Neural Network of the Cerebellum: Temporal
Discrimination and the Timing of Responses
Michael D. Mauk
Dean V. Buonomano
The cerebellum is important for initiating smooth,
directed movements.
Damage to the cerebellum causes severe movement
deficits, including poor ability to time movements in
response to external stimuli or in directed action.
The speculation is that there must be a distinct
biological mechanism within the cerebellum that
encodes time differences between sensory inputs.
Other models of the biological timing mechanism depend on
delays between units, varying time constants, or other
imposed design choices.
The benefits of Mauk’s neural network model:
• Temporal information about the stimulus is encoded in a
subset of units activating the output unit.
• Emulating a conditioned stimulus/unconditioned stimulus
response only takes one training phase to learn the temporal
information before the testing phase
• Information for multiple different stimuli can be encoded.
Multiple unconditioned stimuli over time can be coded for a
single conditioned stimuli. Conditioned stimuli patterns can
span multiple time steps and temporal information can still be
retained.
Some basic cerebellum anatomy:
Mauk uses theories of structure proposed by Marr and Albus in which:
• Climbing Fibers = outside input that contain error signals, modifies Purkinje
cell synapses
• Mossy Fibers = provide sensory stimulus information to the granule and golgi
cells
• Granule cells = encode the “context” in which movements take place
• Golgi cell = provides negative feedback to granule cells to stabilize cell activity
• Purkinje cell = provides the appropriate output, in this case a timed motor
movement in response to a stimuli
Some basic cerebellum anatomy:
Some basic cerebellum anatomy:
Golgi Cell
The model hypothesis:
The structure of interactions between the granule cell layer
and the golgi cell layer allows population subsets of the
granule cell layer to represent physical and temporal
information about the stimulus.
In other words, a subset of granule cells will encode not only a
pattern of activations that identify the unique input pattern
(stimulus), but also how much time has elapsed from the onset
of the input pattern. This is achieved by the mossy fiber
“input” layer seeding the feedback loop between the granule
cell layer and the golgi cell layer.
How does this work?
1.Input comes through the mossy fiber and activates a subset
of granule cells.
2.These granule cells activate a subset of golgi cells on the golgi
cell layer.
3.The activated golgi cells inhibit back to the granule cell layer
in a negative feedback loop, but inhibit a different, overlapping
subset of granule cells than were activated by the initial mossy
fiber input.
4.This negative feedback loop between layers creates a
“dynamic, nonperiodic population vector” of granule cell
activity representing the stimulus pattern, even if the mossy
fiber input is periodic.
5.Changing the weights on a particular granule cell subset that
represents the correct time interval can represent that interval
in purkinje cell activation levels.
• Pyramidal regions depict the subset of granule or golgi cells which a cell on the
other layer is able to contact. Within this subset of cells, the connections are
uniformly distributed.
• White cells in the diagram depict post-synaptic cells that end up receiving input
from the pre-synaptic cell in the other layer.
Specifics of the neural network:
• 10,000 granule cells
• 900 golgi cells
• 500 mossy fiber inputs
• 1 purkinje cell output (graded activation)
• A single granule cell receives excitatory input from 3 mossy
fiber inputs and inhibitory input from 3 golgi cells
• A single golgi cell receives excitatory input from 100 granule
cells and 20 mossy fiber inputs
• The purkinje cell receives input from all the granule cells
A clearer diagram:
PC
Golgi Cell Layer
Granule Cell Layer
Mossy Fiber Input Layer
Activation update mechanism:
“Integrate and fire” cell types:
Vi Go = the voltage of each golgi cell
Thri Go = the threshold voltage for the golgi cell i to fire
Gi Go:leak = current leak from golgi cell I
Gi Go:MF = mossy fiber to golgi cell synaptic current
Gi Go:Gr = granule cell to golgi cell synaptic current
All inputs to the cell are summed together into a single current, which
saturates at 1.0 and decays according to a set decay constant
Activation update mechanism cont.:
This represents the synaptic current of a mossy fiber to golgi cell I with
respect to time.
Sn MF = representation of a spike in a mossy fiber
Wgo MF = synaptic weights for the mossy fiber synapse at the golgi cell
Synaptic currents emulate instantaneous rise in voltage and exponential
decay in that voltage after spiking.
Granule cells are controlled by similar equations, but they have additional
inhibitory versions of the equations from the golgi cells.
Specifics of the neural network cont.:
• Initially all granule cells are connected to the single output
Purkinje cell with the same weights
• When the first stimulus (conditioned stimulus emulation) is
presented, the weights of the granule cells active within that
window to the purkinje cell are decreased
• this simulates LTD produced by co-activation of the
climbing fibers and parallel fibers to the purkinje cell
• The “voltage” of the purkinje cell is a weighted summed
activity of all granule cells in the network with a time constant
of 2.5 msec
• Mossy fiber activation patterns “seed” activation of different
subsets of granule and golgi cells at each time step
Specifics of the neural network cont.:
• In training trials:
• at 200ms, unconditioned stimulus simulated by
decreasing the strength of the all weights projecting from
activated granule cells to the purkinje cell, like in the
conditioned stimulus
• In testing trials:
• the unconditioned stimulus is not simulated, but because
the pattern of activation of granule cells is the same as in
the training period (same initial mossy fiber activation
pattern seed) there is a decrease in activation at the same
time interval
• The model is capable of learning timing for multiple different
stimulus patterns, and stimulus patterns over a series of timesteps
Top line represents the % of mossy fibers active in each time bin. Initial increase in
total mossy fiber activation represents the conditioned stimulus.
The bottom line represents purkinje cell activity in the testing phase. After training
an unconditioned stimulus at 200ms the granule cell subset during that time step
have lower weights, dropping the purkinje cell voltage.
The model is very sensitive to noise. Variance in the mossy
fibers used to signal the conditioned stimulus, variance in the
pre-conditioned stimulus state of the model, and variance in
constants of units such as threshold all have detrimental
effects on the network’s ability to train the timing for the
unconditioned stimulus.
The network here is
trained to respond to
US at 125, 200, and
225ms. The injection
of noise eliminates its
ability to predict the
US after enough has
been injected into the
model.
Additional weaknesses of the model:
• The noise can be decreased by decreasing the influence of
the mossy fibers on the golgi cells, but this also decreases its
ability to retain temporal information.
• The model can discriminate temporal information for
conditioned stimulus and unconditioned stimulus simulations,
where the timing is absolute, but there is no mechanism for
learning relative timing between sensory information patterns.
The rhythm of a song, for example, is learned regardless of the
tempo that the song is played at.