Lecture 6: Biophysics of neurons

Teacher: Grégory Dumont

Action potential

Membrane potential:

difference of potential between the inside and outside of the cell

Resting potential: the neuron is polarized

When the neuron fires ⇒ Spike trains

Post-synaptic potential (PSP) ⟶ temporal integration of all the incoming spikes ⟹ triggers an action potential (AP) (= spike)

Membrane equation

Neuronal electricity

Why do we have a non-zero equilibrium potential?

• Depolarization: when the membrane voltage $V_m$ decreases
• Hyperpolarization: when it increases

Channels in the membrane: difference of ion concentration ⇒ generates an electric current

Let’s focus on potassium from now on.

The cell membrane lets the potassium cross inside and outside to reach an equilibrium ⟶ electro-diffusion through a semi-permeable membrane

Nernst potential

Nernst equation:

\[E_X = \frac{RT}{zF} \ln \frac{[X]_{out}}{[X]_{in}}\]

where

• $E_X$: the equilibrium potential
• $R$: gas constant
• $T$: temperature
• $z$: valence of the integration
• $F$: Faraday constant
• $[X]{out}, [X]{in}$: Ion concentrations

Linear approximation of leak current:

\[I = g_L(V_m - E_L)\]

• $g_L$: leak conductance
• $E_L$: leak or resting potential

The membrane equation

\[C \frac{dV_m}{dt} = \frac{E_L - V_m}{R} + I_{inj}\\ ⟹ τ \frac{dV_m}{dt} = E_L - V_m + RI_{inj}\]

where $τ = RC$ is the membrane time constant (typically $3$-$100$ms)

We integrate the differential equation:

\[V_m(t) = V_∞ + (V_m(0) - V_∞) \exp(-t/τ)\]

where $V_∞ = E_L + RI$

Integrate-and-fire model:

whenever $V$ reaches a threshold $V_t$, it is reset to $V_r$

Synapses

Pre-synaptic neuron ⟶ Post-synaptic one

Tha action potential goes through the pre-synaptic axon, and then neuro-transmitters are released in the synaptic space

Mathematically: the post-synaptic neuron is at equilibrium potential, then, when neurotransmitter arrive: transfer of charge (=ions):

\[Q = C ΔV\\ ⟹ ΔV = Q/C = RQ/τ\] \[τ \frac{dV_m}{dt} = E_L - V_m\\ ⟹ V_m → V_m + \underbrace{RQ/τ}_{≝ w}\]

• $w$: synaptic weight

Then:

\[V_m(t) = E_L + w \exp(-t/τ)\]

A synapse can be either excitatory or inhibitory (they don’t switch behavior).

Many spikes at time $t_i^k$ ($i$=synapse, $k$=spike index):

\[V_m → V_m + w_i\]

so that:

\[V_m(t) = E_L + \sum\limits_{ i, k } PSP_i(t-t_i^k)\]

where

\[PSP_i(t) ≝ \underbrace{H}_{\rlap{\text{Heaviside}}}(t) w_i \exp(-t/τ)\]

When two AP delayed by $d$, the threshold is reached if:

\[\vert d \vert < - τ \, \log\left(\frac{V_t-E_L}{w} - 1\right)\]

Indeed:

\[E_L + w \exp(-d/τ) + w \overset{?}{>} V_t\\ ⟺ d < - τ \, \log\left(\frac{V_t-E_L}{w} - 1\right)\]

⟹ Jeffress model of sound localization

Firing rate

Interspike interval:

\[T_n = t_{n+1} - t_n\]

Frequency:

\[F = \frac 1 {⟨T_n⟩}\]

---

\[τ \frac{dV}{dt} = \underbrace{V_∞}_{= E_L + RI} - V(t)\]

Firing condition:

\[V_∞ = E_L + RI ≥ V_t\\ ⟺ I ≥ \frac{V_t - E_L}{R}\]

Time to threshold: $V(T) = V_t$:

But

\[V(t) = V_∞ + (V(0)-V_∞) \exp(-t/τ)\]

so:

\[T = τ \log\left(\frac{E_L + RI - V_m(0)}{E_L + RI - V_t}\right)\]

Hence:

\[F =\frac 1 T = \left(τ \log\left(\frac{E_L + RI - V_m(0)}{E_L + RI - V_t}\right)\right)^{-1}\]

⟶ $F\text{-}I$-curve: frequency $F$ as a function of the current $I$ (or the voltage $RI$)

Refractory period:

after resetting the voltage at the reset potential, the voltage stays constant during the refractory period (experimentally: around $5$ms)

For a refractory period $Δ$:

\[F = \left(Δ + τ \log\left(\frac{E_L + RI - V_m(0)}{E_L + RI - V_t}\right)\right)^{-1}\]