of 13
Neuron, Volume
87
Supplemental Information
Structured Variability in Purkinje Cell
Activity during Locomotion
Britton A. Sauerbrei, Evgueniy V. Lubenov, and Athanassios G. Siapas
Supplemental Experimental P
rocedures
Task and behavioral measurements
Three male Long-Evans rats (age 3 months) were trained to walk for water reward on a 1.8m-
long linear track (
Figure
1A
). Three LEDs were fixed to the headstage, and animal position
was mo
nitored using an overhead camera with video acquisition timestamped on the same
global clock as electrophysiological recordings. Head pitch and roll were estimated from data
acquired with a headstage-mounted inertial measurement unit using a Kalman smoother.
Bipolar patch EMG electrodes (Loeb and Gans, 1986) were implanted in the acromiotrapezius
muscle (English, 1978; Greene, 1968) and
were differentially amplified to monitor locomotion.
Electrode location was verified by postmortem dissection. Steps were identified by filtering the
EMG with a 100
-1000Hz finite impulse response filter, rectifying and smoothing the
signals
using a 100ms Gaussian kernel, then searching for sequences of peaks (local maxima)
occurring during track traversal (
Figure
S2
). Stepping sequences were required to include at
least four peaks, with consecutive peaks separated by 220-
450 ms. The step
phase was
obtained by linear interpolation, with the EMG peak defined as 0 for cells ipsilateral to the
muscle and on the midline, and as
π
for contralateral cells. All animal procedures were in
accordance with the National Institutes of Health (NIH) guidelines, and with the approval of the
Caltech Institutional Animal Care and Use Committee. The animals wer
e kept on a 12-hour
light cycle, followed by 12 hours of darkness, and were run in two daily sessions: one
approximately four hours into the light cycle, and one approximately four hours into the dark
cycle.
Electrophysiology
Rats were chronically implant
ed with a 24
-tetrode microdrive array. Twenty of the tetrodes
targeted lobules V and VI of the cerebellar vermis, spanning +/- 1mm of the midline
(
Figure
S3
). Four additional tetrodes targeted the visual cortex and the hippocampus. Tetrodes were
lowered to their targets over several days following implantation and subsequently adjusted in
small increments to maintain unit isolation. Signals were buffered on the headstage, amplified,
and acquired as 24-bit samples at 25kHz (National Instruments PXI-4498 cards) using custom
LabView-based acquisition software. A screw in the skull overlying the left cerebellum was
used as a reference for all signals. Spikes were clustered by fitting a mixture model in a 12-
dimensional feature space (3 waveform principal components per tetrode channel) (Calabrese
and Paninski, 2011; Ecker et al., 2014; Tolias et al., 2007
). For cells with large and stable
complex spikes, an additional stage of processing was used to distinguish complex from
simple spikes. The initial, fast segment
of the complex spikes had the same amplitude ratios
across tetrode channels as the simple spikes for the same unit (de Solages et al., 2008
), so
that a single unit cluster for a Purkinje cell contained both simple and complex spikes. A
matched filter for complex spikes was constructed by convolving a 25-30Hz finite impulse
response filter with a decaying exponential function having a time constant of 5ms. This filter
was applied to the raw, broadband data, and the
pe
aks in the resulting signal were registered
with the preceding spike from the corresponding unit cluster. The
broadband waveforms were
extracted within -2ms and 10ms of these putative complex spike events and sorted using
principal component features. Thus
, each spike in the unit cluster was identified as either
simple or complex. For analyses that did not compare simple and complex spike activity, the
cell’s spike train was taken to be the union of spikes of both types. At the end of each
1
experiment, electrolytic lesions were applied at each recording site, and the tetrode locations
were verified in Hoescht-stained tissue sections (
Figure
S3
).
Statistical analysis
The state
-dependent mean firing rates for locomotion, licking, and inactivity were defined as
the total number of spikes occurring in the state divided by the total duration of the state
throughout the dataset. For each Purkinje cell, a circular distribution was fit to the step phases
sampled at the spike times. A second-order generalized von Mises distribution (Gatto and
Jammalamadaka, 2007) was fit for cells with one or two peaks in the step cycle. For cells with
three peaks, a kernel density estimator was used with a wrapped normal distribution of width
(4/3N)
0.2
v, where v is the sample circular standard deviation (Mardia and Jupp, 2000). To
determine whether spiking was locked to the step cycle, Kuiper’s test was used (Mardia and
Jupp, 2000
) for both simple and complex spikes. Lick modulation was determined using the
same procedure, with the lick onset defined to be 0 degrees.
The step
-to
-step variability of spike counts was quantified using the variance-to
-mean ratio, or
Fano factor (
Figure
2D
). For each neuro
n and window length
Δ
t, the Fano factor was
computed as:
where n(
Δ
t) is the number of spikes lying within a window of length
Δ
t ms following the
EMG
peak for each step.
The shape of the step
-locked firing rate curves was investigated by smoothing the spike trains
with a 25ms Gaussian kernel and extracting the curves for each step, with the time axis
normalized so that each curve was parameterized by step phase
θ
, rather than time. Principal
component analysis was performed on the curves for each cell, and the effect of each
component, w(
θ
), was visualized as a perturbation of the mean curve, f(
θ
) (Ramsay and
Silverman, 2005). The extent to which each component was an additive shift, a multiplicative
scaling, or a phase shift was determined by computing bias, amplitude, and phase scores,
respectively. A pure bias shift corresponds to a completely flat component, so the bias score
was defined to be:
with v(
θ
) = 1/
n, where n = 50 is the length of the curve. A pure shift in amplitude corresponds
to a component with the same shape as the mean curve, but rescaled by a multiplicative
constant. The amplitude score was defined using:
S
bias
=
|
w
(
)
v
(
)
|
f
(
)=
f
(
)
μ
f
k
f
(
)
μ
f
k
S
amp
=
|
w
(
)
f
(
)
|
FF
(
t
)=
var
(
n
(
t
))
mean
(
n
(
t
))
2
where
μ
f
is the mea
n of f(
θ
). A pure phase shift of
δ
0 would correspond to a transformation
of f(
θ
) to f(
θ
+
δ
), so the phase score was defined using
:
where
Δ
= 22 degrees. The range of possible offsets was limited to 2
Δ
in order to regularize
the phase score es
timation.
The influence of behavioral variables on Purkinje cell activity across steps was
first determined
by averaging speed, forward acceleration, sin(roll), and sin(pitch), and EMG amplitude
within
each step cycle. Larger values of pitch reflect upward movement of the head (dorsiflexion).
Positive values of roll correspond to rightward rotations of the head for cells on the right side,
and to leftward rotation for cells on the left side and midline. To study the variation in curve
shape with behavior, slicing intervals for behavioral values were determined using an equal-
count algorithm with twelve (
Figure
4B
) or six (
Figure
4C
) intervals and 50% overlap
(Cleveland, 1993). Step
-locked firing rates were then averaged within each interval.
In order to quantify the influence of
behavior on spiking, we defined the firing rate on each step
to be the number of spikes occurring during that step divided by the step duration. We
converted the firing rate and the behavioral variables to z
-scores, and
estimated a multiple
regression model for each cell:
where Z
rate
is the z-score of the step
-locked firing rates
, Z
roll
is the z-score of sin(roll),
Z
pitch
is the
z-score of sin(pitch),
Z
EMG
is the z-score of the EMG amplitude, and
ε
is a normally-distributed
error term. The normality of the residuals was checked using quantile-quantile plots (
Figure
S7A
). Performing the same analysis using rank-transformed data, substituting the z-scored
ranks of the independent and dependent variables for the original linear data, produced nearly
identi
cal parameter estimates (
Figure
S7B
).
In order to study how the relationship between neural activity and behavior varies throughout
the step cycle, we next estimated step-phase
-dependent regression curves (
Figure
4D
). For
each step phase
θ
, we estimated the following model:
where Z
rate
(
θ
) is firing rate at phase
θ
, z-scored with respect to the distribution of firing rates at
θ
, and Z
speed
(
θ
) is
the
z-scored speed at
θ
.
We characterized the shape of these regression curves by computing scores for bias
(S
bias
),
amplitude
(S
amp
), and phase
(S
phase
), by substituting
β
var
(
θ
) /
‖β
var
(
θ
)
for w(
θ
) for each
g
(
)=
f
(
+
)
f
(
)
k
f
(
+
)
f
(
)
k
S
phase
=max
2
(
,
)
\{
0
}
|
w
(
)
g
(
)
|
Z
rate
=
speed
Z
speed
+
acc
Z
acc
+
roll
Z
roll
+
pitch
Z
pitch
+
EMG
Z
EMG
+
Z
rate
(
)=
speed
(
)
Z
speed
(
)+
acc
(
)
Z
acc
(
)+
roll
(
)
Z
roll
(
)+
pitch
(
)
Z
pitch
(
)+
EMG
(
)
Z
EMG
(
)+
3
behavioral variable. This produced an ordered triplet (S
bias
, S
amp
, S
phase
) for each significantly
tuned cell (
β
var
0 at q = .05). For each behavioral variable, we performed an agglomerative
hierarchical clustering analysis of these triplets using a Ward linkage function and a gap
criterion for determining the number of clusters (Everitt, 2011) (
Figure
4E
).
Coordinated activity between pairs of cells was assessed using the partial rank correlations in
the firing rates: the effects of the behavioral variables were first removed using the multiple
regression model, and Spearman’s
ρ
was computed between the residuals. Partial rank
correlations computed using the residuals from the rank-transformed regression model
produced nearly identical correlations (
Figure
S7C
).
The low rate of complex spikes did not permit the analysis of step-locked firing rat
es; instead,
we studied the effects of
behavioral variables on the number of complex spikes, C, within a
350ms window
starting at the onset of each step cycle. For each Purkinje cell with stable
complex spikes, we estimated a Poisson regression model:
where
μ
C
is the mean number of spikes, and Z
speed
, Z
acc
, Z
roll
, Z
pitch
, Z
EMG
and
ε
are defined as in
the previous equation.
For the analysis of step phase and lick modulation, the linear models, and the pairwise
correlations, corrections for multiple comparisons were made by setting the false discovery
rate to q = .05 (Benjamini and Hochberg, 1995). Data
from all animals, Purkinje cells, and
steps were included in the analysis. For parametric tests, the assumption of normality was
checked using normal quantile-quantile plots
.
log
(
E
(
C
|
Z
speed
,Z
roll
,Z
pitch
))=
μ
C
+
speed
Z
speed
+
acc
Z
acc
+
roll
Z
roll
+
pitch
Z
pitch
+
EMG
Z
EMG
+
4
Supplemental References
de Solages, C., Szapiro, G., Brunel, N., Hakim, V., Isope, P., Buisseret, P., Rousseau, C.,
Barbour, B., and Léna, C. (2008). High-frequency organization and synchrony of activity in the
purkinje cell layer of the cerebellum. Neuron
58
, 775-788.
English, A.W. (1978). An electromyographic analysis of forelimb muscles during overground
stepping in the cat. The Journal of experimental biology
76
, 105
-122.
Everitt, B. (2011). Cluster analysis, 5th edn (Chichester, West Sussex, U.K.: Wiley).
Gatto, R., and Jammalamadaka, S.R. (2007). The generalized von Mises distribution.
Statistical Methodology
4
, 341
-353.
Greene, E. (1968). Anatomy of the rat (New York: Hafner Pub. Co.).
Loeb, G.E., and Gans, C. (1986). Electromyography for experimentalists (Chicago: University
of Chicago Press).
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