Computational Neuroscience

Computational neuroscience: neuron and network simulation


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:



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

Passive Membrane Model:


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}


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.



(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):

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