Figure 3: The temporal evolution of the peak of the potential
profiles following a step function in the initial condition.
Below is the stimulus-length relationship. This reflects the threshold
initial condition as a function of the excited region is bounded by
x = LL on the left and x = LR on the right such that for the initial
condition, U(x < LL,0) = 0; U(x > LR,0) = 0; U(LL <= x <= LR,0) = U(ic)
Figure 4: Threshold initial condition potential as a function of
the excited region
Cable Responses: Initial Condition Excitation (rectangular pulse):
gNa = 5, equivalent to fig 5-8 external stim (correction of my
error above (wrong gNa constant)
gNa = 5.0; Excitation Length = 4 (256 dx segments) Rectangular
Initial Condition
gNa = 5.0; Excitation Length = 32 (2048 dx segments) Rectangular
Initial Condition
Cable Responses: Initial Condition Excitation (sinusoidal pulse):
gNa = 5 Length = 8
gNa = 5.0 Exciation Length = 8 (512 dx units) Sinusoidal
Initial Condition
Collapse and Expansion from an initial sinusoidal initial condition
Temporal evolution of max(U(x))
Fit of the observed critical U*(x) to sech**2 and gaussian functions
Cable Responses: Initial Condition Excitation (sinusoidal pulse):
gNa = 5 Length = 16
gNa = 5.0 Exciation Length = 16 (1024 dx units) Sinusoidal
Initial Condition
Collapse and Expansion from an initial sinusoidal initial condition
Temporal evolution of max(U(x))
Fit of the observed critical U*(x) to sech**2 and gaussian functions
gNa = 5.0 Exciation Length = 32 (2048 dx units) Sinusoidal
Initial Condition
Collapse and Expansion from an initial sinusoidal initial condition
Temporal evolution of max(U(x))
Fit of the observed critical U*(x) to sech**2 and gaussian functions
Cable Responses: Initial Condition Excitation (sinusoidal pulse):
gNa = 1 (never delete mistakes)
Beware: the fast function for the sinusoidal
initial condition was u(1-u)(u-a) and thus, when comparing these results
with the external stimulation results (figs 5-8) where I used
5 u(1-u)(u-a) - the critial pulse widths are different - larger (roughly
2x )when the multiplier = gNa = 1 than when the multiplier = gNa = 5.
gNa = 1.0 Exciation Length = 8 (512 dx units) Sinusoidal
Initial Condition
Collapse and Expansion from an initial sinusoidal initial condition
Fit of the observed critical U*(x) to sech**2 and gaussian functions
Cable Responses: External Applied Rectangular Current Excitation,
gNa = 5
Instead of exciting with an initial condition, here we
inject current along a length of cable of length, L. After termination
of the stimulus current the potential profile collapses at the
boundaries with the unexcited regions and starts to increase in the
middle of the excited region. Here I refer
to the threshold of excitability (a = 0.139) as the 0D threshold and the
threshold for propagation (U = 0.21290) as the 1D threshold. Note the
similar slow velocity of evolution as the pulse profile approaches that of
what we speculate is the solution to Uxx + f(U) = 0.
The peak of the zero-velocity profile is determined by the value of U
such that the intergral of f(U) from 0 to Ucrit = 0. So we solve for
the roots of the polynomial:
U**4/4 - 4*(+a)U**3/3 + aU**2/2 = 0
which is equivalent to U**2 - 4U(1+a)/3 + 2a = 0
which has roots: 2(a+1)/3 + sqrt[(4a - 2)(a - 2)]/3 and
2(a+1)/3 - sqrt[(4a - 2)(a - 2)]/3
so when a, the threshold of cellular excitation is 0.139, the two roots
are 0.21290 and 1.30576. The root associated with the zero-velocity profile
is the smaller of the two, 0.21290, which we label as the 1D threshold.
From this relationship, we can see directly how
the media properties alter the zero-velocity profile. Our model assumes
a diffusion coefficient of 1.
This simple analysis (perhaps too simple) indicates that the
amplitude of the zero velocity
profile is dependent ONLY on the media excitability, while the morphology
of the zero velocity profile is influenced by both the threshold of excitation
as well as the diffusion coefficient.
gNa = 5.0 Excitation length = 32 (2048 dx segments)
Figure 5: The tenporal evolution of the potential profile
following excitation by an externally applied (Istim) current for a
duration of 1 time unit and a value near threshold. Note the collapsing region
at the periphery of the excited region and the increase in amplitude at the
center of the excited region.
Here is the plot of the peaks of each pulse, max(U(x,t=0,10,20,30 ...))
for a range of stimulus currents. This graph clearly illustrates the
two-threshold possibility - an initial visit to the threshold of excitation
of an individual cell (a = 0.139) then a transition to the threshold for
propagation (U = 0.21290) - the peak of the zero-velocity profile, U(x).
Figure 6: The tenporal evolution of the peak of the potential
profiles - following pulse excitation (Istim). Note that the peaks appear
to pass through two thresholds, one that coincides with the threshold
of media excitability (a = 0.139, middle root of f(U)) and one
that coincides with the peak amplitude of a zero velocity profile
of a front.
A comparison of fitting the quasi-stationary U(x) profile - a0*(1-tanh**2(x/a1)
and a0*exp(-x**2/a1**2)
Figure 7: Selecting a pulse from figure 5 that appears to be stationary
and fitting the hyperbolic and gaussian functions.
The relationship between stimulus amplitude and excitation length
Figure 8: The relationship between excitation length and
stimulation current for a trigger wave in a cable - evidence that there is
probably not a pure liminal length, but rather a region (our stationary
profile of U(x) whose width has the meaning of liminal length.
Cable Responses: External Applied Sinusodial Current Excitation,
gNa = 5
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Copyright C. Frank Starmer 2002
