Interactive Neuron Simulation

The Hodgkin-Huxley model describes how action potentials in neurons are initiated and propagated through a set of nonlinear differential equations:

\[ \begin{aligned} C_m \frac{dV}{dt} &= -\bar{g}_\text{Na} m^3 h (V - E_\text{Na}) - \bar{g}_\text{K} n^4 (V - E_\text{K}) - \bar{g}_\text{L} (V - E_\text{L}) + I_\text{ext},\\[1mm] \frac{dm}{dt} &= \alpha_m(V)(1-m) - \beta_m(V)m,\\[1mm] \frac{dh}{dt} &= \alpha_h(V)(1-h) - \beta_h(V)h,\\[1mm] \frac{dn}{dt} &= \alpha_n(V)(1-n) - \beta_n(V)n. \end{aligned} \]

where \(V\) is the membrane potential, \(m\), \(h\), and \(n\) are gating variables, and \(I_\text{ext}\) is the external current stimulus.

The Izhikevich model is a simplified neuronal model that can reproduce various firing patterns while being computationally efficient:

\[ \begin{aligned} \frac{dv}{dt} &= 0.04v^2 + 5v + 140 - u + I,\\[1mm] \frac{du}{dt} &= a(bv - u) \end{aligned} \]

with the reset condition:

\[ \text{if } v \geq 30 \text{ mV, then } \begin{cases} v \leftarrow c,\\[1mm] u \leftarrow u + d \end{cases} \]

where \(v\) represents the membrane potential, \(u\) is a recovery variable, and parameters \(a\), \(b\), \(c\), and \(d\) control the firing pattern.