Physics

Introduction [GIM14] Ch. 2 – Structure and function of the nervous system [GIM14] Ch. 3 – Overview of experimental neuroscience methods; Extracellular potentials (LFP) Deterministic dynamics of neurons [GKNP14] Ch. 2.3 / [SGGW11] Ch. 2 – Basis of electrical activity in neurons; ionic channel diversity [GKNP14] Ch. 2.1 / [SGGW11] Ch. 2 – Equilibrium potential; Nernst and Goldman–Hodgkin–Katz equations [GKNP14] Ch. 3 / [SGGW11] Ch. 3 – Hodgkin–Huxley model: action potentials (spikes) and axons [GKNP14] Ch. 3.1–2 – Synapses, synaptic transmission, and dendrites [GKNP14] Ch. 4 – Dimensionality reduction of the Hodgkin–Huxley model; Morris–Lecar and FitzHugh–Nagumo models; phase-plane analysis [S94] Ch. 3, 6, 7, 8 / [GKNP14] – Elements of bifurcation theory for two-dimensional neurons [GKNP14] Ch. 6.1, 6.3.1 – Integrate-and-Fire (IF) neuron models; spike-frequency adaptation Stochastic dynamics of neurons [GKNP14] Ch. 7.1–7.5.2 / [T88] Vol. 2, Ch. 10 – Variability of spike trains as renewal processes [T88] Vol. 2, Ch. 9 / [GKNP14] Ch. 8 – Intrinsic and extrinsic noise [T88] Vol. 2, Ch. 9 / [GKNP14] Ch. 8 – Leaky IF models; diffusion approximation; Fokker–Planck equation [GKNP14] Ch. 8 / [T88] Vol. 2, Ch. 9 – Spike statistics as a first-passage time problem; input–output gain function in stationary regime Networks of neurons and population activity [GKNP14] Ch. 12.1–12.3 – Neuronal populations [GKNP14] Ch. 12.4 – Mean-field approximation [GKNP14] Ch. 13.1–13.4 – Population density approach [GKNP14] Ch. 15.1, 15.3 – Elements of dimensionality reduction and rate models [GKNP14] Ch. 8.2.2, 12.3.4 – Balanced excitation and inhibition Models of cognitive functions [GKNP14] Ch. 19.1 – Hebbian learning; attractor dynamics in networks of IF neurons [GKNP14] Ch. 17 – Models of working (short-term) memory Reference textbook [GKNP14] W. Gerstner, W.M. Kistler, R. Naud, L. Paninski, Neuronal Dynamics, Cambridge University Press, 2014 Optional bibliography [A89] D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge University Press, 1989 [GIM14] M. S. Gazzaniga, R. B. Ivry, G. R. Mangun, Cognitive Neuroscience: The Biology of the Mind, 4th Student Edition, W.W. Norton, 2013 [S94] S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books Publishing, 1994 [SGGW11] D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Principles of Computational Modelling in Neuroscience, Cambridge University Press, 2011 [T88] H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2: Nonlinear and Stochastic Theories, Cambridge University Press, 1988

To successfully follow the course, a good knowledge of the fundamentals of mathematical analysis, linear algebra, and basic statistics is required, as well as familiarity with differential equations, and with the concepts of physics of dynamical systems and stochastic processes. Preliminary familiarity with the basic concepts of neurophysiology and mathematical modeling of complex systems is considered an advantage.

Reference textbook [GKNP14] W. Gerstner, W.M. Kistler, R. Naud, L. Paninski, Neuronal Dynamics, Cambridge University Press, 2014

Attendance highly recommended

Exam methods The examination consists of a single oral test, aimed at assessing the student’s knowledge and understanding of the topics covered during the course, as well as their ability to present the fundamental concepts of the discipline clearly and rigorously. During the exam, the student is required to present a topic of their choice, selected from those included in the course program, demonstrating mastery of theoretical contents and the ability to establish connections with other subjects discussed in the course. Subsequently, the examiner will ask additional in-depth questions on other topics of the program, in order to evaluate the overall completeness of preparation, the consistency of reasoning, and the student’s capacity for critical analysis.

[A89] D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge University Press, 1989 [GIM14] M. S. Gazzaniga, R. B. Ivry, G. R. Mangun, Cognitive Neuroscience: The Biology of the Mind, 4th Student Edition, W.W. Norton, 2013 [S94] S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books Publishing, 1994 [SGGW11] D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Principles of Computational Modelling in Neuroscience, Cambridge University Press, 2011 [T88] H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2: Nonlinear and Stochastic Theories, Cambridge University Press, 1988

Teaching methods Teaching is delivered through lectures, during which the instructor presents the course topics using projected slides. The slides used in class will be made available weekly on the reference website, which will be communicated to students at the beginning of the course. During the lectures, application examples, scientific articles, experimental results and techniques, as well as mathematical demonstrations, will be discussed to support the understanding of the models covered.

Introduction [GIM14] Ch. 2 – Structure and function of the nervous system [GIM14] Ch. 3 – Overview of experimental neuroscience methods; Extracellular potentials (LFP) Deterministic dynamics of neurons [GKNP14] Ch. 2.3 / [SGGW11] Ch. 2 – Basis of electrical activity in neurons; ionic channel diversity [GKNP14] Ch. 2.1 / [SGGW11] Ch. 2 – Equilibrium potential; Nernst and Goldman–Hodgkin–Katz equations [GKNP14] Ch. 3 / [SGGW11] Ch. 3 – Hodgkin–Huxley model: action potentials (spikes) and axons [GKNP14] Ch. 3.1–2 – Synapses, synaptic transmission, and dendrites [GKNP14] Ch. 4 – Dimensionality reduction of the Hodgkin–Huxley model; Morris–Lecar and FitzHugh–Nagumo models; phase-plane analysis [S94] Ch. 3, 6, 7, 8 / [GKNP14] – Elements of bifurcation theory for two-dimensional neurons [GKNP14] Ch. 6.1, 6.3.1 – Integrate-and-Fire (IF) neuron models; spike-frequency adaptation Stochastic dynamics of neurons [GKNP14] Ch. 7.1–7.5.2 / [T88] Vol. 2, Ch. 10 – Variability of spike trains as renewal processes [T88] Vol. 2, Ch. 9 / [GKNP14] Ch. 8 – Intrinsic and extrinsic noise [T88] Vol. 2, Ch. 9 / [GKNP14] Ch. 8 – Leaky IF models; diffusion approximation; Fokker–Planck equation [GKNP14] Ch. 8 / [T88] Vol. 2, Ch. 9 – Spike statistics as a first-passage time problem; input–output gain function in stationary regime Networks of neurons and population activity [GKNP14] Ch. 12.1–12.3 – Neuronal populations [GKNP14] Ch. 12.4 – Mean-field approximation [GKNP14] Ch. 13.1–13.4 – Population density approach [GKNP14] Ch. 15.1, 15.3 – Elements of dimensionality reduction and rate models [GKNP14] Ch. 8.2.2, 12.3.4 – Balanced excitation and inhibition Models of cognitive functions [GKNP14] Ch. 19.1 – Hebbian learning; attractor dynamics in networks of IF neurons [GKNP14] Ch. 17 – Models of working (short-term) memory Reference textbook [GKNP14] W. Gerstner, W.M. Kistler, R. Naud, L. Paninski, Neuronal Dynamics, Cambridge University Press, 2014 Optional bibliography [A89] D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge University Press, 1989 [GIM14] M. S. Gazzaniga, R. B. Ivry, G. R. Mangun, Cognitive Neuroscience: The Biology of the Mind, 4th Student Edition, W.W. Norton, 2013 [S94] S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books Publishing, 1994 [SGGW11] D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Principles of Computational Modelling in Neuroscience, Cambridge University Press, 2011 [T88] H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2: Nonlinear and Stochastic Theories, Cambridge University Press, 1988

To successfully follow the course, a good knowledge of the fundamentals of mathematical analysis, linear algebra, and basic statistics is required, as well as familiarity with differential equations, and with the concepts of physics of dynamical systems and stochastic processes. Preliminary familiarity with the basic concepts of neurophysiology and mathematical modeling of complex systems is considered an advantage.

Reference textbook [GKNP14] W. Gerstner, W.M. Kistler, R. Naud, L. Paninski, Neuronal Dynamics, Cambridge University Press, 2014

Attendance highly recommended

Exam methods The examination consists of a single oral test, aimed at assessing the student’s knowledge and understanding of the topics covered during the course, as well as their ability to present the fundamental concepts of the discipline clearly and rigorously. During the exam, the student is required to present a topic of their choice, selected from those included in the course program, demonstrating mastery of theoretical contents and the ability to establish connections with other subjects discussed in the course. Subsequently, the examiner will ask additional in-depth questions on other topics of the program, in order to evaluate the overall completeness of preparation, the consistency of reasoning, and the student’s capacity for critical analysis.

[A89] D. J. Amit, Modeling Brain Function: The World of Attractor Neural Networks, Cambridge University Press, 1989 [GIM14] M. S. Gazzaniga, R. B. Ivry, G. R. Mangun, Cognitive Neuroscience: The Biology of the Mind, 4th Student Edition, W.W. Norton, 2013 [S94] S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books Publishing, 1994 [SGGW11] D. Sterratt, B. Graham, A. Gillies, D. Willshaw, Principles of Computational Modelling in Neuroscience, Cambridge University Press, 2011 [T88] H. C. Tuckwell, Introduction to Theoretical Neurobiology, Vol. 2: Nonlinear and Stochastic Theories, Cambridge University Press, 1988

Teaching methods Teaching is delivered through lectures, during which the instructor presents the course topics using projected slides. The slides used in class will be made available weekly on the reference website, which will be communicated to students at the beginning of the course. During the lectures, application examples, scientific articles, experimental results and techniques, as well as mathematical demonstrations, will be discussed to support the understanding of the models covered.