lectures 1-2: nonlinear physics and biological...

Jordi Soriano Fradera

Dept. Física de la Matèria Condensada, Universitat de Barcelona

UB Institute of Complex Systems

September 2016

Lectures 1-2:

Nonlinear Physics and

Biological Complexity

■ PhD in scaling laws in statistical Physics (rough fronts).

(imbibition experiments, fractal surfaces, avalanches…)

■ Postdoc in pattern formation during regeneration of polyps.

■ Postdoc in neuroscience and statistical physics.

(percolation, connectivity in neuronal networks)

■ My own research group in Barcelona since 2009:

Collective phenomena in living neuronal networks and biomedicine.

0. Introducing Jordi Soriano’s background

advancing wet front

neuronHydra Vulgaris

“Without a theory,the facts are silent”

Friedrich A. von Hayek

PART 1

■ Linear systems are those in which input and output are related trough

a linear operator H.

H can be any form of linear ordinary differential equations.

One of the simplest examples in Physics is the Hooke’s Law:

1. Linear physical systems

(harmonic oscillator)

Hinput output

A-A

v max

- v max

x





with damping:



Hinput output

A-A

v max

- v max

x

viscous

damping

- c v





with damping:



Hinput output

A-A

v max

- v maxphase diagram

orbits

viscous

damping

- c v

x

angular frequency

trajectories

■ Phase diagrams are very useful to draw the dynamics of the system,

particularly (as we will see later) in nonlinear systems.


Important concept: nullclines and equilibrium points.

The equation is equivalent to the linear o.d.e system:

And that and provide v=h(x) and v=g(x) called nullclines whose

intersection sets the equilibrium points.

x

v

equilibrium point

X-nullcline

V-nullcline

■ Plotting phase diagrams is fundamental to understand the dynamics of

the system Computer assignment!


▫ The physics of the system may help defying some restrictions for the orbit.

Harmonic oscillator: (1/2)kx2 + (1/2)mv2 = ctant (conservation of energy),

i.e. X2 + V2 = CTANT (one and only one circular orbit for a given set of parameters).

▫ Equilibrium points may be stable or unstable. Small perturbations drive the system

back to them (stable) or away (unstable). We will see examples later.

▫ Nullclines help setting flow vectors tangential to the orbit of the system.

▫ The vector f is ⊥ or ‖to the reference system.

▫ The graphical crossing of nullclines

stablish regions with characteristic

directions of the flow.

x

v Harmonic oscillator

x

v

■ Plotting phase diagrams is fundamental to understand the dynamics of

the system Computer assignment!


Damped oscillator

▫ The physics of the system may help defying some restrictions for the orbit.

Harmonic oscillator: (1/2)kx2 + (1/2)mv2 = ctant (conservation of energy),

i.e. X2 + V2 = CTANT (one and only one circular orbit for a given set of parameters).

▫ Equilibrium points may be stable or unstable. Small perturbations drive the system

back to them (stable) or away (unstable). We will see examples later.

▫ Nullclines help setting flow vectors tangential to the orbit of the system.

▫ The vector f is ⊥ or ‖to the reference system.

▫ The graphical crossing of nullclines

stablish regions with characteristic

directions of the flow.


■ More formally, a linear system obey the properties of:

▫ homogeneity: output increases linearly with input

(i.e. no saturation, no threshold,

no feedback…)

▫ superposition (or additivity):

‘output behavior’ = (‘input behavior’)

▫ time invariance:

HH


■ Consequences:

▫ Deterministic (i.e. non stochastic) and predictable:

The state of the system at any moment is known.

▫ Accurate physical modelling from few observations:

Model a few oscillators and you understand all of them!

▫ Universal behavior:

Systems that obey the same linear ODEs behave in the same way.

▫ Analytically solvable.

One of the highest potentials of Physics!

A-A

v max

- v max

General oscillator Series RLC circuit

chargeR = resistance

C = capacitance

L = inductance


■ Are there linear systems in Nature?

YES, and surprisingly close to us:

▫ Wave propagation: sound and light

▫ Elastic structures and materials: from cell membrane to vegetal fibers.

However, linear systems are predominant in industrial and tech. applications.

sound = waves (freq. k)k

2. Nonlinear physical systems

■ Nonlinear systems are ubiquitous in Nature:

▫ Nonlinear behavior is not ad hoc. It may emerge from simple physical laws,

complex interactions, or the intrinsic biochemical functioning of a system.

▫ Nonlinearities often lead to non-trivial, counter-intuitive solutions and

singularities (so be careful!).

Example 1: oscillator with quadratic dumping:

complex damping ~ v2

Counter-intuitive:

at the beginning slows down faster,

but later needs more time to stop!

▫ Nonlinearity is not ignorance: we may know exactly how a system

works, but it may exhibit intrinsic nonlinearities.

▫ Some systems can be linearized to better model them, but that may

suppress their intrinsic richness.

Example 2: Real pendulum:

nonlinear term

(in standard pendulum, )

▫ The standard pendulum is an

example of linearization!

damping


▫ Linearization helps understanding the Physics

and visualize the simplest solutions!

Leads to the same Eqs. as the oscillator!

▫ The real pendulum, with the additional feature of forcing, exhibits another

important nonlinear behavior: chaos.


forcing term

Forcing = 0

C small

- The orbits are very sensitive to the

initial conditions and parameters.

- Hard to numerically integrate

(numerical errors grow exponentially).

- Behavior is unpredictable (though confined!)

Attractors




forcing term

Small forcing

C small






Attractors




forcing term






Attractors

Large forcing

C small

▫ Curiosity: one of the most famous forced pendula in the world is in Spain

(known as ‘Botafumerio’, cathedral of Santiago de Compostela)


The worst accident occurred in 1499, with the pendulum breaking and virtually leaving the church!

▫ The most famous chaotic system is the Lorenz equations (1963).

It was derived as a simplified model for convection, but also describes other

systems such as lasers. 3 coupled nonlinear ODEs.

- Bounded orbits known as strange attractor.

- Very sensitive to parameters and i.c.


Weather predictions fail for > 3-5 days.

mechanical model

(identical ODEs)

CHAOS: aperiodic long-term behavior in a deterministic system that

exhibits sensitive dependence on initial conditions.

3. Chaos and nonlinear dynamics

▫ Stable (●) and unstable (○) points:


■ CHAOS is neither randomness nor violation of physical laws!

■ Nonlinear dynamics and chaos have shown that spatio-temporal

order can spontaneously emerge in a nonlinear (biological) system.

■ Examples: typical features of nonlinear dynamics with implication in

biological systems:

Describe a biological complex system with

a set of equations, and use nonlinear dynamics to

understand its emergence, stability and robustness.

real pendulum

▫ Limit cycles: isolated closed orbits.

▫ Attractors: Structures to which all neighboring trajectories converge.

▫ Strange attractors: those that are

very sensitive to initial conditions.


▫ Bifurcations:

In a biological system, both the equations of

a process and the parameters can change

along time (e.g. external perturbation,

intrinsic fluctuations…)

switching scenarios!

x

r

Lorenz

genes

x

r

▫ Limit cycles: isolated closed orbits.

▫ Attractors: structure to which all neighboring trajectories converge.

▫ Strange attractors: those that are

very sensitive to initial conditions.


▫ Bifurcations:

x

r

Lorenz

genes

x

r

predators

prey

LECTURE 3

PART 2

4. Towards complexity

1 nonlinear element (nle)(one may include linear el.)

nlenle

nle

nle

coupled

nonlinear

elements

“The whole is more

than the sum of its

parts”

complex

system

- Oscillators.

- Neurons (excitable cells).

- Living creatures.

- Lasers.

- Stars, massive cell bodies.

- …

(+ noise)

Coupling:

- Mechanic, Coulomb…

- Biochemistry and wiring.

- Physical/verbal interaction.

- Feedback, wiring in electronics.

- Gravitation.

- …

Complex system

- Chemical reactions.

- Fluids, climate.

- Complex networks.

(brain, human, communications).

- Finance, stock markets.

- Planetary systems, galaxies.

nle

(coupling can be nonlinear!)

Strogatz book: “Nonlinear dynamics and chaos”

▫ Any system formed by a large number of elements with complex interactions

is nonlinear.

Life, weather and climate, geological processes.

Nonlinearities are at the core of the rich spatiotemporal dynamics and

complex behavior of several natural systems.

▫ Simplifications and assumptions are often required. The beauty of modelling

complexity is the ability to increase/decrease the level of detail to understand a

specific problem.

(and keeping its features!)

▫ The scale (temporal and spatial) at

which we look at a problem is

important (social networks, disease

spreading JESÚS GÓMEZ)

5. Towards complexity

▫ Biochemical reactions are intrinsically nonlinear.

▫ Several interactions among elements occurring at the same time (e.g.

electromagnetic, signaling molecules, pressure gradients…)

▫ Living organisms are open systems, i.e. exchange matter and energy with

the environment.

▫ Fluctuations and noise are ubiquitous in practically any scenario.

6. Biological complexity

?But I see order in Nature.

Watch out! Nonlinearity is not randomness!

6. Biological complexity- Several processes simultaneously.

- Efficient in a highly noise environment.

Molecular motors: 3:40


Functional network of genes interactions in Yeast.

- Robust to damage.

- Robust to fluctuations.

- Reliable/efficient.

- Modular.


?

- Reproducible.

- Self-maintained.

- Exquisite variety.





the environment.



?





the environment.



▫ Nonlinearity drives self-organization, emergence of collective

behavior, and symmetry-breaking mechanisms (e.g. attractor

dynamics).

▫ Noise drives the generation of stable patterns, and several

mechanisms are enhanced by noise.

7. Dissecting a complex system

complex

system

For the understanding of any complex system the key point is to build

manageable models that reproduce the observed behavior.

Is it enough? No, the model should also have a predicting power, and

even uncover hidden mechanisms.

Intrinsic nonlinear behavior

in system’s elements

and/or coupling

Exact evolution of the system unknown

Elements and coupling

difficult to describe +

=

Observation /

analysis

Model

Description

and predictionnonlinear analysis

& tools

isolate key features

8. The importance of coupling

■ A system of coupled harmonic oscillators illustrates the emergence

of complex collective behavior, and the key role of coupling.

(go to time 2:50)

(VIDEO)

Harmonic oscillator has

natural frequency

Its dynamics can be

understood as a particle

moving in a unit circle, and

written for convenience as:


▫ The interaction between different oscillators can be written as:

▫ Simplest nonlinear coupling:

And leads to the Kuramoto model:

Can be solved for

You will see its strength in Jesús Gómez models!

coupling linear: superposition of oscillators (not interesting)

Nonlinear: emergence of synchronization

(collective behavior)

coupling constant


▫ Kuramoto is one of the most celebrated models. Captures natural phenomena.

Extensions: - nonlinear oscillators?

- More complex couplings?

Are harmonic oscillators sufficiently rich?

Well.. neurons are not exactly oscillators FitzHugh-Nagumo model.

input outputLINEAR SYSTEM

Higher input, higher response

+++input output

NONLINEAR SYSTEM

“sufficient inputs in Dt to sum up over a barrier”

For a neuron:

integration+ threshold

any?

one or several?

Neurons work by “integrating and firing”

leak

input

threshold?

output

9. Introducing neurons (and excitable cells)

E (+)

I (-)

QUORUM OF INPUTS

“Integrate and fire”


t


Isolated neuron receiving inputs

E (+)QUORUM OF INPUTS



U above a threshold? Fire!

Coupled neuron receiving inputs from other neurons

E (+)

noise

QUORUM OF INPUTS



U above a threshold? Fire!


E (+)

I (-)

QUORUM OF INPUTS


Neuronal dynamics + connectivity + noise = complex collective behavior

Formal description as a nonlinear system

LECTURE 8



■ Cell level: Mutually repressing genes

10. Important examples of biological complexity

▫ Upon a signal, two genes express

proteins that repress each other.

signal signal

repression rates

▫ The system has 1 unstable equilibrium points and 2 stable ones.

▫ Counter intuitive: the cell rapidly switches from one state to another.

In a colony of bacteria, both states coexist, and this may help

to “stay alert” or adapt for changes in the environment.

■ Colony of bacteria level: mutually killing strains

(“rock-paper-scissors” dynamics)


▫ Trivial case: one survivor (3 fixed points)

▫ Actual experiments: limit cycle

(but depends on environment)

reason: range of interaction

is important!


■ Tissue level: a propagating wave of excitation. Tissue is treated as a

continuum system, with direct coupling among cells.

fast variable

(electric potential)

recovery variable

diffusion

Reaction-Diffusion,

propagating wave

Detailed description

of RD systems

LECTURE 3


■ Organism level: pattern formation during embryogenesis.

LECTURE 5

■ Population level: flocks of birds and swarms of insects.

■ Human (socio-technological) level: social networks, disease

spreading (Jesús Gómez).

End of lectures 1-2

Questions and discussion aspects:

- Can complexity be numerically quantified?

- What is more important, nonlinear elements or

nonlinear coupling?

- There exist something called “catastrophe physics”.

Can you image what is it?

TAKE HOME MESSAGE:

- Linear systems are exact. Adding them up retains dynamics.

- Nonlinear systems are in general not exact, and are sensitive to

parameters and initial conditions. But their behavior is bounded

and can be sketched out. Behavior may often scape intuition.

Adding them up increases richness and variability.

- Biological systems are intrinsically nonlinear. Their complex

behavior arises from nonlinear elements and coupling.

References

▫ S. Strongatz, “Nonlinear dynamics and chaos”, Westview Press (2014).

▫ B.E. Kendall, “Nonlinear Dynamics and Chaos”, Encyclopedia of life sciences (2001).

▫ B.R. Smith, “The quadratically damped oscillator: A case study of a non-linear

equation of motion”, Am. J. Phys. (2012).

▫ S. Salcedo-Sanz, “Modern meta-heuristics based on nonlinear physics processes: A

review of models and design procedures”, Physics Reports (2016).

▫ G. Torzo and P. Peranzoni, “The real pendulum: theory, simulation, experiment”, Lat.

Am. J. Phys. Educ. (2009).

“J.A. Acebrón, “The Kuramoto model: A simple paradigm for synchronization phenomena”,

Rev. Mod. Phys. (2005).

▫ J-P. Eckmann et al., “The Physics of Living Neural Networks”, Physics Reports (2007).

▫ A. Schlemmer, “Entropy Rate Maps of Complex Excitable Dynamics in

Cardiac Monolayers”, Entropy (2015).

▫ C. Adami et al., “Evolution of biological complexity”, PNAS (2000).

▫ J.E. Ferrell, “Self-perpetuating states in signal transduction: positive feedback, double-

negative feedback and bistability”, Current Opinion in Chemical Biology (2002).

▫ F. Mosconi et al., “Some nonlinear challenges in biology”, Nonlinearity (2008).

lectures 1-2: nonlinear physics and biological...

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