Northwestern University Departments of 2Learning Sciences and. 3Physiology .... In: Encyclopedia of Life Sciences., Macmillan. ⢠Houk JC, Mugnaini E. (2003) ...
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. 2
One Microscopic Module Attractor network
Cerebellar Cortex
Cerebellar Nucleus
Motor Cortex Sensory cue
Elemental Motor command
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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. 5
Initiation of a movement command 1. Purkinje cell inhibition is off in preparation for movement. 2. Sensory cue is applied into the motor cortical neuron to initiate a movement command. 3. When the attractor network moves into the high state, a constant velocity movement is commanded. 6
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. 7
Abbreviations m: n: Rm: Vm: Rn: Vn: T: f: w: p: bb:
motor cortical neuron nuclear neuron the firing rate of neuron m the membrane potential of neuron m the firing rate of neuron n the membrane potential of neuron n constant time factor activation function the synaptic weight between m and n the firing rate of Purkinje cell bias input to the motor cortical neuron 8
The computational microscopic model
f: sigmoid activation function
The attractor network
T d(Vm)/dt + Vm = wRn - bb
(1)
Rm = f(Vm)
(2)
T d(Vn)/dt + Vn = wRm – p
(3)
Rn = f(Vn)
(4) 9
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 10
Non-linear dynamics (Fixed points) Points where Vm nullcline “intersects” Vn nullcline.
High fixed point (stable - attractor)
Low fixed point (stable - attractor)
Threshold fixed point (source) 11
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 16 lower fixed point.
A time course for turning on/off a motor command p
Vm time
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A simulation of the time course DEMO
Intensity is weak
Intensity is sufficient
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Purkinje cell discharge
Premotor neuron discharge Purkinje cell discharge starts
Summary of the Time Course for a Motor Command
The time it takes for Vm to reach the high state.
motor command exertion
100/1000 ms for Vm to stay at upper state.
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 •
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Ekerot CF, Oscarsson O. (1981) Prolonged depolarization elicited in Purkinje cell dendrites by climbing fibre impulses in the cat. Physiol. Sep;318:207-21. Holdefer RN, Miller LE, Chen LL, Houk JC.(2000) Functional connectivity between cerebellum and primary motor cortex in the awake monkey. J. Neurophysiol.84:585-590. Houk, J.C. and Miller, L.E. (2001) Cerebellum: Movement regulation and cognitive functions. In: Encyclopedia of Life Sciences., Macmillan. Houk JC, Mugnaini E. (2003) Cerebellum. In Larry Squire's Fundamental Neuroscience, V. Motor Systems, Chapter 32. Elsevier Science, pp. 841872. Houk JC (2005) Agents of the Mind, Biological Cybernetics 92: 427 Sarrafizadeh R, Keifer J, Houk JC. (1996) Somatosensory and movement-related properties of red nucleus: a single unit study in the turtle. Exp Brain Res.108(1):1-17. 22
