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: