Sequential Discrimination by Integral FeedbackControl

1
Sequential Discrimination by Integral
FeedbackControl
Paul Miller and Xiao-Jing Wang, Brandeis
University
Experiments by group of R. Romo et al.,
UNAM Nature 399470 (1999), Cereb. Cort. 131196
(2003)
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Delay activity in PFC ties the task together
Tuned neurons
Firing rates
Primary somatosensory cortex
Secondary somatosensory cortex
Premotor cortex
Prefrontal cortex
Romo et al. Philos Trans Roy Soc Biol, 2002
3
Graded short-term memory requires an integrator
Input
Time
Memory activity
Time
4
... but integration yields amplitude x time
Input
Time
Memory activity
Time
5
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Input
cue 1 low
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
t
cue1
delay
cue2
7
Input
cue 2 higher
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
t
cue1
delay
cue2
8
Input
cue 2 lower
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
Threshold not reached
t
cue1
delay
cue2
9
Input
cue 1 high
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
t
cue1
delay
cue2
10
Input
cue 2 lower
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
Threshold not reached
t
cue1
delay
cue2
11
Input
cue 2 higher
t
IDInput -WMD rM
rD
t
t
cue2
delay
cue1
rM
t
cue1
delay
cue2
12
Amplitude dissociated from duration of input
cue 1low
Input
t
rD
IDInput -WMD rM
t
t
cue1
delay
cue2
rM
t
cue1
delay
cue2
13
Amplitude dissociated from duration of input
cue 1longer
Input
t
rD
IDInput -WMD rM
t
t
cue1
delay
cue2
rM
t
cue1
delay
cue2
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Activity of model discriminating neuron for f2 lt
f1.
base
f1
delay
comparison f2
2Hz
10Hz
6Hz
14Hz
18Hz
10Hz
14Hz
22Hz
18Hz
26Hz
22Hz
30Hz
26Hz
34Hz
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Activity of model discriminating neuron for f2 gt
f1.
base
f1
comparison f2
delay
18Hz
10Hz
22Hz
14Hz
18Hz
26Hz
30Hz
22Hz
34Hz
26Hz
38Hz
30Hz
42Hz
34Hz
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Trial-averaged firing rate through time of model
discriminating neuron for different pairs of
stimuli
Comparison, f2
Delay
Base, f1
100
f1 10Hz
f1 22Hz
f2gtf1
f1 34Hz
Firing rate (Hz)
f2ltf1
0
0
4
Time (sec)
3.5
0.5
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Delay tuning
Base tuning
Comparison tuning
f2gtf1
f2ltf1
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Trial-averaged firing rate through time from
experimental data of Romo (prefrontal cortex)
Delay
Comparison, f2
Base, f1
35
f112Hz f120Hz f128Hz
f2gtf1
Firing rate (Hz)
f2ltf1
0
4
Time (sec)
3.5
0.5
0
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PFC cell from Romo's data
Initial tuning ve to f1 final tuning to f2-f1
Comparison, f2
Delay
Base, f1
60
f2gtf1
f110Hz f122Hz f134Hz
Firing rate (Hz)
f2ltf1
0
0
4
Time (sec)
3.5
0.5
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PFC cell from Romo's data
Initial tuning -ve to f1 final tuning to f1-f2
Comparison, f2
Delay
Base, f1
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f110Hz f122Hz f128Hz
f2ltf1
Firing rate (Hz)
f2gtf1
0
0
4
Time (sec)
3.5
0.5
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Two models presented today readily distinguished
by data (A,B) this talk (C,D) Machens and Brody
(Science, 2005)
Fit rate, r(t) r(t) a(t) f1 b(t) f2
c(t) plot a(t) and b(t) to see tuning to
f1and f2.
Dashed diagonal is discrimination (f1-f2) or
(f2-f1)
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Acknowledgments
Grateful for Funding NIH-NIMH for K25 Award
(PM) NIH and Swartz
Foundation (XJW) Ranulfo Romo (monkey
experimental data) Carlos Brody (monkey
data analysis and discussion) Members of
Brandeis University's Volen Center and Wanglab
for discussions, suggestions and other help,
especially Alfonso Renart Albert
Compte Caroline Geisler Eugene Carter
Larry Abbott
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A noisy integrator performs a random walk
Input
Time
Memory activity
Time
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(No Transcript)
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(No Transcript)
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Analysis Separate time-scale for ring switching
In stable UP state
Preceding a switch down
Turnover
Turnover
No. of active subunits, single ring
Total no. of active subunits
Time (hrs)
Time (hrs)
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Analysis Separate time-scale for ring switching
Goal Rapid speed-up by converting system to 1D
and solving analytically. Method Essentially a
mean-field theory. Justification Changes to and
from P0 (unphosphorylated state) are slow.
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Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
30
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
31
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
32
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
33
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
34
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
35
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
36
Analysis Project system to 1D
1) Number of rings on with any activation,
n. 2) Assume average number, P, of subunits
phosphorylated for all rings on. 3) Calculate
reaction rates for one ring, assuming contibution
of others is (n-1)P. 4) Calculate average
time in configurations with these reaction
rates. 5) Hence calculate new value of P. 6)
Repeat Step 2 until convergence. 7) Calculate
rate to switch on, rn, and off, r-n. 8)
Continue with new value of n.
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Analysis Solve 1D model exactly
rn-1
rn1
rn
N1
N0
n1
n-1
n
n2
r-n2
r-n1
r-n
Time to hop from N0 to N1
Use rn Tn 1 r-n1Tn1 for N0 n lt N1
rn Tn r-n1Tn1 for n lt N0
Tn 0 for n N1 Average total time for
transition, Ttot ?Tn
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Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes
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Optimal system lifetime is a trade-off between
lifetimes of UP and DOWN states
10 yrs
DOWN state lifetime
1 yr
Average lifetime of state
UP state lifetime
1 mth
1 day
Number of PP1 enzymes
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Integral feedback control
Input
Firing-rate analysis ? drD -rD - wMD rM
Input dt ? drM -rM wMM rM wDM rD ? wDM
rD dt
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Total current, I(total) external current
feedback current
Why bistability (A) is easier to obtain than a
continuum of stable states (B).
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Decision-making 2-choice task,
winner-takes-all competition
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Standard decision task vs. Working Memory
Task (moving dots left/right)
(vibration on finger) Inputs concurrent
Inputs separated in time Inputs from
antagonistic, Stimuli activate same
opposing stimuli afferents
?
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Negatively monotonically tuned neuron
Romo et al. Nature 1999
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Neuronal activity correlated with the
decision.
Romo et al. (2002) Nat Neurosci 51217
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Tuning during first stimulus
Tuning during second stimulus
f2 gt f1
f2 lt f1
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Network model using leaky integrate-and-fire
neurons
threshold
reset
Presynaptic Spike
Postsynaptic Current
Spike
Membrane potential, V
Isyn(t) ?gisi(t)(V-Vi) ? dsi -si(t)
(1-si)??(t-tspike) dt
?? dV -gLV(t)-VL Isyn(t) dt
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Discrete Integrator
Continuous Integrator
Integrates everything, so with feedback
inhibition, suppresses everything.
Threshold for integration, only suppresses
inputs above threshold.