There is something gratifying to view a Parkinsonian tremor as a super critical hopf bifurcation.

This point is made in a lecture by the great Steven Strogatz, who while teaching a class on non-linear dynamics at Cornell, likened the tremor associated with parkinson’s disease to a bifurcation leading to the creation of a stable limit cycle. He further speculates that the bifurcation could arise potentially from an alteration in the signal delay in the corresponding neural circuitry which controls the movement of the tremoring hand. For Strogatz, the proposed mechanism reaches beyond mere professional interest, as he goes on to mention how his father suffered from the debilitating consequences of Parkinson’s disease. At an emotional juncture in the otherwise nerdy lecture, Strogatz recounts how his father used to be an avid golfer and tennis player before Parkinson’s struck and, in his own words, ‘messed him up real bad towards the end of his life’. For someone who has family members who’ve gone through the torments of neurological disorders, this was a crucial insight to ponder upon.

It is common knowledge that the brain is an elaborate circuit consisting of spiking units called neurons. Each neuron has the capacity to receive inputs from a multitude of neurons, parse through the contents of those messages and take a decision on whether it needs to pass on that message to other multitude of neurons. This can be likened to a system of pendulums connected through rods or springs. Each pendulum receives mechanical inputs from all the other pendulum(mediated by the spring/rod) and without any delay passes on the information to other pendulum. It is easy to see how the resulting global dynamics cascades. The same goes for neurons, except 1.they don’t convey information continuously and 2. don’t directly pass information back to the neurons they receive information from. Crucial as they seem, but we can safely discount the differences for our purposes. What’s of import here is that, given the system lies in a well defined parameter space, it is capable of displaying interesting deterministic dynamics. Imagine a system of connected pendulum where individual pendulum sways at a phase that is synchronized to some of the other pendulum connected to it. Still further, imagine a scenario where pendulum sway at a fixed frequency in a sequential manner, recruiting more and more pendulum along the chain .Now connect the end point of that chain to a muscle group and what we have is a rudimentary way of moving that hand at a certain desired frequency. Ofcourse, we’ve stripped down this description considerably so that what remains is just the primary intuition.

So in order to achieve a certain desired state, the brain instantiates a circuit that has the desired target as a fixed point of the system. Let’s assume that I’ve asked you to hold your right hand perpendicular to your torso. After listening to my instructions your brain will instantiate a circuit which is connected to the muscles that are responsible for lifting your right hand. Envisioned as a dynamical system, the network configuration that keeps your hand stable in the perpendicular position, would correspond to a stable fixed point of the network dynamical equations. Now, for some reason(pathology, too much coffee etc.), let’s assume that the fixed point loses stability and gives way to a limit cycle solution. What you have now is a low frequency tremor. Hence, we’ve successfully envisioned a clinical syndrome as a bifurcation of a dynamical system. Similar reasoning can be applied to understanding another ‘dynamical disease’ called epilepsy. Due to various reasons like genetic malformation, injury or even dehydration, some networks of the brain lose stability and enter into a hypersynchronous state.