Study Notes on Neuronal Dynamics

This is a study note for Neuronal Dynamics (Gerstner et al.).

The passive membrane

The passive membrane model can be described using an RC (resistor–capacitor) circuit, where

\[I=I_R+I_C\]

and

\[I_C=\frac{dQ}{dt}=C\cdot\frac{du}{dt}\] \[I_R=\frac{1}{R}(u-u_{rest})\]

so

\[R\cdot C\cdot \frac{du}{dt}=-(u-u_{rest})+R\cdot I(t); R\cdot C=\tau\]

If we remove the battery:

\[V=u-u_{rest}\] \[\frac{dV}{dt}=\frac{d}{dt}(u-u_{rest})=\frac{du}{dt}\] \[\tau \cdot \frac{dV}{dt}=-V+R\cdot I(t)\]

The pulse input is modelled using Dirac’s delta function:

\[I(t)=q\cdot \delta(t-t_0)\]

The total charge q is the area under the I vs. t curve.

Its effect on \(u(t)\) is to cause it jump from \(u_{rest}\) to a larger value ( \(\frac{q}{C}\) ) and then decay exponentially.

The properties of Dirac’s delta function:

\[\int_{t_0-a}^{t_0+a}\delta(t-t_0)dt=1\]

and:

\[\int_{t_0-a}^{t_0+a}f(t)\delta(t-t_0)dt=f(t_0)\]

Next we want to solve for the membrane response \(u(t)\) to the input current:

For a singe pulse, suppose the duration is short:

\[\Delta=t-t_0<<\tau\]

The rise component is dominated by the first terms in the Taylor expansion of the exponential function, so:

\[u(t)=u_{rest}+\frac{q}{C}e^{-\frac{t-t_0}{\tau}}\]

For arbitrary inputs, the response is as follows:

\[u(t)=u_{rest}+\int_{-\infty}^{t}\frac{1}{C}e^{-\frac{t-t^\prime}{\tau}}I(t^\prime) dt^\prime\]

But this assumes that the input before \(t_0\) has little contributions, otherwise

\[u(t)=u_{rest}+[u(t_0)-u_{rest}]e^{-\frac{t-t_0}{\tau}}+\int_{t_0}^{t}\frac{1}{C}e^{-\frac{t-t^\prime}{\tau}}I(t^\prime) dt^\prime\]

I&F Model

The previous section discussed the subthreshold regime. To simulate firing, we basically need a passive membrane + threshold.

In LIF model,

\[\tau\cdot \frac{du}{dt}=-(u-u_{rest})+R\cdot I(t)\]

If firing ( \(u(t)=\vartheta\) ), then

\[u\rightarrow u_r\]

In nonlinear I&F model,

\[\tau\cdot \frac{du}{dt}=F(u)+R\cdot I(t)\]

The \(F(u)\) can be quadratic or exponential. And the intrinsic threshold depends on the input. (Although we still have a reset condition.)

And we can draw an F (frequency)-I curve or gain function of the neuron.

Seems that the EIF model fits the experimental data (layer-5 pyramidal cells; Badel et al., J Neurophysiol., 2008) best:

\[F(u)=-(u-u_{rest})+\Delta_T e^{\frac{u-\vartheta}{\Delta_T}}\]

We didn’t consider the adaptation, noise, and other biophysical details yet.