, an action potential is evoked: the system follows a counterclockwise trajectory, quickly approaching the
-nullcline, following the right stable branch until reaching the right knee, quickly reaching the left stable branch, and then returning to rest. (Right) When
,
, the critical value above which an action potential cannot be evoked by a brief perturbation from the steady-state.
Table 1.
Hodgkin-Huxley model current parameters.
Table 2.
Synaptic current parameters.