We develop the theory of firing rates, showing how it appears in tonically spiking neurons. We then review some applications of firing rates, such as tuning curves, rate coding, f-I curves, and effective (mean-field) models of neural networks. We review some interesting results obtained from firing rates, such as the Hopfield theory, Wilson-Cowan model applications, and our proposed rate-variance coding scheme.

Lecture 1: Neurons and neural dynamics

• Basic mathematical and computational tools: differential equations, linear algebra, probability, statistics, and numerical methods.
• Integrate-and-fire neuron models.
• Basic neuronal dynamics and electrophysiology.
• Conductance-based models and neuroelectronic principles.
• Neural coding and firing rates.

Lecture 2: From neurons to neural networks

• Synapses and interactions between neurons.
• Firing-rate models and population dynamics.
• Relationship between spiking models and firing-rate descriptions.
• Neural coding and information theory.
• Introduction to brain networks and network representations of brain activity.

Lecture 3: Learning, plasticity, and network organization

• Synaptic plasticity and learning.
• Hopfield networks and associative memory.
• Criticality and collective dynamics in neural systems.
• Brain networks and large-scale organization of brain activity.
• Connections between neural computation, dynamical systems, and network science.

Bibliography

1. P. Miller (2018) An Introductory Course in Computational Neuroscience. The MIT Press.
2. Gerstner W, Kistler WM, Naud R, Paninski L (2014) Neuronal Dynamics: From single neurons to networks and models of cognition. Cambridge University Press.
3. Dayan, P. and Abbott, L.F. (2001) Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems. The MIT Press.
4. Izhikevich E.M. (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT press
5. M Girardi-Schappo, L Maler, A Longtin (2026) Optimal rate-variance coding due to firing threshold adaptation near criticality. arXiv:2509.04106 [q-bio.NC]
6. D D Caprioti, M Haas, C F Vasconcelos, M Girardi-Schappo (2026) Learning About Learning: A Path from Spin Glasses to Artificial Intelligence. arXiv:2601.07635 [cond-mat.dis-nn]