Point Neuron and Neural Mass Models:
an Introduction to Computational Neuroscience

Course given as part of FEE CTU, BAM31MOA 6/13

Abstract

This course is designed as an introduction to computational neuroscience. First, it will explain why computational neuroscience is desirable for complementing experimental resources and for gaining a deeper understanding of the brain at different scales. Next, the necessary biological concepts for understanding the course will be introduced: neuronal anatomy, the concept of excitability, synaptic connections, brain complexity, etc. Following this, the concept of mathematical models in neuroscience will be introduced, both for a single neuron and for a population of neurons. A brief historical overview will be provided, followed by a cataloging of these models within the context of point neuron models and/or neural mass models. Once the context is established, a first example will be studied with the class to understand how to process a mathematical model in order to extract information about its dynamics (nullclines, fixed points, stability). The concept of bifurcation diagrams will also be introduced to provide a preview of the subject. Finally, a concrete example of research also carried out by the BRADY/COBRA project will be explained in broad outline in order to put context on what has been seen in class.

The practical component will cover the same material as the lectures: the first exercise will require applying a phase-plane analysis of a simple model, then using pre-established code to study the effect of its parameters. Next, students will study code for the FitzHugh-Nagumo model to understand its diverse dynamics. Finally, they will code the Wilson-Cowan model independently.

Literature

• Izhikevich, Eugene M. Dynamical systems in neuroscience. MIT press, 2007. 

• Dayan, Peter, and Laurence F. Abbott. Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT press, 2005. 

• Kandel, Eric R., et al., eds. Principles of neural science. Vol. 4. New York: McGraw-hill, 2000. 

• Nicholls, John G., et al. From neuron to brain. Vol. 271. Sunderland, MA: Sinauer Associates, 2001. 

• Hodgkin, Alan L., and Andrew F. Huxley. "A quantitative description of membrane current and its application to conduction and excitation in nerve." The Journal of physiology 117.4 (1952): 500. 

• Rieke, Fred, et al. Spikes: exploring the neural code. MIT press, 1999. 

• Wilson, Hugh R., and Jack D. Cowan. "Excitatory and inhibitory interactions in localized populations of model neurons." Biophysical journal 12.1 (1972): 1-24. 

• Strogatz, Steven H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity). Vol. 1. Westview press, 2001. 

• Rinzel, John. "Excitation dynamics: insights from simplified membrane models." Federation proceedings. Vol. 44. No. 15. 1985. 

• Guckenheimer, John, and Philip Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Vol. 42. Springer Science & Business Media, 2013. 

• Girier, Guillaume, et al. "Ion Dynamics Underlying the Seizure Delay Effect of Low-Frequency Electrical Stimulation." bioRxiv (2025): 2025-04.