Aims

Over the past decades, hard sciences such as Chemistry and Biology witnessed the emergence of counterpart "computational" disciplines, like Computational Chemistry and Computational Biology. These are all branches of Computational Science (known also as Scientific Computing), a discipline that uses math and computers to answer scientific questions. As in daily weather forecasts or global-warming computer simulations, scientists use equations and computer to solve and analyse them, so that they can make predictions of real experiments.

Computational Neurosciences is a multidisciplinary field that studies brain functions and dysfunctions and how information is processed by cells and circuits, by ultimately building mathematical models and numerically simulating them on computers, aiming at tackling scientific questions that are relevant for the understanding of the brain. These models are often, but not always, intimately grounded in first-principle biology, biochemistry, physics, and electromagnetism, so that the processes within individual nerve cells, synapses, and networks are described like a real pendulum can be studied by pen and paper, by a simple formula and a series of parameters.

This is not a course in computer programming, but it features hands-on exercises and training, allowing students to build and get familiar with models and computer simulations. This is also not a course in math and physics, but it builds on high-school or bachelor-level elementary concepts (i.e. calculus, algebra, electricity, etc.).

Students will be presented with a selection of elementary topics in Computational Neuroscience, understanding the motivations behind a mechanistic dissection on how the mammalian nervous system works. They will increase their knowledge in quantitative approaches in Neurosciences and will increasingly be able to apply their knowledge to grasp modelling components of cutting-edge scientific publications.

Students will increasingly start making quantitative judgement of cell electrophysiology and biophysics, analysing how significant discoveries could be made in these domains and where the availability of solid theoretical and computational tools revealed to be extremely fruitful. Students will become fluent in this language and skills.

Students will be presented by a quantitative style of analysis of neural systems, as an opportunity to expand their learning skills towards "synthesis", "quantitative analysis", and "in silico investigation" in Neurobiology.

Schedule

  • Week 0
  • Week 1: live class on Oct 4th 2023 (10:00 - 12:45)

  • Week 2
    • live class on Oct 11th 2023 (10:00 - 12:45)
    • live class on Oct 13th 2023 (9:00 - 11:45)
  • Week 3: live class on Oct 20th 2023 (9:00 - 11:45)

  • Week 4: live class on Oct 27th 2023 (9:00 - 11:45)

  • Week 5: holidays break

  • Week 6
    • live class on Nov 8th 2023 (10:00 - 12:45)
    • live class on Nov 10th 2023 (9:00 - 11:45)

Blended learning: at the start of each week, the professor invites via Discord all students to access and study extra material, presented as a self-paced video (~1h/week). Such blended-learning content introduces, explains, or refreshes certain topics, whose preliminary knowledge is required for attending forthcoming classes.

In class: classes start at 9:00am (sharp) or 10:00am (sharp), and break for 10-15min, every 45-50min.

There is no stupid question. Read it again, please! The professor does NOT judge you or even remember/care of your questions at the final exam. His task is to make sure that you understand the material. If you do not understand something, it is likely that others do not understand it either, but maybe they are shy to ask. Ask your questions! The professor is available for questions during the class or its breaks, as well as before its start and after its end.

Office hours: no explanation is provided by the professor over email. Office hours are available to all students only in person (SISSA, office 441) and upon prior appointment (or on Zoom if you are abroad). Consider posting your question in public, on our Discord server: other students are encouraged to try answering to their peers' questions.

content

0. Mathematical Preliminaries (watch 📺 here)
  • Mathematical functions and their graphs
  • Derivatives and Integrals
  • The Dirac's Delta and the Convolution Integral
  • Ordinary differential equations and their solution
  • Numerical solutions and the Euler forward method
1. Introduction
  • Definition of Computational Neuroscience
  • The Concept of a (mathematical) model
  • Large-scale projects in Neuroscience
2. Neuroelectronics
  • Charge carriers in (semi)conductors and in electrochemical solutions
  • Silver-chloride junction: formula, electrical equivalent model, use
  • Elementary Neurobiology: cellular excitability and the membrane bilayer
  • Lipid bilayer: organization, electrical equivalent, definition of capacitance
  • Ionic membrane permeability, electrical equivalent, definition of resistance
  • Definitions and conversions between density and concentrations
  • Ionic gradients across a biological membrane
  • Mobility of a particle in an aqueous medium, under an external force field
  • Definition of 1-dimensional flux, under a force field
  • Diffusion force field and Fick's law of diffusion
  • Conservation of mass: 1-dimensional Equation of (electro) diffusion
  • Drift force field, under external electrical potential gradient
  • 2-compartments, permeable membrane, sigle ion specie, equilibrium: Nernst equation, electrical equivalent circuit element
  • 2-compartments, permeable membrane, single ion specie, non-equilibrium: ohmic approximation
  • 2-compartments, permeable membrane, single ion specie, non-equilibrium: Goldman equation and non-ohmic approximation
  • 2-compartments, permeable membrane, several ion species, non-equilibrium: the existence of a membrane potential, the Goldman-Hodgkin-Katz equation
3. The Hodgkin-Huxley model
  • Phenomenology of an Action Potential
  • Full equations of the HH model and their intuitive understanding
  • Ions exchanged during an Action Potential (AP)
  • Ion channels: gate model, chemical reactions, mass action law, and kinetic schemes
  • Fast-inactivating sodium channel and its gate model
  • Delayed rectifier potassium channel and its gate model
  • Relationship between injected current and time-to-spike
  • Potassium calcium-dependent ion currents: their impact on neuronal firing
  • Dependency on the extracellular potassium concentration on a AP
  • Effect of selective toxins on a AP
4. Synaptic Transmission
  • Electrical synapses: features and electrical equivalent circuit element
  • Chemical synapses: features and electrical equivalent circuit element
  • Synaptic receptors: nomenclature, ionotropic, metabotropic receptors, modelling
  • Two-state gate-model of chemical synaptic transmission: definition and analysis
  • NMDA-receptor and its membrane potential dependence: definition and modeling
  • Simplification of the two-state model by Dirac's Delta neurotransmitter pulses
  • Elementary analysis of input-output properties of a synapse
5. Short-term synaptic plasticity
  • Short-term depression and facilitation: features and frequency-dependent impact
  • Model of short-term plasticity and simplest model of short-term depression
6. Simplified Spiking Neuronal Models
  • Motivations for reducing complexity
  • Integrate-and-Fire model neurons: analysis and simulation
  • Frequency versus Current (F-I) curve: analytical derivation
  • Absolute refractory period and its effect on the F-I curve
  • Simple model of spike-frequency adaptation
  • Connecting two Integrate-and-Fire neurons by a chemical model synapse
  • Conductance-driven and Current-driven model synapses
7. Firing-Rate Population Models
  • Mean-field approximation of incoming synaptic current, in homogeneous networks
  • Rate model of homogeneous excitatory populations
  • Amplification by feedback
  • Neuronal integrator
  • Attractors and persistent activity
  • Analysis of equilibrium point and their stability: graphical method
  • Persistent activity with two subpopulations
  • Oscillations with two subpopulations
  • Extended dynamical description of the F-I curve: attenuation and phase
  • Self-consistent oscillations in recurrent excitatory and inhibitory networks

Discord & Code of Conduct

We built a Discord server as an (online, real-time) virtual meeting place and as an (offline, asynchronous) forum for questions and answers, for discussions on topics of the course, as well as for the students to offer mutual assistance during their study process. Access is reserved only to students attending the course.

If you do qualify as a legitimate member of our community, simply click on the icon above. Watch out!! After clicking, you will get an error. Then, please edit the URL address you navigate to upon the first click, and add at its very end an extra "five" as a number. This precaution should make life slightly difficult for automated spam BOTs and scammers.

Before joining, please do take a serious look at our Code of Conduct, below:

Code of Conduct of our Class/Discord Server

We are committed to creating a collaborative, open, and inclusive teaching and learning environment. All students, teaching assistants, affiliated faculty, organizers and contributors are expected to adhere to this Code of Conduct.

Participants or affiliates who are asked to stop any inappropriate behaviour are expected to comply immediately. This applies to any events and platforms, either online or in-person. If a participant engages in behaviour that violates this Code of Conduct, the organisers may warn the offender, ask them to leave the event or platform, or engage UniTs/SISSA’s Ombuds Offices to investigate the Code of Conduct violation and impose appropriate sanctions.

Violations of the Code of Conduct should be reported to MG.

1. Be inclusive

We welcome and support people of all backgrounds and identities. This includes, but is not limited to members of any sexual orientation, gender identity and expression, race, ethnicity, culture, national origin, social and economic class, educational level, color, immigration status, sex, age, size, family status, political belief, religion, and mental and physical ability.

2. Be considerate

We all depend on each other to produce the best work we can as an organization. Your decisions will affect students, teaching assistants, and colleagues around the world, and you should take those consequences into account when making decisions.

3. Be respectful

We won’t all agree all the time, but disagreement is no excuse for disrespectful behavior. We will all experience frustration from time to time, but we cannot allow that frustration become personal attacks. An environment where people feel uncomfortable or threatened is not a productive or creative one.

4. Choose your words carefully

Always conduct yourself professionally. Be kind to others. Do not insult or put down others. Harassment and exclusionary behavior aren’t acceptable. This includes, but is not limited to:

  • Threats of violence
  • Insubordination
  • Discriminatory jokes and language
  • Sharing sexually explicit or violent material via electronic devices or other means
  • Personal insults, especially those using racist or sexist terms
  • Unwelcome sexual attention
  • Advocating for, or encouraging, any of the above behavior.

5. Don’t harass

In general, if someone asks you to stop something, then stop. When we disagree, try to understand why. Differences of opinion and disagreements are mostly unavoidable. What is important is that we resolve disagreements and differing views constructively.

6. Make differences into strengths

We can find strength in diversity. Different people have different perspectives on issues, and that can be valuable for solving problems or generating new ideas. Being unable to understand why someone holds a viewpoint doesn’t mean that they’re wrong. Don’t forget that we all make mistakes, and blaming each other doesn’t get us anywhere. Instead, focus on resolving issues and learning from mistakes.

7. Act honestly and with academic integrity

We expect you to respect basic academic integrity principles and take academic integrity to mean adherence to the following values:

  • Honesty
  • Trust
  • Fairness
  • Respect
  • Responsibility
  • Courage.

More information on academic integrity and these values can be found at the International Center of Academic Integrity.

Be honest in your applications and in your potential reasons for missing classes, or project assignments. Take responsibility for your mistakes and work to remedy them. Don’t take the course under someone else’s name or identity.

Resources

GitHub icon
GitHub Repository YouTube icon
Video Material Hyperlink icon
Mathematical Preliminaries Hyperlink icon
Introductory Lecture