From d59f471befbcc62aa178b46e2aaad99d58e350f1 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Thu, 29 May 2014 23:17:46 -0400 Subject: tonight's lectures --- lectures/lec13_bistability4 | 20 ++++++++++++++++++++ lectures/lec14_bistability5 | 25 +++++++++++++++++++++++++ 2 files changed, 45 insertions(+) create mode 100644 lectures/lec13_bistability4 create mode 100644 lectures/lec14_bistability5 diff --git a/lectures/lec13_bistability4 b/lectures/lec13_bistability4 new file mode 100644 index 0000000..f0c6363 --- /dev/null +++ b/lectures/lec13_bistability4 @@ -0,0 +1,20 @@ + +Bistability in 2 variable systems + +review: mutual activation (positive feedback loop) + and mutual inhibition (negative feedback loop) + +[review of simple mutual activitation and inhibition physical systems] + +when doing analysis, often want to plot nullclines in phase (aka, variable) +space. the nullcline of a variable is a curve in phasespace where the time derivative of the given variable is zero. + +find these by expressing the differential of the variable (w/r/t time) as a +symbolic expression (probably involving both variables) and solving for equals +0. then we will analyse the intersecting points (which are equilibria, though +not necessarily stable). might need to plot nullcline for varying "other"/free +variables to find a state where there are 3x (or more) intersections, which are +bistable systems. + +overall a bit confused; shouldn't this lecture have come earlier, before the +stability analysis? oh, no, that was a single variable system. diff --git a/lectures/lec14_bistability5 b/lectures/lec14_bistability5 new file mode 100644 index 0000000..e53785f --- /dev/null +++ b/lectures/lec14_bistability5 @@ -0,0 +1,25 @@ +Stability analysis for multi-variate systems + +could do a crude thing: chose points (on nullclines) near an equilibrium and +integrate to see if there is convergence or divergence. + +jacobians! horray. the eigenvalues determine stability. stable if both real +parts of eigenvalues are negative; otherwise unstable. + +claims that analytical evaluation of jacobians is hard but numerical is easy. +hmm. + +the above is the algebaic way. + +geometric way is to calculate gradient w/r/t to time in the areas around and +between nullclines (by calculating at zero/zero and/or very far away at +inf/inf, then "flipping" every time a null cline is crossed (which makes sense +because these are points were derivative is zero). the arrows/gradient "points +to" the stable intersections. + +aka, vector looks like: | d[A]/dt | + | | + | d[B]/dt | + +can also do one-dimensional analysis on individual nullcline lines, again using +"flip if crossing (other) nullclines" trick -- cgit v1.2.3