Single Neuron Models

Hodgkin-Huxley Model

Biological principles: ion channel openness

$$ I(t)= I_\mathrm{cap}(t)+\sum_{k}I_k(t) $$

This equation summarize the current run over the all ion channels. Three types of ion channels are considered: sodium channels, potassium channels, and unspecific leaky channels.

$$ C\frac{du}{dt}=-\sum_kI_k(t)+I(t) $$

$C\frac{du}{dt}$ denotes the current across the capacitor.

$$ \sum_kI_k(t) = g_{Na}m^3h(u-E_{Na})+g_kn^4(u-E_k)+g_L(u-E_L) $$

This equation denotes the ionic channel currency, where m , n, and h could be analyzed with master equation.

$$ \dot m= \alpha_m(u)(1-m)-\beta_m(u)m $$

$$ \dot n= \alpha_n(u)(1-n)-\beta_n(u)n $$

$$ \dot h= \alpha_h(u)(1-h)-\beta_h(u)h $$

Advantages: represent real biological process Disadvantage: computation-expensive

2D Model

Leaky Integrate-and-Fire Model

$$ I(t)=\frac{u(t)}{R}+C\frac{du}{dt} $$

where $\tau_m=RC$ is the leaky integrator, so it yields this equation:

$$ \tau_m\frac{du}{dt}=-u(t)+RI(t) $$

Nonlinear Integrate-and-Fire Model

Spike Response Model

no equations available

Izhikevich Model