Computational Dynamical Systems

The over-arching theory of the whole brain is something I like to call "Computational Dynamical Systems". CDS is essentially the theories that allow for computation through systems of differential equations. Everything about the brain can be modeled as a system of differential equations - and this can be taken down to the smallest details of the brain. We could model the states of every atom and channel with systems of differential equations. However, we won't need to go into that much detail. In the theoretical direction, we need to explore how you put together systems of differential equations in order to do computation.

There are certain concepts from dynamical systems that will be important in engineering computational dynamical systems. Computation requires a fine balance of reproducibility and malleability. We must be able to produce the same outputs for the same inputs. However, if the input information is changed only by a single bit, the system must be able to transition to a completely different state. Local-minima will be the driving causes of these states, and local-maxima will define the transitions between states. To be clear, local-minima and maxima do not need to be 1-dimensional points. I would like to consider stable limit-cycles as local minima, and unstable limit cycles as local-maxima as well.

If you consider the energy metaphor for dynamical systems - something like a marble rolling around a vast landscape of mountains and valleys (in high-dimensional space), then we can begin to draw a picture of how computation may be performed by dynamical systems. The state of the systems will be defined by the location of the marble. Computational time will be the time it takes for the marble to settle into a local-minima (or stable limit-cycle). But the actual computation will have to be done by changing the landscape. When these systems are doing computation, they are not directly changing the state of the system (we do not directly place the marble in certain locations). Rather, they are changing the landscape and the marble rolls down the hill to a new state. The landscape is defined by tons of parameters, and setting these parameters and how they change with different inputs will drive the computational process - the programming of these systems is setting these parameters. So if we want to transition to a new state, we must alter the landscape to form a new local-minima and make sure there is a gradient that drives the marble from its current state to the new local-minima.

Now the brain is a specific implementation of a computational dynamical system. The generalization of the theory leads to such a vast paramter space and possibilities that it is almost impossible to really consolidate everything into a unified thoery of CDS. For scope, every cell in the body is also a computational dynamical system. The DNA and protein networks that exist within all of your cells can also be thought of in this light. The mechanisms and parameter space of this system could be vastly different than those of the brain, but they each would fall under the realm of CDS theory. It will be important to keep these concepts in the back of our minds and try to make some theoretical progress, but this could be extremely hard and maybe even impossible due to the arbitrary nature of computation.