Thursday, November 3, 2011

The Brain as a Dynamical System

So, I linked to state space (or here, for a different flavor) in passing, but it's time to explicitly bring it up.  The human brain is a dynamical system.  What this means is that we come up with some n-dimensional mathematical space where each brain state is mapped to a different point. That is, any difference in what's going on in the brain is represented as moving around in this space. The next state is determined by the current state and sensory inputs (and possibly noise). All "thoughts" and "feelings" are included in the state. If you want to respond with a counterexample where one state sometimes leads to multiple states given the same conditions "depending on other factors", then ask yourself what these factors are, and then refine your definition of state to include those too.

For purposes here, trying to get to the level of detail of actually starting to write equations too precisely will get us bogged down to the point of being useless. Proceeding without at least a way to visualize, think, and talk about the dynamics, however, is foolish as well. I aim to show a framework to guide your thoughts.

One of the big things here is to be able to draw the system boundary at different sizes. Family therapists might draw it around the family, but I'm most interested in the individual level and smaller. This means looking at subspaces of the entire system and modeling the effect of other dimensions as input.

I like to visualize the state transition as a 2 dimensional vector field (this only depicts values of two out of many dimensions at a time, but the insights are generalizable) where you can Euler integrate and find trajectories after summing an input over the field.

Some of the interesting features you might find are attractors, bifurcation points, and limit cycles. These can come in all sorts of interesting varieties, and sometimes only meet the strict definition when you shrink the system boundary and ignore input disturbances.

This is very simple stuff, but the implications can protect you from a large class of stupidity. For example, if you mind your trajectory and the surrounding landscape, you just don't become an addict, no matter how fun the drug, for the same reason you tend not to fall in physical holes.

Another extremely common failure mode is to directly oppose the presenting problem instead of the causal structure that underlies it. In general, if you're using will power to solve something, you're doing it wrong. If you draw the system boundary around your "subconscious" and model your will power as a input, then if theres any significant problem, you're stuck in a deep attractor. Trying to quit smoking by "trying really hard to stop" is just pushing weakly up a steep hill - it doesn't always work, and when it does it's not fun. If you do manage to get far enough out of the attractor to reach a bifurcation point, then you win... until you fall back in.

A better way to handle it is to sneak out in the next dimension. Our knots don't hold in 4 dimensions. Of course, you still have to not fall back in.

An even better way is to do some landscaping and remove the attractor. Of course, you have to make sure there's no loop outside that in which the smoking attractor existing in the landscape is an attractor for the landscape itself. Solve the outermost control loop, and the rest will solve itself. For permanent.

1 comment:

  1. Very interesting analogies, but more examlpes could by helpful.