Mathematical modeling of neuronal dynamics: from theoretical pattern formation to data-driven connectivity inference

Understanding how the brain works is one of the greatest challenges in science. This PhD thesis contributes to the understanding of emergent brain dynamics by studying how neurons, the brain s nerve cells, activate and interact with each other.

The research is divided into two parts. In the first part, we develop a mathematical model to simulate a row of neurons. This model helps us predict and understand different patterns of neuronal activity, such as waves that stay in place, waves that travel, and waves with a more complex oscillatory pattern. By adjusting certain parameters in the model, we can control the neuronal dynamics, which allows us to mimic various biological processes. Our simulations confirm our theoretical predictions and reveal that some complex patterns can occur with types of connections between neurons that extend those previously considered necessary. Our findings challenge existing assumptions and open new avenues for research.

In the second part, we analyze data recordings from four types of neurons in brain areas associated with movement and decision-making. Our goal is to gain insights into possible functional connectivities among these types of neurons and how they collectively work to support behavior. We first apply a method that was previously shown (in other context) to accurately mathematical rules from the data. This approach doesn t yield reliable results in reproducing the whole neuronal network while providing evidence on a sufficient connectivity for each neuron. Then, we modify the preliminary model of the small network and optimize it to fit the recorded data. The resulting model successfully captures the patterns observed in the data and provides insights into the connections and interactions between the neurons.

This research advances our theoretical understanding of neuronal dynamics and provides practical tools for interpreting brain data recorded in vivo, potentially opening new venues in the field of diagnosing and treating neurological disorders.