Stable Neural Population Dynamics in the Regression Subspace for Continuous and Categorical Task Parameters in Monkeys

Abstract

Neural population dynamics provide a key computational framework for understanding information processing in the sensory, cognitive, and motor functions of the brain. They systematically depict complex neural population activity, dominated by strong temporal dynamics as trajectory geometry in a low-dimensional neural space. However, neural population dynamics are poorly related to the conventional analytical framework of single-neuron activity, the rate-coding regime that analyzes firing rate modulations using task parameters. To link the rate-coding and dynamic models, we developed a variant of state-space analysis in the regression subspace, which describes the temporal structures of neural modulations using continuous and categorical task parameters. In macaque monkeys, using two neural population datasets containing either of two standard task parameters, continuous and categorical, we revealed that neural modulation structures are reliably captured by these task parameters in the regression subspace as trajectory geometry in a lower dimension. Furthermore, we combined the classical optimal-stimulus response analysis (usually used in rate-coding analysis) with the dynamic model and found that the most prominent modulation dynamics in the lower dimension were derived from these optimal responses. Using those analyses, we successfully extracted geometries for both task parameters that formed a straight geometry, suggesting that their functional relevance is characterized as a unidimensional feature in their neural modulation dynamics. Collectively, our approach bridges neural modulation in the rate-coding model and the dynamic system, and provides researchers with a significant advantage in exploring the temporal structure of neural modulations for pre-existing datasets.

Full text

Cognition and Behavior
Stable Neural Population Dynamics in the
Regression Subspace for Continuous and
Categorical Task Parameters in Monkeys
He Chen,
1,p
Jun Kunimatsu,
2,3,p
Tomomichi Oya,
4,5
Yuri Imaizumi,
6
Yukiko Hori,
7
Masayuki Matsumoto,
2,3
Takafumi Minamimoto,
7
Yuji Naya,
1,8,9
and Hiroshi Yamada
2,3
https://doi.org/10.1523/ENEURO.0016-23.2023
1
School of Psychological and Cognitive Sciences, Peking University, Beijing 100805, People’s Republic of China,
2
Division of Biomedical Science, Institute of Medicine, University of Tsukuba, Tsukuba 305-8577, Japan,
3
Transborder
Medical Research Center, University of Tsukuba, Tsukuba 305-8577, Japan,
4
The Brain and Mind Institute, University
of Western Ontario, London N6A 3K7, Canada,
5
Department of Physiology and Pharmacology, University of Western
Ontario, London N6A 3K7, Canada,
6
Medical Sciences, University of Tsukuba, Tsukuba 305-8577, Japan,
7
Department of Functional Brain Imaging, National Institutes for Quantum Science and Technology, Chiba 263-8555,
Japan,
8
IDG/McGovern Institute for Brain Research at Peking University, Beijing 100805, People’s Republic of China,
and
9
Beijing Key Laboratory of Behavior and Mental Health, Peking University, Beijing 100805, People’sRepublicof
China
Abstract
Neural population dynamics provide a key computational framework for understanding information processing
in the sensory, cognitive, and motor functions of the brain. They systematically depict complex neural popula-
tion activity, dominated by strong temporal dynamics as trajectory geometry in a low-dimensional neural
space. However, neural population dynamics are poorly related to the conventional analytical framework of
single-neuron activity, the rate-coding regime that analyzes firing rate modulations using task parameters. To
link the rate-coding and dynamic models, we developed a variant of state-space analysis in the regression
subspace, which describes the temporal structures of neural modulations using continuous and categorical
task parameters. In macaque monkeys, using two neural population datasets containing either of two standard
task parameters, continuous and categorical, we revealed that neural modulation structures are reliably cap-
tured by these task parameters in the regression subspace as trajectory geometry in a lower dimension.
Furthermore, we combined the classical optimal-stimulus response analysis (usually used in rate-coding analy-
sis) with the dynamic model and found that the most prominent modulation dynamics in the lower dimension
were derived from these optimal responses. Using those analyses, we successfully extracted geometries for
both task parameters that formed a straight geometry, suggesting that their functional relevance is character-
ized as a unidimensional feature in their neural modulation dynamics. Collectively, our approach bridges neural
Significance Statement
Our results differ from earlier studies and suggest that our state-space analysis in the regression subspace
provides a mechanistic neural population structure for visual recognition of items when monkeys perceived
continuous and categorical task parameters. The neural population dynamics obtained from different brain
regions using different behavioral tasks were similar and may share some common underlying information
processing in a neural network. Our approach provides a simple framework for incorporating the single-neu-
ron approach into the dynamic model as a procedure for describing neural modulation dynamics in the
brain. This analytic extension gives researchers a significant advantage in that all types of pre-existing data
for single-neuron activity are useful for easily exploring their dynamics in a low-dimensional neural modula-
tion space.
July 2023, 10(7) ENEURO.0016-23.2023 1–20
Research Article: New Research

modulation in the rate-coding model and the dynamic system, and provides researchers with a significant ad-
vantage in exploring the temporal structure of neural modulations for pre-existing datasets.
Key words: dimensional reduction; monkey; neural population dynamics; regression subspace
Introduction
Recent innovations in state-space analyses have pro-
vided deep insights into the dynamic structure of informa-
tion processing in neural population activity (Brendel et
al., 2011;Churchland et al., 2012;Mante et al., 2013). The
identified fine temporal structures are known as neural
population dynamics and are considered to reflect under-
lying computations occurring in a neural network in the
sensory, cognitive, and motor domains (Churchland et al.,
2012;Raposo et al., 2014;Murray et al., 2017;Aoi et al.,
2020;Osako et al., 2021;Rossi-Pool et al., 2021). Neural
population dynamics provide a different perspective from
conventional analytical frameworks such as the rate-cod-
ing model, which usually analyzes neural modulations by
task parameters. However, dynamic and conventional
rate-coding models have rarely been compared with ana-
lyze neural activity. Thus, an important question remains
regarding how these two approaches reflect putatively
different or shared aspects of information processing
used by neurons in the underlying neuronal network.
An early analytical framework for information process-
ing was developed to describe the functional role of sepa-
rately recorded single-neuron activity (Mountcastle and
Henneman, 1949;Hubel and Wiesel, 1959;Evarts, 1968;
Wurtz, 1968), mostly using the rate-coding model (Dayan
and Abbott, 2001). Neuronal discharge rates were as-
sumed to be modulated by mathematical parameters
controlled in an experimental task. Examples of these
task parameters are listed, such as a Gabor function in
the visual cortex (Tolhurst and Movshon, 1975;Jones and
Palmer, 1987), movement direction and muscle force in
the motor cortex (Fetz and Cheney, 1980;Georgopoulos
et al., 1982), reward value in the parietal cortex (Platt and
Glimcher, 1999), and spatial awareness during navigation
in the hippocampus (HPC; O’Keefe and Dostrovsky,
1971). Recently, large-scale multichannel recording tech-
nology (Buzsáki et al., 2015;Jun et al., 2017) has been de-
veloped to understand information processing used by
neural networks. It has enabled the observation of single-
neuron activities in the tens of thousands, reinforcing the
need for a novel theoretical framework for neural compu-
tations (Yuste, 2015), one of which is neural population
dynamics provided by state-space analysis (Timothy and
Bona, 1968). State-space analysis has emerged with a
greater focus on temporal changes in neuronal activity com-
bined with dimensional reduction techniques (Elsayed and
Cunningham, 2017;Aoi and Pillow, 2018;Keemink and
Machens, 2019;Saxena and Cunningham, 2019;Vyas et
al., 2020). Both analytical frameworks have described brain
functions in various functional domains. However, the rela-
tionship between the dynamic and conventional rate-coding
models remains poorly understood.
Some remarkable studies have attempted to find the
link between the rate-coding and dynamic models. For
example, demixed principal component analysis (dPCA;
Brendel et al., 2011;Kobak et al., 2016) decomposed
neural population data into latent components related to
discrete task parameters, such as stimulus and binary
choices. In another approach, targeted dimensionality re-
duction (TDR) analysis (Mante et al., 2013;Aoi et al., 2020)
incorporated linear regression into neural population dy-
namics with respect to task parameters, such as the stim-
ulus, binary choice, and task context, and identified the
encoding axis of these task parameters in a multidimen-
sional neural space. These state-of-the-art techniques
visualize complex datasets with multiple task parameters
and depict the temporal structures of task-related neural
modulation. However, they have several limitations when
applied to actual data and require the user to precisely
understand these elaborate procedures involving complex
multistep mathematical processes (Kobak et al., 2016;Aoi
et al., 2020). Although these analytical tools describe the
temporal structure of neural modulation as a latent compo-
nent in the multidimensional neural space (Kobak et al.,
2016, their Fig. 3B; Aoi et al., 2020, their Fig. 3A) the ex-
tracted modulation structures do not simply correspond to
the results from the rate-coding analyses. Thus, it is worth-
while to develop a simple, user-friendly method to extract
temporal structures of neural modulations that are compati-
ble with the rate-coding model.
We previously developed a variant of state-space anal-
ysis for continuous parameters (Yamada et al., 2021) that
describes how a neuronal population dynamically repre-
sents some cognitive task parameters in a regression
subspace. This analysis simply extracts the temporal
structures of neural modulations in the following two
steps using standard statistical software: (1) an estimation
Received January 13, 2023; accepted June 21, 2023; First published June 29,
2023.
The authors declare no competing financial interests.
Author contributions: H.C., Y.N., and H.Y. designed research; H.C., Y.I.,
Y.H., M.M., T.M., Y.N., and H.Y. performed research; H.C. and H.Y.
contributed unpublished reagents/analytic tools; H.C., Y.I., Y.H., and H.Y.
analyzed data; H.C., J.K., T.O., Y.I., Y.H., M.M., T.M., Y.N., and H.Y. wrote the
paper.
This research was supported by Japan Society for the Promotion of Science
KAKENHI (Grant JP 22H04832), the Research Foundation for the Electrotechnology
of Chubu, Japan Science and Technology Administration Moonshot R&D Grant
JPMJMS2294 (H.Y.) and JPMJMS2295 (T.M.), and the National Natural Science
Foundation of China (Grant 31871139; Y.N.).
Acknowledgments: We thank Yasuhiro Tsubo, Tomohiko Takei, and Kazuyuki
Samejima for valuable comments; and Takashi Kawai, Ryo Tajiri, Yoshiko Yabana,
Yuki Suwa, and Shiho Nishino for technical assistance. We also thank the National
BioResouce Project “Japanese Monkeys”of the Ministry of Education, Culture,
Sports, Science and Technology of Japan for providing Monkey FU.
*H.C. and J.K. contributed equally to this work.
Correspondence should be addressed to Hiroshi Yamada at h-yamada@
md.tsukuba.ac.jp.
https://doi.org/10.1523/ENEURO.0016-23.2023
Copyright © 2023 Chen et al.
This is an open-access article distributed under the terms of the Creative
Commons Attribution 4.0 International license, which permits unrestricted use,
distribution and reproduction in any medium provided that the original work is
properly attributed.
Research Article: New Research 2 of 20
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of regression coefficients for neural modulations by task
parameters across time and neurons, which provides a
regression matrix representing the extent of neural modu-
lation as a function of time in a neural population; and (2)
application of PCA to the regression matrix, which pro-
vides neural dynamics in the regression subspace (i.e.,
temporal structures of neural modulations by the task pa-
rameters). Our previous study successfully captured neural
modulation dynamics using continuous task parameters re-
lated to value-based decision-making. However, the analy-
sis was only performed for continuous task parameters, and
it is still unclear whether this approach could be expanded
to neural population activity modulated by noncontinuous
categorical task parameters.
In the present study, we developed an analysis method
that could be applied to existing datasets using a typical
factorial design for conventional single-neuron recordings
(Fig. 1), that is, categorical task parameters, from the hip-
pocampus of monkeys while memorizing a visual item
and its location (Chen and Naya, 2020). Thereafter, we
compared neural modulation dynamics between continu-
ous and categorical task parameters for visual stimulus
modulations. In particular, we have described neural dy-
namics within a short time period of 0.6 s when a visual
cue appears, and neurons encode the information of the
categorical or continuous task parameter conveyed by
the visual stimulus. This approach successfully provided
the temporal structures of neural modulations for both
types of task parameters as trajectory geometry in the low-
dimensional neural modulation space. The extracted geo-
metries for both task parameters form a straight geometry,
suggesting that their functional relevance is characterized
as a unidimensional feature in their neural modulation dynam-
ics. Thus, analyzing the neural modulation dynamics for all
types of pre-existing data is beneficial, allowing researchers
to incorporate the rate-coding model into a dynamic system.
Materials and Methods
Subjects and experimental procedures
Four macaque monkeys were used in two experiments
[Experiment 1 (Exp. 1): monkey SUN, Macaca mulatta,
7.1 kg, male; monkey FU, Macaca fuscata, 6.7 kg, female;
Exp. 2: monkey A, M. mulatta, 9.3 kg, male; monkey D, M.
mulatta, 9.5 kg, male). All experimental procedures were
approved by the Animal Care and Use Committee of the
University of Tsukuba (Exp. 1, protocol #H30.336), and
the institutional animal care and use of laboratory animals,
approved by Peking University (Exp. 2, project #Psych-
YujiNaya-1). All procedures were performed in compli-
ance with the US Public Health Service Guide for the Care
and Use of Laboratory Animals.
Behavioral task
Cued lottery tasks in Experiment 1
The animals performed one of two visually cued lottery
tasks: a single-cue or choice task. Neuronal activity was
only recorded during the single-cue task.
The animals performed the task under dim lighting condi-
tions in an electromagnetically shielded room. Eye move-
ments were measured using a video camera system at
120 Hz (EyeLink, SR Research). Visual stimuli were gener-
ated by a liquid crystal display at 60 Hz placed 38 cm from
the face of the monkey when seated. At the beginning of the
single-cue task trials, the monkeys had 2 s to align their
gaze within 3° of a 1° diameter gray central fixation target.
After fixation for 1 s, a pie chart was presented for 2.5 s to
provide information regarding the probability and magnitude
of rewards in the same location as the central fixation target.
The probability and magnitude of the rewards were associ-
ated with the number of blue and green 8° pie chart seg-
ments, ranging from 0.1 to 1.0 ml in 0.1 ml increments for
magnitude, and from 0.1 to 1.0 in 0.1 increments for proba-
bility. Following a 0.2 s interval from the removal of the pie
chart,eithera1ora0.1kHztoneof0.15sdurationwaspro-
vided to indicate reward or no-reward outcomes, respec-
tively. After a 0.2 s interval following the high tone, a fluid
reward was delivered, whereas no rewards were delivered
following the low tone. An intertrial interval of 4–6swas
used. During the choice task, the animals were instructed to
choose one of two peripheral pie charts, each of which indi-
cated either the probability or magnitude of an upcoming re-
ward. The two target options were presented for 2.5 s at 8°
to the left or right of the central fixation location. The animals
received a fluid reward, indicated by the green pie chart of
the chosen target, with the probability indicated by the blue
pie chart. Otherwise, no reward was delivered.
A total of 100 pie charts were used in the experiments,
which were composed of 10 levels of probability and
Figure 1. Schematic depictions of the state-space analysis in the regression subspace. The state-space analysis in the regression sub-
space provides the neural dynamics for activity modulations. In a particular experimental condition, neural population activity composed
of multiple neurons is modulated by the task variables [e.g., location (L1, L2) and stimulus (S1, S2; left)]. By applying PCA (principal com-
ponent analysis) for neural modulations (left), predominant components of the neural modulations are extracted (middle) and depicted as
the neural trajectory (right). Typical examples of neural population structures observed previously are straight, curvy, and rotational struc-
tures. imp/s indicates impulse per second. PC1 to 3 indicates first to third principle components, respectively.
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magnitude of rewards. In the single-cue task, the 100 pie
charts were presented once in random order. In the
choice task, two pie charts were randomly allocated to
the two options. During one session of electrophysiologi-
cal recording, ;30–60 trial blocks of the choice task were
interleaved with 100–120 trial blocks of the single-cue
task.
Item-location-retention task in Experiment 2
The animals performed the task under dim light condi-
tions in an electromagnetically shielded room. The task
started with an encoding phase, which was initiated by
the animal pulling a lever and fixating on a white square
(0.6°) presented within one of four quadrants at 12.5°
(monkey A) or 10° (monkey D) from the center of the touch
screen (17 inches; MicroTouch Display M1700SS, 3M),
situated ;28 cm from the subjects. Eye position was
monitored using an infrared digital camera with a sam-
pling frequency of 120 Hz (model ETL-200, ISCAN). After
fixation for 0.6 s, one of the six items (radius: 3.0° for mon-
key A; 2.5° for monkey D) was presented in the same
quadrant as a sample stimulus for 0.3 s, followed by an-
other 0.7 s fixation on the white square. If fixation was
successfully maintained (typically, ,2.5°), the encoding
phase ended with the presentation of a single drop of
water.
The encoding phase was followed by a blank interphase
delay interval of 0.7–1.4 s during which no fixation was re-
quired. The response phase was initiated using a fixation
dot presented at the center of the screen. One of the six
items was then presented at the center for 0.3 s as a cue
stimulus. After another 0.5 s delay period, five disks were
presented as choices, including a blue disk in each quad-
rant and a green disk at the center. When the cue stimulus
was the same as the sample stimulus, the animal was
required to choose by touching the blue disk in the same
quadrant as the sample (i.e., the match condition). Otherwise,
the subject was required to choose the green disk (i.e., the
nonmatch condition). If the animal made the correct choice,
four to eight drops of water were provided as a reward; oth-
erwise, an additional 4 s was added to the standard intertrial
interval (1.5–3 s). The number of reward drops was in-
creased to encourage the animal to maintain a good per-
formance in the latter phase of a daily recording session,
which was typically conducted in blocks (e.g., a minimal set
of 60 trials with equal numbers of visual items presented in a
match/nonmatch condition). During the trial, a large gray
square (48° on each side) was presented at the center of the
display, as a background. After the end of the trial, all stimuli
disappeared, and the entire screen displayed a light red
color during the intertrial interval. The start of a new trial was
indicated by the reappearance of a large gray square on the
display, at which point the monkey could pull the lever, trig-
gering the appearance of a white fixation dot. In the match
condition, sample stimuli were pseudorandomly chosen
from six well learned visual items, and each item was pre-
sented pseudorandomly within four quadrants, resulting in
24 (6 4) configuration patterns. In the nonmatch condition,
the location of the sample stimulus was randomly chosen
from the four quadrants, and the cue stimulus was randomly
chosen from the remaining five items that differed from the
sample. The match and nonmatch conditions were ran-
domly presented at a ratio of 4:1, resulting in 30 (24 16)
configuration patterns. The same six stimuli were used
during all recording sessions.
Electrophysiological recordings and data
preprocessing
Experiment 1
We used conventional techniques to record single-neu-
ron activity from the central part of the orbitofrontal cortex
(cOFC; area 13 M). A tungsten microelectrode (1–3MV;
FHC) was used to record single-neuron activity. The elec-
trophysiological signals were amplified, bandpass filtered
(50–3000 Hz), and monitored using a recording system
(model RZ5D, Tucker-Davis Technologies). Single-neuron
activity was isolated manually based on the online spike
waveforms. The activity of all single neurons was sampled
when the activity of an isolated neuron demonstrated a
good signal-to-noise ratio (.2.5). The signal-to-noise
ratio was calculated online as the ratio of the spike ampli-
tude and the range of the baseline voltage on the oscillo-
scope. Blinding of recorded neurons was not performed.
The sample sizes required to detect the effect sizes (the
numbers of recorded neurons, recorded trials in a single
neuron, and monkeys) were estimated based on previous
studies (Yamada et al., 2013;Chen and Stuphorn, 2015;
Yamada et al., 2018). Neural activity was recorded during
the 100–120 trials of the single-cue task. Neural activity
was not recorded during the choice trials. We recorded
the cOFC of a single right-side hemisphere in each of the
two monkeys in the experiment, with a total of 190 cOFC
neurons (98 SUN and 92 FU). In Exp. 1, only a single-neu-
ron recording was performed online.
Experiment 2
To record single-unit activity, we used a 16-channel vec-
tor array microprobe (model V1 X 16-Edge, NeuroNexus), a
16-channel U-Probe (Plexon), a tungsten tetrode probe
(Thomas RECORDING), or a single-wire tungsten microelec-
trode (Alpha Omega). The electrophysiological signals were
amplified, bandpass filtered (200–6000 Hz), and monitored.
Single-neuron activity was isolated based on the spike
waveforms online or offline. For both clustering and offline
sorting, the activity of all single neurons was sampled when
the activity of an isolated neuron demonstrated a good sig-
nal-to-noise ratio (.2.5). We visually checked the signal-to-
noise ratio by calculating the range of background
noise against the spike amplitude, monitored online
using the OmniPlex Neural Data Acquisition System
orofflineusingtheOfflineSortersoftware(Plexon).
Blinding of recorded neurons was not performed. The
sample sizes required to detect the effect sizes (the
numbers of recorded neurons, recorded trials in a sin-
gle neuron, and monkeys) were estimated based on
previous studies (Naya et al., 2003;Naya and Suzuki,
2011). Neural activity was recorded during the 60–240
trials of the item-location-retention (ILR) task. We re-
corded 590 HPC neurons, among which the recording
sites appeared to cover all its subdivisions (i.e., the
dentate gyrus, CA3, CA1, and subicular complex).
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Statistical analysis
For the statistical analyses, we used the statistical soft-
ware packages R (Exp. 1) and MATLAB (Exp. 2). All statis-
tical tests for the neural analyses were two tailed.
Behavioral analysis
No new behavioral results were included, but the proce-
dure for the behavioral analysis are as follows.
Regarding Exp. 1, we previously reported that monkey
behavior depends on expected values, defined as the
probability time magnitude (Yamada et al., 2021). We de-
scribed the analysis steps to check whether the behavior
of the monkey reflected the task parameters, such as re-
ward probability and magnitude. Importantly, we showed
that the choice behavior of the monkey reflected the
expected values of the rewards (i.e., probability multi-
plied by magnitude). For this purpose, the percentage
of choosing the right-side option was analyzed in the
pooled data using a general linear model with the fol-
lowing binomial distribution:
PchoosesR¼1=ð11ezÞ;
where the relationship between Pchooses
R
and Zwas
given by the logistic function in each of the following three
models: number of pie segments (M1), probability and
magnitude (M2), and expected values (M3), as follows:
M1 :Z¼b01b1NpieL1b2NpieR;
where b
0
is the intercept and Npie
L
and Npie
R
are the
number of pie segments contained in the left and right pie
chart stimuli, respectively. Values of b
0
to b
2
were free pa-
rameters and were estimated by maximizing the log likeli-
hood, as follows:
M2 :Z¼b01b1PL1b2PR1b3ML1b4MR;
where b
0
is the intercept; P
L
and P
R
are the probability of
rewards for left and right pie chart stimuli, respectively,
and M
L
and M
R
are the magnitude of rewards for left and
right pie chart stimuli, respectively. Values of b
0
to b
4
were free parameters and estimated by maximizing the
log likelihood, as follows:
M3 :Z¼b01b1EVL1b2EVR;
where b
0
is the intercept and EV
L
and EV
R
are the ex-
pected values of rewards as probability multiplied by
magnitude for left and right pie chart stimuli, respectively.
Values of b
0
to b
2
were free parameters and estimated by
maximizing the log likelihood. We identified the best
model to describe the behavior of the monkeys by com-
paring their goodness-of-fit values based on Akaike in-
formation criterion and Bayesian information criterion
(Burnham and Anderson, 2004).
Regarding Exp. 2, we previously reported that two mon-
keys learned to retain item and location information of the
sample stimulus (Chen and Naya, 2020). Here, we de-
scribed the analysis steps to check whether the monkey
used both item and location information to perform the
task.
To examine the above point, we compared the actual
correct rates of animals during the recording to the ran-
dom correct rates (
x
2
test). The ILR response phase has
five options, resulting in a 20% random correct rate. If the
animal uses a wrong strategy, such as only retaining the
location information of the sample stimulus and ignoring
the item information, the correct rate for the match condi-
tion would be 100% and the correct rate for the nonmatch
condition would be 0. Based on the above considerations,
we examined the actual correct rates of the two animals in
the match and nonmatch conditions, respectively. In gen-
eral, the average correct rates of both animals in the
match and nonmatch conditions were well above the
chance level after training.
Neural analysis
Peristimulus time histograms were constructed for each
single-neuron activity, aligned at the onset of the visual
stimulus. Average activity curves were smoothed for vis-
ual inspection using a Gaussian kernel (Exp. 1,
s
¼50
ms; Exp. 2,
s
¼20 ms), while the Gaussian kernel was not
used for the statistical tests.
To achieve comparisons between the two different da-
tasets that were as fair as possible, we used the same
criteria to analyze the neural activity. For the neural anal-
yses, we used the following four criteria: (1) same analy-
sis window size; (2) visual response within a short time
0.6 s; (3) neural modulations detected at the same signif-
icance level (p,0.05); and (4) by using a general linear
model(linearregression,Exp.1.;ANOVA,Exp.2).The
details of these analysis procedures are shown below for
the rate-coding and dynamic models.
Rate-coding model: conventional analyses to detect
neural modulations in each neuron
We analyzed neural activity for the 1 s time period (0–1
s after cue onset, Exp. 1) and 0.92 s time period (0.08–1s
after sample onset, Exp. 2). These activities were used for
the conventional analyses described below. No Gaussian
kernel was used.
Experiment 1
Neural discharge rates (F) were fitted using a linear
combination of the following parameters:
F¼b01bpProbability 1bmMagnitude;(1)
where Probability (P) and Magnitude (M) denote the
probability and magnitude of the rewards indicated by
the pie chart segments, respectively. b
0
is the inter-
cept, and if b
p
and b
m
were not 0 at p,0.05, the dis-
charge rates were regarded as significantly modulated
by that variable. The regression coefficients for proba-
bility and magnitude were plotted as a time series for
visual inspection. These results have been reported
previously (Yamada et al., 2021).
Based on linear regression, activity modulation pat-
terns were categorized into the following several types:
Probability type with a significant b
p
and without a sig-
nificant b
m
; Magnitude type without a significant b
p
and
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with a significant b
m
; Both type with significant b
p
and
b
m
.
Experiment 2
For neural responses during the encoding phase after
the sample presentation, we evaluated the effects of
“item”and “location”for each neuron using a two-way
ANOVA (p,0.05 for each). We analyzed neurons that
were tested in at least 60 trials (10 trials per stimulus and
15 per location). On average, we conducted 100 trials per
neuron (n¼590). The results of the ANOVA were previ-
ously reported (Chen and Naya, 2020). The statistical sig-
nificance level was set at p,0.05 in Exp. 2, same as that
in Exp. 1 (p,0.05). The regression table was extracted
and plotted as a time series for visual inspection.
Based on the ANOVA, activity modulation patterns
were categorized into several types: Item type, with a
significant main effect of the item and without Location
effect; Location type, with a significant main effect of
the location and without item effect; and Both type, with
significant effects of Item and Location or with a signifi-
cant interaction.
Dynamic model: population dynamics for representing
neural modulations
We analyzed neural activity during a 0.6 s time period
from cue onset (Exp. 1) and sample onset (Exp. 2). To ob-
tain a time series of neural firing rates within this period,
we estimated the firing rates of each neuron for each 0.02
s time bin (without overlap) during the 0.6 s period. No
Gaussian kernel was used. For this dynamic analysis, the
standard statistical software R was used to apply (1) a re-
gression analysis or ANOVA and (2) PCA. No other mathe-
matical applications were used.
Regression subspace
We used a general linear model to determine the proba-
bility and magnitude of rewards (Exp. 1) and item and lo-
cation (Exp. 2) affecting the activity of each neuron in the
neural population. Each neural population was based on
all recorded neurons in each brain region.
In Exp. 1, we first set probability and magnitude to 0.1
and 1.0, and 0.1–1.0 ml, respectively. We then described
the average firing rates of neuron iat time tas a linear
combination of the probability and magnitude in each
neural population, as follows:
Fði;t;kÞ¼b0ði;tÞ1b1ði;tÞProbabilityðkÞ1b2ði;tÞMagnitudeðkÞ;
(2)
where F
(i,t,k)
denotes the average firing rate of neuron iat
time ton trial k, Probability
(k)
denotes the probability of the
reward cued to the monkey in trial k, and Magnitude
(k)
is
the magnitude of the reward cued to the monkey in trial k.
The regression coefficients b
0(i,t)
to b
2(i,t)
describe the
degree to which the firing rates of neuron idepend on
the mean firing rates (hence, firing rates independent
of task parameters), probability of rewards, and mag-
nitude of rewards respectively, at a given time tduring
the trials. To construct the regression matrix, this infor-
mation was obtained in the regression table for linear
regression, which represents the coefficient for the
probability and magnitude.
In Exp. 2, we first set six items and four locations as cat-
egorical parameters. We then described the average firing
rates of neuron iat time tas a linear combination of item
and location in each neural population, as follows:
Fði;t;kÞ¼b0ði;tÞ1b1ði;tÞItemðkÞ1b2ði;tÞLocationðkÞ;(3)
where F
(i,t,k)
denotes the average firing rate of neuron iat
time tin trial k, Item
(k)
denotes the type of item cued to the
monkey in trial k, and Location
(k)
is the type of location
cued to the monkey in trial k. Each of the regression coef-
ficients b
0(i,t)
,b
1(i,t)
, and b
2(i,t)
describe the degree to which
the firing rates of neuron idepend on the mean firing
rates (b
0
, firing rates independent of task parameters),
the degree of the firing rate in each item relative to the
mean firing rates (b
1
, composed of six coefficients for
corresponding items), and the degree of firing in each
location relative to the mean firing rates respectively
(b
2
, composed of four coefficients for corresponding
locations), at a given time tduring the trials. The inter-
action term was not included in the model because of
the linear assumption in the state-space analysis. To
construct the regression matrix, this information was
obtained in the regression table for ANOVA, which rep-
resents the averaged activity differences in both item
and location from their mean.
In Exp. 1 and Exp. 2, we used the regression coeffi-
cients (regression table in the case of ANOVA) described
in Equations 2 and 3to identify how the dimensions of the
neural population signals were composed of information
related to probability and magnitude (Exp. 1) and to item
and location (Exp. 2). This step constructed an encoding
model in the subspace, where the regression coefficients
could be represented by the temporal structure of the
neural modulation of two continuous parameters (Exp.
1) or two categorical parameters (Exp. 2) at the popula-
tion level. In this study, our orthogonalized task design
allowed us to reliably project neural firing rates into a
regression subspace. In Exp. 1., two continuous pa-
rameters for probability and magnitude (10 10) were
orthogonalized in the data. In Exp. 2., two categorical
parameters for item and location (6 4) were ortho-
gonalized in the data. Our procedures were analogous
to the state-space analysis performed by Mante et al.
(2013), in which regression coefficients were used to
provide an axis (or dimension) of the parameters of in-
terest in a multidimensional state space obtained
through PCA. However, our analysis was different from
that of Mante et al. (2013), in terms of describing the
temporal structures of neural modulations (i.e., neural
dynamics in the regression subspace).
Principal component analysis
We used PCA to identify the dimensions of the neural
modulation signal in the orthogonal spaces composed of
the probability and magnitude of rewards in Exp. 1 and
items and locations in Exp. 2 for each neural population.
In each neural population, we first prepared a two-dimen-
sional data matrix Xof size N
(n)
M
(CT)
; regression
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coefficient vectors b
1(i,t)
and b
2(i,t)
in Equations 2 and 3,
whose rows correspond to the total number of neurons (n)
in each neural population, while the columns correspond
to C, the total number of conditions (i.e., probability and
magnitude in Exp. 1, six items and four locations in Exp.
2). Tis the total number of analysis windows; hence, time
(i.e., 30 bins:0.6 s divided by the window size bin 0.02 s).
A series of eigenvectors of the covariance matrix of Xwas
obtained by applying PCA one time to the data matrix Xin
each neural population. We used the prcomp () function in
the R software. The two-dimensional regression matrix X
was not normalized. The principal components (PCs) of
this data matrix consisted of vectors v
(a)
of length N
(n)
and
the total number of recorded neurons, if M
(CT)
is greater
than N
(n)
; otherwise, the length was M
(CT)
. The PCs
were indexed from those explaining most of the variance
to the least. We did not include the intercept term b
0(i,t)
to
focus on neural modulation using the parameters of
interest.
Eigenvectors
When we applied PCA to the data matrix X, we obtained
the eigenvectors and eigenvalues of the covariance matrix
of X. Each eigenvector had a corresponding eigenvalue.
In our analysis, the eigenvectors at time trepresented
vectors in the space of probability and magnitude in Exp.
1 and six items and four locations in Exp. 2. The eigenval-
ues at time tfor probability and magnitude in Exp. 1 and
items and locations in Exp. 2, respectively, indicate the
extent of variance in the data of that vector. Therefore, the
first PC is the eigenvector with the highest eigenvalue
over time. We analyzed the eigenvectors for the first three
PCs (PC1 to PC3). We applied PCA one time to each neu-
ral population; thus, the total variance contained in the
data differed among the neural populations.
Analysis of eigenvectors
We evaluated the characteristics of eigenvectors for
PC1 to PC3 in each neural population in terms of vector
angle, size, and deviance in the space of probability and
magnitude in Exp. 1, and in terms of items and locations
in Exp. 2, respectively. The angle is the vector angle from
the horizontal axis of 0°, from 180° to 180° against the
main PCs. The size is the length of the eigenvector, and
the deviance is the difference between vectors. We esti-
mated deviance from the mean vector for each neural
population. These three characteristics of the eigenvec-
tors were compared in each neural population at p,0.05,
using the Kruskal–Wallis and Wilcoxon rank-sum tests.
The vector during the first 0.1 s was extracted from these
analyses.
Shuffle control for PCA
We performed three shuffle controls to examine the sig-
nificance of the population structures described by PCA.
A two-dimensional data matrix Xwas randomized by
shuffling in three ways. In shuffle condition 1, matrix X
was shuffled by permutating the allocation of neuron iat
each time t. This shuffle provided a data matrix Xof size
N
(n)
M
(CT)
, eliminating the temporal structure of neural
modulation by condition Cin each neuron but retaining
neural modulations at time tat the population level. In
shuffle condition 2, matrix Xwas shuffled by permutating
the allocation of time tin each neuron i. This shuffle pro-
vided a data matrix Xof size N
(n)
M
(CT)
, eliminating the
neural modulation structure under condition Cmaintained
in each neuron, but retaining neural modulation in each
neuron at the population level. In shuffle condition 3, ma-
trix Xwas shuffled by permutating the allocation of both
time tand neuron i. In these three shuffle control condi-
tions, matrix Xwas estimated 1000 times. PCA perform-
ance was evaluated by constructing the distributions
of the explained variances for PC1 to PC12. The statis-
tical significance of the variances explained by PC1
and PC3 was estimated based on the 95th percentile
of the reconstructed distributions of the explained var-
iance or bootstrap SEs (i.e., the SD of the reconstructed
distribution).
Preference ordering
In Exp. 2, each neuron had a preferred item and loca-
tion, as in the conventional rate-coding model analysis.
We defined the preferred item and location in each neuron
to construct matrix X. We constructed Xwith and without
rank order. Items 1–6 were rank ordered from most to
least preferred, defined as the mean firing rate during a
whole analysis time window from 0.08 to 1 s. Thus, Item
(k)
was the rank-ordered item cued to the monkey on trial k.
Similar to the definition of Item, Location
(k)
was the rank-
ordered location cued to the monkey during trial k. This
preference ordering did not change over time tfor each
neuron i.
Optimal stimulus response analysis in the dynamic model
As typical single-neuron analyses assess optimal and
nonoptimal conditions (i.e., the best and worst), we linked
the optimal response analysis to the dynamic analysis of
the neuronal population as follows.
We used only two columns in each time bin, the most
and least preferred, for each condition C, item, and loca-
tion. Thus, matrix Xwas Xof size N
(590)
M
(4 30)
. This
corresponded to the conventional analysis used in the
rate-coding model, which compares neuronal responses
between the most and least preferred conditions, but for
the evaluation of neural modulation dynamics in these
two conditions. We evaluated the percentage explained
by the model between the original and restricted matrices
in the HPC for categorical task parameters.
Data availability
All data and analysis codes in this study are available
from the corresponding authors.
Results
We proposed and tested a dynamic analysis method
that could be applied to existing datasets using a typical
factorial design for conventional single-neuron recordings
(Fig. 1; Exp.1; Chen and Naya, 2020), which is in line with
the compatible analysis we have previously developed for
the continuous task parameters (Exp. 2; Yamada et al.,
2021).
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Tasks, behavior of monkeys, and datasets
During the cued lottery task in Exp. 1 using two continu-
ous task parameters (Fig. 2A), the monkeys estimated the
expected value of the lottery, defined as a multiplicative
combination of probability and magnitude (Fig. 2B;
probability, 0.1–1.0 in 0.1 increments; magnitude, 0.1–
1.0 ml in 0.1 ml increments) and chose the option with
higher expected values when two options were pre-
sented (Yamada et al., 2021). This choice behavior
was observed and analyzed separately from the neural
recordings (see Materials and Methods). We analyzed
neuronal activity recorded from the cOFC (Fig. 2C)in
the nonchoice condition, where a single lottery cue
and its outcome were provided to the monkeys (Fig.
2A). For neural analyses, information on the probability
and magnitude of rewards provided by the cue stimu-
lus was used for continuous task parameters.
In Exp. 2 using two categorical task parameters, the
monkeys retained the visual items and their locations
during the encoding phase, after which they indicated
whether the sample item matched the cued items by
choosing the memorized location (Match trial) or not by
choosing the central green disk (Nonmatch trial; Fig. 2D;
see Materials and Methods). Six visual items and four lo-
cations were used (Fig. 2E). After completing training, av-
erage correct rates were well above the chance level
(Chen and Naya, 2020). We analyzed neural activity re-
corded from the HPC (Fig. 2F) when the sample stimulus
was presented to the monkeys during the encoding
phase. For neural analyses, information on the items
provided by the sample stimulus and the locations were
used as categorical task parameters, whereas the loca-
tion information had already been provided to the mon-
keys before the sample appearance. Details of the
behavioral training, learning progress, and behavioral per-
formance of the animals in the cued lottery (Exp. 1;
Yamada et al., 2021) and in the ILR tasks (Exp. 2; Chen
and Naya, 2020) have been reported previously.
We constructed two different sets of pseudopopula-
tions of neurons using the single-neuron activity of the
cOFC (Fig. 2C, 190 neurons) in a 0.6 s time period with re-
spect to the lottery cue onset in the single-cue task (Fig.
2A, gray bar) and HPC (Fig. 2F, 590 neurons) in a 0.6 s
time period with respect to the sample onset in the ILR
task (Fig. 2D, gray bar). We denote that the HPC popula-
tion data in Exp. 2 were analyzed and previously reported
using a rate-coding model (Chen and Naya, 2020), but
this has not been analyzed using a dynamic model. The
cOFC population data from Exp. 1 was analyzed and re-
ported previously using both rate-coding and dynamic
models (Yamada et al., 2021), and we repeated the same
analysis with a shorter analysis time period (2.7 s was
used previously) for comparisons between the neural
modulation dynamics in Exp.1 and Exp. 2.
Neural recordings and data preprocessing
To achieve the fairest possible comparisons between
the two distinctive datasets, we adhered to the following
criteria for neural recordings and analyses. First, for the
recordings and preprocessing of the neural data, datasets
Figure 2. Behavioral task and recording location of neurons. A, The sequence of events during the single-cue task in Exp. 1. A sin-
gle visual pie chart containing green and blue pie segments was presented to the monkeys. Neural activity was analyzed during the
initial 0.6 s after cue onset, that is, for the same duration as in Exp. 2. B, Payoff matrix: each of the magnitudes was fully crossed
with each probability resulting in a pool of 100 lotteries. C, Illustration of neural recording areas based on coronal magnetic reso-
nance (MR) images for the cOFC (13 M, medial part of area 13) at the A31–A34 anterior–posterior level. D, The sequence of events
during the ILR task in Exp. 2. The cue stimulus during the response phase was the same as the sample stimulus during the encod-
ing phase in the match trial, whereas the two stimuli differed in the nonmatch trial. Neural activity was analyzed during the initial 0.6
s after sample onset, that is, for the same duration as in Exp. 1. E, Six visual item stimuli and spatial composition for the sample
stimulus. F, Coronal MR images from monkey A for the HPC population showing the recording area at A16–A10.5, depicted in pur-
ple within the red box. Awas published previously in the study by Yamada et al. (2021).D–Fwas published previously in the study
by Chen and Naya (2020).
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in Exp. 1 and Exp. 2 were individually prepared, because
each experimenter has their own criteria for single-neuron
activity. To ensure that the dataset differences were reli-
able, we provided the detailed procedure for recording
single-neuron activity in the Materials and Methods, in-
cluding bandpass filtering, signal-to-noise ratio, and on-
line or offline sorting. Second, for the neural analyses we
used a similar size of analysis window and statistical pro-
cedure because analysis window size is critical for detect-
ing neural modulations in both conventional and dynamic
analyses. A window of ;1.0 s was used for the conven-
tional rate-coding analyses with a fixed time window (Exp.
1, 0–1.0 s; Exp. 2, 0.08–1.0 s). A 0.02 s time bin was used
for the analysis of time-dependent neural modulation 0.6
s after the visual stimulus appeared in both experiments.
Third, for statistical procedures, we used the general line-
ar model at p,0.05 (linear regression for the continuous
task parameter in Exp. 1 and ANOVA for the categorical
task parameter in Exp. 2). Based on these criteria, we de-
scribed the neural response and its dynamics in fine time
resolution (0.02 s) during a short time period (0.6 s) after
the appearance of visual stimulus, during which continu-
ous and categorical task parameters were processed by
the neurons.
Conventional analyses for detecting neural
modulations by task parameters
We first applied common conventional analyses, such
as the general linear model typically used in rate-coding
models (linear regression in Exp. 1 and ANOVA in Exp. 2;
see Materials and Methods). In Exp. 1, we examined how
the probability and magnitude of rewards were encoded
and integrated by the activity of OFC neurons immediately
after cue presentation. Linear regression analysis re-
vealed that cOFC neurons encode both probability and
magnitude to some extent after cue onset, as shown in an
example neuron (Fig. 3A,B;n¼119 trials; coefficient/in-
tercept: 0.74; t¼0.72, p¼0.47; probability: 8.55,
t¼6.91, p,0.001; magnitude: 11.1, t¼8.95, p,0.001).
In the cOFC populations, approximately half of the neu-
rons were modulated by the probability and magnitude
of rewards during the 1 s time window (0–1 s after cue
onset; probability: 44%, 84 of 190; magnitude: 49%, 94 of
190). Modulations for either or both probability and mag-
nitude were found (Fig. 3C; Both: 30%, 57 of 190; P:
14.2%, 27 of 190; M: 19.5%, 37 of 190; Nonmodulated
(NO): 36.3%, 69 of 190). The analysis with 0.02 s time bins
showed that the percentage of neurons modulated by
these two parameters increased, reached a maximum
percentage at ;0.25 s, and then gradually decreased dur-
ing the 1.0 s after the onset of the lottery cue (Fig. 3D).
In Exp. 2, we analyzed neural modulations using the fol-
lowing two categorical task parameters: the item and lo-
cation after the sample presentation. The analyses were
aimed at examining how the item information was en-
coded and integrated with the location information by
the neurons immediately after the sample presentation.
ANOVA revealed that HPC neurons encoded both item
and location information to some extent, as shown in an
example neuron (Fig. 3F,G; two-way ANOVA, n¼120
trials; item: F
(5,96)
¼52.1, p,0.0001; location: F
(3,96)
¼
4.4, p¼0.006). In the HPC population, neurons were
modulated by these two factors (0.08–1 s after sample
onset; Item, 39%, 232 of 590; Location, 39%, 233 of 590).
Modulations for either or both factors were found (Fig. 3H;
Both: 23%, 138 of 590; Item: 16%, 95 of 590; Location:
16%, 94 of 590; NO: 45%, 263 of 590). The analysis with
the 0.02 s time bins showed that the percentage of neu-
rons modulated by these two factors increased, reached
a maximum percentage at ;0.25 s, and then gradually
decreased during the 1.0 s after the onset of the sample
stimulus (Fig. 3I). We note that the percentage of neural
modulation by the presented location did not change with
time because the monkeys already knew the item location
before the sample presentation (Fig. 3I, see blue), where-
as the percentage change in neural modulations was be-
cause of an increase in item information (Fig. 3I, see
green). Thus, for both neural populations, typical changes
in neural modulation were observed as a percentage in-
crease, followed by a decrease after visual stimulus onset
in the conventional analysis.
We then examined a regression coefficient that de-
scribes the extent of neural modulations in each neural
population. In Exp. 1, the regression coefficients for the
probability and magnitude of rewards in the case of con-
tinuous variables were visualized in the cOFC population
(Fig. 3E). The extent of neural modulations increased be-
tween 0.2 and 0.4 s after cue onset, as seen in the larger
positive or smaller negative regression coefficients for
probability and magnitude. For the categorical parame-
ters in Exp. 2, we plotted the regression coefficients rep-
resented in the ANOVA table for the best and worst items
detected in each neuron because we mainly focused on
the neural modulation by the visual item (Fig. 3J; data not
shown for all visual items). The distribution of coefficients
for the best items seemed to be wider at 0.2–0.5 s (see x-
axis, especially larger positive values), compared with
that just after the sample onset (0–0.02 s plots within a
range of 20 to 20). Thus, for both neural populations, an
increase in neural modulation was observed in the
regression coefficients within a reasonable time period,
compatible with the changes in the proportion of sig-
nificant neural modulations. These changes in regres-
sion coefficients were used for further analyses of
neural modulation dynamics.
Collectively, we showed that the general linear model
usually used in the rate-coding model detects neural
modulations using the same continuous and categorical
parameters as a standard analysis procedure. This con-
ventional approach can provide temporal changes in the
selected metrics (e.g., proportion and extent of neural
modulations; Fig. 4, top row), but they cannot provide tra-
jectory geometry at the lower dimension. We note that de-
tailed results from these conventional analyses have been
previously reported (Chen and Naya, 2020, their Figs. 2,
5; Yamada et al., 2021, their Fig. 2E,F,K,O).
Before describing the dynamic analysis, we also noted
other task differences. First, information on two variables
was simultaneously provided to the monkeys in Exp. 1,
while the item information was provided to the monkeys
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Figure 3. Example activity of neurons during the single-cue and ILR tasks. A, An example activity histogram of a cOFC neuron
modulated by the probability and magnitude of rewards during the single-cue task. Activity aligned with cue onset is represented
for three different levels of probability (P, 0.1–0.3, 0.4–0.7, 0.8–1.0) and magnitude (M, 0.1–0.3 ml, 0.4–0.7 ml, 0.8–1.0 ml) of re-
wards. Gray hatched areas indicate the 1 s time window used to estimate the neural firing rates shown in B. Histograms smoothed
using a Gaussian kernel (
s
¼50 ms). B, An activity plot of the cOFC neuron during the 1 s time window shown in Aagainst the
probability and magnitude of rewards. C, The percentage of neural modulation types detected in 1 s time window shown in A: the
P, M, Both, and NO. D, Percentages of neural modulation type detected in the 0.02 s time bins during the 1.0 s after cue onset.
Calibration: 0.2 s. E, Regression coefficient plots for the probability and magnitude of rewards estimated for all cOFC neurons in
Exp. 1. Regression coefficients in the 0.02 s time bin shown every 0.1 s during the 0.6 s after cue onset (0–0.02 s, 0.10–0.12 s,
0.20–0.22 s, 0.30–0.32 s, 0.40–0.42 s, 0.50–0.52 s, and 0.58–0.60 s). Filled gray indicates significant regression coefficient for either
Probability or Magnitude at p,0.05. F, An example of an HPC neuron showing sample-triggered sample–location signals and item
signals. A 0.08–1.0 s time window after sample onset was used to estimate the neural firing rates shown in G. Histograms are
smoothed using a Gaussian kernel (
s
¼20 ms). G, An activity plot of the HPC neuron during the time window shown in Fagainst
item and location. H, The percentage of neural modulation types detected in the 0.08–1.0 s window shown in F; Item, Location,
Both, and NO. I, Percentages of neural modulation types detected in the 0.02 s time bins during the 1.0 s after sample onset. J,
Regression coefficient plots for the best and worst items estimated for all HPC neurons in Exp. 2. Filled gray indicates significant re-
gression coefficient for item at p,0.05 using ANOVA without interaction term. The location modulation was not shown because we
showed changes of neural modulation by the sample stimulus, whereas the location had already been provided to the monkeys. A,
B, and Dwere published previously in the study by Yamada et al. (2021).
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Figure 4. Graphic methods for the conventional rate-coding analysis and state-space analysis in the regression subspace.
Conventional analysis (top and middle rows): in each single neuron, activity modulations by task variables are detected in the fixed
time window (top row) using linear regression and ANOVA for continuous (left, Exp. 1) and categorical (right, Exp. 2) task parameters
(Fig. 2, see for the task details), respectively. The same analyses were applied in a fine time resolution in Exp. 1 and Exp. 2 (middle
row). The conventional analyses using a general linear model (linear regression and ANOVA) provide the extent of neural
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following the presentation of location information in Exp.
2. Second, the monkeys simply looked at the visual cue in
Exp. 1, while the monkeys were required to make choices
using the item and location information later in the task in
Exp. 2. Third, the monkeys made eye movements without
choices in Exp. 1, while the monkeys made arm move-
ment to choose in Exp. 2. Our further analyses provide a
temporal structure of neural modulation signals, which
links these rate-coding results with a dynamic system
perspective.
State-space analysis for detecting the temporal
structure of neural modulation at the population level
We previously developed a variant of state-space
analysis, which extracts temporal structures of neural
modulations by task-related continuous parameters
using standard statistical software (see Yamada et al.,
2021). Here, we extend this analysis to neural modula-
tions using categorical parameters to describe how
the HPC neural population dynamically represents
item and location information.
First, we used a general linear model to project a time
series of each neural activity into a regression subspace
composed of task parameters as continuous and cate-
gorical, as shown in the regression equations in Figure
4(middle row, step 1; for details, see Materials and
Methods). This step captures the across-trial variance
caused by task-related parameters moment-by-mo-
ment at a population level, which demonstrates the ex-
tent of neural modulations by the task parameters
across time. This corresponds to the estimation of the
regression coefficients shown in Figure 3, Eand J(for
all time bins and conditions), which constructs the re-
gression matrices detected in each neural population
withafinetimeresolution(Fig. 4,middlerow,step1,X).
Second, we applied PCA one time to the time series of
neural activity in the regression subspace in each neural
population (Fig. 4, bottom row, step 2). This step deter-
mines the main features of the neural modulation signal
across time in the predominant dimensions as trajec-
tory geometry. These two steps identify how neural
modulations by task parameters change as a time se-
ries of eigenvectors in the regression subspace.
We evaluated the properties of the extracted time series
of the eigenvectors in the lower-dimensional space: the
first three principal components (PC1 to PC3) in each neu-
ral population, in terms of vector angle, size, and deviance
(Extended Data Fig. 4-1). The angles and sizes provide
trajectory geometry that describes how neural modulation
evolves after the visual presentation. Deviance indicates
the stability of neural modulation between vectors. We
compared two neural populations recorded during two
different cognitive tasks in terms of these vector proper-
ties. We note that the typical population dynamics previ-
ously found were straight, curvy, and rotational structures
(Fig. 1, right).
Neural population dynamics representing continuous
and categorical parameters
We first qualitatively explained how our state-space
analysis describes neural population dynamics in the re-
gression subspace in cOFC (Fig. 5) and HPC (Fig. 6) pop-
ulations during the perception of visual stimuli. Using the
identical analysis window size between the two experi-
ments, we ensured that the neural population structures
were comparable between the cOFC and HPC popula-
tions, with continuous and categorical parameters.
We first confirmed the performance of the state-space
analysis, as indicated by the percentage of variance ex-
plained in the cOFC population (Fig. 5A). In the cOFC
population, .40% of the variance was explained by PC1
and PC2 (gray arrowhead). We then characterized the en-
tire structure of the cOFC population by plotting its eigen-
vectors moment-by-moment in their temporal order. As
shown in Figure 5, Band C, the eigenvectors for PC1 and
PC2 evolved ,0.2 s after the onset of the cues in both
probability and magnitude. The eigenvectors then short-
ened after ;0.3 s. The angle of the eigenvectors evolved
in the 45° direction in a stable manner in the PC1–PC2
plane (Fig. 5C, top), whereas the vectors in PC3 changed
in the opposite direction, from positive to negative, over
time (Fig. 5B,C, bottom). These stable structures in the
top two dimensions were consistent with our previous re-
sults even when the analysis window sizes differed: a
whole cue period of 2.7 s previously, but here, only the ini-
tial 0.6 s were analyzed (Yamada et al., 2021, their Fig.
7B). These straight geometric changes (i.e., stable vectors
in angle) indicate that almost equal neural modulations by
probability and magnitude (Fig. 3E) evolved across the 0.6
s time period at the neural population level. Thus, the two
neural modulation ratios were kept similar.
For the HPC population modulated by the two categori-
cal parameters, the performance of the state-space anal-
ysis was lower than that in Exp. 1 (Fig. 6A). The first two
continued
modulations as the coefficients in the analysis table of the statistical software. These are neural modulations in a fine time resolution
observed at the level of population. irepresents the number of neurons in each neural population (Exp. 1, 190 neurons; Exp. 2, 590
neurons). trepresents the number of time bins (30 for both Exp. 1 and Exp. 2). In our state-space analysis as step 1, the time series
of neural population activity was projected onto a regression subspace composed of probability and magnitude (left, Exp. 1) and
item and location (right, Exp. 2). The middle row, therefore, represents the neural population activity in the regression subspace X.
By applying PCA to X, eigenvectors for probability and magnitude were extracted and plotted after coordinate transformation
against PC1 and PC2 (step 2, left, Exp. 1). Eigenvectors for item and location were plotted after coordinate transformation against
PC1 and PC2 (right, Exp. 2). A series of eigenvectors was obtained by applying PCA once to the cOFC and HPC populations, re-
spectively. The number of eigenvectors obtained by PCA was 0.6 s, divided by the analysis window size, 0.02 s, for P and M; in
total, 30 eigenvectors for each (left, Exp. 1), and for the six items and four locations; in total, 30 eigenvectors for each. Extended
Data Figure 4-1 represents detail of the vector analyses.
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PCs only explained ;10% of the variance (Fig. 6A, gray
arrowhead). Because the percentage of modulated neu-
rons was similar between the two neural populations (Fig.
3C,H), this would not be because of the weaker influence
of task parameters in Exp. 2 on the recorded HPC popula-
tion. This may be partly because of the larger regression
matrix composed of 10 vectors at each time point (six
items and four locations) and a larger neural population
containing 590 neurons: total Xof size N
(590)
M
(300)
,as
in our previous study of PCA performance, where var-
iance captured by PCA tended to decrease as the matrix
size increased (Yamada et al., 2021, their Fig. 7A).
The eigenvectors in the first three PCs appeared to de-
scribe neural population dynamics in the HPC. For
example, the extracted eigenvectors for each visual item
evolved within a reasonable range of time with an increase
and a subsequent decrease at ;0.2–0.5 s (Fig. 6B), which
is consistent with our findings using typical conventional
analyses, ANOVA (Fig. 3H,I). In contrast, the eigenvectors
for locations did not exhibit clear trends over time (Fig.
6C), as location information was provided to the monkeys
before the sample presentations. This is also consistent
with our findings using ANOVA, in which percentages
modulated by locations did not change over time (Fig. 3I,
blue). If we plotted eigenvectors in the space of the first
three PCs, they consistently evolved in one direction in
the PC1 and PC2 spaces (I2, I3, and I6), or PC3 spaces
(I1, I4, and I5; Fig. 6D, left). In contrast, the eigenvectors
Figure 6. Temporal structure of neural modulation in the HPC population. A, Cumulative variance explained by PCA in the HPC
population. The arrowhead indicates the percentages of variances explained by PC1 and PC2. B, Time series of eigenvectors for
six items in the HPC population. The top three PCs are shown. C, Time series of eigenvectors for four locations. D, A series of ei-
genvectors for PC1 to PC3 are plotted against PC1 and PC2, and PC2 and PC3 dimensions in the HPC population. a.u., Arbitrary
unit. Extended Data Figure 6-1 represents shuffled control results.
Figure 5. The state-space analysis provides a temporal structure of neural modulation in the cOFC. A, Cumulative variance ex-
plained by PCA in the cOFC population. The arrowhead indicates the percentage of variance explained by PC1 and PC2. B, Time
series of eigenvectors, PC1 to PC3 in the cOFC population. C, A series of eigenvectors for PC1 to PC3 are plotted against PC1 and
PC2, and PC2 and PC3 dimensions in the cOFC population. Plots at the beginning and end of the series of vectors are labeled as
start (s) and end (e), respectively. a.u., Arbitrary unit.
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for locations were positioned in a constant location
over time (Fig. 6D,topright).Unambiguously,thear-
rangements of the eigenvectors for item and location
were orthogonalized, as seen in item representations
in the second and fourth quadrants and in location rep-
resentations in the first and third quadrants (Fig. 6D,
top row). The increase and subsequent decrease in
vector sizes at ;0.2–0.5 s were similar to those in the
neural modulations (Fig. 3J). Thus, our state-space
analysis in the regression subspace may capture the
neural modulation dynamics in HPC populations similar
to that in cOFC populations, while reflecting continuous
and categorical parameters in their neural modulations.
To summarize, the main components of the neural
modulation signals for categorical task parameters were
evolved for item and remained stable for location, similar
to the results using ANOVA. These temporal patterns of
neural modulations as trajectory geometry may be able to
visualize the strength of the main neural modulations and
how they change over time as the eigenvectors. Next, we
quantitatively examined whether the temporal patterns of
geometric changes are statistically significant.
Effect of shuffle control on PCA performance
To validate the statistical significance of these findings,
we used in three ways a shuffle control procedure (for de-
tails, see Materials and Methods) that determines the
number of available dimensions in the neural population.
In shuffled conditions 1 and 2, information on task-related
parameters was partially shuffled in the regression sub-
space matrix X. In shuffle condition 1, a random permuta-
tion of neuron iwas performed at each time t, eliminating
the temporal neural modulation structure by condition C
across each neuron but retaining the effect of neural mod-
ulation at each time tat the population level. In shuffle
condition 2, a random permutation of time twas per-
formed for each neuron i, eliminating the temporal neural
modulation structure by condition Cin each neuron, but
retaining the effect of neural modulation in each neuron i
at the population level. In shuffled condition 3, random
permutations of both time tand neuron iwere performed.
We evaluated the performance of PCAs for each condition
in each experiment.
AsshowninExtendedDataFigure 6-1,thethree
shuffle control procedures reproduced different distur-
bances in neural populations. In shuffle conditions 1
and3(ExtendedDataFig. 6-1A, left, right; see black
boxplot), the explained variance decreased compared
with that from the original PCA without shuffles (red
circle) in the cOFC population. In shuffle condition 2, a
considerable amount of variance was explained by PCA
(Extended Data Fig. 6-1A, middle, black boxplot). These
effects are consistent with those of our previous study
(Yamada et al., 2021, their Fig. 5A,E,I). Since the eigen-
vectors were very stable across time in the cOFC popula-
tion (Fig. 5B,C), the shuffle within each neuron largely did
not reduce PCA performance. Indeed, some stable popu-
lation structures were observed under the shuffle within
each neuron (Extended Data Fig. 6-1C, middle) located in
the first and fourth quadrants through a trial, although the
trajectory geometry observed without shuffles (Fig. 5C,
top) was impaired. In the first and third shuffle conditions,
the neural population structures were completely im-
paired (Extended Data Fig. 6-1C, left, right, respectively),
indicating that the shuffle across neurons strongly af-
fected neural modulation structures in Exp. 1.
The same effects of shuffle controls were observed in
the HPC population, for which categorical parameters
were used (Extended Data Fig. 6-1B, black boxplot). A
considerable amount of variance was explained by PCA
in shuffled condition 2 (Extended Data Fig. 6-1B, middle),
although the maintained population structures were un-
clear (Extended Data Fig. 6-1D, middle) compared with
those in Exp. 1 (Extended Data Fig. 6-1C, middle). When
examining the details of the decreased performance in
each experiment, the performances of the first 3 and 12
PCs were better than those in the shuffled control condi-
tion 2 in Exp. 1 and Exp. 2, respectively (shuffle condition
2, p,0.05). Thus, while the total number of available di-
mensions differed between the experiments, all three
shuffles revealed the significance of the lower-dimension-
al neural modulation space. Note that the smaller variance
explained by the top-two dimensions in Exp. 2 compared
with that in Exp. 1 must be partly because of the availabil-
ity of the higher dimensions in Exp. 2 (Exp. 1 explains
;50% of variances by the top 3 PCs; Exp. 2 explains
.30% by top 12 PCs).
Activity preference and neural population dynamics
In the conventional rate-coding analysis, the preferred/
nonpreferred activity of neurons is accumulated to evalu-
ate the extent of neural modulations at the population
level. In Exp. 2, each neuron possesses a particular pref-
erence for item and location (Fig. 7A, left), and we de-
tected the best and worst conditions in addition to all
remaining conditions by ordering the activity magni-
tudes (Fig. 7A, right). In this way, we reconstructed the
regression subspace in line with the conventional per-
spective, such as activity preference to task conditions,
item and location (see Materials and Methods). This an-
alytic approach allowed us to examine how the conven-
tional analytic framework affects the neural population
dynamics.
We analyzed the most to least preferred conditions for
six items and four locations in each neuron, in which item
and location were remapped to the most to least preferred
activity in each condition of item and location, individually.
The neural preference was defined using whole analyzed
activity in the 0.08–0.6sanalysiswindowineachneu-
ron. The regression subspace comprised the same size,
total Xof size N
(590)
M
(300)
; however, condition Cwas
changed to the most to least preferred items and most
to least preferred locations.
The percentage variance explained by the model for
PC1 and PC2 was almost the same in the preference-or-
dering analysis (Fig. 7B, 12%) compared with the original
analysis (Fig. 6A, 10%). The composition of the eigenvec-
tors for items was also similar between the analyses in the
PC1 and PC2 dimensions (Fig. 7C, top left), located in the
second and fourth quadrants from the most preferred (Ib,
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best item) to least preferred (Iw, worst item). However, the
eigenvectors for items were clearly different in the PC3 di-
mension, as observed for the most preferred item (Ib; Fig.
7C, left bottom). The composition of eigenvectors for lo-
cations was not clearly changed by the preference order-
ing (Figs. 6D, right, 7C, right). These results suggest that
preference ordering may affect eigenvector compositions
at higher dimensions equal to or greater than PC3, and
the entire compositions of eigenvectors seemed main-
tained at the lower dimension.
In summary, our analysis can extract temporal structures
of item and location modulations with the preference or-
dering in the lower-dimensional subspace, describing how
preferred and nonpreferred neural modulation change over
time. The predominant temporal structures maintained
with and without preference ordering (compare Figs. 7C,
top row, 6D, top row) indicate that the lower dimensional
feature may be unrelated to the relocation of the population
activity by their preference. Under this condition, the PC1
and PC2 may represent the stimulus feature according to
the preferences. We note that the presented information
for the best item on PC3 remains unclear and further exam-
ine these properties in the Optimal response analysis
subsection.
Quantitative comparisons of two neural population
dynamics
To quantitatively examine and compare these neural
population structures using geometry, we compared the
properties of the eigenvectors by estimating the vector
size, angle, and deviance in each neural population (Fig.
8). We analyzed rank-ordered HPC data shown in Figure
7as well as cOFC data shown in Figure 5. For the rank-or-
dered data, we evaluated the best and worst conditions
as the typically used conditions in conventional rate-cod-
ing analyses.
First, the vector size evaluation provided clear time-de-
pendent structures in both cOFC and HPC populations
for probability and magnitude (Fig. 8A) and for the best
and worst items (Fig. 8B). Such time-dependent changes
were not clearly observed in the eigenvectors for the best
and worst locations (Fig. 8B, right and second to right col-
umns), because location information already has been
provided to the monkeys before the samples appeared.
The vector sizes during the period 0.1–0.6 s after the
onset of the lottery cue were not significantly different be-
tween the two continuous parameters, probability and
magnitude of rewards (Fig. 8C; Wilcoxon rank-sum test;
PC1 to PC2: n¼52, df ¼51, W ¼330, p¼0.892; PC2 to
PC3; n¼52, df ¼51, W ¼341, p¼0.964). In contrast, the
vector sizes during the period 0.1–0.6 s after the onset of
the sample significantly differed between the best and
worst items (Fig. 8D; Wilcoxon signed-rank test; PC1 to
PC2, item: n¼52, df ¼51, W ¼502, p¼0.002; PC2
to PC3, item: n¼52, df ¼51, W ¼588, p,0.001; PC1
to PC2, location: n¼52, df ¼51, W ¼600, p,0.001;
PC2 to PC3, location: n¼52, df ¼51, W ¼542, p,
0.001). This is because the regression coefficients for
the best item were considerably different from their
mean neural modulation (Fig. 3J,x-axis) in contrast to
those for the worst item (y-axis). Thus, the vector sizes
captured temporal changes in neural modulation at the
population level, which is consistent with the results ob-
tained from the rate-coding analyses (Fig. 3).
The analyses of vector angles showed that all eigenvec-
tors were stable in both populations in the top two dimen-
sions (Fig. 8E,F, top; Wilcoxon rank-sum test; cOFC, PC1
Figure 7. Effects of preference ordering on the HPC categorical
data. A, Three examples of HPC neurons for preference order-
ing. The activities were ordered by their preference to the items
and locations (right, shown best to worst), while their activity
have a preference to item or location during the 0.08–0.6 s after
sample presentation. B, Cumulative variance explained by PCA
in the HPC population when item and location were arranged in
the order of their activity preferences (see Materials and
Methods). The arrowhead indicates the percentages of varian-
ces explained by PC1 and PC2. C, Series of eigenvectors for
PC1 to PC3 when item and location were arranged in the order
of their preferences, plotted against the PC1 and PC2, and PC2
and PC3 dimensions in the HPC population. Ib and Iw indicate
the best and worst items, respectively. I2 to I5 indicate the sec-
ond to fifth best items. Lb and Lw indicate the best and worst
locations, respectively. L2 and L3 indicate the second and third
best locations, respectively. NA, No significant difference
using ANOVA at p,0.05. *p,0.05, ***p,0.001. Extended Data
Figure 7-1 represents results for optimal response analysis.
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to PC2: n¼52, df ¼51, W ¼62, p,0.001; HPC, PC1 to
PC2, item: n¼52, df ¼51, W ¼520, p,0.001; HPC, PC1
to PC2, location: n¼52, df ¼51, W ¼0, p,0.001). The
angles in the PC2–PC3 plane were not stable (Fig. 8E,F,
bottom; Wilcoxon rank-sum test; cOFC, PC2 to PC3:
n¼52, df ¼51, W ¼343, p¼0.935; HPC, PC2 to PC3,
item: n¼52, df ¼51, W ¼321, p¼0.765; HPC, PC2 to
PC3, location: n¼52, df ¼51, W ¼312, p¼0.643). Both
neural populations showed some vector deviance (,0.1),
with some statistical differences (Fig. 8G,H; Wilcoxon
rank-sum test; cOFC, PC1 to PC2: n¼52, df ¼51,
W¼361, p¼0.683; cOFC, PC2 to PC3: n¼52, df ¼51,
W¼300, p¼0.496; HPC, PC1 to PC2, item: n¼52,
df ¼51, W ¼459, p¼0.027; HPC, PC2 to PC3, item:
n¼52, df ¼51, W ¼581, p,0.001; HPC, PC1 to PC2, lo-
cation: n¼52, df ¼51, W ¼352, p¼0.807; HPC, PC2 to
PC3, location: n¼52, df ¼51, W ¼384, p¼0.408).
Collectively, our state-space analysis in the regression
subspace described neural modulation dynamics in the
cOFC and HPC during two different cognitive tasks com-
posed of continuous and categorical parameters. These
dynamic structures, evaluated qualitatively (Figs. 5-7) and
quantitatively (Fig. 8), reflected the neural modulation
properties described by the conventional rate-coding
analyses (Fig. 3). The straight dynamics observed in both
cOFC and HPC populations were captured by a combina-
tion of changes in vector size and stable vector angle
across time, which cannot be captured by the conven-
tional rate-coding framework.
Optimal-stimulus response analysis in the dynamic
model
The rate-coding model typically analyzes optimal and
nonoptimal responses of neurons at the population level,
which was used to evaluate the extent of neural modula-
tions without dynamics. For the last comparison between
the rate-coding and dynamic models, we described neu-
ral population dynamics for optimal/nonoptimal activity
and compared this with the dynamics obtained from
the whole activity (Fig. 7). We reconstructed matrix Xof
the HPC population by extracting the best and worst
Figure 8. Quantitative evaluations of eigenvector properties in the cOFC and HPC populations. A, Time series of vector size esti-
mated in the cOFC population for P and M of rewards. Vector sizes are estimated in the PC1–PC2 plane (top) and PC2–PC3 plane
(bottom), respectively. a.u., Arbitrary unit. The solid-colored lines indicate interpolated lines using a cubic spline function to provide
a resolution of 0.005 s. B, Time series of vector size estimated in the HPC population for the best and worst items. C, Boxplots of
vector size estimated in the cOFC population for probability and magnitude of rewards. D, Boxplots of vector size in the HPC popu-
lation for the best and worst items and locations. E,F, Boxplots of vector angle estimated in the cOFC (E) and HPC (F) populations.
G,H, Boxplots of vector deviance from the mean estimated in the cOFC (G) and HPC (H) populations. In C–H, data after 0.1 s are
used. *p,0.05, ***p,0.001.
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conditions for item and location, whereas the regression
matrix from the other conditions, second to fifth preferred
items and second to third preferred locations, were re-
moved. The regression matrix was reduced from the origi-
nal large size, total X, of size N
(590)
M
(10 30)
to N
(590)

M
(4 30)
.
In this optimal and nonoptimal regression matrix, PCA
captured a greater percentage variance (Extended Data
Fig. 7-1A,;16% of the variance explained by PC1 and
PC2). Similar to the original matrix, straight dynamics
were observed (Extended Data Fig. 7-1B, left column)
with a slight difference in vector size between the best
and worst items. The PCs appeared to be rotated at an
angle of ;135° from the original on the PC1–PC2 plane
(Fig. 7C, top row, Extended Data Fig. 7-1B, top row). The
percentage variance explained by PCA clearly differed
from those in the shuffled conditions for the top three PCs
(Extended Data Fig. 7-1C, circles), and they significantly
differed from the shuffled control even in condition 2
(Extended Data Figs. 7-1C,6–1B, middle, boxplot, circles,
p,0.05), suggesting that some neural population struc-
tures in higher dimensions might be removed in this se-
lected matrix. Thus, the straight dynamics in the lower
dimension were clearly derived from this restricted opti-
mal and nonoptimal activity, which represents the main
neural modulations in the neural population.
We note that in this optimal response analysis, the
first PC represented location information in a stable
manner (Extended Data Fig. 7-1B, right column), where-
as the second and third PCs represented item informa-
tion (Extended Data Fig. 7-1B, left bottom), in contrast
to the preference ordering analysis with the full regres-
sion matrix (Fig. 7C,leftbottom).Wealsonotethatthis
reduction of the regression matrix caused relocations of
whole vectors because the entire structure of the matrix
largely changed.
To summarize, using two standard continuous and cat-
egorical task parameters we found stable evolutions of
neural modulation structures as the straight geometry
over a relatively short period, 0.6 s, when the monkeys
perceived the visual items. We incorporated the activity
preference analysis used in the rate-coding model and
showed that the straight geometry was derived from the
preferred/nonpreferred activity modulations. While the
tasks and examined brain regions were different in the
two neural populations, these straight geometries with a
unidimensional feature of neural modulations indicated
that the modulation structures by the task parameters re-
tained similarity across time in the lower-dimensional sub-
space (i.e., similarity in mathematics).
Discussion
Temporal structures of neural modulations in the
regression subspace
In the present study, we developed a state-space anal-
ysis in the regression subspace for categorical task pa-
rameters and captured neural modulation dynamics for
item and location, as an extension of our previous study
(Yamada et al., 2021). Thus, our analysis covers all types
of task parameters typically used in the rate-coding
model, continuous and categorical. We achieved the fol-
lowing two goals in this article: (1) we newly developed
the state-space analysis in the regression subspace for
the categorical task parameters; and (2) we found that the
HPC neural population exhibited straight dynamics. Fair
comparisons of neural modulation dynamics of the two
parameters (Figs. 5-8) indicated that straight dynamics
observed at the lower dimension exhibited a gradual de-
velopment (Fig. 8A,B) and stable composition of the neu-
ral modulation structures at different angles (Fig. 8E,F,
top). Thus, we conclude that the neural population struc-
tures obtained from different brain regions using different
behavioral tasks were similarly stable in terms of geomet-
ric changes.
To bridge the rate-coding and dynamic models, we
incorporated a standard rate-coding approach into the
dynamic analysis, such as activity preference analysis
in the categorical task parameters (Fig. 7). We also
performed the additional optimal/nonoptimal response
analysis using the best and worst conditions for item
and location in line with the rate-coding analysis.
These analyses showed that straight dynamics were
derived from the optimal and nonoptimal activity of
neurons (Fig. 7C, Extended Data Fig. 7-1B). In the
rate-coding analysis, the time course of neural modu-
lation by item and location was observed in the per-
centages of modulated neurons (Fig. 3I)andinthe
magnitudes of regression coefficients (Fig. 3J). In the
dynamic analysis, these characteristics were observed
onthetimecourseofthevectorsize(Fig. 8A,B), while
the neural modulation structures were evaluated in
terms of their similarity across time. Thus, the classical
rate-coding analysis was well incorporated into the dy-
namic analyses, which specifically captured their neu-
ral population geometries.
In the rate-coding analysis, the extent of neural modula-
tions at the level of the population is analyzed by relocat-
ing their activity using activity preference, such as the
preferred direction of movement, preferred location, or
preferred stimulus. In our dynamic analysis incorporating
the activity preference approach (Fig. 7), we have ob-
served similar structures in neural dynamics between
preference-ordered (Fig. 7) and nonordered (Fig. 6) neural
population activity at the lower dimensional. What does
this comparison provide as an aspect for understanding
information processing? The neural population structures
relocated by the preference ordering should affect the
neural population dynamics, although the entire PC1–
PC2 spaces were similar (Figs. 6D, top row, 8C, top row)
and third PC changed (Fig. 7C, left bottom). Further, the
elimination of the population activity except for the best
and worst activity changed the trajectory at the lower di-
mensional spaces, but the straight geometry remained
apparent (Extended Data Fig. 7-1). These observations in-
dicated that the straight unidimensional feature in their
dynamics at the lower dimension is derived from the activ-
ity relationship between optimal and nonoptimal condi-
tions, while the whole allocation of geometry depends on
the entire structure of neural population activity. The
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extent of neural modulation might be represented in the
lower dimensional neural modulation space after the pref-
erence ordering.
Both cOFC and HPC neural populations consisted of
separately recorded single-neuron activities after visual
stimulus presentations, and we extracted both neural
modulation dynamics using an original state-space analy-
sis. These straight geometries with unidimensional fea-
tures of neural modulations indicated that modulation
structures by task parameter remained similar across
time (Fig. 3E,J). We would like to caution researchers
against using nonmodulated task parameters in PCA (i.e.,
task parameters that show minimal neural modulation), as
the PCs derived in such cases were less biologically
meaningful (Björklund, 2019). Our simple extractions of
neural modulation dynamics indicate that all types of data
can be easily reanalyzed and evaluated to find the trajec-
tory geometry of neural modulations, which can be com-
pared with pre-existing rate-coding evidence.
TDR and dPCA analyses
Two predominant dynamics analyses for task-related
parameters exist, TDR and dPCA. The dPCA was aimed
at detecting latent components related to discrete task
parameters (Brendel et al., 2011;Kobak et al., 2016)by
decomposing neural population data. In contrast, TDR
was aimed at identifying the encoding axis of the task pa-
rameters in a multidimensional neural space (Mante et al.,
2013;Aoi et al., 2020) by incorporating linear regression
into neural population dynamics with respect to task pa-
rameters. Both methods are the state of the art and shed
light on the link between the rate-coding and dynamic
models by visualizing complex datasets with multiple task
parameters. Moreover, Elsayed and Cunningham (2017)
compared the neural population responses with surrogate
data that simultaneously preserve the temporal correla-
tion of discharge rates, signal correlations across neu-
rons, and tuning to the experimental parameters of the
task.
These procedures have several limitations when precisely
applied to data and they require multiple mathematical
steps for the user to apply these elaborate methods. In con-
trast, our method is simpler and more intuitive and directly
extracts the temporal structures of neural modulations by
the task parameters. If we compare the dPCA and our PCA
in the regression subspace (PCArs), the dPCA decomposes
allneuralsignals(Fig. 9,toprow),butPCArsprojectsthe
neuronal activity into the regression subspace at the popula-
tion level by removing the average activity in each moment
(Fig. 9, bottom row). Our methods are similar to the dPCA,
but not the same. For example, we did not include the inter-
action component since the state-space analysis assumes
a linear system. PCArs only focuses on neural dynamics in
the subspace and requires only two steps (Fig. 4, middle,
bottom rows) and minimal assumptions (orthogonality in the
task design). Our PCArs is beneficial in terms of being sim-
ple and intuitive as it is an extension of the conventional
rate-coding approach. Thus, our state-space analysis pro-
vides researchers with a significant advantage to find modu-
lation dynamics in lower-dimensional subspaces for pre-
existing datasets.
Another analysis of regression subspace, TDR, has been
performed in a limited number of studies (Mante et al., 2013;
Aoi et al., 2020). These studies aimed to detect the regression
subspace within a neural structure at a constant time point
(Mante et al., 2013). The detected modulation axis was as-
sumed to be projected orthogonally and sometimes stably
through a task trial (Aoi et al., 2020). In contrast, we directly
detected neural population dynamics in the regression sub-
space, which was a straight dynamic for both continuous and
categorical task parameters. Because the axis of neural mod-
ulation remains stable in these straight dynamics (Figs. 5,6),
our results support assumptions regarding the stability of the
regression subspace, which were not examined in previous
studies.
Two different types of task parameter yield
comparable dynamics in regression subspaces
In our state-space analysis, neural population activity was
projected onto the regression subspace, reflecting the
across-trial variance caused by task-related parameters at
Figure 9. Thematic depiction of the difference between dPCA and PCArs. Illustration of the procedure to construct the neural population
dynamics by dPCA and ours (PCArs). dPCA decomposes the neural activity into the dynamics as a linear summation of the multiple
components for categorical variables (Kobak et al., 2016). Our method first projects the neural population dynamics into a regression
subspace that removes the activity change other than the neural modulations by task parameters and demonstrates the modulation dy-
namics. dPCA figures are represented in the study by Kobak et al. (2016, their Fig. 8). PSTH indicates peristimulus time histogram.
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the population level. In this step, both continuous and cate-
gorical task parameters were reliably used within the frame-
work of the general linear model. However, it was reliably
performed with one limitation; the conditions in any parame-
ter would be orthogonalized as the experimental design
(Grafen and Hails, 2002). For the continuous task parame-
ters in Exp. 1, the 10 10 conditions for probability and
magnitude were orthogonalized. In the case of categorical
task parameters in Exp. 2, 6 4 conditions for item and lo-
cation were orthogonalized. For both experiments, the num-
ber of recorded trials for each neuron was almost the same
for all conditions. Thus, the projections of neural activity to
the regression subspace were nearly orthogonal.
In the linear system assumed here, the concept of or-
thogonality is critical in terms of statistics and to avoid
skewed projection of neural activity into the regression
subspace, which is part of the entire neural activity, re-
flecting activity modulation by the task parameters of in-
terest (Fig. 9). However, one concern for the comparison
between the two distinctive datasets is observed in the
percentages of the variance explained by the PCA (Figs.
5A,6A). The smaller percentages of the variance ex-
plained by the model in the HPC might be because of the
larger matrix size. While the dynamics observed in the
HPC population were not meaningless, the descriptive
ability at the lower dimension is different from that in the
cOFC.
In this study, we evaluated neural modulation dynamics
in terms of vector size, angle, and deviance in the regres-
sion subspace (Extended Data Fig. 4-1) that composes
possible trajectory geometries. The combination of vector
size and angle describes our straight geometry (i.e.,
constant angle), whereas deviance also reflects vector
stability over time. The straight geometry indicated that
modulation structures by the task parameters kept sim-
ilarity across time in the multidimensional neural sub-
space. If neural modulations are completely stable
across time (i.e., all vectors are the same across time),
then all three parameters become very stable (i.e., con-
gruence in mathematics: constant vector angle, con-
stant vector size, and very small deviance). Thus, our
analysis of vector properties (Fig. 8) can quantitatively
evaluate trajectory geometry in neural modulations.
Stable and fluctuating signals in neural modulation
dynamics
In this study, we observed stable straight neural modu-
lation dynamics in both cOFC and HPC populations, a di-
vergent excursion with a straight trajectory in a subspace
plane. Although these tasks were designed using different
task parameter types, both brain regions showed similar
stable-modulation structures during the visual perception
(Figs. 5-8). One possible explanation for these two brain
regions exhibiting stable modulation dynamics is that they
may share similar information processing when accessing
memories for expected values as a combination of proba-
bility and magnitude (as reflected in Exp. 1) and the asso-
ciation between stimulus and position for future decisions
(Exp. 2). Access to some memories may stabilize the neu-
ral population activity. More stable structures have been
observed in the dorsolateral prefrontal cortex during a typi-
cal working memory task (Murray et al., 2017), although this
activity was highly stable through memory maintenance.
Comparisons of the stability in neural modulations would
not be easy using conventional analyses based on the rate-
coding model. Another important aspect is whether behav-
ioral change (or policy) affects trajectory geometry, since
neurons in the dorsolateral prefrontal cortex have shown un-
stable dynamics during a choice task (Mante et al., 2013). In
our experiments, this test was not performed because of the
no-choicedatainExp.1.
In our previous study, fluctuating neural population signals
were observed in the dorsal striatum (DS) and medial OFC
(mOFC) because of signal instability or weakness (Yamada
et al., 2021, their Fig. 5A,B; Imaizumi et al., 2022). As the sig-
nal carried by the mOFC population was weak (Yamada et
al., 2021, their Fig. 8, bottom row), eigenvector fluctuation in
the mOFC population reflected the weak signal modulations
by the probability and magnitude of rewards. In this case,
moment-by-moment vector fluctuations were observed be-
cause there was no clear neural modulation structure in the
mOFC population. Conversely, the fluctuating DS signal
seemed to reflect the functional role of the DS neural popu-
lation in detecting and integrating the probability and magni-
tude of rewards related to the control of some actions
(Balleine et al., 2007;Enomoto et al., 2020;Inokawa et al.,
2020). In the DS population, structural changes in eigenvec-
tors occurred over time (Yamada et al., 2021,theirFig.8).
Future studies are required to explore neural modulation ge-
ometry to elucidate how neural circuitry operates and com-
putes (Ebitz and Hayden, 2021;Humphries, 2021).
Conclusions
Rate-coding models have provided mounting evidence
that neural modulation is associated with task parameters in
many regions of the brain (e.g., movement direction, muscle
force, place, and reward value). A recently developed dy-
namic model approach appears promising to account for
the different aspects of neural information processing.
However, its relationship with rate-coding models remains
unclear. Although some studies have sought a connection
between these two advances (Mante et al., 2013;Chen and
Stuphorn, 2015;Kobak et al., 2016;Murray et al., 2017;Aoi
et al., 2020), direct comparisons are necessary to link the
two analytical results. Our results allow us to consider
whether neural modulation dynamics observed in neural
population ensemble activities are compatible with rate-
coding models. Specifically, trajectory geometries provided
temporal dynamics of neural modulation that could bring
about new insights into neural processing and conceptual
advances, such as straight geometry with a unidimensional
feature. Thus, our simple approach encourages research
aimed at incorporating traditional rate-coding models into
dynamic systems as neural modulation dynamics.
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