From detailed models to formal spiking neurons

Reduction of the Hodgking-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 Hodgking-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 integrte-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