1. Response dynamics and phase oscillators in the brainstem
Eric Brown
Jeff Moehlis
Mark Gilzenrat
Phil Holmes
Jonathan Cohen
Princeton Univ.
Program in Neuroscience
Psychology / Applied and
Computational Mathematics
Ed Clayton
Janusz Rajkowski
Gary Aston-Jones
Univ. of Pennsylvania
Dept. of Psychiatry
Laboratory of Neuromodulation
and Behavior
FOR NEUROSCI RETREAT TALK … open up with a sentence that says “From previous talk, we saw that … in context of decision processes, there may be time-dependent effects that help to optimize responses. I’d like to discuss with you the locus c, which may be responsible for implementing some of these transient optimization effects.”
cognitive control: changes in behavoiur to adapt to changing circumstances
SIAM: 25 mins

2. LC consists of ~ 30,000 neurons ...
Neuromodulatory nucleus locus coeruleus (LC)
LOCUS COERULEUS: ROLE IN MODULATION OF COGNITIVE PERFORMANCE
(Usher et al. 1999, Aston-Jones et al. 1994)
SAY …
NE traditionally seen as “arousal juice …” setting overall level of attention. Now, role proposed in modulation of cog performance (on finer timescale) …
let’s look at some of the evidence for this.
MAKE SURE … say IMPLICATED not implemented
LC consists of ~ 30,000 neurons ...

3. OUTLINE … from spikes to speed-accuracy
Experimental data
Model of LC dynamics
Role of LC in cognitive performance

4. Cognitive performance is correlated with LC firing rate (1)
2 distinct modes of LC operation
Tonic (T): fast baseline firing, poor performance
Phasic (P): slow baseline firing, good performance
Usher et al. (1999)
LC firing rate:
FAST=Tonic mode
SLOW=Phasic mode T P
Task errors:
HIGH=Tonic mode
LOW=Phasic Mode

5. Larger LC responses occur at lower baseline firing rates (2)
Just saw...
2 distinct modes of operation
Tonic (T): fast baseline firing, poor performance
Phasic (P): slow baseline firing, good performance
Also...
Tonic : rel. weak response to task stimulus
Phasic: rel. strong response to task stimulus
Tonic
Stim.
Hz.
Phasic
Hz.
Stim. 8 8
NE has complex effects…but in some cases has been shown to increase population-level responsivity in cortex. GET ref from GAJ. Thus, increases gain, which (S+S 1990), increases computational efficiency.
UNITS on the Usher et al PSTHs. As per 8/1/2002 communcation, multiply units by 2 to get Hz 4 4
t (ms)
t (ms)
LINK TO COGNITIVE PERFORMANCE?
NE release (Usher et al. 1999, Servan-Schreiber et al. 1990)

6. Role of LC in cognitive performance
Experimental data:
PHASIC and TONIC LC modes have differing …
baseline firing rates
responses to stimulus
levels of task performance
Model of LC dynamics
Role of LC in cognitive performance
Above summarizes the phenomelology …
now, we talk about modeling LC, looking to understand in particular the different modes of LC operation

7. Modeling LC neurons Data from slice Solution to HR equations
The Hindmarsh-Rose model includes A current and is reduced to 2 effective variables:
J.T. Williams measured transient potassium current (“A current”) in rat LC neurons, responsible for slow “pacemaker” firing
Rose and Hindmarsh,Proc. R. Soc. Lond. B., 1989.
Data from slice
Solution to HR equations
~80 mV
~100 mV
Currents …
FIGURE OUT!!!! WHAT IS CONDITION FOR BASELINE FIRING IN SLICE???? Is in “control solution”
Persistent Ca current depolarizes --> ~-45 mV. This is leak current in HR equations (vl=-17). Then Na current generates spike.
Fast K current repolarizes. Ca current starts depolarizing again, but is resisted by relatively slow (call “transient”) Ca-dependent K current (the A current) which is responsible for slow firing …
A current turns on when hyperpolarized. “de-inactivates”
AP in LC recordings is approx mV (mean 82.5) above “jumping off point” at approx. -45 mV. Spike chopped in pic. The HR equations give a ~100 mV spike, “on the high end”
Say … timescales are different here; all I’d like to say is that HR equations capture qualitatively the slow recovery as well as fact that spike occupies v. small fraction of overall firing cycle. See next page for HR neuron at faster firing rate, looks more like the slice.
ARE NEURONS really oscillators? See from Fig 2 of Janusz’s “Check it” get a couple of rings clear after selecting spikes in 10 ms window. Good enough : noise + dist freq (recall recordings over extended interval of time).
30 mV mV
3 s
in vivo: additional (noisy) inputs

8. Winfree ‘74, Guckenheimer ‘75
Reduction of neurons to phases V t q
SAY … this reduction works for systems that continue to fire periodically … assume that this is the case for neurons we model. Weak coupling, stim, etc, seems to work out. Well, for SNIPER neurons … the invariant circle is always there.
so ...
“SAY… as log as external inputs are not too great, this basic periodic pattern of trajectories in phase space persists.”
Reduced dimensional model of individual neuron dynamics
View of an AP trace, comment: actually would observe sim. Cycles in the other 4-D coordinates
Let’s mimic with projection fast/slow variables; tractably observe excitability (fig. P 134 K+S), fast/slow rep (nts.). No need to mention F-N: as a separate model: it’s just this projection
Let’s reduce once more. From 2-D to circle map. Will observe how, in presence of g, excitability preserved: two domains of activity, as in the traces of APs going at different rates above
1) Firing at repetitively
2) Noise perturbs over thresh ® single firing
EMPH: Lim cycle preserved! V
fire q V
ALERT: Interesting methods here! q 2p
Winfree ‘74, Guckenheimer ‘75

9. Ermentrout, Neural Comp., 1996
Simple (and accurate) phase description
sensory input
I(t)
coupling
“noise”
natural frequency
phase response curve
w x z(q)
SAY: depending on noise model, may have O(sigma^2) correxn term above too. q
fire q=0
fast p
slow q
Movie!
Ermentrout, Neural Comp., 1996

10. Ermentrout, Neural Comp., 1996
Simple (and accurate) phase description
sensory input
I(t)
coupling
“noise”
natural frequency
phase response curve
w x z(q)
SAY: depending on noise model, may have O(sigma^2) correxn term above too. q
Ermentrout, Neural Comp., 1996

11. Actually have many oscillators rotating at once
firing rate(t) = averaged rate at which oscillators cross 0
the current movie, NOsc3Hz.avi, is
250 oscillators, no noise, no coupling
Running at 5 x bio speed
Stimulus comes on in and synchronizes oscillators.
Phase resetting! Winfree (1964), Tass (1999) ...

12. Actually have many oscillators rotating at once
f. rate baseline
f. rate peak
Introduce this terminology …
Phase resetting! Winfree (1964), Tass (1999) ...

13. Take continuum limit and discover...
DESCRIBE LC BY DENSITY of phases
(w/o noise, coupling)
r(q,t)
cf. Fetz and Gustaffson, 1983 ; Hermann and Gerstner 2002
FIRE q=0
EMPHASIZE!!! IT IS ONLY changes in baseline freq. between above two cases…all else is EXACTLY the same
Say…quite intuitive … stim makes more of a difference at lower freq .. this type of an effect has been certainly been noticed before by other authors using other models.
NOT FOR THIS TALK...
SAY: depending on noise model, may have O(sigma^2) correxn term above too.

14. Take continuum limit and discover...
DESCRIBE LC BY DENSITY of phases
r(q,t)
(w/o noise, coupling)
cf. Fetz and Gustaffson, 1983 ; Hermann and Gerstner 2002
FIRE q=0
(3 Hz base)
(1 Hz base)
snap
EMPHASIZE!!! IT IS ONLY changes in baseline freq. between above two cases…all else is EXACTLY the same
Say…quite intuitive … stim makes more of a difference at lower freq .. this type of an effect has been certainly been noticed before by other authors using other models.
NOT FOR THIS TALK...
SAY: depending on noise model, may have O(sigma^2) correxn term above too.

15. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
Say … while this ringing is interesting (e.g. provides mechanism for persistent activity resulting from short stimulus), it is not what is observed to happen in the in vivo LC.

16. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
FREQUENCY DRIFT Hz

17. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
FREQUENCY DRIFT
NOISE Hz

18. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
FREQUENCY DRIFT
NOISE

19. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
FREQUENCY DRIFT
add decay line to LH plot --- this grabbed from tonic mode decay of CNS talk …
dist freq and noise as in paper around 3 hz
NOISE
cf. Tass (1999)

20. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
THEORY/ SIMS.
Hz.
Hz.
t (s)
t (s) 1
BUT: data returns to base w/o periodic ringing. Add:
NOISE
FREQUENCY DRIFT
COUPLING Hz
cross-cor.
data
model

21. Possible mechanism for different tonic vs. phasic LC responses12
Stim. 12
Tonic
Phasic
Hz.
Hz.
DATA
(3 Hz base)
(2 Hz base) 12 12
SIMS: freq. drift, noise, coupling
Hz.
Hz.
t (s)
t (s) 1 1
So … how is this baseline freq. actually SET in biology?
Could be baseline levels of inhibition …
OR decreased inputs from brain regions afferent to LC, such as medullary nuclei.
The “contributes” phrase means … this is not the whole story … other mechanisms, such as varying noise levels, coupling changes, etc. also contribute
Above: % increase in phasic spikecount during response pulse over tonic. See 5/3/03 notes in rmag folder
Baseline frequency contributes to the different responses in tonic vs. phasic modes
cf. Alvarez and Chow (2001), Usher et al. (1999)

22. Experimental data
Model of LC dynamics
Role of LC in cognitive performance

23. Devilbiss and Waterhouse, Synapse (2000)
LC emits norepinephrine, which can enhance responsivity (gain) in cortex
Devilbiss and Waterhouse, Synapse (2000)
Responses of single neurons in rat cortex to glutamate
# spikes
(percent of control)
iontophoretically applied norepinephrine to rat whisker barrel field cortical neurons. responses of individual layer V neurons shown to iontophoretic pulses of suprathreshold levels of glutamate
say: constant glu in each experiment -- like repeating same I(t) over and over again.
CURRENT MEANS (GAJ):
The electrical current that is applied to the base of the NE pipette to create voltage difference which leads to flow of NE onto the slice. This current is related to NE flow rate in a complicated nonlinear way...

24. Two-alternative choice task
Firing rates (y1, y2) of competing neural pops...
[Usher + McClelland, 1999]
y1, y2 approach fg(input).
gain
fg(input) g
input

25. Two-alternative choice task
Firing rates (y1, y2) of competing neural pops...
Decision 1 or 2 made when firing rate y1 or y2 crosses threshold
thresh. 2
y1, y2 approach fg(input).
gain
thresh. 1
fg(input) g g g
input

26. Two-alternative choice task
Firing rates (y1, y2) of competing neural pops...
FEF spike rates vs. time neural evidence for crossing fixed threshold prior to response
y1, y2 approach fg(input).
gain
fg(input)
J. Schall, V. Stuphorn, J. Brown, Neuron, 2002 g
input

27. LC sets gain in decision network
Firing rates (y1, y2) of competing neural pops...
(Servan-Schreiber, Printz, Cohen 1990, Usher et al., 1999, Usher + McClelland, 1999)
approach values fg(input), modulated by gain. g
fg(input) g
input

28. Expect speed-accuracy tradeoff between
Avg. Reaction Time (<RT>) and Error Rate (ER) “in general”

29. Observe speed-accuracy tradeoff between
Avg. Reaction Time (<RT>) and Error Rate (ER)
as gain is increased
<RT> g ER g g g g t t t

30. In computational model, Gilzenrat et al
In computational model, Gilzenrat et al. (2002) beat the speed-accuracy tradeoff via the LC and TRANSIENT gain
Given this …
what is the POINT of the tonic mode, anyway? “See mark’s poster.” Tonic mode is a more exploratory state. In this state, one is NOT committed to a single interpretation of the incoming information (e.g. want to pay attention to a stimulus that steps on, etc…)

31. LC becomes more “phasic”
In computational model, Gilzenrat et al. (2002) beat the speed-accuracy tradeoff via the LC: TRANSIENT gain
<RT> ER
Given this …
what is the POINT of the tonic mode, anyway? “See mark’s poster.” Tonic mode is a more exploratory state. In this state, one is NOT committed to a single interpretation of the incoming information (e.g. want to pay attention to a stimulus that steps on, etc…)
LC becomes more “phasic” g g g t t t
General trend agrees with Aston-Jones et al. (1994,1999)
behavioral + physiological data

32. SUMMARY 1) Have developed biophysical model of LC population:
Phase density formulation shows 1/w scaling of response
Helps explain phasic vs. tonic LC modes
CURRENT WORK:
Study coupling effects
Study population responses of other neuron types (nonintuitive results!)
2) Decision model shows that LC (in phasic mode) can beat speed accuracy tradeoff
Study dynamic gain effects

33. Jeff Moehlis Ed Clayton Mark Gilzenrat Janusz Rajkowski Phil Holmes
Jonathan Cohen
Princeton Univ.
Prog. Neurosci./Dept. of Psychology
Prog. in Applied and Computational Mathematics
Ed Clayton
Janusz Rajkowski
Gary Aston-Jones
Univ. of Pennsylvania
Dept. of Psychiatry
Laboratory of Neuromodulation and Behavior
For general background, the parts of the brain we consider are … ACC and LC
cognitive control: changes in behavoiur to adapt to changing circumstances
Study three aspects of neural modeling: effects of random perturbations, role of symmetry, and tendency of neurons to synchronized.
Population level of analysis: one variable describes evolution of a whole group of neurons.
Cellular level: one variable per cell
Attitude of the presentation: not focused on exactly reproducing what’s observed neurobiologically, but on understanding how behaviors can emerge from simple models of the biophysics.
At same time, link back to biology not complete for several of the models I will discuss today--can think of as mathematical ‘toy’ models motivated by a real problem.

34. Extend results to non-LC neurons:
Scaling of PRC encodes sensitivity of population response
Form of PRC encodes time-course of population response
e.g. “population rebound excitation” that (always) exceeds stimulus excitation for HH HH
Hz.
stim
FIRING RATE PEAK
t (ms)
General mechanism for action of neuromodulators -- sensitivity of neural populations and “dynamic” f-I curves?
Mention previous authors on extension to other neurons (I+F)
Mention application…switching in and out of slow firing modes by e.g. modulated a current or whatever Hansel’s mechanism is. This, then, switches neural population from one in which max response comes after stimulus (detecting end of stim) to during stim itself (detecting stim itself)

35. Coupling
LC has gap junctions,
as well as slow inhibitory synapses.

36. Both gap junctions and inhibitory synapses (partially) synchronize
fe, fs contain:
mostly just first Fourier
harmonics...
so only in-phase state is
stable (e.g. Okuda 1992) N N
add slide … the effects of these coupling terms is to sync oscillators,
period … weak coupling model, in-phase state stable, all clustered states NOT stable
see fe, fs curves … sync effect is “same” no matter what combo you add (cf Tim Lewis)
Both gap junctions and inhibitory synapses (partially) synchronizes heterogeneous, noisy population of HR neurons X X