09.Reduction of the Hodgkin-Huxley model type II
Reduction of the Hodgkin-Huxley model ‘type II’
Reduction of the Hodgkin-Huxley model type II
Another way of approximation, compare to two phase analysis
Reduction
- Hodgkin and Huxley model:
$$C\frac{du}{dt}=-\Sigma I_{k}(t)+I(t)$$
$$\Sigma I_{k}=g_{Na}m^{3}h(u-E_{Na})+g_{K}n^{4}(u-E_{k})+g_{L}(u-E_{L})$$
$$\dot{m}=\alpha_{m}(u)(1-m)-\beta_{m}(u)m$$
$$\dot{n}=\alpha_{n}(u)(1-n)-\beta_{n}(u)n$$
$$\dot{h}=\alpha_{h}(u)(1-h)-\beta_{h}(u)h$$
SRM: $ u(t) = \eta(t-\hat t) + \int_0^{t-\hat{t}} \kappa(t-\hat t_i,s) I^{ext}(t-s) 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 Hodgkin-Huxley model, let: $$I(t)=c\frac{q_0}{\Delta}\Theta(t)\Theta(\Delta-t)$$ we can get $u(t)$, then use $u(t)$ to get $\eta(t-\hat{t})$ $$\eta(t-\hat(t))=[u(t)-u_{rest}]\Theta(t-\hat{t})$$
$\kappa(t-\hat{t})$ weak input current, slight perturbed Input: strong plus at $\hat{t}$, weak plus at $t$, $(t>\hat{t})$ $$\kappa(t-\hat{t},t)=\frac{1}{c}[u(t)-\eta(t-\hat{t})-u_{rest}]$$
$\vartheta$ threshold for spike fixed use different value in different cases
Scenarios
time-dependent input
the metrics: $$\Gamma=\frac{1}{C}\frac{N_{coinc}-{\langle}{N_{coinc}}{\rangle}}{\frac{1}{2}(N_{SRM}+N_{full})}$$ $\langle{N_{coinc}\rangle}=2\nu\Delta{N_{full}}$ $C=1-2\nu\Delta$ if Possison process: $$\Gamma=0$$ if two model fit perfect: $$\Gamma=1$$ 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: $u_i(t)=\eta(t-\hat{t_i})+\sum\limits_{j}w_{ij}\sum\limits_{f}\epsilon(t-\hat{t_i},t-t_{j}^{(f)})+u_{rest}$
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 integrate-and-fire model
Reduction
$$C\frac{du}{dt}=-\sum{I_{k}(t)}+I(t)$$
$$\sum{I_{k}}=g_{Na}m^{3}h(u-E_{Na})+g_{K_{slow}}n^{4}_{slow}(u-E_{K})+g_{K_{fast}}n^2_{fast}(u-E_{K})$$
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 integrate-and-fire model with spike-time-dependent time constant
second step
integrate the model, get $\eta$ and $\kappa$
third step
choose appropriate spike-time-dependent threshold $\vartheta$
Scenarios
constant input
better with dynamic threshold
fluctuating input
the accuracy is more stable than nonlinear integrate-and-fire model
Limitations
- even $\Gamma$ of the multi-current integrate-and-fire model is far below 1
- time-dependent threshold of SRM is import to achieve generalize over a broad range of different inputs
- time-dependent threshold seems to be more important for the random-input task than the nonlinearity of function $F(u)$
- in the immediate neighborhood of the firing threshold, nonlinear integrate-and-fire model performs better than SRM
Table of Contents
Current Ref:
- snm/09.from_detailed_models_to_formal_spiking_neurons.md