Reduction of the HodgkinHuxley model type II
Another way of approximation, compare to two phase analysis
Reduction
 Hodgkin and Huxley model:
 SRM: $ u(t) = \eta(t\hat t) + \int_0^{t\hat{t}} \kappa(t\hat t_i,s) I^{ext}(ts) ds+u_{rest} $ we need to define $\eta(t\hat{t})$, $\kappa(t\hat{t})$, $\vartheta$

$\eta(t\hat{t})$ action potential is stereotyped when triggered the spike In HodgkinHuxley model, let: we can get $u(t)$, then use $u(t)$ to get $\eta(t\hat{t})$

$\kappa(t\hat{t})$ weak input current, slight perturbed Input: strong plus at $\hat{t}$, weak plus at $t$, $(t>\hat{t})$
 $\vartheta$ threshold for spike fixed use different value in different cases
Scenarios
timedependent input
the metrics:
$\langle{N_{coinc}\rangle}=2\nu\Delta{N_{full}}$
$C=12\nu\Delta$
if Possison process:
if two model fit perfect:
if $\kappa$ does not depend on last firing time, $\Gamma$ will be lower (lower accuracy)
constant input
different $\vartheta$ make big differences
step current input
same three zones also show inhibitory rebound
spike input
use $\epsilon $ to substitute external input: $u_i(t)=\eta(t\hat{t_i})+\sum\limits_{j}w_{ij}\sum\limits_{f}\epsilon(t\hat{t_i},tt_{j}^{(f)})+u_{rest}$
Reduction of a cortical neuron
type I
SRM can also be used as a quantitative model of cortical neurons.
cortical neurons has continuous gain function
Reduction to a nonlinear integrateandfire model
Reduction
first step
define:
 $\vartheta$
 $\Delta_{abs}$
 $u_{r}$
 $m_{r}$
 $h_{r}$
 $n_{slow}$
 $n_{fast}$
we get multi integrate and fire model
second step
 fast variables: replace with steady state values (function of u)
 slow variables: replace with constant $m \rightarrow m(u)$ $n_{fast} \rightarrow n_{0,fast}$ $n_{slow} \rightarrow n_{slow, average}$ $h \rightarrow h_{average}$
we get nonlinear integrate and fire model
Scenarios
constant input
fluctuating input
Reduction to SRM
Reduction
aim: find $\eta$, $\kappa$, $\vartheta$
first step
reduce the model to and integrateandfire model with spiketimedependent time constant
second step
integrate the model, get $\eta$ and $\kappa$
third step
choose appropriate spiketimedependent threshold $\vartheta$
Scenarios
constant input
better with dynamic threshold
fluctuating input
the accuracy is more stable than nonlinear integrateandfire model
Limitations
 even $\Gamma$ of the multicurrent integrateandfire model is far below 1
 timedependent threshold of SRM is import to achieve generalize over a broad range of different inputs
 timedependent threshold seems to be more important for the randominput task than the nonlinearity of function $F(u)$
 in the immediate neighborhood of the firing threshold, nonlinear integrateandfire model performs better than SRM