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

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 ...

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

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

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)

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

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 75 - 100 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” email: 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

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

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

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

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) ...

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

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.

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.

Possible mechanism for different tonic vs. phasic LC responses 12 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.

Possible mechanism for different tonic vs. phasic LC responses 12 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

Possible mechanism for different tonic vs. phasic LC responses 12 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

Possible mechanism for different tonic vs. phasic LC responses 12 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

Possible mechanism for different tonic vs. phasic LC responses 12 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)

Possible mechanism for different tonic vs. phasic LC responses 12 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

Possible mechanism for different tonic vs. phasic LC responses 12 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: 18.3 % 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)

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

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...

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

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

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

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

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

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

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…)

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

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

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.

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)

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

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