robustness in neurons & networks - yale university · robustness in dynamical systems...
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Robustness in Neurons & Networks
Mark Goldman
(H.S. Seung, D. Lee)
eye position representedby location alonga low dimensional
manifold (“line attractor”)
saccade
Many-neuron Patterns of Activity Represent Eye Position
Activity of 2 neurons
Line Attractor Picture of the Neural Integrator
Decay along directionof decaying eigenvectors
No decay or growth along direction of eigenvector with eigenvalue = 1
Axes = firing rates of 2 neurons
“Line Attractor” or “Line of Fixed Points”
Geometrical pictureof eigenvectors:
r1
r2
v1
v2I2
I1M12
M21
0.8
Examples
=
0.8 =
Q: Can you guess what input pattern Iwill be amplified most?(i.e. eigenvector with largest λ)
Which will be compressed most?(i.e. eigenvector with smallest λ)
A: [1 1] is amplified most amplifies common input[1 -1] is compressed most attenuates differences
Effect of Bilateral Network Lesion
Bilateral Lidocaine: Remove Positive FeedbackControl
Unstable Integrator
Human with unstable integrator:
E
dEd t
( 1)dE w Edt
= −
Issue: Robustness of Integrator
1biodr r r Id
wt
τ = − + +Integrator equation:
~ 10 secnetworkτ
Experimental values:
Synaptic feedback w must be tuned to accuracy of:
| | %w ~ 11 bio
network
ττ
− =
~ 100 msbioτSingle isolated neuron:
Integrator circuit:
|1 |wbio
networkττ =−
W
r(t)I
w
Weakness: Robustness to PerturbationsImprecision in accuracy of feedback connections severely
compromises performance (memory drifts: leaky or unstable)
10% decrease in synaptic feedbackModel:(Seung et al., 2000)
Robustness in Dynamical Systems
Robustness refers to:A. Low sensitivity of a system to perturbationsB. Ability to recover, over time, from a perturbation (e.g. plasticity, drug tolerance)
Issues to consider:
1) Time scale for robust behavior
2) What perturbations is a system robust against?-Design systems to resist the most common perturbations
3) What features of a system’s output are robust to a particular perturbation?
4) What are the signatures of a system exhibiting variousrobustness mechanisms?
Learning to Integrate
How accomplish fine tuning of synaptic weights?
IDEA: Synaptic weights w learned from “image slip”
E.g. leaky integrator:
Eye slip 0dEdt
<
Imageslip = dE
dt−
w Need to turn up feedback w dEdt
Δ ∝ −
(Arnold & Robinson, 1992)
Experiment: Give Feedback as if Eye is Leaky or Unstable
Magnetic coilmeasures
eye position
E(t)
Compute image slip −
that would resultif eye were
leaky/unstable
v dEdt
= −
E = dEdt
−
dEd t
dEd t
Integrator Learns to Compensate for Leak/Instability!
Control (in dark):
Give feedbackas if unstable Leaky:
Give feedbackas if leaky Unstable:
Previous Example:Error signal to tune network is due to sensory error
(image slip on retina)
Question:• Might systems have intrinsic monitors of activity
to accomplish tuning?• What might be the signatures of a system that utilizes
such a mechanism?
Pattern Generating Network:Stomatogastric ganglion (STG) of crab/lobster stomach
Controls digestive rhythm using recurrent inhibitory network:
Conductance-based neuron models
ENa=V C
Outside of cell
Na+
Kd+
Inside of cell
gNa gKd
EKd
Electrical circuit model of neuron:
C = Σ gmax,i popen,i(V) (Ei - V)dVdt
i = conductance type = Na, Ca, A, KCa, Kdpopen(V) = probability channel is open
i
Inward currents(increase V)
Na (fast)
Ca (slower)
Outward currents(decrease V)
Kd (fast)
A (slower)
KCa (slowest)
-60
-50
-40
-30
Vm
(mV
) -60
-30
0
30
60
-60
-30
0
30
Time (ms)0 500 1000
-90-60-30
03060
Silent(synaptic output = 0)
1 Spike Burster(synaptic output = 729 mV-ms)
5-Spike Burster(synaptic output = 776 mV-ms)
Tonic(synaptic output = 176 mV-ms)
Model of crustacean STG neuronsbased on data of Turrigiano et al., 1995
Sample of Firing States Observed
Identified neurons: >Same location, morphology, function >Traditional view:
Data:
KCa/3Kd A
Pea
k C
ondu
ctan
ce (n
S)
0
1
2
3
4
Can different conductances give similar firing?
(Crab IC neuron; Golowasch et al., 1999)
3 outward K+ conductances
- Same conductances- Each conductance has unique role
Identified neurons, yet different conductances
Ca (*50)
A(*5)
KCaNa(*0.5)
Kd
g max
(mS/
cm2 )
0
50
100
150
200
250
300
350
400
SilentTonicBursting
Single Conductances Do Not Determine Firing State
-90
-60
-30
0
30
60
time (ms)0 500 1000
-90
-60
-30
0
30
600
70
140
210
280
350
0
70
140
210
280
350
Ca(*50)
A(*5)
KCa
Col 5: --
Vm
(mV
)
g max
(mS
/cm
2 )
Similar firing, different conductances:
-90
-60
-30
0
30
60
time (ms)0 500 1000 1500 2000
-90
-60
-30
0
30
60
X 50,5,1
0
60
120
180
240
300
0
60
120
180
240
300Vm
(mV
)
g max
(mS
/cm
2 )
Ca(*50)
A(*5)
KCa
Different firing, similar conductances:
# of spikes/burstwithin bursting region
Firing State Diagram:Combinations of Conductances Better Determine Firing State
Dynamic clamp gmaxCa (nS)-75 -50 -25 0 25 50 75 100D
ynam
ic c
lam
p g m
axA
(nS
)
-2000
-1000
0
1000
2000
Real Data
Day 1
Day 4
Day 2
SilentTonicBurstingCultured cell
Neurons regulate firing to achieve rhythmic pattern
Change in firing during development:(Turrigiano et al., 1995)
Day 1
Day 4
Day 2
Homeostatic learning rule to recover activity
Idea: Cell may have target levels of activity on differenttime scales
1. Monitor Ca++ entry on:a) fast (action potential) time scaleb) medium (burst) time scalec) slow (average voltage) time scale
2. Feed back the error from target onto:a) fast (action potential-generating) channelsb) Medium time-scale generating channelsc) Burst rate-controlling channels
Extra slides: Models for Robustness of the Integrator
Geometry of Robustness& Hypotheses for Robustness on Faster Time Scales
Plasticity on slow time scales:Reshapes the trough to make it flat
1) Extrinsic Perturbations (in external inputs to system):-Only input component along integrating mode persists-Strategy: make integrating mode orthogonal to perturbations
2) Intrinsic Perturbations (in weights):-Need eigenvalue of integrating mode = 1:
Many different network structures can obey this condition.Goal: find structures that resist common perturbations in weights
Geometry of Robustness
3a) Add “friction” by effectively “putting system in viscous fluid”-Sliding friction: if a separate circuit monitors integrator slip,
it can feed back an opposing input:
w estimao
ebi
tdr r r I drAddt t
τ = − + + −
w( )biodrA Id
rt
rτ + = + +−
W
r(t)externalinput
w
ddt
-
Control theory: Integral feedback can make perfect adaptation/derivativeDerivative feedback can make perfect integrator
But….how does one make a perfect derivative???
Geometry of Robustness
Trough of energy function:
3b) Add “friction” by “roughening” energy surface
W
r(t)externalinput
w
Need for fine-tuning in linear feedback models
Fine-tuned model:wneuron rdr external inpur t
dtτ = +− +
decay feedback
Leaky behavior Unstable behavior
r
r (decay)wr (feedback)
dr/dtr
r (decay)wr (feedback)dr/dt
time time
IDEA: Can bistability add robustness to persistent neural activity?
Bistability in firing rate relationships Jumps in firing rate not observed experimentally
Integrator neuron:
Dendritic bistability distributed across multiple independent dendrites
1) Dendritic bistability has been observed experimentally- due to the self-sustaining properties of NMDA, NaP, or Ca++ channels:
Evidence for dendritic bistability & independence
Ca++ channelsactivate
Ca++ channelsinactivate
decreasing I
increasin
g I
Model compartment I-V curve (Adapted from spinal motoneuronmodel of Booth & Rinzel, 1995)
2) Anatomically realistic models suggest that different dendritic branchesmay behave approximately independently (Koch et al., 1983; Poirazi et al., 2003)
offr
Simplified analytic model
( , ) ( , ) ( )recdD r t D r t h r
dtτ = − +
Dendrite dynamics:
r = presynaptic input firing rate
D(r, t) = dendritic compartment activation
onr r
h(r)
1
where
τrec = time scale for dendrite to reach steady state activation
h(r) = steady-state dendritic compartment activation
Network of N neurons, each with N identical dendrites:
, ,1( ) [ ( , ) ]N
i ij j ton i com ijr t W D r t r r +=
= + +∑recurrent
inputbackground
inputeye movement
commands
Network with bistable dendrites
ri
rton,i
rcom,i
rjWij
D(rj)
Result 1: Firing rate is linear in eye position
, ,[ ]ˆi i ton i com ir r rEξ += + + E
ir
iξ
,ton i
i
rξ
−
Due to successive activation of dendrites w/increasing ˆ:E
E
iη
,( )off ton i
i
r rξ− ,( )on ton i
i
r rξ−
Activation of dendrite i
dendrite i+1dendrite idendrite i-1
E≡
ij i jW ξη=
, ,1() [ ],( )N
j ji i ton i cj om iD r tr t r rηξ= += + +∑
Simplify weight matrix to 1-D (rank 1) form:
Graphical solution of balanced leak and feedback
,1
ˆ( )ˆ ˆ
N
i i ton irei
c h E rEdEdt
η ξτ=
+= +− ∑
,1
ˆ( )N
i i ton ii
h E rη ξ=
+∑ˆ ,E
3η
E,3
3
( )off tonr rξ− ,3
3
( )on tonr rξ−
No external input:
N →∞
Hysteretic band of stability
EmaxE
width of band% mistuningtolerated ~
maxE
Fixations Are Robust to Mistuning of Feedback
Comparison with no-hysteresis models:
E
decay > feedback
E
decay within feedback band
Biophysical ModelFi
ring
rate
(Hz)
Som
atic
vol
tage
(mV
)
time (sec)
(Simulations by Joseph Levine)
Spiking model somaCalcium plateau mediated bistabilitySoma and dendrites ohmically coupled
Network of 20 recurrently connected neurons, each with: