Simple Manifold Explains how Saccade Kinematics are Encoded by Purkinje Cells

A plethora of studies have determined that the cerebellar cortex, and the Purkinje cells in particular encode information about movement kinetics. One fruitful set of studies focuses on trained eye saccades in monkeys. It has been shown in many studies that Purkinje cell firing rate encodes peak velocity (PV) of eye movements, higher firing rate correlates with a faster saccade. It also correlates with a longer duration of the saccade, meaning the eye moves a further distance for a longer time.

Purkinje cells are not just tonically firing simple spikes. They also have dendritic spikes. These are characteristic increases in firing rate followed by a brief pause in spiking.

It turns out this burst-pause encodes the similar information as simple spike frequency, with stronger burst frequency correlated with both peak velocity (PV) and duration. It appears that the Purkinje is indeed encoding this information, albeit its unclear what the underlying structure is. (Fig1)

Fig1. Left: Increases in Purkinje cell simple spike firing rate correlates with both peak velocity (PV) and saccade duration. Right: Likewise increases in burst-pause firing rate are correlated with both peak velocity (PV) and saccade duration.

That's where Dr. Markanday, Dr. Hong, Dr. Inoue, Dr. De Schutter, and Dr. Thier come in with there study, Multidimensional cerebellar computations for flexible kinematic control of movements. They use a low dimensional manifold to decipher the underlying encoding principles of the Purkinje cells.

A manifold is a low-dimensional mathematical representations of the complex and often high-dimensional variability of individual Purkinje cells. In other words for vast array of possible activities of individual Purkinje cells, there exists a 2-dimensional subspace (ie manifold) that the dynamics are largely constrained to (see this excellent video for more) . Variation within this 2-dimensional space is sufficient to explain the peak velocity (PV) and duration of saccades.

Here Dr. Markanday et al. calculated the manifold as an oval shaped path that starts from a single point (here a white circle) and seemingly orbits along the manifold and returns to the starting point. Along the way, the peak velocity (PV) occurs lowest point (here marked with the triangle along the orbit. Whats fascinating about this is changing the size of the oval shaped orbit only changes the peak velocity (PV) of the saccade, while changing the rotation speed around the orbit changes the duration of the saccade. (Fig2)

Fig2. Left Column:  cartoon kinematics of saccades, green is a correlated change in both peak velocity (PV) and duration. Orange is only a change in peak velocity (PV). Blue is only a change in duration of saccades. Center Column: The manifolds calculated from Purkinje cells. The shapes and rotation speed of the manifold change as one of the kinematic parameters change. For peak velocity (PV), the orange manifold gets bigger in size as the peak velocity (PV) increase. Likewise, the rotation speed along the orbit increases the longer the saccade. Right: Shows that both manifold size and rotation speed are independent parameters that map independently to both peak velocity (PV) and duration.

Even more striking is that error correcting codes from the climbing fibers (so called complex spikes), induce learning in the PC cells. Some complex spikes only encode errors in peak velocity (PV), other complex spikes only encode errors in duration. While complex spikes induce changes in both simple spike firing rates, and burst-pause dynamics, they only change one variable in the manifold.  The peak velocity error complex spike only changes the manifold size, while the saccade duration error complex spike only changes the rotation speed of the manifold. This implies that error correcting codes used by the cerebellum are modifying this manifold.

To summarize, Purkinje cells use simple spike firing rates and burst-pause dynamics to encode saccade kinematics. Dr. Markanday et al. Have shown there is a a 2-dimensional manifold that encodes peak velocity and saccade duration. Moreover error correcting codes implemented by complex spikes adjusts this manifold, and allows the animal to perform its task better.

Author: Alexander J. White

Original paper: Markanday, A., Hong, S., Inoue, J. et al. Multidimensional cerebellar computations for flexible kinematic control of movements. Nat Commun 14, 2548 (2023). https://doi.org/10.1038/s41467-023-37981-0