A Computational Model of a Microzone of the Cerebellum

1
A Computational Model of a Microzone of the
Cerebellum

• Sule Yildirim1, Greg Dam2, Jim Houk3
• 1Department of Computer Science, Hedmark
  University College, Norway
• Northwestern University Departments of 2Learning
  Sciences and 3Physiology, Northwestern
  University, Evanston IL

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The Purpose of the Model

• The main purpose of this model is to explore how
  Purkinje cells exert control over the intensity
  of firing rate of elemental motor commands.
• Understand how motor commands are set and turned
  off.

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One Microscopic Module
Cerebellar Cortex
Cerebellar Nucleus
Attractor network
Motor Cortex
Elemental Motor command
Sensory cue
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Purkinje cell

• Purkinje cell (PC) controls intensity, velocity
  and duration of a movement.
• Bi-stability of PC It acts like a switch that
  can turn on or off a motor command with also the
  help of a sensory cue.

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The states of the attractor network

• The attractor network has two stable states that
  it can be drawn to depending on
• the firing rate of the Purkinje
• the strength of an input signal into the motor
  cortical neuron
• High state There is sustained maximum positive
  feedback in the attractor network and a movement
  is initiated.
• Low state There is not any positive feedback in
  the attractor network and the movement stops.

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Initiation of a movement command

• Purkinje cell inhibition is off in preparation
  for movement.
• Sensory cue is applied into the motor cortical
  neuron to initiate a movement command.
• When the attractor network moves into the high
  state, a constant velocity movement is commanded.

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The method for computational modeling

• Draw phase portraits, place phase points and
  observe their behavior under the effect of action
  potentials of Purkinje cell, nuclear neuron and
  motor cortical neuron.
• Phase points move to the closest stable fixed
  point. Phase points move away from unstable fixed
  points.

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Abbreviations

• m motor cortical neuron
• n nuclear neuron
• Rm the firing rate of neuron m
• Vm the membrane potential of neuron m
• Rn the firing rate of neuron n
• Vn the membrane potential of neuron n
• T constant time factor
• f activation function
• w the synaptic weight between m and n
• p the firing rate of Purkinje cell
• bb bias input to the motor cortical neuron

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The computational microscopic model
f sigmoid activation function
T d(Vm)/dt Vm wRn - bb (1) Rm
f(Vm) (2) T d(Vn)/dt Vn wRm p (3) Rn
f(Vn) (4)
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Non-linear dynamics(Nullclines)

• Vm nullcline Vm wRn - bb where d(Vm)/dt 0
• Vn nullcline Vn wRm - p where d(Vn)/dt 0

Vm nullcline
Vn nullcline
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Non-linear dynamics(Fixed points)

• Points where Vm nullcline intersects Vn
  nullcline.

Threshold fixed point (source)
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Runge Kutta
Upper trajectory with initial state Vm 3.6 and
Vn -6.6
Lower trajectory with initial state Vm -2 and
Vn 8
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Vector fields

• For each initial state (Vm, Vn), following is
  calculated
• d(Vm)/dt (wRn bb Vm ) / T
• d(Vn)/dt (wRm p Vn) / T
• The values of d(Vm)/dt and d(Vn)/dt define the
  flow in state space followed by the attractor
  network at any given initial state (Vm, Vn).
• DEMO OF A STATE SPACE FOR A GIVEN p VALUE.

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Adjustment of p value

• The current vector fields can be changed by the
  adjustment of p value for a given initial (Vm,
  Vn) state.
• DEMO FOR A GIVEN INITIAL POINT and p VALUE IS
  VARIED.

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Bifurcation

• Sometimes the adjustment of p value can result in
  the removal of a fixed point which is called a
  bifurcation.

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States of the attractor network(the model )

• High state Upper fixed point
• Low state Lower fixed point
• Threshold Middle fixed point

• Sufficient intensity Appears over threshold.
• Maximum intensity occurs at the upper fixed
  point.
• Lowest intensity occurs at the lower fixed point.
  

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A time course for turning on/off a motor command
time
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A simulation of the time course
DEMO
Intensity is sufficient
Intensity is weak
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Purkinje cell discharge Premotor neuron
discharge
Summary of the Time Course for a Motor Command
Purkinje cell discharge starts
The time it takes for Vm to reach the high state.
100/1000 ms for Vm to stay at upper state.
motor command exertion
Purkinje cell firing starts
motor command exertion decays
The time it takes for Vm to reach the low state.
motor command exertion stops
Vm is at its low state.
a sensory input might appear now.
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Simulation results
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Regulation of movement direction
Purkinje cell discharge Premotor neuron discharge
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References

• Ekerot CF, Oscarsson O. (1981) Prolonged
  depolarization elicited in Purkinje cell
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  Encyclopedia of Life Sciences., Macmillan.
• Houk JC, Mugnaini E. (2003) Cerebellum. In Larry
  Squire's Fundamental Neuroscience, V. Motor
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• Houk JC (2005) Agents of the Mind, Biological
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