question related to Hodgkin-Huxley model

r norman rsn_ at
Tue Dec 23 09:01:38 EST 2003

On Tue, 23 Dec 2003 08:04:55 GMT, "k p  Collins"
<kpaulc@[----------]> wrote:

>"r norman" <rsn_ at> wrote in message
>news:4bveuvkjbubflf7dqmmil11sp6dvrp4rst at
>> > [...]
>> [...]
>> I don't really know of any way to calculate the
>> threshold, even knowing the Hodgkin-Huxley
>> equations.  It is usually just found by trial and
>> error in a simulation or an experiment.

>And, =not= to 'criticize' but to only offer a perspective
>on stimulus-response continuity:
>It's my analysis that the ionic flow is always
>continuous. Even though the direction of the ionic
>flow changes at threshold, it's still continuous.
>To see a crude example of what I mean, fill your
>kitchen sink and take a collander [spaghetti strainer]
>and alternatingly partially submerse and lift it up.
>The flow of the water into and out of the collander
>is continuous, even though its directionality changes.
>Why this matters with respect to nervous system
>function is that the ability of a nervous system to
>calculate the g'zillions of things that it calculates in
>real 'time' derives in the inherent continuity of the
>ionic dynamics.
<snip some other discussion>

Ken, your inimitable style and unconventional train of thought makes
it rather difficult to follow some of your argument.  Still, the point
you raise about discontinuities is one that does come up often.

Put aside the "quantal" detail that the actual membrane current is
made of the discrete sudden opening and closing of a finite number of
membrane channels and assume, as in the Hodgkin-Huxley model, that ion
current is continuous.  The laws governing ion current across the
membrane as a function of m, n, h and V are continuous, the laws
governing the state of the ion channel, m, n, and h as a funtion of
alpha and beta are continuous and the laws governing the variation of
alpha and beta as a function of V are continuous.  Further, if you put
the membrane (or the equations) in a voltage clamp situation, the
calculated and the observed membrane currents do vary continuously
with voltage.

However under normal circumstances (current clamp) the simultaneous
set of differential equations produces a discontinuity.  There is a
singular point in the "phase space" that can be used to describe the
set of equations and follow the solutions.  When you stimulate the
membrane, the equations trace out a trajectory in this phase space, a
closed loop. The "action potential" is the behavior of the solution if
the trajectory encloses the singular point.  "Electrotonic potentials"
are the behavior if the trajectory does not enclose the singular
point.  There is no half-way or in-between.  The solution cannot cross
the singular point -- it is singular.  It must go around it one way or
the other.  One way is the action potential, the other way is none.
The behavior of the set of simultaneous equations shows a mathematical
discontinuity even though all the underlying processes are continuous.

The situation is much easier studied in a simpler system, the
FitzHugh-Nagumo equation, which mimics the nerve membrane in many
qualitative respects and is much studied.  You can google on
FitzHugh-Nagumo to get all the details.


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