This project consists in the implementation and comparison of two abstract neuronal models, by deriving their sub- and supra-threshold behaviour and computing their gain functions.

The LIF model is inspired by the famous integrate-and-fire (IF) reduced neuronal model, but it adds a leakage channel. This model is able to display only tonic spiking activity.

$$C_{m}\frac{dV_{m}}{dt}=-I_{leak}+I_{stim}$$

where $I_{leak}=g_{leak}\bigl(V_{m}-E_{leak}\bigr)$ is the leakage current and $I_{stim}$ is the stimulation current, which might be substituted by the synaptic current $I_{syn}$ as well.

The IFB model, on the other hand, extends the LIF one and enables the simulation of bursting activity patterns as well, by taking into account T-type low voltage activated $\text{Ca}^{2+}$ channels.

$$C_{m}\frac{dV_{m}}{dt}=-I_{leak}-I_{T}+I_{stim}$$

$$\tau_{h}\frac{dh}{dt}=-h+h_{\infty}$$

where $I_{leak}=g_{leak}\bigl(V_{m}-E_{leak}\bigr)$ is the leakage current, $I_{T}=g_{T}h\bigl(V_{m}-E_{T}\bigr)\Theta\bigl(V_{m}-V_{h}\bigr)$ and $I_{stim}$ is the stimulation current, which might be substituted by the synaptic current $I_{syn}$ as well. Moreover, $h_{\infty}=\Theta\bigl(V_{h}-V_{m}\bigr)$ and $\tau_{h}=\tau_{h}^{-}\Theta\bigl(V_{m}-V_{h}\bigr)+\tau_{h}^{+}\Theta\bigl(V_{h}-V_{m}\bigr)$. Finally, $\Theta\bigl({\cdot}\bigr)$ is the Heaviside step function.