Small signals die down, but if signal is above a threshold, then the neuron fires a spike.

Voltage-gated ion channels

ions flow through pores in membrane called ion cahnnels. Simple model assumes linear dependence of current I on V (Ohm's law).

So for a single ion channel S, $J_S=q_S (V-V_S)$, where $V_S$ is the Nernst potential.

Conductance: $g=1/R$. $g$ will depend on $V$.

Ions transported across ion channels that are not always open. Proportion of open and closed channels depend on V

channels open with rate $\alpha(V)$, and close with rate $\beta(V)$.

$n(t)$ is the proportion of open channels at time t.

$\frac{dn}{dt}=\alpha(V)(1-n) -\beta(V) n$

This can be rewritten as

$\tau_n(V) \frac{dn}{dt} = n_\infty (V)=n$

$n_\infty = \frac{\alpha(V)}{\alpha(V)+\beta(V)}$ is equilibrium value of $n$.

$\tau_n(V)=\frac{1}{\alpha(V)+\beta(V)}$ is the time scale of equilibation.

each ion channel, conductance $g=g^{\text{max}} n$

$C\frac{dV}{dt}+g^{\text{max}} n(V-V_S)=0$

$\frac{dn}{dt}=\alpha(V)(1-n) -\beta(V) n$

The idea of the Huxley-Hodgkin equations

Can generalize to channels with multiple identical subunits (gates).

Suppose each channel has 2 gates, either open or closed. Ion channel open only if all gates are open.

Denote $S_i$, $i\in \{0,1,2\}$ denotes proportion of channels with exactly $i$ gates open. Rate of going from $S_0$ to $S_1$ is $2 \alpha(V)$ because can open either of the two gates, and similarly for $S_2$ to $S_1$ being $2\beta(V)$.