Computational neuroscience utilizes mathematical models to understand the complex dynamics of neuronal activity. Among various neuron models, the Izhikevich model stands out for its ability to combine biological fidelity with computational efficiency. Developed by Eugene Izhikevich in 2003ꜛ, this model simulates the spiking and bursting behavior of...

An important step beyond simplified neuronal models, like the previously discussed Integrate-and-Fire model or the FitzHugh-Nagumo model, is the Hodgkin-Huxley model. This model is based on the experimental data of Hodgkin and Huxley, who received the Nobel Prize in 1963 for their discoveries concerning the ionic mechanisms involved in excitation and...

In the previous post, we analyzed the dynamics of Van der Pol oscillator by using phase plane analysis. In this post, we will see, that this oscillator can be considered as a special case of another dynamical system, the FitzHugh-Nagumo model. The FitzHugh-Nagumo model is a simplified model used to describe the dynamics of the action potential in neurons....

After we have discussed the basics of phase plane analysis in the previous post, we will now apply this method to the Van der Pol oscillator. The Van der Pol oscillator is a non-conservative oscillator with nonlinear damping, which was first described by the Dutch electrical engineer Balthasar van der Pol in 1920. It was originally used to describe...

After introducing phase plane analysis in the previous post, we will now apply this method to the Rössler attractor presented earlier. We will investigate the system’s nullclines and fixed points, and analyze the attractor’s dynamics in the phase space. Nullclines and fixed points Let’s recall the system of ordinary differential equations (ODEs)...

When it comes to understanding the behavior of dynamical systems, analyzing the system’s behavior directly from its differential equations can often be too complex. In such situations, phase plane analysis can be a powerful tool for gaining insights into the system’s dynamics. By plotting the system’s state variables against each other, this method...