Lecture 6: Hodgkin-Huxley model

Cell membrane = semi-permeable

• capacitor effect: \(\frac{d Q}{dt} = C_m \frac{V_m}{dt}\)

• ion channels: \(i_j = g_j (V - E_j)\)

Charge conservation:

\[c_m \frac{V_m}{dt} = \underbrace{-i_m}_{\text{leaky term}} + \frac{I_e}{A}\]

Integrate-and-Fire

LIF: Spiking mechanism:

\[V > V_{th} ⟹ V = V_r\]

⟹ Adaptation: the firing rate $ν$ is logarithmic with respect to $I_e$

Saturation: With a refractory period $τ_{ref}$ ⟹ $ν$ is bounded by $1/τ_{ref}$

Experimentally: the inter-spike interval decreases after the first spikes and then becomes constant

Hodgkin-Huxley

At the beginning: a phenomenological model ⇒ they were trying to find a mathematical model to fit the data

⟹ Nobel prize: they developed also a whole bunch expermiental techniques to measure the voltage (at first: on a squid)

Channel dynamics

Membrane channels open/close stochastically

The current going through each channel is very low ⟶ order of magnitude: $\text{pA} = 10^{-12} \text{ A}$ VS input (external) current at the membrane: in $\text{nA}$

Hypothesis: Channels are independent with one another & Deterministic incoming current if you add up a lot of channels

Channels:

• are selective for different ions
• stick and ball model (activation gate ⟺ stick / inactivation gate ⟺ ball)

Potassium $K$ current

• $n$: activation variable: the fraction of open channels that are selective to $K$

\[\frac{dn}{dt} = \underbrace{α_n(V)}_{\text{opening rate}}(1-n) - \underbrace{β_n(V)}_{\text{closing rate}}n\]

NB: $α$ and $β$ are determined experimentally

It can be rewritten as:

\[\underbrace{\frac{1}{α_n + β_n}}_{≝ \, τ_n(V)}\frac{dn}{dt} = -n + \underbrace{\frac{α_n}{α_n + β_n}}_{≝ \, n_∞(V)}\]

Persistent current:

\[I_K = \underbrace{\bar{g}_K}_{\text{effective conductance}} n^4(V, t)(V - E_K)\]

NB: $n^4$: to fit the data, better with $n^5$ but there wasn’t enough computing power when it was designed

“Persistent” current because:

\[I_K \propto n^4 \xrightarrow[t \to +∞]{} n_∞^4\]

Sodium $N_a$ current

• $m$: activation (fast activation rate)
• $h$: inactivation (slow inactivation rate)

\[I_{N_a} = \bar{g}_{N_a}(V,t) m^3(V,t) h(V,t)(V- E_{N_a})\]

NB: so the activation gate must be active and the inactivation one inactive

Voltage clamp technique ⟹ change the command voltage (plateau-like function), and then measure the spikes

$K$ and $N_a$ counterbalance one another ⇒ interplay between $n_∞, m_∞$ and $h_∞$

The model

Charge conservation:

\[c_m \frac{V_m}{dt} = \underbrace{-i_m}_{\text{leaky term}} + \frac{I_e}{A}\]

\[i_m = \bar{g}_L (V- E_L) + \bar{g}_K n^4(V, t)(V - E_K) + \bar{g}_{N_a}(V,t) m^3(V,t) h(V,t)(V- E_{N_a})\] \[\begin{cases} \frac{dn}{dt} = α_n(V)(1-n) - β_n(V)n \\ \frac{dm}{dt} = α_m(V)(1-n) - β_m(V)m \\ \frac{dh}{dt} = α_h(V)(1-n) - β_h(V)h \\ \end{cases}\]

Euler method:

\[\vec{X}_{i+1} = \vec{X}_i + Δt \cdot [f(\vec{X}_i)]\\ \vec{X} = \begin{pmatrix} V \\ n \\ m \\ h \\ \end{pmatrix}\]

Application of Hodgkin-Huxley to Paramecium

Paramecium: one-cell protist ⟹ we can link the membrane current directly to its behavior (with Hodgkin-Huxley with calcium instead of sodium)

See Romain Brette’s work

---