Upon successful completion of this course, students will be able to: 1. Grasp the fundamental building blocks of mathematical analysis, including real numbers, functions, limits, and continuity. Students will develop a solid understanding of domains, ranges, and elementary functions such as polynomials, exponential and trigonometric functions. 2. Develop proficiency in calculating derivatives and integrals. Students will learn to interpret derivatives as rates of change, apply derivatives to solve problems involving tangent lines and linear approximations, and utilize the Fundamental Theorem of Calculus for integration to solve problems related to areas and other applications. 3. Acquire skills to solve simple differential equations and analyze their solutions, understanding concepts like fixed points and stability. This includes applications to modeling problems in neuroscience. 4. Engage with higher-order derivatives, Taylor expansions, and numerical approximations to understand local and asymptotic behavior of functions, including extrema, concavity, and behavior at infinity. 5. Understand and apply concepts of linear algebra, including matrix operations, systems of linear equations, vector spaces, and linear transformations. Students will also master eigenvalues and eigenvectors, diagonalization, and applications to stability analysis. 6. Delve into complex numbers, exploring their algebraic properties, geometric interpretation, and applications. Understand and apply concepts of orthogonality, inner product spaces, and orthogonal bases, critical for various applications in mathematics and physics. By the end of this course, students will have developed a comprehensive mathematical toolkit, enabling them to tackle complex problems in neuroscience. This will include a proficiency in visualizing mathematical concepts in multiple dimensions and applying mathematical theory to practical and theoretical problems.

K2: Understands and applies the most appropriate mathematical, statistical and computational tools within Neuroscience, appropriately using spreadsheets for data management, and appropriate graphical representations for data presentation. S1: Uses a variety of techniques to find, manage, integrate and critically evaluate available information for the development of professional activities in Neuroscience, especially in the digital sphere S5: Appropriately uses the scientific and technical vocabulary of the different subfields within Neuroscience. C2: Apply knowledge about the organisation, structure and function of the Central Nervous System (CNS) to contribute to the evolution and improvement of technologies and systems for computing, data handling and analysis. C3: Apply knowledge about technologies for the study of the Nervous System and the brain (Medical Imaging, brain-machine interfaces) to develop new systems for diagnosis and treatment, as well as and other applications within Neuroscience (Artificial Intelligence, Robotics) with the aims of improving the quality of life and furthering social progress. C4: Uses advanced mathematical, statistical and computational tools to increase and improve knowledge in neuroscience and its applications. C5: Apply your neuroscience knowledge in a unifying and integrated fashion as part of a multidisciplinary team (pharmaceutical sector, health industry, diagnostic techniques, health information technologies, government agencies and regulatory bodies. C6: Apply the results of your comprehensive training to your everyday professional activities, combining Neuroscience knowledge with a solid foundation of ethical responsibility and respect for fundamental rights, diversity and democratic values. C7: Apply the scientific and technical principles you acquired during your undergraduate training, together with your own natural learning capabilities, to better adapt to novel opportunities arising from scientific and technological development.

1. Building blocks - Introduction to real numbers - Functions - Basic definitions of functions: input, output, domain, range - Limits, continuity - Elementary functions (polynomials, exponential, trigonometric) 2. Derivatives - What is the derivative? Rate of change - Introduction to differential equations - Properties of the derivative - Tangent line - Linear approximations - Basic derivatives 3. Higher order derivatives - Interpretation - Taylor expansions - Introduction to power series and Taylor series - Numerical approximations 4. Local and asymptotic behavior of functions - Relative extrema - Concavity and convexity - Limits at infinity, asymptotes 5. Integration - Calculating areas: the Riemann integral - Properties of the integral - Fundamental Theorem of Calculus - Basic integration methods 6. Differential equations - Solutions of differential equations - Fixed points, stability - Linear stability analysis 7. Linear Functions in many variables - Linear Differential Equations as motivation - Systems of linear equations - Linear Transformations - Matrix representation 8. Vector spaces - Vector spaces - Bases and Linear Combinations 9. Matrix algebra - Inverse - Determinant 10. Eigenvalues and eigenvectors - Definition - Eigenvectors as a new basis - Diagonalization 11. Linear differential equations - Equilibrium points - Linear stability analysis 12. Complex numbers - Complex eigenvalues as motivation - Binary representation - Modulus, argument - Geometric interpretation - Polar representation - Euler's formula 13. Orthogonality - Inner product - Norm, distance - Orthogonal sets - Orthogonal bases 14. Fourier series (optional) - Introduction, motivation - Periodic functions - Derivation - Fourier series as change of basis

The learning methodology will include: - Attendance to master classes, in which core knowledge will be presented that the students must acquire. The recommended bibliography will facilitate the students' work. - Resolution of exercises by the student that will serve as a self-evaluation method and to acquire the necessary skills. - Exercise classes, in which problems proposed to the students are discussed. - Partial exams. - Final Exam. - Tutorial sessions. - The instructors may propose additional homework and activities.