Abstract
Cerebellar learning appears to be driven by motor error, but whether or not error signals are provided by climbing fibers (CFs) remains a matter of controversy. Here we show that a model of the cerebellum can be trained to simulate the regulation of smooth pursuit eye movements by minimizing its inputs from parallel fibers (PFs), which carry various signals including error and efference copy. The CF spikes act as “learn now” signals. The model can be trained to simulate the regulation of smooth pursuit of visual objects following circular or complex trajectories and provides insight into how Purkinje cells might encode pursuit parameters. In minimizing both error and efference copy, the model demonstrates how cerebellar learning through PF input minimization (InMin) can make movements more accurate and more efficient. An experimental test is derived that would distinguish InMin from other models of cerebellar learning which assume that CFs carry error signals.
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Abbreviations
- A :
-
Amplitude of input sine waves
- B c :
-
Base firing rate of climbing fibers (CFs)
- b :
-
Bin number for CF spike histogram
- c :
-
CF signal
- \({\dot{{\bf C}}}\) :
-
Total velocity command from frontal eye fields (FEF)
- C :
-
Integrated velocity (position) command from FEF
- d :
-
Delay due to visual system processing
- \({{\bf e},\dot{{\bf e}}}\) :
-
Eye position and velocity
- f :
-
Base frequency of input sine waves
- g r :
-
Gain for smooth eye movement (r)
- g s :
-
Gain for saccades (s)
- g :
-
Purkinje cell (PC) gains specific to a microzone (MZ)
- g i :
-
Gain of individual PC
- Δg :
-
Perturbation specific to an MZ
- \({\left\|\Delta{\bf g}\right\|}\) :
-
Scale of perturbations
- \({\gamma_{\smallint}}\) :
-
Decay rate for brainstem integrator
- γl :
-
Decay rate for long parallel fiber (PF) activity integrator
- γs :
-
Decay rate for short PF activity integrator
- h :
-
Activity vector of all PFs assigned to an MZ
- \({\bar {h}_{\rm l}}\) :
-
Smoothed PF activity from the long integrator
- \({\bar {h}_{\rm s}}\) :
-
Smoothed PF activity from the short integrator
- \({\Delta\bar{h}}\) :
-
Change in PF activity
- i :
-
Generic index
- j :
-
Index of winning PC in self-organizing map (SOM) learning
- L c :
-
Competitive (SOM) learning rate
- L p :
-
Perturbative learning rate
- L d :
-
Delta rule learning rate
- M :
-
Masking matrix for delta-rule learning
- m :
-
Motor command fed into eye plant
- m :
-
Combined output of a single cerebellar MZ
- n :
-
Quantities of various things (PCs, PFs, etc)
- \({{\bf o},\dot{{\bf o}}}\) :
-
Object position and velocity
- \({\phi}\) :
-
Phase lead of input sine waves
- p :
-
Output of the PCs in a specific MZ (before scaling by g)
- P :
-
Combined output of cerebellum following push–pull processing
- \({{\bf r},\dot{{\bf r}}}\) :
-
Retinal position (error), retinal velocity (slip error)
- \({\dot{{\bf s}}}\) :
-
Saccade velocity
- Δt :
-
Duration of one discrete time-step in simulation
- T Δ :
-
PF activity change threshold
- T s :
-
Saccade threshold
- τe :
-
Eye plant time constant; also used to scale velocity command \({\dot{{\bf C}}}\)
- u:
-
Random number in [0,1] drawn from uniform distribution
- W :
-
PF to PC weight matrix
- \({{\bf x},\dot{{\bf x}}}\) :
-
Position and velocity sensitivity vectors
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Acknowledgements
Supported in part by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.
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Rothganger, F.H., Anastasio, T.J. Using input minimization to train a cerebellar model to simulate regulation of smooth pursuit. Biol Cybern 101, 339–359 (2009). https://doi.org/10.1007/s00422-009-0340-7
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DOI: https://doi.org/10.1007/s00422-009-0340-7