IUBio

question related to Hodgkin-Huxley model

Xiaoshen Li xli6 at gmu.edu
Tue Dec 23 06:19:04 EST 2003



r norman wrote:
> On Tue, 23 Dec 2003 08:04:55 GMT, "k p  Collins"
> <kpaulc@[----------]earthlink.net> wrote:
> 
> 
>>"r norman" <rsn_ at _comcast.net> wrote in message
>>news:4bveuvkjbubflf7dqmmil11sp6dvrp4rst at 4ax.com...
>>
>>>>[...]
>>>
>>>[...]
>>
>>>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.
> 
> 
> 
>  
> 
Dear Dr. Norman:

Thank you so much for your reply. I found that your understanding with 
Action Potential is very deep. I cannot understand singular point and 
the difference between action potential and electrotonic potentials. Do 
you mind if I ask that how I can catch up your comment? I have a very 
weak knowledge of differential equations. Should I work on it so I can 
understand the trajectory in the phase space? Thank you very much.

Best Regards
Xiaoshen




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