# Computational and Mathematical Methods for Neuroscience
## 2 Fields of Neuroscience
### Theoretical Neuroscience  
Theoretical neuroscience focuses on developing mathematical, computational, and statistical models to represent neural processes across multiple scales, from an individual neuron to the brain. Mathematical neuroscience applies mathematical theories, models, and equations to describe and analyze the mechanisms of the nervous system at various levels, from single neurons to whole-brain dynamics. This branch of neuroscience aims to build theoreticalframeworks for neural activity, capturing phenomena such as neuronal electrical properties, network dynamics, and brain connectivity patterns. Some of the most widely used neural models include the Hodgkin-Huxley (HH) [1], FitzHugh-Nagumo (FHN) [2; 3], Hindmarsh-Rose (HR) [4], Wilson-Cowan (WC) [5], and Izhikevich [6] models. The HH model provides a detailed description of neuronal electrical behavior based on ion channel dynamics, offering a foundation for understanding neuron excitability. The FHN and HR models, simplified versions of the HH model, are commonly used to simulate excitable systems due to their computational efficiency. The WC model, on the other hand, captures the collective dynamics of populations of excitatory and inhibitory neurons, making it useful for studying large-scale neural networks. The Izhikevich model combines biological realism with computational efficiency, enabling the simulation of a wide range of spiking and bursting patterns observed in neurons. In addition to these continuous-time models, various discrete-time models are employed in theoretical neuroscience, such as the Leaky Integrate-and-Fire (LIF) model [7] and the Rulkov map [8]. The LIF model approximates biological neurons by simulating membrane potential decay in the absence of input spikes, while the Rulkov map generates spike patterns through the interplay of membrane potential dynamics and recovery variables. The Rulkov model also incorporates a reset mechanism, allowing neurons to recover after firing. These mathematical models play a crucial role in predicting and explaining phenomena such as neural oscillations, wave propagation in the brain, and the synchronization of neuronal activity, processes essential for neural communication and understanding neurological conditions like epilepsy [9]. These models enable theoretical neuroscience to uncover fundamental principles of neural behavior, enhancing our capacity to analyze and interpret the intricate dynamics of neural systems. Computational neuroscience aims to develop quantitative tools to analyze neural data and predict neural system dynamics, helping to uncover the principles that govern  
Figure 1: Fields and subfields of neuroscience. brain function. Computational neuroscience involves developing and using computer simulations, algorithms, and artificial neural network (ANN) models to investigate the functioning of the nervous system. It bridges theoretical models with experimental data, often serving as a testing ground for hypotheses. Computational neuroscience focuses on simulating neural circuits, analyzing large datasets from neural recordings, and predicting brain activity and behavior. Popular examples of computational techniques are neural network simulations, machine learning (ML), and data-driven models. Former methods simulate networks of neurons to study how they encode, process, and retrieve information. ML models are used to classify patterns in brain data, such as electroencephalography (EEG), magnetoencephalography (MEG), Magnetic Resonance Imaging (MRI), functional Magnetic Resonance Imaging (fMRI), and Positron Emission Tomography (PET), to model learning and adaptation in neural systems. Finally, data-driven models use real experimental data to create models of complex phenomena like sensory processing, decision-making, or motor control. Computational neuroscience helps us understand brain function (e.g., sensory processing, memory, and emotions), design brain-machine interfaces (BMIs), and develop treatments for neurological diseases through predictive modeling.