On Tue, 23 Dec 2003 11:19:04 +0000, Xiaoshen Li <xli6 at gmu.edu> wrote:
>>>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
I apologize for getting carried away with the abstract math. You can
really understand what is going on during an action potential without
all that. If you do want to go into biophysics, though, you will need
the "higher" mathematics.
The "singular point", loosely speaking, is a point where a function
doesn't behave properly, although it is "nice" all around it. More
simply, it is a more formal way of getting at the idea of the
threshold in abstract terms. The "electrotonic" potential is one of
the names for the subthreshold behavior of the membrane. So below
threshold, you get one kind of response, a graded (or electrotonic)
response where the response amplitude varies smoothly (continuously)
with the stimulus amplitude. Above threshold, you get an entirely
different response, the action potential. The threshold is the cutoff
point.
The important idea for physiology, though, is what this means in terms
of the ion channels. Above threshold, you have the positive feedback
loop (usually called the Hodgkin cycle) where depolarization leads to
sodium activation and inward sodium current which leads to more
depolarization which leads to more sodium activation ... etc. Once
this cycle catches, it just builds up stronger and stronger until the
sodium channels are completely activated or until sodium inactivation
starts to kick in. Below threshold, either the sodium channels are
not really activated or the inward current is insufficient to overcome
the spread of current along the axon and the outward passive potassium
and leakage current. As a result, you don't get positive feedback.
All of that you can understand without going into the details of the
mathematics.