Life and several natural phenomena are complex systems governed by nonlinear laws. Interestingly, the intrinsic nonlinear nature of these systems is precisely what allows them to exhibit a broad repertoire of behaviors, from rich spatiotemporal dynamics to the emergence of patterned structures and collective phenomena. In this series of lectures, I provide an overview of Complexity in Living Systems, with special attention to the brain and neuronal networks, and I describe the fundamental tools from nonlinear Physics that help describing and understanding them.
The lectures start with a presentation of the concepts of nonlinear dynamics and biological complexity, and with substantial examples to illustrate how the simplest nonlinear systems already manifest extraordinary behaviors, often surprisingly counterintuitive. They next jump to the description of Turing patterns to illustrate how nonlinearities allow for the emergence of organized structures out from disorder. They then introduce the importance of noise and fluctuations in biological systems, and to assert their fundamental role in driving complexity. These lectures set the basis for understanding two problems of special interest: multicellular organization and the brain. The first one is focused on the metazoan Hydra, an animal model for self-organization, regeneration and biological pattern formation. The second one initially covers the state of the art for brain functioning and its challenges, to later explore specific paradigms. Concretely, and in the context of complex networks and experiments in small living neuronal networks, the lecture explores the problem of describing a neuron and an assembly of connected neurons, to next treat the inference of structural and functional connectivity, the resilience to damage, and neuronal network collective dynamics. The lectures finish by illustrating the ability of Physics and Mathematics to continuously supply elegant concepts and analysis tools for understanding complex system, and show how concepts as different as percolation and information-theory are helping to extract connectivity traits in living neuronal networks.

Lectures 01-02:
Nonlinear Physics and Biological Complexity

Lectures 03:
Emergence of Order from Disorder: Turing Patterns

Lectures 04:
Importance of Noise and Fluctuations

Lectures 05:
Multicellular Organization and Hydra Regeneration

Lectures 06:
Brain Functioning and its Challenges

Lectures 07:
Networks in the Context of Neuroscience

Lectures 08:
Neurons as Nonlinear Systems: FitzHugh-Nagumo and Collective Dynamics

Lectures 09:
Percolation in Living Neuronal Networks

Lectures 10:
Information Theory and Connectivity Inference