Lecture 21
Neural Modeling II
Martin Giese
Aim of this Class
Account for experimentally observed effects in
motion perception with the simple neuronal
model presented in Lecture 18:
•
•
•
•
•
•
Dependence on stimulus strength
Stochastic nature of perception
Psychometric function
Bistable motion perception
Hysteresis
Perceptual switching
Model Neuron
Model Neuron
Population of real neurons
Similar tuning properties:
–
–
–
position
direction
speed
Visual Field
Simplified Neuron Model
Model for single local motion detector
(consisting of many neurons with similar tuning)
“Time
constant”
du (t )

 u (t )  s (t )  h
dt
Rate of change of
average membrane
potential
Resting level
parameter
Stimulus intensity
(input signal)
Average membrane potential
(over all neurons)
Phase Diagram
du
dt
du (t )

 u (t )  s  h
dt
attractor
(u stays constant)
u*
u
u increases
“Phase diagram”:
u decreases
u*
du
dt
u
Attractor
Follow the arrows in the phase diagram:
u(t) changes always in the direction of the attractor u*
u(t)
u(t)
u*
u(t)
u*
u*
t
t
t
u*
u
attractor
du
dt
With Noise
State fluctuates near attractor
u(t)
Probability
u*
u
t
u*
u
attractor
du
dt
With Noise (for Experts)
Exact mathematical description:
Stochastic Process
(Ornstein-Uhlenbeck)
u(t)
u*
t
du (t )

 u (t )  s  h   (t )
dt
white noise
Single Neuron Model
Activation and Perception
Assumption:
Motion is seen only when u(t) is larger
than a threshold value T.
u
T
u
T
u(t)
t
No stimulus
seen
u(t)
t
Stimulus
t seen
No Stimulus s = 0
du
dt
du (t )

 u (t )  0  h
dt
motion seen
attractor
u*
-h
“Phase diagram”:
u
T (threshold)
u*
du
dt
u
Large Stimulus s > 0
du
dt
du (t )

 u (t )  s  h
dt
motion seen
attractor
u*
T
“Phase diagram”:
u
s-h
u*
du
dt
u
Psychometric Function
Stimulus:
-h
motion seen
T
u
P
no
u
u
u
*
Psychometric
function:
P(seen)
1
P
weak
u
u
u*
0
P
strong
u
u
*
u
s
Two Neuron Model
Inhibition
Motivation
Bistable perception by lateral inhibition between
motion detectors !
Inhibition
Two Uncoupled Neurons
Neuron 1:
Neuron 2:
du1 (t )

 u1 (t )  s1 (t )  h
dt
du 2 (t )

 u 2 (t )  s2 (t )  h
dt
No inhibition.
2D Phase Diagram
u
Attractor:
pair of activations
[u1*, u2*]
u2
attractor
u*
u*
du
dt
u
*
du
dt
u
u1
Synaptic Coupling
Neuron for
“horizontal”
Neuron for
“vertical”
Inhibition
Coupling through nonlinear
threshold function !
(Katz & Miledi,
1967)
Two Coupled Neurons
Inhibition of neuron 1
by neuron 2
du1 (t )
 u1 (t )  s1 (t )  h  w1 f (u2 (t ))
Neuron 1: 
dt
du2 (t )
Neuron 2: 
 u2 (t )  s2 (t )  h  w2 f (u1 (t ))
dt
f(u)
Inhibition of neuron 2
by neuron 1
u
2D Phase Diagram
Two attractors !
2 Attractors (for Experts)
Linear system theory predicts for
purely linear interaction:
• not more than one isolated attractor
• if multiple attractors, they are not
asymptotically stable (in all
directions in phase space)
vertical motion seen
2 Attractors
horizontal motion not seen
Two attractors !
vertical motion
not seen
horizontal motion seen
Bistability
basin of attraction for vertical
 perceptual
ambiguity
basin of attraction
for horizontal
Perceptual Switching
Fluctuations can push state
in the other basin of
attraction
 perceptual switch
Influence of the Aspect Ratio
Change of aspect ratio:
•
•
different motion
detectors activated
change of the mutual
inhibition
Stimulus
Inhibition
Relative Stability
 horizontal and
vertical equally
stable
 horizontal more
stable than vertical
Hysteresis
State can not follow change of attractor immediately
 tendency to stay in the same basin of attraction
 hysteresis
Switching Probability
Medium aspect ratio
 switch possible
Extreme aspect ratio
 switch very unlikely
Programming
NeuDyn.m
Simulates neural model with 1or 2 neurons
and returns the time series of activation.
ut = NeuDyn(tsim, u0, S, tau, Q, W);
NeuDyn.m
Simulates neural model with 1or 2 neurons
and returns the time series of activation.
simulation time
steps
time constant
strength of
fluctuations
ut = NeuDyn(tsim, u0, S, tau, Q, W);
NeuDyn.m
Simulates neural model with 1or 2 neurons
and returns the time series of activation.
ut = NeuDyn(tsim, u0, S, tau, Q, W);
initial activation
stimulus strength
(number / vector)
NeuDyn.m
Simulates neural model with 1or 2 neurons
and returns the time series of activation.
ut = NeuDyn(tsim, u0, S, tau, Q, W);
simulated activity
time series
(vector / matrix)
interaction
(vector / matrix)
NeuDyn.m
Simulates neural model with 1or 2 neurons
and returns the time series of activation.
simulation time
steps
time constant
strength of
fluctuations
ut = NeuDyn(tsim, u0, S, tau, Q, W);
initial activation stimulus strength
simulated activity
interaction
time series
Literature
Additional readings (for interested people):
Hock, H.S., Kelso, J.A.S., Schöner, G. (1993). Bistability and
hysteresis in the organization of apparent motion patterns.
Journal of Experimental Psychology: Human Perception
and Performance, 19(1), 63-80.
Giese, M.A. (1999). Dynamic Neural Field Theory of Motion
Perception, Kluwer, Dordrecht, NL.
Wilson, H.R. (1999). Spikes, Decisions, and Actions. Oxford
University Press, Oxford, UK.