Spiral waves in disinhibited mammalian neocortex.
ABSTRACT Spiral waves are a basic feature of excitable systems. Although such waves have been observed in a variety of biological systems, they have not been observed in the mammalian cortex during neuronal activity. Here, we report stable rotating spiral waves in rat neocortical slices visualized by voltage-sensitive dye imaging. Tissue from the occipital cortex (visual) was sectioned parallel to cortical lamina to preserve horizontal connections in layers III-V (500-mum-thick, approximately 4 x 6 mm(2)). In such tangential slices, excitation waves propagated in two dimensions during cholinergic oscillations. Spiral waves occurred spontaneously and alternated with plane, ring, and irregular waves. The rotation rate of the spirals was approximately 10 turns per second, and the rotation was linked to the oscillations in a one-cycle- one-rotation manner. A small (<128 mum) phase singularity occurred at the center of the spirals, about which were observed oscillations of widely distributed phases. The phase singularity drifted slowly across the tissue ( approximately 1 mm/10 turns). We introduced a computational model of a cortical layer that predicted and replicated many of the features of our experimental findings. We speculate that rotating spiral waves may provide a spatial framework to organize cortical oscillations.
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ABSTRACT: Neural field equations are integro-differential systems describing the macroscopic activity of spatially extended pieces of cortex. In such cortical assemblies, the propagation of information and the transmission machinery induce communication delays, due to the transport of information (propagation delays) and to the synaptic machinery (constant delays). We investigate the role of these delays on the formation of structured spatiotemporal patterns for these systems in arbitrary dimensions. We focus on localized activity, either induced by the presence of a localized stimulus (pulses) or by transitions between two levels of activity (fronts). Linear stability analysis allows to reveal the existence of Hopf bifurcation curves induced by the delays, along different modes that may be symmetric or asymmetric. We show that instabilities strongly depend on the dimension, and in particular may exhibit transversal instabilities along invariant directions. These instabilities yield pulsatile localized activity, and depending on the symmetry of the destabilized modes, either produce spatiotemporal breathing or sloshing patterns.02/2014;
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ABSTRACT: We study the effects of additive noise on traveling pulse solutions in spatially extended neural fields with linear adaptation. Neural fields are evolution equations with an integral term characterizing synaptic interactions between neurons at different spatial locations of the network. We introduce an auxiliary variable to model the effects of local negative feedback and consider random fluctuations by modeling the system as a set of spatially extended Langevin equations whose noise term is a $Q$-Wiener process. Due to the translation invariance of the network, neural fields can support a continuum of spatially localized bump solutions that can be destabilized by increasing the strength of the adaptation, giving rise to traveling pulse solutions. Near this criticality, we derive a stochastic amplitude equation describing the dynamics of these bifurcating pulses when the noise and the deterministic instability are of comparable magnitude. Away from this bifurcation, we investigate the effects of additive noise on the propagation of traveling pulses and demonstrate that noise induces wandering of traveling pulses. Our results are complemented with numerical simulations.01/2014;
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ABSTRACT: We study existence and stability of 2 - bump solutions of the one - population homogenized Wilson - Cowan model, where the heterogeneity is built in the connectivity functions by assuming periodic modulations in both the synaptic footprint and in the spatial scale. The existence analysis reveals that the generic picture consists of two bumps states for each admissible threshold value for the case when the solutions are independent of the local variable and the firing rate function is modeled as a Heaviside function. A framework for analyzing the stability of 2 - bumps is formulated, based on spectral theory for Fredholm integral operators. The stability method deforms to the standard Evans function approach for the translationally invariant case in the limit of no heterogeneity, in a way analogous to the single bump case for the homogenized model. Numerical study of the stability problem reveals that both the broad and narrow bumps are unstable just as in the translationally invariant case when the connectivity function is modeled by means of a wizard hat function. For the damped oscillating connectivity kernel, we give a concrete example of a 2 - bump solution which is stable for all admissible values of the heterogeneity parameter.Physica D Nonlinear Phenomena 01/2014; 271:19 - 31. · 1.67 Impact Factor
horizontal connections in layers III–V (500-?m-thick, ?4 ? 6 mm2). In such tangential slices, excitation waves propagated in two
dimensions during cholinergic oscillations. Spiral waves occurred spontaneously and alternated with plane, ring, and irregular waves.
The rotation rate of the spirals was ?10 turns per second, and the rotation was linked to the oscillations in a one-cycle–one-rotation
manner. A small (?128 ?m) phase singularity occurred at the center of the spirals, about which were observed oscillations of widely
A spiral wave in the broadest sense is a rotating wave traveling
outward from a center. Such spiral waves have been observed in
many systems (Winfree, 2001; Murray, 2003), including biolog-
ical systems, such as heart ventricular fibrillation (Davidenko et
al., 1992), retinal spreading depression (Gorelova and Bures,
1991), and glial calcium waves in cortical tissue culture (Verkh-
observed in many systems from invertebrates to mammals (Er-
mentrout and Kleinfeld, 2001), spiral waves of neuronal activity
have not been confirmed in mammalian brain despite consider-
Demonstrating a true spiral wave requires that the medium
result of spatial undulations in the excitable medium and its
properties (e.g., human EEG cannot detect true spirals if re-
corded on a scale larger than a single gyrus or from the scalp). If
“phase singularity” should be observed at the center that distin-
2001). The most rigorous demonstration of spiral wave forma-
tion in cortex that we are aware of is the finding of phase singu-
larities in optical imaging of turtle visual cortex, which demon-
strated circular waves persisting for up to four rotations (Prechtl
et al., 1997).
Although circular waves were predicted from early models of
cortical activity (Beurle, 1956), true spiral wave formation was
not observed until the more sophisticated Wilson–Cowan for-
ing simulation strategies (Milton et al., 1993). Our experimental
work was inspired by such theoretical considerations. Neverthe-
less, a close link between computational models of spiral wave
formation in cortex and experiment has not been attempted
In this report, we present evidence for stable spiral waves (up
to 30 cycles) in rat neocortical slices with robust phase singulari-
ties. We also introduce a computational model of a cortical layer
that predicts and replicates many of the features of our experi-
mental findings. Our results suggest the possibility that spiral
related to sensory and motor events.
Tangential slice. Neocortical slices were obtained from Sprague Dawley
rats (postnatal days 21–35). Tangential slices were cut with a vibratome
on the rostrocaudal and mediolateral coordinates of bregma ?2 to ?8
mm and lateral 1–6 mm, respectively (see Fig. 1, left). The first cut was
made 300 ?m deep from the pial surface, and the tissue was discarded.
The second cut was made 500 ?m deeper to obtain a 500-?m-thick slice
of middle cortical layers. The slice was perfused with artificial CSF
(ACSF) containing the following (in mM): 132 NaCl, 3 KCl, 2 CaCl2, 2
MgSO4, 1.25 NaH2PO4, 26 NaHCO3, and 10 dextrose (saturated with
95% O2and 5% CO2at 28°C for 1 hr before experiments). When the
This work was supported by National Institutes of Health Grants R01NS036447 (J.-Y.W.) and K02MH01493
TheJournalofNeuroscience,November3,2004 • 24(44):9897–9902 • 9897
lations (4–15 Hz) occurred spontaneously, and the activity appeared as
spiral and other waves in the voltage-sensitive dye imaging. These activ-
ities lasted as long as the preparation was perfused with carbachol and
bicuculline, similar to coronal slices (Lukatch and MacIver, 1997; Bao
and Wu, 2003).
Voltage-sensitive dye imaging. An oxonol dye, NK3630 (Nippon
Kankoh-Shikiso Kenkyusho, Okayama, Japan) was used as an indicator
of transmembrane potential. Slices were stained with 5–10 ?g/ml of dye
dissolved in ACSF for 60–120 min (26°C) and perfused in a submersion
chamber during the experiment (28°C). Imaging was performed with a
photodiode array on an upright microscope with transillumination (ab-
sorption) arrangement (Wu et al., 1999; Jin et al., 2002). Data were
displayed in the form of traces (see Figs. 1, 2B, 3) (digitally filtered 2–50
Hz) and pseudocolor images (see Figs. 2A, 3A) (digitally filtered 3–30
Hz). Supplementary movies S1–S4 (supplemental material, available at
www.jneurosci.org) are made with consecutive pseudocolor images.
Phase analysis. The phase map of the spiral (see Fig. 2C) was made
using programs written in the MATLAB (Mathworks, Natick, MA) ac-
cording to methods described by Prechtl et al. (1997). Briefly, raw data
(filtered 2–50 Hz) recorded by each photodetector (see Fig. 2C, Raw
frequency (see Fig. 2C, Filtered). Phase was then assigned to the filtered
signal (see Fig. 2C, Phase). The phase value at each location at a given
was composed of colors at all locations.
a, but, as in our experiments, no inhibition:
?u?x, y, t?
? ? u?x, y, t? ??
w?x, y, p, q? f ?u?p, q, t? ? ??dpdq ? a?x, y, t?
??a?x, y, t?
? ?u?x, y, t? ? a?x, y, t?,
where w represents the connectivity between neurons as a function of
position (x, y). We have taken w to be a product of the following form:
w?x, y, p, q? ? w1???x ? q?2? ?y ? p?2? g?q, p?,
where w1???x ? q?2? ?y ? p?2? is a symmetric decreasing Gaussian-
type function, and g(q,p) represents a mild anisotropy. In polar coordi-
f is represented by either a sigmoid or Heaviside function (equal to 0 for
variable whose rate of increase is proportional to ?u, and ? is the time
constant for change in a relative to change in u. For efficiency, we used a
Fourier transform method (Laing and Troy, 2003) to transform the sys-
tem into an equivalent system of partial differential equations. These
were solved using a finite difference scheme with Neumann (free)
boundary conditions. Spirals and ring waves were initiated with both
homogenous and mildly inhomogeneous coupling.
Oscillations of 4–15 Hz developed in neocortical tissue when
perfused with 100 ?M carbachol and 10 ?M bicuculline (Lukatch
and MacIver, 1997). The oscillations were organized as epochs,
each epoch containing 10–50 cycles. The epochs occurred spon-
by infrequent electrical stimulation (?3 min intervals).
Voltage-sensitive dye signals from these oscillations have
waveforms similar to the local field potential from the same lo-
cation (Fig. 1, right) (Bao and Wu, 2003). The signal-to-noise
ratio of the dye signal was high (5–10) (Fig. 1), and no signal
The voltage-sensitive dye signal reflects the fluctuation of trans-
membrane potential summed over a population of neurons
464 (25 ? 25 array) optical detectors to image a field of 3.2 or 4
mm diameter. In each slice, we recorded 300–2000 oscillation
cycles during 10–30 spontaneous or evoked epochs in a fixed
field of view.
ing waves (Golomb and Amitai, 1997; Wu et al., 1999; Bao and
Wu, 2003). In tangential slices (Fig. 1, left) (Fleidervish et al.,
1998), such oscillations develop into two-dimensional waves.
Four wave patterns were observed: spiral, plane, ring, and irreg-
ular (Fig. 2A) (movies 1–4, available at www.jneurosci.org as
supplemental material). Spiral waves appeared as a wave front
rotating around a center (Fig. 2A, top row). Each cycle of the
available at www.jneurosci.org as supplemental material). Plane
(movie 2, available at www.jneurosci.org as supplemental mate-
rial). The plane waves appeared to evolve from ring waves as the
wavefront curvature decreased when propagating outward from
the center of the ring (movie 4, available at www.jneurosci.org as
supplemental material). Irregular waves had multiple simulta-
neous wavefronts with unstable directions and velocities (Fig.
2A, bottom row) (movie 3, available at www.jneurosci.org as
within an oscillation epoch. Irregular waves usually occurred at
the beginning and the end of the oscillation epochs; the plane,
ring, or spiral waves typically occurred in the middle of the
epochs (Fig. 2B) and were relatively stable, i.e., similar wave
patterns repeated with each cycle of the oscillation. Spiral waves
able at www.jneurosci.org as supplemental material).
Spiral waves were observed in 48% of trials (116 of 242 trials
66 had at least four rotations (57%) and 10 had ?10 rotations
(9%). Both clockwise and counterclockwise rotations were seen
each emergent spiral always rotated in the same direction.
tex. The slice was 6 ? 8 mm (horizontal dimensions) and 500 ?m thick, containing cortical
line). Optical and electrical recordings from one location (black dot) are shown on the right.
Right, Optical signals (O) and local field potential (E) during rest and oscillation. Calibration:
9898 • J.Neurosci.,November3,2004 • 24(44):9897–9902Huangetal.•SpiralWavesinMammalianNeocortex
To distinguish the spirals from other types of rotating waves, we
analyzed the spatial phase distribution of the spirals (Fig. 2C).
During the entire period of the spiral, the phase distribution
spiral (Fig. 2C, white dots). The presence within such a phase
gradient of a phase singularity would be the hallmark of a true
spiral wave (Ermentrout and Kleinfeld, 2001; Winfree, 2001;
observed as a small region containing oscillating neurons with
nearly all phases represented between ?? and ?. Such phase
mixing would result in amplitude reduction in the optical signal.
In the experiment in Figure 3, we used
higher spatial resolution to search for the
singularity. Using a 25 ? 25 hexagonal ar-
ray with 464 elements, each detector cov-
ered a circular area 128 ?m in diameter
(total field of view, 3.2 mm in diameter).
amplitude oscillations before the forma-
tion of spirals (Fig. 3A, traces a–e, before
the first broken vertical line). During spi-
ral waves, the phase singularity drifted
The four detectors, a–d, alternately re-
corded reduced amplitude as the spiral
center approached each detector in turn.
Such amplitude reduction was localized at
the spiral center, and this reduced ampli-
tude propagated with drift of the spiral
center (Fig. 3A, traces a–d). In locations
distant from the spiral center (Fig. 3A, lo-
cation e), the amplitude remained high
during all rotations of the spiral. In plane
or ring waves, no localized region of oscil-
latory amplitude reduction was seen.
To further confirm that the amplitude
reduction was caused by superposition of
anti-phased oscillations, we examined the
signals surrounding a spiral center. Figure
3B shows the signals from a group of de-
tectors when a spiral center drifted over
the center detector. As the spiral center
hovered briefly (?200 msec, two rota-
the amplitude was reduced (Fig. 3B, trace
C). Simultaneously, the six surrounding
detectors (1–6) did not show amplitude
reduction, but the oscillation phases ex-
actly opposed each other symmetrically
across the center (Fig. 3B, traces 2, 5).
These results indicate that the area of am-
plitude reduction was less than or equal to
(128 ?m diameter). When we added the
the averaged waveform showed a similar
(Fig. 3B, AVG). This combined signal was
nearly identical to the central signal, as
demonstrated by the small residual when
strongly support that the amplitude reduction was not caused by
inactivity but rather by the superimposition of multiple widely
center was fully confined within the area of the central detector.
In all of the 63 cases of spiral waves examined by the 25 ? 25
array, we found that localized amplitude reduction always oc-
curred at the center of the spiral. Such phase singularities were
not anchored to a fixed location within the tissue but were ob-
served to drift while rotating.
We modeled the spiral waves in cortical slices with a homoge-
neous and isotropic two-dimensional excitable medium. The
Huangetal.•SpiralWavesinMammalianNeocortexJ.Neurosci.,November3,2004 • 24(44):9897–9902 • 9899
Cowan equations (Wilson and Cowan, 1972, 1973). Later, modifi-
wave propagation in excitatory disinhibited neural networks. We
points in a continuum that has excitation and recovery but, as in
our experiments, no inhibition. Such a model represents the
qualities of a disinhibited network dominated by fast excitation
ery adaptation that combines the refractory effects of spike inac-
tivation and voltage- and calcium-activated potassium currents.
Point stimulation of this model can successfully reproduce ring
and plane waves propagating away from the point of stimulation
(Fig. 4A, left and center).
The spirals in the model were initiated by breaking a wave-
front with an inhibitory stimulus applied where the wave meets
the boundary of the medium, creating a free end of sufficiently
high curvature to initiate spiral rotation. Such spiral waves
emerged macroscopically as a property of the network in the
absence of pacemaker or oscillatory microscopic dynamics and
were sustained as a prolonged and periodic activity in the net-
supplemental material). The model also showed irregular
waves with multiple wave fronts and annihilation after colli-
sions. These behaviors were consistent with the waves ob-
served in the cortical slices. When virtual detectors were
placed at different locations in the field of the rotating spiral
waves (Fig. 4A, right, a–d), amplitude reductions were ob-
served near the spiral center but not at other locations, con-
sistent with the observations in slices (Fig. 3).
Our results support the existence of true spiral waves of cortical
neuronal activities in four respects. First, a phase singularity
(Figs. 2, 3) was observed in spiral waves but not in other wave
patterns. Second, the oscillation amplitude was reduced at the
spiral center (Fig. 3A), and the reduction only occurred when a
the field of view of a single photodetector at our highest spatial
resolution (Fig. 3B). Last, spirals were not an artifact of the
boundary constraints from the edge of the slices because nonro-
ically alternate with rotating spiral waves.
organization within the phase singularity. The observed spiral
center was small, ?100 ?m in diameter (Fig. 3B). We speculate
that, at the center of the spiral, intracellular measurement would
show high-amplitude oscillations but that neighboring neurons
would have different phases. Future experiments with simulta-
neous intracellular measurements from neurons near the phase
singularity would be required to fully characterize such
Neuronal population wave activity in slices cut normal (perpen-
dicular) to the surface of the cortex produce unidirectional trav-
eling waves as if the cortical circuits were functionally one-
al., 1999, 2001; Bao and Wu, 2003). Population oscillatory wave
dervish et al., 1998) can generate activities that suggest that the
cortical circuits are functionally two-dimensional. Oscillation
patterns in two-dimensional networks can be complex, and at
least four types were described in this report. These patterns oc-
curred spontaneously and alternatively in the same tissue, sug-
gesting that the patterns are organized dynamically rather than
associated with particular anatomical structures within the
Golomb and Amitai (1997) have proposed that the speed of
related to the spread of horizontal connections. In our results,
icant role in the propagating velocity, direction, and the curva-
Amplitude reduction. A, Amplitude snapshots. The spirals started at approxi-
9900 • J.Neurosci.,November3,2004 • 24(44):9897–9902 Huangetal.•SpiralWavesinMammalianNeocortex
ture of the wave front, because, within a given set of anatomical
connections, different wave patterns occur (Fig. 2) (movies 1–4,
available at www.jneurosci.org as supplemental material). Inter-
estingly, all of the patterns were associated with the oscillation in
the same manner: one-cycle–one-wave for nonrotating waves
and one-cycle–one-rotation for spirals. This is consistent with
previous characterizations of one-dimensional waves in coronal
slices (Wu et al., 1999; Bao and Wu, 2003).
Although oscillations are commonly observed in sensory
tional (Pesaran et al., 2002) cortices, little is known about the
spatial organization that accompanies such oscillatory activity. It
has been shown in visual cortex that sensory-evoked oscillations
can demonstrate intercolumnar coherency (Eckhorn et al., 1988;
namic stability of spirals might extend the duration of evoked
logical conditions, might contribute to seizure generation. Spiral
waves might serve as emergent population pacemakers to gener-
ual cellular pacemakers. Spirals might be used for coordinating
oscillation phases over a population of neurons, serving func-
storage in working memory.
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