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Mathematics, Department of
Faculty Publications, Department of
University of Nebraska - Lincoln Year 
A Simple Model of Cortical Dynamics
Explains Variability and State
Dependence of Sensory Responses in
Urethane-Anesthetized Auditory Cortex
Carina Curto∗ Shuzo Sakata† Stephan Marguet‡
Vladimir Itskov∗∗ Kenneth D. Harris††
∗ University of Nebraska - Lincoln, [email protected]
† Rutgers, The State University of New Jersey
‡ Rutgers, The State University of New Jersey
∗∗ Columbia University,, [email protected]
†† Rutgers, The State University of New Jersey, [email protected]
This paper is posted at DigitalCommons@University of Nebraska - Lincoln.
10600 • The Journal of Neuroscience, August 26, 2009 • 29(34):10600 –10612
A Simple Model of Cortical Dynamics Explains
Variability and State Dependence of Sensory Responses
in Urethane-Anesthetized Auditory Cortex
Carina Curto, Shuzo Sakata, Stephan Marguet, Vladimir Itskov, and Kenneth D. Harris
Center for Molecular and Behavioral Neuroscience, Rutgers, The State University of New Jersey, Newark, New Jersey 07102
The responses of neocortical cells to sensory stimuli are variable and state dependent. It has been hypothesized that intrinsic cortical
dynamics play an important role in trial-to-trial variability; the precise nature of this dependence, however, is poorly understood. We
show here that in auditory cortex of urethane-anesthetized rats, population responses to click stimuli can be quantitatively predicted on
a trial-by-trial basis by a simple dynamical system model estimated from spontaneous activity immediately preceding stimulus presen-
tation. Changes in cortical state correspond consistently to changes in model dynamics, reflecting a nonlinear, self-exciting system in
synchronized states and an approximately linear system in desynchronized states. We propose that the complex and state-dependent
pattern of trial-to-trial variability can be explained by a simple principle: sensory responses are shaped by the same intrinsic dynamics
that govern ongoing spontaneous activity.
Introduction chronized and desynchronized states are traditionally considered
Cortical responses can vary substantially between presentations discrete, cortical activity during behaviors such as quiet resting
of an identical sensory stimulus. The factors underlying this vari- shows an intermediate pattern in which downstates of reduced
ability are incompletely understood. Whereas one contribution length and depth are observed (Petersen et al., 2003; Luczak et al.,
may be stochastic noise (Faisal et al., 2008), variability may also 2007, 2009; Poulet and Petersen, 2008). Under anesthesia, the
arise from deterministic interactions of sensory responses with cortex usually operates in the synchronized state. However, un-
spontaneous activity produced in the absence of sensory stimu- der some anesthetics (such as urethane), desynchronized periods
lation (Arieli et al., 1996; Tsodyks et al., 1999; Kenet et al., 2003; may occur spontaneously (Clement et al., 2008), or be induced by
Petersen et al., 2003; Castro-Alamancos, 2004; DeWeese and a tail pinch (Duque et al., 2000) or electrical stimulation of
Zador, 2004; Harris, 2005; Yuste et al., 2005; Hasenstaub et al., areas such as the pedunculopontine tegmental nucleus (PPT)
2007; Lakatos et al., 2008). (Moruzzi and Magoun, 1949; Vanderwolf, 2003). The magni-
The structure of cortical spontaneous activity varies with tude, tuning, and dynamics of sensory responses varies across the
brain state. The classical picture holds that cortical state is a func- sleep cycle, between behavioral states and between synchronized
tion of the sleep cycle: during waking or rapid eye movement and desynchronized states under anesthesia (Worgotter et al.,
sleep, the cortex operates in the desynchronized (or activated) 1998; Edeline, 2003; Castro-Alamancos, 2004; Hentschke et al.,
state, characterized by low-amplitude, high-frequency local field 2006).
potential (LFP) patterns; during slow-wave sleep, the cortex op- How do sensory responses interact with spontaneous cortical
erates in the synchronized (or inactivated) state, characterized by activity? The simplest possibility is linear summation, whereby a
larger, lower-frequency LFP patterns organized around an alter- stereotyped response is added onto ongoing background activity
nation of “upstates” of generalized activity and “downstates” of (Arieli et al., 1996). Other work indicates a nonlinear interaction,
network silence (Steriade et al., 1993, 2001). Although the syn- at least in synchronized states, with different studies reporting
larger or smaller responses in the upstate versus downstate
Received April 30, 2009; revised June 10, 2009; accepted July 1, 2009.
(Kisley and Gerstein, 1999; Massimini et al., 2003; Sachdev et al.,
This work was supported by National Institutes of Health Grants MH073245 and DC009947. C.C. was supported by 2004; Haslinger et al., 2006; Haider et al., 2007; Hasenstaub et al.,
a Courant Instructorship. S.S. was supported by the Japanese Society for the Promotion of Science. V.I. was sup- 2007). Furthermore, sensory stimuli may themselves trigger bi-
ported by the Swartz Foundation. K.D.H. is an Alfred P. Sloan fellow. We thank members of the Harris laboratory for directional transitions between upstates and downstates (Shu et
valuable comments on this manuscript.
Correspondence should be addressed to Kenneth D. Harris, Center for Molecular and Behavioral Neuroscience,
al., 2003; Hasenstaub et al., 2007), effectively changing the course
Rutgers, The State University of New Jersey, 197 University Avenue, Newark, NJ 07102. E-mail: of ongoing activity.
[email protected] Here, we show that auditory cortical population dynamics in
C. Curto’s present address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, urethane-anesthestetized rats can be well approximated by a fam-
New York, NY 10012.
V. Itskov’s present address: Center for Theoretical Neuroscience, Columbia University, New York, NY 10032.
ily of low-dimensional dynamical system models. These models,
DOI:10.1523/JNEUROSCI.2053-09.2009 with parameters estimated from spontaneous activity preceding a
Copyright © 2009 Society for Neuroscience 0270-6474/09/2910600-13$15.00/0 stimulus, can quantitatively predict the structure of the subse-
Used by permission. http://www.jneurosci.org/
Curto et al. • Dynamics in Auditory Cortex J. Neurosci., August 26, 2009 • 29(34):10600 –10612 • 10601
quent sensory response. Thus, observed patterns of trial-to-trial highly synchronized state, whereas values close to 0 correspond to desyn-
variability can be understood as a natural consequence of sensory chronization. Synchronization ranges 1–5 (see Fig. 2 and supplemental
responses evolving according to the same dynamics that govern Figs. 1– 4, available at www.jneurosci.org as supplemental material) cor-
previous spontaneous activity. We used the model to characterize respond to intervals 0 – 0.2, 0.2– 0.4, 0.4 – 0.6, 0.6 – 0.8, and 0.8 –1,
cortical dynamics as a function of brain state and found that the
Additional details pertaining to Figure 2. Here, we provide additional
synchronized state corresponds to a nonlinear, self-exciting sys- details pertaining to Figure 2 and supplemental Figures 1– 4 (available at
tem, whereas dynamics in the desynchronized state are close to www.jneurosci.org as supplemental material). For each trial, initial and
linear. persistent responses were computed as the number of spikes of all cells in
the initial (10 –35 ms) and persistent (40 –135 ms) response periods. The
Materials and Methods value of w at the time of the stimulus (0 ms) is the “past activity” associ-
Experimental methods. All procedures for animal care and experimenta- ated with each trial, denoted w0. To pool data across animals, we normal-
tion were approved by the Institutional Animal Care and Use Committee ized (i.e., multiplied by a scale factor) the initial response, persistent
of Rutgers University. Adult Sprague Dawley rats (200 – 430 g) were anes- response, and past activity values w0 separately for each recording so that
thetized with 1.5 g/kg urethane. Additional doses of urethane ( 0.2 g/kg) 1 was equal to the mean plus 2 SDs within each recording (results for each
were given when necessary. The animal was placed in a custom naso- animal individually are shown in supplemental Fig. 1, available at www.
orbital restraint that left the ears free and clear. Body temperature was jneurosci.org as supplemental material). To create Figure 2, the normal-
maintained with a heating pad. After reflecting the temporalis muscle, ized data were then pooled across animals and divided into the five
left auditory cortex was exposed via craniotomy, and a small durotomy synchronization ranges. Note that on a small number of outlying trials,
was carefully performed. Neuronal activity in the auditory cortex was re- normalized values were 1 and were therefore not included in the pooled
corded extracellularly with 16- or 32-channel silicon probes (NeuroNexus data plots. In supplemental Figure 2 (available at www.jneurosci.org as sup-
Technologies). Extracellular signals were high-pass filtered (1 Hz) and plemental material), a similar analysis was performed, but for “phantom”
amplified (1000 ) using a 64-channel (Sensorium) or a 32-channel stimulus controls (i.e., times when no click stimulus was presented). The
(Plexon) amplifier; recorded at a 20 kHz, 14-bit resolution using a per- lack of negative correlation in the synchronized state demonstrates that
sonal computer-based data acquisition system (United Electronic Indus- the “flipping” of state is indeed generated by click stimuli, rather than
tries); and stored on disk for later analysis. Spike detection and sorting occurring spontaneously. Supplemental Figures 3 and 4 (available at
was software based, using previously described semiautomatic clustering www.jneurosci.org as supplemental material) show a similar analysis com-
methods (Harris et al., 2000). Acoustic stimuli were generated digitally puted with 20 –200 ms, in 20 ms steps. This shows that the pattern
(sampling rate, 97.7 kHz; TDT3/RP2.1; Tucker-Davis Technologies) and observed in supplemental Figure 2 (available at www.jneurosci.org as sup-
delivered free-field through a calibrated electrostatic loudspeaker (ES1) plemental material) for phantom trials is not sensitive to changing the value
located 10 cm in front of the animal, in a single-walled soundproof of used for this control condition.
chamber (Industrial Acoustics Company) with the interior covered by 3 Fitting the model to data. Cortical activity was modeled with a dynam-
inches of acoustic absorption foam. Calibration was conducted using a ical system given by the FitzHugh-Nagumo (FHN) equations:
pressure microphone (ACO-7017; ACO Pacific) close to the animal’s
˙ a3v 3 a 2v 2 a 1v bw I (t) (1)
right ear. Single clicks (square pulses, 1 and 3 ms) were played at ampli-
tudes ranging from 20 to 100 dB sound pressure level (SPL). Only w
˙ v w)/ . (2)
presentations with loud clicks ( 60 dB SPL) were considered for this
analysis. The location of the electrodes was estimated to be primary au- The FHN model is a simple, well-studied dynamical system that admits
ditory cortex by stereotaxic coordinates, vascular structure (Sally and the possibility of up to two stable fixed points, depending on parameters,
Kelly, 1988; Doron et al., 2002; Rutkowski et al., 2003), and tonotopic in addition to linear phase portraits (supplemental Fig. 5, available at
variation of frequency tuning across recording shanks assessed by pre- www.jneurosci.org as supplemental material). Although in neuroscience
sentation of 50 ms tone pips. these equations are usually considered as models for action potential
PPT stimulation. In experiments using PPT stimulation, the bone generation in single neurons, we use them here to model the activity of a
above the PPT was removed, and a concentric bipolar stimulation elec- population. To fit the parameters of the model, v and w were computed
trode (SNE-100; David Kopf Instruments) was implanted into the PPT directly from the experimental data as described above. Computation of
(7.5 mm posterior from the bregma, 1.8 mm lateral from the midline, w independent of model parameters is possible because of the particular
6.5–7.0 mm deep from the dorsal surface of the brain). A 1 s pulse train form of Equation 2; this makes methods such as the Kalman filter unnec-
(100 Hz, 200 s duration, 50 –100 A) was applied to induce the desyn- essary for determining the most likely evolution of the “hidden” variable
chronized state. w, rendering the fitting procedure more computationally efficient. The
Obtaining v and w from multiunit activity. Multiunit activity (MUA) model parameters a1, a2, a3, b, and I that appear in Equation 1 were
was obtained by accumulating the spike trains of all recorded cells to- estimated individually for the 3 s windows preceding (but not including)
gether in 0.8 ms time bins and used to compute two time series, vt and wt. each sensory response, by a procedure that minimizes the squared resid-
To compute v, the MUA was filtered with a causal half-hanning window ual 2(t)dt (see Fig. 3d). Because the coefficient of the cubic term a3 is
of 16 ms width. The value vt in the tth time bin is therefore a weighted particularly vulnerable to overfitting, a stepwise procedure was used. For
average of MUA in the previous 20 time bins. To allow for comparison fixed values of a3, the parameters a1, a2, b, and I were fit via linear regres-
between recordings with different numbers of cells, v was normalized sion of Equation 1 with the measured time series v, v 2, v 3, v, and w (with
such that the maximum value in any recording was 0.5. w was computed v computed by one-step differencing). The best value of a3 was then
from v by solving the differential equation w ˙ 1/ (v w). This was determined by exhaustive search on the set { 2, 1.9, 1.8, . . . , 0.1,
achieved using the corresponding IIR filter and is equivalent to convolv- 0} to minimize integrated square error using fivefold cross-validation
ing v causally with a decaying exponential having time constant (i.e., w (values of a3 0 always yielded poor fits and hence were not considered
is the output of a leaky integrator driven by v). The value of (100 ms) in the automated search). Note that the model-fitting procedure, includ-
was chosen by maximizing the correlation of w with persistent response ing cross-validation, used only spontaneous activity data before stimulus
in a single rat (rat 1); the same value gave high correlations in the other onset and never used data from the stimulus response period. The pa-
rats (see Fig. 2 and supplemental Fig. 1, available at www.jneurosci.org as rameters for the model fits in Figure 4 are as follows: (a1 0.0271, a2
supplemental material). 0.394, a3 1, b 0.0374, I 0.00217) for the synchronized data and
Degree of synchronization. The degree of synchronization at the time of (a1 0.00119, a2 0.00344, a3 0, b 0.0671, I 0.00653) for the
a stimulus event was computed by taking the ratio between total 0 –5 Hz desynchronized case. A variety of possible phase diagrams and corre-
power and total 0 –50 Hz power of the MUA trace in the 1 s interval sponding parameters for the FHN model are shown in supplemental
immediately preceding the stimulus. Values near 1 correspond to a Figure 5 (available at www.jneurosci.org as supplemental material).
10602 • J. Neurosci., August 26, 2009 • 29(34):10600 –10612 Curto et al. • Dynamics in Auditory Cortex
Generating simulated data from the model. Simulated data can be gen- spontaneous and click-evoked activity across a range of synchro-
erated from the model for a given time series of “kicks” { (t)} by simply nized and desynchronized states and focused our analysis on
evolving the model equations (for a fixed set of parameters) with (t) as smoothed MUA obtained by pooling together all spikes from
a driving force input at each time t. We have done this in two situations: simultaneously recorded neurons (see Materials and Methods).
(1) to generate simulated spontaneous activity data, as in Figure 4 and
We began by visualizing how click responses vary with cortical
supplemental Figure 5 (available at www.jneurosci.org as supplemental
material), and (2) to generate simulated stimulus-evoked activity, as in
state. In the neocortex, the word “state” is used with two different
Figures 5 and 6. Note that v is allowed to go negative when we integrate meanings, corresponding to two different time scales. We shall
the FHN equations; for comparison with experimental data, we later refer to the state of the cortex, in the sense of the dynamics of
threshold the values [v] so that negative values are replaced with 0. network activity on a time scale of seconds or more and as re-
In situation 1, we obtained a “noise” time series (t) by first sampling flected in the LFP power spectrum, as its “dynamic state”; the
independently from a lognormal distribution (with a mean of 25 and synchronized and desynchronized states are examples of dy-
variance of 100) and then filtering the resulting white noise time series so namic states. The word “state” is also used to refer to fluctuations
that its power spectrum matched that of the residuals obtained when the in instantaneous network activity at time scales of the order hun-
model was fit on desynchronized data. Residuals of desynchronized dreds of milliseconds, as in the case of upstates and downstates.
rather than synchronized data were used for the power spectrum because
We will use the term “activity state” to describe cortical states that
the residuals from synchronized data cannot be accurately determined
during downstates, as the thresholded values of v match the data exactly
persist on these shorter time scales.
during these periods. A lognormal distribution was used as the residuals To illustrate how sensory responses can depend on activity
had heavier tails than a Gaussian, to which the lognormal distribution state at the time of stimulus presentation, Figure 1a shows pop-
allowed a good approximation. ulation activity before and after six presentations of a click stim-
In situation 2, stimulus-evoked responses were simulated by evolving ulus in a recording that was consistently in the synchronized
the model according to initial conditions (v0, w0) at the time of the state. It has been reported that in somatosensory cortex, whisker
stimulus and an -function (t) (to simulate the stimulus) given by the stimulation can “flip” neural activity from downstate to upstate
formula (t) (t t0)e(t0 t)/ . The parameters t0, , and were and vice versa (Hasenstaub et al., 2007). Visual examination of
chosen to be constant on an experiment-wide basis, reflecting repeated pre- our data suggested that click stimuli presented during the syn-
sentations of loud noise-click stimuli within the same recording. In Figure 6b
chronized state can evoke a similar flip in auditory cortex. When
(rat 3), we had t0 10 ms, h 0.018, 5 ms, and 0.0036e ms 1,
where h /e is the maximum height of the -function. In Figure 6c (rat
the stimulus arrived during a downstate, an upstate frequently
1), we had t0 10 ms, h 0.021, 6 ms, and 0.0035e ms 1. ensued (Fig. 1a, trials 1 and 2). Similarly, stimuli that arrived during
Evaluating the performance of the model. To measure the performance upstates could trigger downstates (Fig. 1a, trials 5 and 6). Note that
of the model on any given trial, we first fit the parameters of the model on the initial response to the click (10 –35 ms; dark gray shading) was
the 3 s of spontaneous activity data immediately preceding the stimulus. approximately similar across trials, whereas the persistent response
The performance of the model in predicting the structure of the subse- (40 –135 ms; light gray shading) was more strongly modulated by
quent response was then quantified as follows. Using Equation 1 together ongoing cortical activity at the time of the stimulus.
with the time series {v(t), w(t)} for the first 300 ms of the response period, To illustrate how dynamic state can affect click responses,
we derived the error term (t) that would be required at each time step as Figure 1b shows data from a recording session whose dynamic
a driving force in order for the model, with the given parameters, to
state spontaneously varied from highly synchronized (trials 1 and
exactly reproduce the stimulus-evoked response. The “prediction error”
for the model in predicting this response is then defined as 1 2(t)dt,
2) to highly desynchronized (trials 5 and 6) activity. In these data,
where T 300 ms is the length of the response period. Note that the synchronized and desynchronized are not discrete cortical states
prediction error does not depend on comparison with a particular sim- but, rather, extremes in a continuum. In the more synchronized
ulated response; rather, it is a measure of how naturally the real response cases (Fig. 1b, top), responses follow a pattern similar to that
trajectory follows the flow lines of the model’s phase diagram. To com- shown in Figure 1a. In desynchronized states, however (Fig. 1b,
pute the “fit error” (supplemental Fig. 8, available at www.jneurosci.org bottom), the persistent response appears to be weakly modulated
as supplemental material), the same metric was used with (t) obtained by the stimulus, instead returning to a baseline firing rate that
as the residuals from fitting the model to spontaneous activity data. matches the average firing rate preceding the stimulus.
Quantitative assessment of model characteristics. To assess the linearity
of a particular fit model f(v, w), we obtained the model’s linear approxi-
Persistent activity (anti-)correlates with past activity in a
mation flin(v, w) from the Jacobian evaluated at the fixed point closest to
the origin (usually there is only one fixed point). We define the “degree of
state-dependent manner
nonlinearity” to be the log of the normalized norm of the difference To quantify the above observations, we began with a correlational
analysis. We define two “mean field” variables, v and w, that
log( f flin / f ), where f 2
[0,0.4] [0,0.25] f(v, w) 2dvdw summarize activity state at each instant (see Materials and Meth-
ods). The variable v measures average population firing rate and
and the integral is computed on a fine grid. (Note that almost all of the is represented by the red trace in Figure 1 and throughout the
data is confined to the region [0,0.4] [0,0.25] of the v w plane.) paper. The variable w measures the integrated recent activity of
To quantify the variability of dynamic state throughout a recording,
the network and is obtained by convolving v with a causal, decaying
we computed the summed point-wise variance
exponential filter of time constant 100 ms (i.e., passing v through
a leaky integrator) (Fig. 2a). This value of was chosen to maximize
0,0.4 0,0.25 Var f(v, w)dvdw
the (anti-)correlation of w with persistent responses (see below).
of the vector fields f(v, w) computed over all model fits (one for each To examine how sensory responses depend on the combina-
sensory stimulus) in a given recording. tion of dynamic state and activity state, we divided the trials into
five dynamic state categories ranging from the most synchro-
Results nized to the most desynchronized, based on the fraction of 0 –50
We recorded populations of 50 –100 cells, together with LFPs, Hz power that comes from the 0 –5 Hz interval (see Materials and
from auditory cortex of urethane-anesthetized rats using silicon Methods). Figure 2b shows a raster representation of population
microelectrodes (see Materials and Methods). We recorded both responses to clicks for each of these dynamic state categories,
Curto et al. • Dynamics in Auditory Cortex J. Neurosci., August 26, 2009 • 29(34):10600 –10612 • 10603
Figure 1. Trial-to-trial variability across a range of cortical states. a, Six examples of population responses to click stimuli, from a rat that exhibited stable dynamic state throughout the recording.
Vertical green lines denote stimuli (time 0); LFP (black trace), activity of simultaneously recorded single neurons (rasters), and smoothed MUA (red trace) all show a pattern of population activity
characteristic of the synchronized state. The right column shows an expanded view of the smoothed MUA in the response period for each trial; gray shaded areas denote “initial” (10 –35 ms; dark
gray shading) and “persistent” (40 –135 ms; light gray shading) response periods. The stimulus may arrive during a downstate (trials 1 and 2), at the beginning of an upstate (trials 3 and 4), or well
into an upstate (trials 5 and 6). Whereas preceding activity does not have a clear effect on peak activity levels in the initial response period, the timing of the stimulus relative to up/down transitions
appears to modulate activity in the persistent response period. b, Same conventions as in a; all data are selected from a different recording session that showed variable dynamic state. In the
synchronized (synch) state (trials 1 and 2), persistent responses are anticorrelated with activity levels in the 200 –300 ms preceding the stimulus. In intermediate states (trials 3 and 4), the stimulus
induces a large initial response followed by a transient downstate. In the most desynchronized (desynch) states (trials 5 and 6), responses exhibit a small but reliable initial response followed by a
return to baseline, with no discernible persistent response.
accumulated across all recordings (a similar analysis for each rather, a return to the prestimulus baseline, as evidenced by a
recording separately is shown in supplemental Fig. 1, available at similar positive correlation in the phantom trial data (supple-
www.jneurosci.org as supplemental material). Within each plot, mental Figs. 2– 4, available at www.jneurosci.org as supplemental
trials are further sorted by the value of w at the time of stimulus material). The initial response period showed a smaller modula-
presentation. Examination of plots corresponding to the most tion by prior activity than the persistent period, with a strong
synchronized states (Fig. 2b, top) agreed with the visual impres- negative correlation seen only in the most synchronized states.
sion conveyed by Figure 1a, indicating that click stimuli could flip
upstates and downstates and that activity state at the time of click A simple model of cortical state
presentation had more effect on persistent responses than on The previous results (Fig. 2) indicate a complex dependence of
initial responses. For lower levels of synchronization (Fig. 2b, sensory responses on dynamic and activity states at the time of
middle), the effect of prior activity state on persistent responses stimulus presentation. We now aim to show that this apparently
decreased. In the most desynchronized trials (Fig. 2b, bottom), complex relationship can be explained by a simple principle: sen-
there did not appear to be a clear persistent response. sory responses are shaped by the same dynamics that generate
Figure 2, c and d, shows, for each dynamic state category, the spontaneous activity before stimulus presentation. We will do
correlation of initial (10 –35 ms) and persistent (40 –135 ms) fir- this by showing that a dynamical system model, with parameters
ing rate responses with the normalized past activity w at the estimated from spontaneous activity preceding a stimulus, can
time of stimulus presentation. As expected, robust negative quantitatively predict the subsequent sensory response.
correlations were seen between past activity and persistent re- Because dynamic state can vary over a time scale of several
sponse periods in synchronized states (Fig. 2c,d, top). This nega- seconds, to gain an accurate “snapshot” of ongoing dynamics we
tive correlation did not simply reflect the tendency of upstates will need to estimate model parameters from segments of only a
and downstates to alternate in the absence of sensory stimuli, as few seconds of data. We found that a simple family of self-
confirmed by its absence in control analyses centered on ran- exciting dynamical systems yields good approximations for the
domly chosen times when no stimulus was presented (phantom different dynamic states observed in our data. There are many
trials) (supplemental Figs. 2– 4, available at www.jneurosci.org as families of self-exciting system models (Izhikevich, 2007). We
supplemental material). For more desynchronized states, the have chosen the FHN equations (Fitzhugh, 1955), as they are of
negative correlation was less pronounced. For the most desyn- simple form, are sufficiently flexible to allow a wide range of
chronized states (Fig. 2c,d, bottom) a positive correlation was linear and nonlinear dynamics (supplemental Fig. 5, available at
seen, although this did not reflect a true persistent response but, www.jneurosci.org as supplemental material), and allow rapid
10604 • J. Neurosci., August 26, 2009 • 29(34):10600 –10612 Curto et al. • Dynamics in Auditory Cortex
Figure 2. Persistent activity correlates with past activity in a state-dependent manner. a, Schematic of how we obtain a smoothed MUA trace (v; red) and integrated past activity trace (w; green)
from recorded spikes. b, Population responses to click stimuli. Trials from four different recording sessions were pooled and divided into five groups corresponding to different levels of synchroni-
zation (see Materials and Methods). Each box shows a pseudocolor representation of MUA rate (v; red) for each trial within the corresponding synchronization range, with trials within each box sorted
according to the value of the past activity variable w at the time of stimulus onset. c, d, For each trial, the strengths of the initial and persistent responses to the click were quantified by counting spikes
in the period 10 –35 and 40 –135 ms poststimulus, respectively. Each box shows the correlation of response strength with prior activity w, for the corresponding synchronization range. Note that
initial, persistent, and past activity values have all been normalized to better compare across recordings (see Materials and Methods). In highly synchronized states (top row), the persistent response
is anticorrelated with past activity; this correlation grows weaker and eventually reverses with increasing desynchronization. Initial response shows less correlation with past activity than does
persistent response. e, Regression slopes for initial and persistent responses as a function of synchronization range. synch, Synchronized; desynch, desynchronized.
and robust parameter estimation from as little as 3 s of MUA data In this scheme, the dynamical variables v(t) and w(t) correspond
(Fig. 3). Although in neuroscience these equations are usually to the activity state of the network at any instant, whereas the
considered as models for action potential generation in single model parameters (a1, a2, a3, b, and I ) that specify how v and w
neurons, they offer flexibility to model a wide range of self- evolve in time correspond to its dynamic state. In our case, v
exciting systems. The equations (identified previously as Eqs. 1 represents the mean firing rate of cortical neurons and is esti-
and 2) are as follows: mated by the smoothed MUA as descri

Use: 0.1088