1. Mechanisms and Models of Persistent Neural Activity:
Linear Network Theory
Mark Goldman
Center for Neuroscience
UC Davis

2. Outline Neural mechanisms of integration: Linear network theory
Critique of traditional models of memory-related activity & integration, and possible remedies

3. r Issue: How do neurons accumulate & store signals in working memory?
In many memory & decision-making circuits, neurons
accumulate and/or maintain signals for ~1-10 seconds
stimulus
neuronal activity
(firing rates)
accumulation
storage (working memory)
time
Most neurons intrinsically have brief memories
Puzzle:
synapse r
tneuron
synaptic
input
firing rate r
Input stimulus
~ ms

4. Neural Integrator of the Goldfish Eye Movement System
(from lab of David Tank)

5. The Oculomotor Neural Integrator
Eye velocity coding
command neurons
excitatory
inhibitory
persistent activity: stores running total of input commands
Integrator
neurons:
“Tuning curve”
Firing rate
Eye position:
Eye position
1 sec
time
(data from Aksay et al., Nature Neuroscience, 2001)

6. & eye movement commands
Network Architecture
(Aksay et al., 2000)
outputs
Firing rates: R L
Eye Position
Left side neurons
firing rate
100 R L
Eye Position
100
Right side neurons
firing rate
Wipsi
Wcontra
inputs
midline
4 neuron
populations:
Inhibitory
Excitatory
Pay attention to low-threshold neurons vs. high-threshold. Define low-threshold as active for all eye positions. don’t comment on model yet!.
Recurrent
(dis)inhibition
Recurrent
excitation
inputs
background inputs
& eye movement commands

7. Standard Model: Network Positive Feedback
1) Recurrent
excitation
(Machens et al., Science, 2005)
2) Recurrent
(dis)inhibition
Command
input:
This architecture has also suggested how firing can be maintained in the absence of input:
Typical story: 2 sources of positive feedback, rec. excit. & mutual inhibition.
Typical isolated single
neuron firing rate:
time
tneuron
Neuron receiving
network positive feedback:

8. Many-neuron Patterns of Activity Represent Eye Position
Activity of 2 neurons
saccade
Eye position represented
by location along
a low dimensional
manifold (“line attractor”)
(H.S. Seung, D. Lee)

10. Line Attractor Picture of the Neural Integrator
Geometrical picture
of eigenvectors: r2
No decay along direction
of eigenvector with
eigenvalue = 1 r1
Decay along direction of
eigenvectors with eigenvalue < 1
“Line Attractor” or “Line of Fixed Points”

11. Next up… 1) A nonlinear network model of the oculomotor integrator,
and brief discussion of Hessians and sensitivity analysis
2) The problem of robustness of persistent activity
3) Some “non-traditional” (non-positive feedback) models of integration
a) Functionally feedforward models
b) Negative-derivative feedback models
Note: skipped issue of noise: see Lim & Goldman, Neural Comp. and Ganguli/Huh/Sompolinsky PNAS
[4) Projects:
a) Cellular mechanisms of persistence & “bistability neuromodulators”
b) Discretized vs. continuous attractors, and noise: tradeoffs
c) “Integrate-and-forage” model of ant colony decision making
(relates to Pillow talk on fitting methods & Baccus synapse model)
d) Learning/Fitting attractor models (many possibilities)
(could relate to Pillow talk on coupling through spike-history filter,
and potential instability in these GLM methods) ]

12. The Oculomotor Neural Integrator
Eye velocity coding
command neurons
excitatory
inhibitory
persistent activity: stores running total of input commands
Integrator
neurons:
Firing rate
Eye position:
Eye position
1 sec
time
(data from Aksay et al., Nature Neuroscience, 2001)

13. & eye movement commands
Network Architecture
(Aksay et al., 2000)
outputs
Firing rates: R L
Eye Position
Left side neurons
firing rate
100 R L
Eye Position
100
Right side neurons
firing rate
Wipsi
Wcontra
inputs
midline
4 neuron
populations:
Inhibitory
Excitatory
Pay attention to low-threshold neurons vs. high-threshold. Define low-threshold as active for all eye positions. don’t comment on model yet!.
Recurrent
(dis)inhibition
Recurrent
excitation
inputs
background inputs
& eye movement commands

14. Network Model Coupled nonlinear equations: Mathematically intractable?
Outputs
Wcontra
Wipsi
Wij = weight of connection
from neuron j to neuron i
Burst commands
& Tonic background inputs
Firing rate dynamics of each neuron:
Firing rate changes
Intrinsic
leak
same-side
excitation
opposite-side inhibition
Bkgd.
input
Burst
command
input
explain all terms in model, note that also (not shown) have nonlinearity that enforces rates > 0
Coupled nonlinear equations:
Mathematically intractable?

15. Network Model Integrator! Coupled nonlinear equations:
Outputs
Wcontra
Wipsi
Wij = weight of connection
from neuron j to neuron i
Burst commands
& Tonic background inputs
Firing rate dynamics of each neuron:
Firing rate changes
Intrinsic
leak
same-side
excitation
opposite-side inhibition
Bkgd.
input
Burst
command
input
Will focus on steady-state behavior (not showing dynamics of approach to these steady-state values)
For persistent activity: must sum to 0
Integrator!
Coupled nonlinear equations:
Mathematically intractable?

16. total inhibitory current received
Fitting the Model
Fitting condition for neuron i:
Current needed
to maintain
firing rate r
total excitatory
current received
total inhibitory current received
Background
input
Knowns:
-r’s: known at each eye position (tuning curves)
-f(r): known from single-neuron experiments (not shown)
For each neuron, equation is of the form f = W.s + T (could write this on the board)
Unknowns:
-synaptic weights Wij > 0 and external inputs Ti
-synaptic nonlinearities sexc(r), sinh(r)
Assume form of sexc,inh(r)  constrained linear regression for Wij, Ti
(data = rates ri at different eye positions)

17. Model Integrates its Inputs and
Reproduces the Tuning Curves of Every Neuron
Example model
neuron voltage trace:
Network integrates its inputs
…and all neurons precisely
match tuning curve data
Firing rate (Hz)
Time (sec)
At end of slide: We next wanted to know what the microcircuit architecture of this network is. Before showing you our results, I want to show you the key experiment constraining this architecture.
gray: raw firing rate
(black: smoothed rate)
green: perfect integral
solid lines: experimental tuning curves
boxes: model rates (& variability)

18. Inactivation Experiments Suggest Presence
of a Threshold Process
Experiment: Remove inhibition
Record
Inactivate
stable at high rates
drift at low rates
firing rate
Model:
time
Persistence maintained at high firing rates:
 These high rates occur when inactivated side would be at low rates
 Suggests such low rates are below a threshold for contributing

19. Mechanism for generating persistent activity
Network activity when eyes directed rightward:
Left side
Right side
Implications:
-The only positive feedback LOOP is due to recurrent excitation
-Due to thresholds, there is no mutual inhibitory feedback loop
mutual excitatory positive feedback, or could be intrinsic cellular processes that kick in only at high rates
Excitation, not inhibition, maintains persistent activity!
Inhibition is anatomically recurrent, but functionally feedforward 19

20. Model Constrains the Possible Form of Synaptic Nonlinearities…
2-parameter set of tested
synaptic nonlinearities s(r)
Best-fit performance
for different nonlinearities
Networks with saturating synapses
can’t fit data for any choice of weights Wij
At end of slide: We next wanted to know what the microcircuit architecture of this network is. Before showing you our results, I want to show you the key experiment constraining this architecture.
Fisher et al., Neuron, 2013

21. …But Many Very Different Networks Give Near-Perfect Model Performance
Global excitation
Local excitation
Circuits with
very different
connectivity…
right
exc.
inh.
left
right
exc.
inh.
left
…but nearly
identical performance:
right side
left side
both of above are for the same synaptic nonlinearity.

22. Sensitivity Analysis:
Which features of the connectivity are most critical?
Cost function curvature is described
by “Hessian” matrix of 2nd derivatives:
Cost function surface:
cost C W1 W2
Insensitive
direction
(low curvature)
Sensitive
direction
(high curvature)
diagonal elements: sensitivity to
varying a single parameter
off-diagonal elements: interactions

23. Sensitivity Analysis:
Which features of the connectivity are most critical?
Cost function curvature is described
by “Hessian” matrix of 2nd derivatives:
Cost function surface:
cost C W1 W2
diagonal elements: sensitivity to
varying a single parameter
off-diagonal elements: interactions
between pairs of parameter
eigenvectors/eigenvalues: identify
patterns of weight changes to which
network is most sensitive
Insensitive
direction
(low curvature)
Sensitive
direction
(high curvature)
3 most
important
components!

24. Sensitive & Insensitive Directions in Connectivity Matrix
Sensitive directions (of model-fitting cost function)
Insensitive
Eigenvector 1:
make all connections
more excitatory
Eigenvector 2:
strengthen excitation
& inhibition
Eigenvector 3:
vary high vs. low
threshold neurons
Eigenvector 10:
Offsetting changes
in weights
more exc.
less inh.
exc
inh
exc
inh
exc
inh
perturb
perturb
perturb
perturb
We could compute the shape of the cost function and find the most sensitive directions/patterns of synaptic connectivity. Note that showing for 1 example network (top row is class 2)
right side avg.
Fisher et al., Neuron, 2013

25. insensitive eigenvectors
Diversity of Solutions: Example Circuits Differing Only in Insensitive Components
Two circuits with different connectivity…
…differ only in their
insensitive eigenvectors
…but near-identical performance…
right side
left side

26. W Issue: Robustness of Integrator w Integrator equation: I r(t)
Experimental values:
Single isolated neuron:
Integrator circuit:
Synaptic feedback w must be tuned to accuracy of:

27. W Need for fine-tuning in linear feedback models w wr (feedback)
Fine-tuned model: W
r(t)
external
input w
decay
feedback
Leaky behavior
Unstable behavior r
r (decay)
wr (feedback)
dr/dt
r (decay)
wr (feedback)
dr/dt r
rate
time (sec)
rate
time (sec)

28. Geometry of Robustness
& Hypothesis for Robustness on Faster Time Scales
1) Plasticity on slow time scales:
Reshapes the trough to make it flat
2) To control on faster time scales:
Add ridges to surface to add
“friction”-like slowing of drift
-OR-
Fill attractor with viscous
fluid to slow drift
(Koulakov et al. 2002;
Goldman et al. 2003)

29. Questions: Are positive feedback loops the only way
to perform integration? (the dogma)
2) Could alternative mechanisms describe
persistent activity data?

30. Challenging the Positive Feedback Picture
Part 2: Corrective Feedback Model
Fundamental control theory result:
Strong negative feedback of a signal produces an output equal to the
inverse of the negative feedback transformation
Claim: x +
x – f(y)
g(x – f(y)) y g -
f(y) f
Equation:

31. Integration from Negative Derivative Feedback
Fundamental control theory result:
Strong negative feedback of a signal produces an output equal to the
inverse of the negative feedback signal x + y g -
Integrator!

32. Persistent Activity from Negative-Derivative Feedback
Math:
Picture:
(-) corrective
signal
(+) corrective
firing rate
time

33. Negative derivative feedback arises
naturally in balanced cortical networks
slow
fast
Derivative feedback arises when:
1) Positive feedback is slower
than negative feedback
2) Excitation & Inhibition are balanced
Lim & Goldman, Nature Neuroscience, in press

34. Negative derivative feedback arises
naturally in balanced cortical networks
slow
fast
Derivative feedback arises when:
1) Positive feedback is slower
than negative feedback
2) Excitation & Inhibition are balanced
Lim & Goldman, Nature Neuroscience, in press

35. Networks Maintain Analog Memory and Integrate their Inputs

36. Robustness to Loss of Cells or Intrinsic or Synaptic Gains
Change:
-intrinsic
gains
-synaptic
-Exc. cell
death
-Inh. cell

37. Balanced Inputs Lead to Irregular Spiking
Across a Graded Range of Persistent Firing Rates
Spiking model structure:
Model output (purely derivative feedback) :
Experimental distribution of
CV’s of interspike intervals:
time (sec)
time (sec)
time (sec)
(Compte et al., 2003)
Lim & Goldman (2013); see also Boerlin et al. (2013)

38. Working memory task not easily explained by traditional feedback models
5 neurons recorded during a PFC delay task (Batuev et al., 1979, 1994):

39. Response of Individual Neurons in Line Attractor Networks
All neurons exhibit
similar slow decay:
Due to strong coupling that
mediates positive feedback
Time (sec)
Neuronal
firing rates
Summed
output
Problem: Does not reproduce
the differences between
neurons seen experimentally!
Problem 2: To generate stable activity
for 2 seconds (+/- 5%) requires
10-second long exponential decay

40. Feedforward Networks Can Integrate!
(Goldman, Neuron, 2009)
Simplest example:
Chain of neuron clusters that
successively filter an input

41. Feedforward Networks Can Integrate!
(Goldman, Neuron, 2009)
Simplest example:
Chain of neuron clusters that
successively filter an input
Integral of input!
(up to duration ~Nt)
(can prove this works analytically)

42. Same Network Integrates Any Input for ~Nt

43. Improvement in Required Precision of Tuning
Feedback-based Line Attractor:
10 sec decay to hold 2 sec of activity
Feedforward Integrator
2 sec decay to hold 2 sec of activity
Time (sec)
Neuronal
firing rates
Summed
output
Neuronal
firing rates
Time (sec)
Summed
output
Time (sec)

44. Feedforward Models Can Fit PFC Recordings
Line Attractor
Feedforward Network

45. Recent data: “Time cells” observed in rat hippocampal recordings during delayed-comparison task
MacDonald et al.,
Neuron, 2011:
feedforward
progression
(Goldman, Neuron, 2009)
(See also similar data during spatial navigation memory tasks by Pastalkova et al. 2008; Harvey et al. 2012)

46. Generalization to Coupled Networks:
Feedforward transitions between patterns of activity
Feedforward
network
Recurrent (coupled)
network
Map each neuron to
a combination of
neurons by applying
a coordinate rotation
matrix R
(Schur decomposition)
Connectivity
matrix Wij:
Geometric picture:
(Math of Schur: See Goldman, Neuron, 2009; Murphy & Miller, Neuron, 2009; Ganguli et al., PNAS, 2008)

47. Responses of functionally feedforward networks
activity patterns…
Feedforward
network activity patterns
& neuronal firing rates
Effect of stimulating pattern 1:

48. Math Puzzle: Eigenvalue analysis does not predict long time scale of response!
Line attractor
networks:
Eigenvalue
spectra:
Neuronal
responses:
Imag(l)
Real(l) 1
persistent mode
Feedforward
networks:
Real(l)
Imag(l) 1
no persistent mode???
(Goldman, Neuron, 2009; see also: Murphy & Miller, Neuron 2009; Ganguli & Sompolinsky, PNAS 2008)

49. Math Puzzle: Schur vs. Eigenvector Decompositions

50. Answer to Math Puzzle: Pseudospectral analysis
( Trefethen & Embree, Spectra & Pseudospectra, 2005)
Eigenvalues l:
Pseudoeigenvalues le:
Set of all values le that satisfy
the inequality: ||(W – le1)v|| <e
Govern transient responses
Can differ greatly from eigenvalues
when eigenvectors are highly
non-orthogonal (nonnormal matrices)
Satisfy equation: (W – l1)v =0
Govern long-time asymptotic
behavior
Black dots: eigenvalues;
Surrounding contours:
colors give boundaries of
set of pseudoeigenvals.,
for different values of e
(from Supplement to Goldman, Neuron, 2009)

51. Answer to Math Puzzle: Pseudo-eigenvalues
Normal networks:
Eigenvalues
Neuronal responses
Imag(l)
Real(l) 1
persistent mode
Feedforward networks:
Real(l)
Imag(l) 1
no persistent mode???

52. Answer to Math Puzzle: Pseudo-eigenvalues
Normal networks:
Eigenvalues
Neuronal responses
Imag(l)
Real(l) 1
persistent mode
Feedforward networks:
Pseudoeigenvals
Real(l)
Imag(l)
transiently acts like
persistent mode 1 -1 1
(Goldman, Neuron, 2009)

53. Summary Short-term memory (~10’s seconds) is maintained by persistent
neural activity following the offset of a remembered stimulus
Possible mechanisms
Tuned positive feedback (attractor model)
2) Negative derivative feedback
-Features:  Balance of excitation and inhibition, as observed
 Robust to many natural perturbations
 Produces observed irregular firing statistics
3) Feedforward network (possibly in disguise)
-Disadvantage: Finite memory lifetime ~ # of feedforward stages
-Advantage: Higher-dimensional representation can produce
many different temporal response patterns
-Math: Not well-characterized by eigenvalue decomposition;
Schur decomposition or pseudospectral analysis better

54. Summary Short-term memory (~10’s seconds) is maintained by persistent
neural activity following the offset of a remembered stimulus
Tuned positive feedback (attractor model)
Possible mechanisms
2) Negative derivative feedback
-Features:  Balance of excitation and inhibition
 Robust to many natural perturbations
 Produces observed irregular firing statistics
Modeling issue: Degeneracy of model-fitting solutions
-Key Question: Does this model degeneracy reflect lack
of experimental constraints…or patterns of
connectivity that may differ from animal to animal?
-Hypothesis: (some) model degeneracy is real & provides
redundancy that allows the system to robustly
re-tune itself

55. Theory (Goldman lab, UCD)
Acknowledgments
Theory (Goldman lab, UCD)
Experiments
Itsaso Olasagasti (U. Geneva)
Dimitri Fisher (Brain Corp.)
Sukbin Lim (U. Chicago)
David Tank (Princeton Univ.)
Emre Aksay (Cornell Med.)
Guy Major (Cardiff Univ.)
Robert Baker (NYU Medical)

56. Extra Slide(s)

57. Two Possible Threshold Mechanisms Revealed by the Model
Synaptic nonlinearity s(r) & anatomical connectivity Wij for 2 model networks:
Mechanism 2
• High-threshold cells dominate
the inhibitory connectivity
Mechanism 1
• Synaptic thresholds
synaptic
activation
firing rate
synaptic
activation
firing rate
Exc
Right side neurons
Left side
neurons
Inh
Exc
Weight matrix: gives the synaptic weight from each presynaptic neuron onto each postsynaptic neuron. Left side neurons are 1-50; right side First 25 of each of these is excitatory, second 25 is inhibitory. So, e.g. upper left block is left side-excitatory neurons onto left-side excitatory neurons, etc.
Inh
Left side
neurons
Right side
neurons
low-threshold
inhibitory neurons
CLARIFY THRESHOLD IDEA? 57