AHP conductance descriptions
Lyle J. Borg-Graham
lyle at ai.mit.edu
Sat Jan 15 12:05:04 EST 1994
Ilya, I developed a model for I_ahp for hippocampus that is
described in:
Borg-Graham, L., {\it Modelling the Somatic Electrical Behavior of
Hippocampal Pyramidal Neurons}. MIT AI Lab Technical Report 1161,
1989 (290pp).
This can be obtained from the MIT AI Lab Publications office
(richter at ai.mit.edu).
A description of the general extended H-H channel model (including V
and Ca dependencies) that I used is found in:
Borg-Graham, L., {\it Modelling the Non-Linear Conductances of
Excitable Membranes}. Chapter in {\it Cellular Neurobiology: A
Practical Approach}, edited by J.\ Chad and H.\ Wheal, IRL Press at Oxford
University Press, 1991.
I have lifted some of the Latex material that I have on AHP and
included it below - I hope it can be of some use!
Note that the shell.1 and shell.2 [Ca] that are referred to below are
from a proposal in TR1161 that $I_C$ channels are co-localized with Ca
channels, and that $I_{AHP}$ channels are evenly distributed across
the somatic membrane. The motivation for this inhomogeneous
arrangement was to account for the apparent fast and transient
Ca-dependence of $I_C$, in contrast with the slower, more intergrative
Ca-dependence of $I_{AHP}$. Simulations with a simple 3-compartment
model of Ca accumulation (shell.1 being a fraction of the submembrane
space in the vicinity of the co-localized CA and C channels, shell.2
being the remainder of the submembrane space which in turn supplies
[Ca] for the AHP channels, and a constant [Ca] core compartment) show
that this arrangement reproduces the Ca-dependent behavior of the fAHP
(from $I_C$) and the slow AHP (from $I_{AHP}$).
The parameters listed for the Ca-dep $w$ particle result in a
(increasing/decreasing) sigmoidal dependence of the (steady-state/time
constant) on the log[Ca], with a half point at 0.004mM and (0.1/0.9)
points on the curves at about (0.002/0.009) mM. The maximum tau is
100ms. Also, the specific parameter values referenced below should be
used mainly as guidelines; they are currently under revision.
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\noindent
Summary of $I_{AHP}$, extracted from Chapter 7 of:\\
\noindent
Borg-Graham, L., {\it Modelling the Somatic Electrical
Behavior of Hippocampal Pyramidal Neurons}. MIT AI Lab Technical
Report 1161, 1989 (290pp).
.
.
.
.
In the case of $I_{C}$ and $I_{AHP}$, little voltage clamp data is
available for either their steady state or temporal properties of any
presumed activation/inactivation parameters. In addition, describing
these currents is complicated by the fact that they are presumably
mediated by intracellular $Ca^{2+}$. Little quantitative data is
available on this interaction for either current, and there is at
present no consensus among workers in this field as to the mechanisms
involved. As introduced in the previous chapter and which shall be
expanded upon later, I have made the simple assumption (like that used
by other workers, e.g. Tra-Lli-79) that $I_{C}$ and $I_{AHP}$
are dependent on a power of the concentration of $Ca^{2+}$ either
directly beneath the membrane or in a secondary ``compartment''. This
is a highly speculative model, as discussed in the previous chapter.
The parameters of this description are based primarily on heuristics,
specifically the simulation of the fAHP and the AHP that is observed
in HPC. Making the situation more difficult is the fact that there are
no protocols to date in which $I_C$ or $I_{AHP}$ are activated without
the concomitment presence of other currents, thereby inextricably
tying the behavior of any set of estimated parameters for these
currents to those of other currents.
To a first approximation, the actions of $I_C$ and $I_{AHP}$ are
independent of one another. $I_C$ is transient over a time span of a
few milliseconds during the spike, and the evidence indicates that
this a large current. On the other hand, $I_{AHP}$ activates more
slowly, is small, and may last from 0.5 to several seconds. However,
since both these currents are dependent on $Ca^{2+}$ entry, their
estimation was tied to the description of $I_{Ca}$ and the mechanisms
regulating $[Ca^{2+}]_{shell.1}$ and $[Ca^{2+}]_{shell.2}$. Therefore,
while the behavior of the $I_C$ or $I_{AHP}$ descriptions could be
evaluated independently, whenever the $Ca^{2+}$ mechanisms were
modified to alter one of the current's action, the effect of the
modification on the other current had to be checked.
\section{$Ca^{2+}$-Mediation of $K^+$ Currents by $Ca^{2+}$~-~binding Gating Particle $w$}
In order to cause $I_C$ and $I_{AHP}$ to be mediated by intracellular
$Ca^{2+}$, I incorporated a $Ca^{2+}$-binding gating particle in the
expressions for both of these currents. Several workers have
postulated mechanisms for such an interaction between intracellular
$Ca^{2+}$ and different ion channels, ranging from complex multi-state
kinetic models based on experimental data to very simple descriptions
for modelling studies (Tra-Lli-79).
In light of the paucity of quantitative data on such mechanisms in
HPC, my goals for the description of a putative, generic
$Ca^{2+}$-binding gating particle were as follows:
\begin{itemize}
\item Relationship between $Ca^{2+}$ concentration and
particle activation allowing for non-degenerate kinetics considering
the range of $Ca^{2+}$ concentrations during various cell responses.
\item Binding kinetics based on a simple but reasonable model.
\item Kinetic description that could be easily modified to yield
significantly different behavior, that is a description that could be
modified to suit a wide range of desired behaviors.
\end{itemize}
To this end the following description for a $Ca^{2+}$ -binding gating
particle, $w$, was used. Each $w$ particle can be in one of two
states, open or closed, just as the case for the Hodgkin-Huxley-like
voltage-dependent activation and inactivation gating particles. Each
$w$ particle is assumed to have $n$ $Ca^{2+}$ binding sites, all of
which must be bound in order for the particle to be in the open state.
Binding is cooperative in a sense that reflects the two states
available to a given particle, i.e. either a particle has no
$Ca^{2+}$ ions bound to it, and therefore it is in the closed state,
or all $n$ binding sites are filled, and the particle is in the open
state. The state diagram for this reaction is as follows:
$$ w_{closed} + n\, Ca^{2+}_{in}
\buildrel \alpha, \beta \over \rightleftharpoons w_{open}^*$$
\noindent
where the $*$ notation means that the particle is bound to
all $n$ (intracellular) $Ca^{2+}$ ions. $\alpha$ and $\beta$ are the
forward and backward rate constants, respectively.
This scheme results in the following differential equation for $w$,
where now $w$ is the fraction of particles in the open state, assuming
that the concentration of $Ca^{2+}$ is large enough that the reaction
does not significantly change the store of intracellular $Ca^{2+}$:
$$ { {\rm d}w \over {\rm dt}} = (\alpha (1 - w)[Ca^{2+}]_{in})^n - \beta w$$
The steady state value for $w$ ( the fraction of particles in the open
state) as a function of the intracellular $Ca^{2+}$ concentration is
then:
$$ w_{\infty} = {(\alpha [Ca^{2+}]_{in})^n \over (\alpha [Ca^{2+}]_{in})^n + \beta} $$
The time constant for the differential equation is:
$$ \tau_w = ((\alpha [Ca^{2+}]_{in})^n + \beta)^{-1} $$
The order of the binding reaction,$n$, that is the number of $Ca^{2+}$
binding sites per $w$ particle, determines the steepness of the
previous two expressions, as a function of $ [Ca^{2+}]_{in}$. Given
the constraints on the range for $[Ca^{2+}]_{shell.1}$ and
$[Ca^{2+}]_{shell.2}$ during single and repetitive firing, $n$ was set
to three for both the $I_C$ $w$ particle and the $I_{AHP}$ $w$
particle. On the other hand, as shall be presented shortly, the range
of $Ca^{2+}$ concentrations for which the $I_{AHP}$ $w$ particle is
activated is set to about one order of magnitude lower than that for
the $I_C$ $w$ particle, since $I_C$ was exposed to the larger
$[Ca^{2+}]_{shell.1}$ .
\section{AHP Potassium Current - $I_{AHP}$}
$I_{AHP}$ is a slow, $Ca^{2+}$-mediated $K^+$ current that underlies
the long afterhyperpolarization (AHP). Typically the AHP is about 1
to 2 millivolts and lasts from 0.5 -- 3 seconds after a single spike.
Adding $Ca^{2+}$ blockers or noradrenaline to the extracellular medium
eliminates the AHP, and likewise markedly reduces the cell's
accommodation to tonic stimulus.
Since most of the data on the proposed $I_{AHP}$ is derived from
various current clamp protocols, the model description of this current
is based on that used in other models (Koch and Adams, 1986) and from
heuristics derived from the properties of other currents, in
particular $I_{Ca}$ and $I_{DR}$. The important relationship between
the $I_{AHP}$ and $I_{DR}$ parameters arose when I attempted to
simulate both the mAHP (mediated by $I_{DR}$) and the AHP according to
data from Storm (). In addition, since $I_{AHP}$ is dependent on
$Ca^{2+}$ entry, the derivation of this current and the dynamics of
$[Ca]_{shell.1}$ and $[Ca]_{shell.2}$ was done simultaneously. In
fact, it was determined that in order for the activation of $I_{AHP}$
to be delayed from the onset of the spike, it was necessary to
introduce the second intracellular space (shell) that was described in
Chapter 6. Such a relationship between $Ca^{2+}$ influx and the
subsequent delayed activation of $I_{AHP}$ has been suggested in the
literature (Lan-Ada-86).
\subsection{Results}
I propose that the conductance underlying $I_{AHP}$ is dependent both
on $Ca^{2+}$ and voltage. The $Ca^{2+}$ dependence of this current is
clearly demonstrated since the AHP is removed when $Ca^{2+}$ blockers
are added, and construction of a reasonable model of $Ca^{2+}$
dynamics such that $I_{AHP}$ may be dependent on this is possible.
The mechanism that I use for $Ca^{2+}$-mediation of $I_{AHP}$ is
similar to that for $I_C$, that is the $I_{AHP}$ channel includes a
single $Ca^{2+}$-binding $w$ particle, with the same binding reaction
as shown in Equation x.
Voltage-clamp studies (Lan-Ada-86) indicate that there is no
voltage-dependent activation of $I_{AHP}$, however. This puts a
greater constraint on the $Ca^{2+}$-mediated mechanism for this
current since the activation necessary to underly the long, small
hyperpolarization after a single spike is significantly less than that
required to squelch rapid spikes after some delay in response to tonic
stimulus. In particular, these requirements provided rather restricted
constraints on the buildup of $Ca^{2+}$ during each spike in region of
the $I_{AHP}$ channels, $shell.2$, and likewise the dependence of the
$I_{AHP}$ $w$ particle on this localized concentration of $Ca^{2+}$ .
On the other hand I have included two inactivation gating particles,
$y$ and $z$. The rationale for the $y$ particle is based on two pieces
of evidence. First, it has been reported that $Ca^{2+}$ spikes are
insensitive to noradrenaline in protocols where $I_{DR}$ and $I_A$
have been blocked by TEA and 4-AP, respectively (Segal and Barker).
The fact that these spikes are unchanged with the addition of
noradrenaline implies that under this protocol $I_{AHP}$ is
inactivated by some other mechanism, since presumably $I_{AHP}$ has
not been disabled. Since the protocol involves a long (approximately
30 milliseconds) depolarization of the cell before the $Ca^{2+}$
spike, it was possible to include an inactivation particle for
$I_{AHP}$ that was (a) fast enough to disable $I_{AHP}$ under these
conditions, but (b) was slow enough so that normal spiking did not
cause the $y$ particle to change states.
A second indication for the voltage-dependent inactivation particle
$y$ is consistent with the previous evidence, that is the amplitude
and rate of rise of action potentials singly or in trains appears
independent of the presence $I_{AHP}$. In particular, the size of the
$I_{AHP}$ conductance necessary to repress repetitive firing is large
enough to significantly effect the spike once threshold is achieved if
this conductance remained during the spike. Such a role for $I_{AHP}$
has not been demonstrated. $y$ therefore causes $I_{AHP}$ to shut off
during an action potential so that this current does not reduce the
amplitude of the spike.
The second inactivation particle, $z$, was included to account for the
delayed peak seen in the large afterhyperpolarization that occurs
after a long (greater than 100 ms) stimulus (Madison and Nicoll, 1982
and others). At rest, $z$ is partially closed. With a large, lengthy
hyperpolarization the $z$ particle becomes more open, thereby slowly
increasing $I_{AHP}$ and the magnitude of the sAHP, until the
$Ca^{2+}$ in $shell.2$ eventually drains down to its resting level and
subsequently shutting off $w$. The time constant for $z$ was set very
slow above rest so that it did not change appreciably during firing.
Below about -75 mV, however, the time constant approaches 120
milliseconds so that the desired role of $z$ during the sAHP is
obtained.
No voltage-dependence for $I_{AHP}$ has been noted in the literature.
However, the dependence of $I_{AHP}$ on $Ca^{2+}$ influx may have
precluded voltage-clamp experiments which might verify the
voltage-dependencies indicated by the simulations.
With the present formulation for $I_{AHP}$, this current plays an
important role during repetitive firing by shutting off the spike
train after several hundred milliseconds. This occurs primarily
through the dependence of $I_{AHP}$ on $[Ca]_{shell.2}$, which slowly
increases during repetitive firing. Eventually the rise of
$[Ca]_{shell.2}$ causes $I_{AHP}$ to provide sufficient outward
rectification for counter-acting the stimulus current and thus stop
the cell from firing (Figure~\ref{f:ahp-spks}). The fact that
$I_{AHP}$ is strongly activated by this protocol is indicated by the
long hyperpolarization at the end of the stimulus (Madison and Nicoll,
1982, and see simulation of their results in Figure~\ref{f:ahp-spks}).
Madison and Nicoll, 1982 Mad-Nic-82 report that noradrenaline
blocks accommodation by selectively blocking $I_{AHP}$.
The characteristics demonstrated by the model $I_{AHP}$ are in
qualitative agreement with many of the characteristics reported in the
literature (e.g. Lan-Ada-86 , Seg-Bar-86),, including
the increased activation of $I_{AHP}$ with increasing numbers of
spikes in a single train, delayed activation from onset of $Ca^{2+}$
influx, the role of $I_{AHP}$ in modulating repetitive firing, time
constant for inactivation/deactivation of greater than one second, the
apparent voltage insensitivity (the transition of $y$ and $z$ with
sub-threshold depolarization is slow, and once $x$ is activated
deactivation takes several seconds.
The equation for $I_{AHP}$ is -
$$I_{AHP}= y_{AHP}^2 \, z_{AHP}\, w_{ahp}\, (V - E_K)$$
\noindent where
$$\overline g_{AHP} = 0.4 \, \mu \rm S$$
\begin{table} \centering \begin{tabular}{|c||c|c|c|c|c|c|c|}\hline
Gating Variable & $z$ & $\gamma$ & $\alpha_0$ & $V_{{1 \over 2}}\,$(mV) & $\tau_0\,$(ms) &
$\alpha_{Ca}^*$ & $\beta_{Ca}^{**}$ \\ \hline\hline
$y$ (inactivation)& -15 & 0.2 & 0.01 & -50.0 & 1.0 & - & - \\ \hline
$z$ (inactivation)& -12 & 1.0 & 0.0002 & -72.0 & 100.0 & - & - \\ \hline
$w$ ($Ca^{2+}$-activation) & - & - & - & - & - & $50$ & 0.01 \\ \hline
\end{tabular}\caption[Parameters of $I_{AHP}$ Gating Variables]
{Parameters of $I_{AHP}$ Gating Variables.
$* = (ms^{-1/3}mM^{-1})$, $** = (ms^{-1})$}\label{t:ahp}\end{table}
Table \ref{t:ahp} lists the parameters for the $I_{AHP}$ gating variables. These are the rate functions for the
activation variable, $x$, of $I_{AHP}$-
$$\alpha_{y,AHP} = 0.015 \exp \biggl({(V + 50) 0.8 \cdot -15 \cdot F \over R T} \biggr)$$
$$\beta_{y,AHP} = 0.015 \exp \biggl({(-50 - V) 0.2 \cdot -15 \cdot F \over R T} \biggr)$$
These are the rate functions for the activation variable, $y$, of $I_{AHP}$-
$$\alpha_{z,AHP} = 0.0002 \, (\gamma = 0)$$
$$\beta_{z,AHP} = 0.0002 \exp \biggl({(-72 - V) \cdot -12 \cdot F \over R T} \biggr)$$
Again, each $w$ particle was assumed to have three non-competitive
$Ca^{2+}$ binding sites, all of which were either empty (corresponding
to the closed state) or filled (corresponding to the open state).
\end{document}
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