Neuron

The neuron is the most common component in our brain, we have nearly 86 billion neurons and 1000 trillion synaptic connections, estimates to a computer with a 1 trillion bit per seconds processor

Computational Neuroscience

Is the field of study in which we measure and simulate the neurons process. Our brain is a complex machine and its behavior is non-linear.

We need previous knowledge of electronics, ODE’s, neurobiology, chemistry, and programming.

Integrate-and-fire model

We start with the easiest model that tries to interpret the neuronal behavior.

We can consider the neuron an electronic circuit:

$I(t)=I_{R}(t)+I_{C}(t)$

$I_{R}(t)=u_{R}/R=\frac{u(t)-u_{rest}}{R}$

$C=q/u\Rightarrow I_{C}(t)=C\frac{du}{dt}$

Passive Membrane Model:

$RI(t)=[u(t)-u_{rest}]+\tau_{m}\frac{du}{dt}$

We solve the linear differential equation:

$u(t)=\begin{cases} u_{rest}+\Delta ue^{-\frac{t-t_{0}}{\tau_{m}}} & \text{if }I(t)=0\\ u_{rest}+RI_{0}\left(1-e^{-\frac{t}{\tau_{m}}}\right) & \text{if }I(t)=I_{0} \end{cases}$

where $\Delta u=u(t_{0})-u_{rest}$

Time-dependent

In absence of $\vartheta$ (Green’s Function):

(1) $\begin{equation*} u(t)=u(t_{0})+\frac{R}{\tau_{m}}\int_{t_{0}}^{t}e^{-\frac{t-t'}{\tau_{m}}}I(t')dt' \end{equation*}$

where $I(t)$ is an input current.

$I(t)=\begin{cases} I_{0}\sin(\omega t) & \text{sinusoidal}\\ I_{0}\Theta(t) & \text{heaviside step}\\ \dots \end{cases}$

Add threshold $\vartheta$ :

Let $q_{r}$ be the removing charge, $q_{r}=C(\vartheta-u_{r})$ every

time we hit the threshold.

$I_{r}=-q_{r}\delta(t-t^{(f)})$ the reset corresponds to a short current pulse.

$I_{r}=-q_{r}\sum_{f}\delta(t-t^{(f)})=-C(\vartheta-u_{r})S(t)$

$I_{T}(t)=I(t)+I_{r}(t),$

(2) $\begin{equation*} u(t)=u_{rest}+\sum_{f}(u_{r}-\vartheta)e^{-\frac{t-t^{(f)}}{\tau_{m}}}+\frac{R}{\tau_{m}}\int_{0}^{\infty}e^{-\frac{s}{\tau_{m}}}I(t-s)ds \end{equation*}$

Here we have the simulation of a neuron using a heavystep current input first equation and Leaky-integrate and fire (LIF)

The image below shows the different Leaky-integrate and fire methods and the input current gived for the neuron.

Given an aleatory input for the neuron, we obtain the next simulation for voltage potential ( $u_rest=0$ , time-scale constant, $\tau:8e-3$ , $R=1e-7$ ):

Neuron

Computational Neuroscience

Integrate-and-fire model

Passive Membrane Model:

Time-dependent

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