School on Physics Applications in Biology

• William Bialek (Princeton University, USA)

1.Statistical mechanics for networks of real neurons (or, Thinking about a thousand neurons)

We have made some significant progress here:  (a) Showing that we can build Ising models for patterns of activity deep in the mammalian brain – models that connect to many details of the real data, and much more successfully than more complex, “biologically motivated” models. (b) Developing RG-inspired approaches to more complex systems, including methods for analyzing experiments on 1000+ neurons, and seeing hints that real networks are described by non-trivial fixed points.

2.Coding and information flow in a small genetic network 

• Andrea Cavagna (Roma ISC-Sapienza, Italy)

The physics of flocking: correlation as a compass from experiments to theory

• George Sugihara (UC San Diego, USA)

An Introduction to Empirical Dynamics: Equation-Free (Minimalist) Nonlinear Mathematics for a Data-driven Understanding of Nature — Transforming Observations to Insights
My aim will be to provide a particular perspective that may be of special relevance as we move away from simple 20th century reductionist toy models based on fundamental principles, toward trying to understand how messy natural systems behave. All this is being made possible by the era of Big Data. 21st century holistic science is being enabled by a boon in available data. The math itself is not especially challenging, however the resonance of understanding that can be achieved with a deeper understanding of the implications of simple assumptions like equilibrium, linearity etc. can be significant.

Two key points of emphasis are:
1. Detecting causation in natural nonlinear dynamic systems to uncover mechanism
2. Forecasting as a rigorous way to validate understanding.

Here is some ancillary background text:

1) EDM: Empirical dynamic modeling (EDM) is an emerging data-driven framework for modeling nonlinear dynamic systems. EDM is based on the mathematical theory of reconstructing system attractors from time series data (Takens 1981). Many scientific fields use models as approximations of reality in order to test hypothesized mechanisms, explain past observations and predict future outcomes. In most cases these models are based on hypothesized parametric equations or known physical laws that describe simple idealized situations such as controlled single-factor experiments, but do not apply to more complex natural settings. For example, while we can easily write down an accurate equation for diffusion of gases in a test tube, modeling oxygen concentrations at depth in a large lake (where biology, complex chemistry and physical currents intervene) is impractical with explicit equations. Empirical models, which infer patterns and associations from the data (instead of using hypothesized equations), represent an alternative and highly flexible approach.

2) Taken’s theorem: The basic underlying goal of EDM is to reconstruct the behavior of dynamic systems from time series data. This approach is based on mathematical theory developed initially by (Takens 1981), and expanded on by others (Casdagli et al. 1991, Sauer et al. 1991, Deyle and Sugihara 2011). Because these methods operate with minimal assumptions, they are particularly suitable for studying systems that exhibit non-equilibrium dynamics and nonlinear state-dependent behavior (i.e. where interactions change over time and as a function of the system state).

3) Despite the ubiquity of nonlinear dynamics in nature nearly all attempts to understand them in applied contexts (outside of formal studies of turbulence) have used incorrect linear statistical tools (static analytical tools based on a classical linear paradigm). This paradigm based on stable, stationary equilibrium points or cyclic equilibrium dynamics allows systems to be studied piecewise as a decomposable sum of independent parts; a tractable approach that applies robustly in designed engineering contexts. As a consequence, an extensive methodological tool chest has evolved for analyzing linear (separable) systems. Indeed the availability of linear tools seems to be the main reason why these methods and concepts continue to be used in non-engineering contexts, despite the obvious problem that they do not match our current views of how most real non-engineered systems are structured (interdependently) and actually behave (i.e, exhibiting non-stationary, non-equilibrium and non-separable state dependence)

4) There is growing consensus that most living systems are rarely at equilibrium. Rather these systems will exhibit dynamical behavior that in many cases is nonlinear. Nonlinearity in living systems means that its parts are interdependent – variables do not act in a mutually independent manner; rather they interact, and as a consequence correlations between them will change as the overall system state changes. Thus, for many biological systems from neurons to ecosystems, behavior is driven not by a few factors acting independently, but by complex interactions between many components acting together as sequenced events in time – nonlinear dynamics.

EDM Readings and Resources:

I. Two Main Summary Readings:

1) Causation: http://www.sciencemag.org/content/338/6106/496.full?keytype=ref&siteid=sci&ijkey=GzlL9h2cAY51A

2) Model-free:

http://www.wired.com/2015/10/is-it-foolish-to-model-natures-complexity-with-equations/

or   PNAS Commentary:

http://www.pnas.org/content/112/13/3856.short

II. Essential Resource:

The rEDM package on CRAN is a key resource that contains a tutorial that many have found useful.  It consists of code and vignettes with data that will help implementing EDM analysis on the data sets students want to analyze in class.

rEDM package: https://cran.-project.org/web/packages/rEDM/index.html

III. Additional Readings that describe the two main EDM forecasting methods and a short note addressing time lags in CCM.

1) Simplex (1990):

http://www.nature.com/nature/journal/v344/n6268/abs/344734a0.html

2) S-maps original paper. (1994)

http://rsta.royalsocietypublishing.org/content/348/1688/477

3) S-maps. Deyle et al (2016)  – contrary to classic model assumptions shows for the first time that ecological interactions are episodic!

http://rspb.royalsocietypublishing.org/content/283/1822/20152258

4) CCM with time lags. Ye et al (2015): https://www.nature.com/articles/srep14750

IV. Mathematical-Theoretical Background Readings

1)     Embedology: Sauer, Yorke, Casdagli (1991). An excellent summary of state space reconstruction results

https://link.springer.com/article/10.1007%2FBF01053745

2)     Deyle and Sugihara (2011):

http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0018295

3) Ye and Sugihara Science 2016: Multiview Embedding   (information leverage)

http://science.sciencemag.org/content/353/6302/922/tab-figures-data

V. Supplementary Readings with Case Examples

1)    Dixon et al Science (1999): http://science.sciencemag.org/content/283/5407/1528.full

http://deepeco.ucsd.edu/~george/publications/01_noise_nonlinearity.pdf

2)    McGowan et al Ecology (2017)   SIO Red tides: http://onlinelibrary.wiley.com/doi/10.1002/ecy.1804/full

3)    Hsieh et al Nature (2006):  https://www.nature.com/articles/nature03553

4)    Ye et al PNAS (2015): http://www.pnas.org/content/112/13/E1569

5)    Glasser et al (2014): http://onlinelibrary.wiley.com/doi/10.1111/faf.12037/full

6)    Deyle et. al. (2013): Climate effects on pacific sardines. Explains scenario exploration with EDM.  http://www.pnas.org/content/110/16/6430.full

7)    Deyle et. al. (2016):Global Environmental Drivers of Flu. An interesting use of scenario exploration to discover a temperature threshold (75F) below which absolute humidity inhibits flu transmission and above which it promotes flu transmission.   http://www.pnas.org/content/113/46/13081.full

VI. Ancillary:

YouTube Playlist of 3 one-minute Animations (from the supplement of the Causality paper in Science 2012):

https://www.youtube.com/watch?v=fevurdpiRYg&list=PL-SSmlAMhY3bnogGTe2tf7hpWpl508pZZ