PNAS Nexus, 2022, 2, 1–13
https://doi.org/10.1093/pnasnexus/pgac286
Advance access publication date: 10 December 2022

Research Report

Information maximization explains state-dependent
synaptic plasticity and memory reorganization during
non-rapid eye movement sleepa,b,* and Taro Toyoizumi

a,b,

*

a

Laboratory for Neural Computation and Adaptation, RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo
113-8656, Japan∗
To whom correspondence should be addressed: Emails: kensuke_yoshida@mist.i.u-tokyo.ac.jp, taro.toyoizumi@riken.jp
Edited By: Stephen Flemingb

Abstract
Slow waves during the non-rapid eye movement (NREM) sleep reflect the alternating up and down states of cortical neurons; global andlocal slow waves promote memory consolidation and forgetting, respectively. Furthermore, distinct spike-timing-dependent plasticity(STDP) operates in these up and down states. The contribution of different plasticity rules to neural information coding and memory
reorganization remains unknown. Here, we show that optimal synaptic plasticity for information maximization in a cortical neuronmodel provides a unified explanation for these phenomena. The model indicates that the optimal synaptic plasticity is biased toward
depression as the baseline firing rate increases. This property explains the distinct STDP observed in the up and down states. Furthermore, it explains how global and local slow waves predominantly potentiate and depress synapses, respectively, if the backgroundfiring rate of excitatory neurons declines with the spatial scale of waves as the model predicts. The model provides a unifying accountof the role of NREM sleep, bridging neural information coding, synaptic plasticity, and memory reorganization.
Keywords: spike-timing-dependent plasticity, slow wave, efficient coding hypothesis, normative model, learning rule

Significance Statement
Some memories are consolidated, and others are forgotten during non-rapid eye movement sleep. This reorganization process
is considered to involve synaptic plasticity in the presence of slow waves characterized by alternating up and down states atdistinct spatial scales. Previous theories do not consider such patterns of slow waves and give only partial explanations of thereorganization process. Here, we report that the information maximization principle provides a unifying account for two essential
experimental findings: the distinct synaptic plasticity during up and down states in slow waves and the opposite memory reorganization effects during global and local slow waves. The theory bridges neural coding, synaptic plasticity, and memory reorganizationin non-rapid eye movement sleep.

Sleep is an essential physiological process and is widely conserved
across species. One proposed role of sleep is to reorganize memory
by regulating synaptic plasticity; some memories of awake experiences are consolidated, whereas others are forgotten (1–3). Multiple studies have explored the mechanism behind memory consolidation and forgetting by focusing on slow waves observed during non-rapid eye movement (NREM) sleep. Slow waves are lowfrequency (<4 Hz) waves in electroencephalography (EEG) and local field potential (LFP), and their presence distinguishes NREM
sleep from awake or rapid eye movement (REM) sleep. Each cortical neuron shows low-frequency transitions between the up (depolarized membrane potential) and down (hyperpolarized membrane potential) states synchronously to slow waves (4). Both thecorrelational and causal relationships between slow waves andmemory consolidation have been established. It has been reported

that slow-wave activity is correlated with task performance after
sleep (5), and boosting slow waves can enhance memory consolidation (6–9).
Several studies have suggested that slow waves should be separated into distinct classes (10–16). Although different classification schemes have been used in previous studies, one of theclasses is more global, while the other is more local (10, 12–14).
A recent study further suggested that these two classes of slow
waves have opposite effects on memory reorganization; the global
and local classes promote memory consolidation and forgetting,respectively (13). These studies suggest that memory reorganization is induced depending on the subtle sleep states, such as theup and down states of global and local slow waves.
One possible explanation for how these sleep states differentially modulate the memory reorganization is that synaptic plas-

Competing interest: The authors declare no competing interest.
Received: November 4, 2022. Accepted: December 6, 2022

C The Author(s) 2022. Published by Oxford University Press on behalf of National Academy of Sciences. This is an Open Access article
distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits
unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kensuke Yoshida

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Results
Optimal synaptic plasticity is biased toward
depression in high firing rates
To consider optimal synaptic plasticity in different sleep states, wefirst considered a feedforward network model with a postsynaptic neuron and multiple excitatory presynaptic neurons, which wecall the single-neuron model. In this model, presynaptic spikes atsynapse j evoked excitatory postsynaptic potentials (EPSPs) withthe amplitude wj and exponential decays with a time constantof 25 ms. The membrane potential of the postsynaptic neuron

was computed as u(t) = ur + j wj hj (t), where ur is the resting
membrane potential and hj is the EPSP time-course from presynaptic neuron j with an instantaneous increment of 1 after eachpresynaptic spike. The postsynaptic neuron emits spikes with firing probability density gE (u(t))R(t), where gE (u) is a softplus activation intensity function, and refractory factor R(t) models thetransient suppression of the postsynaptic firing rate after a postsynaptic spike (see the “Methods” section). Following the infomax
approach, we derive the optimal synaptic plasticity rule for maximizing information transmission while synaptic weights are constrained by their cost. We assumed that the synaptic weights wj
change following the gradient of the utility function, which is themutual information between the presynaptic and postsynaptic
spikes minus the synaptic weight cost (24, 25). Thus, the synaptic

dw

dI
d
weight changes were described by dt j ∝ dw
− λ dwwith mutualjjinformation I and synaptic weight cost  (see the “Methods” section). The cost term  includes the square of the synaptic weight,
which eliminates the synaptic weights that do not contribute to
information transmission. Coefficient λ controls the importanceof the synaptic cost term relative to the information term. We
omitted the homeostatic term assumed in the previous studies
because it did not contribute to our results, where the postsynaptic firing rate was kept on average within the homeostatic range.
The gradient of mutual information dI/dwj was explicitly derived
and can be computed in real-time using only the variables observable at synapse j, namely, the EPSP time-course hj , postsynaptic spikes informed by a back-propagating action potential, postsynaptic activation intensity gE (u) as a function of postsynaptic
membrane potential u, refractory factor R, and mean activation
intensity ḡ (see the “Methods” section). In addition, the gradientof the cost term decreased the synaptic strength by λwj for every
presynaptic spike of neuron j.
We ran simulations imitating the experimental STDP protocols
in vivo (20) in the single-neuron model. To mimic the experimental setup, we divided the presynaptic neurons into the stimulated
and nonstimulated neurons (Fig. 1A). Twenty stimulated neurons
synchronously emitted a spike upon external presynaptic stimulation, and their synaptic weights changed according to the infomax rule. One hundred nonstimulated neurons spontaneously
emitted Poisson spikes at 2.0 and 0.1 Hz in the up and down states,
respectively, while their synaptic weights were fixed for simplicity.
The mean postsynaptic activation intensity, ḡ, was computed bytaking the average of gE (u) in each state. This yielded ḡ(u) ≈ 5.9 Hz
and ḡ(d) ≈ 0.5 Hz for the up and down states, respectively.
We first characterized synaptic changes induced by pre-post
stimulation, where a presynaptic spike was induced 10 ms before the postsynaptic spike. Representative traces of a synaptic
weight from a stimulated neuron in the up and down states areplotted in Fig. 1(B). The increase in gE (u)/ḡ after presynaptic
stimulation was greater when the mean firing rate was low, indicating that a postsynaptic spike can transmit a greater amountof information at a lower mean firing rate. Consequently, theinformation term caused a greater synaptic potentiation in thedown state than that in the up state. The amount of synaptic potentiation due to the information term was roughly proportional to log(1 + g/ḡ)/(ḡ + g), where g = gE (u) − ḡ represents the increment in activation intensity due to the presynaptic stimulation (see the “Methods” section for details). Intuitively, g measured the reliability of a synapse for transmitting
the signal, and ḡ represented the noise level that quantifies thefrequency of the postsynaptic spikes in the background. Thus,
g/ḡ corresponds to the signal-to-noise ratio. By contrast, thechange in the synaptic weight by the cost term was −λwj after
every presynaptic spike of neuron j, regardless of the mean firing
rate.
Fig. 1(B) displays the representative traces of synaptic weight;
however, synaptic changes also depended on other postsynaptic spikes and presynaptic spikes from nonstimulated neurons,
which can occur randomly. Below, we quantify the average synaptic changes induced by three kinds of stimulation: pre-only stimulation and post-pre stimulation (a presynaptic spike was induced
10 ms after an induced postsynaptic spike), in addition to the prepost stimulation explored above. We started by simulating thepre-only stimulation. Experimentally, pre-only stimulation in thedown state did not significantly change the synaptic weights ofstimulated neurons (20). To reproduce this experimental result,
we set the coefficient of the cost term to λ = 0.32 (mV)−2 , so that

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ticity is modulated depending on the sleep state. Previous studies
have shown that neuronal activity patterns in the awake state arereactivated within the slow waves during NREM sleep (17–19). Although the synaptic plasticity rule during NREM sleep is largely
unknown, a recent experimental study using anesthetized young
mice in vivo has suggested that the spike-timing-dependent plasticity (STDP) during up states is biased toward depression compared with down states (20). Consistently, another experimental
study using acute brain slices demonstrated that the subthreshold inputs during up but not down states induce synaptic weakening (21). These findings suggest that neuronal reactivation can
induce different synaptic plasticity in the up and down states.
This difference might be the key to understanding memory reorganization during NREM sleep and raises two further issues
worth exploring theoretically. First, what is the benefit of modulating the synaptic plasticity rule depending on the up and downstates? Because the nervous system has evolved to work efficiently, the efficiency of neuronal coding might be enhanced bythis modulation. Second, how does the state-dependent synaptic plasticity reorganize memories in global and local slow
waves?
To understand these issues, we adopted a normative approachbased on the information maximization (infomax) principle (22,23) and derived a synaptic plasticity rule for a spiking neuronmodel (24, 25) that achieves efficient information transmission.
We found that the baseline firing rate is an important parameter of the infomax rule. An increased baseline firing rate biases
the synaptic plasticity towards depression, consistent with the reported difference in STDP between the up and down states. We
then constructed a neuronal network model exhibiting global andlocal slow waves and showed that four states (up and down statesof global and local slow waves) have distinct STDP owing to different baseline firing rates. Finally, we suggest that the difference insynaptic plasticity in global and local slow waves can set a balance between memory consolidation and forgetting, consistent
with the previous experimental findings (see Fig. 6 for a schematic
summary).

Yoshida and Toyoizumi | 3

the changes in the stimulated synapses were on average zero in
the down state (Fig. 2A). This value of λ was used throughout thispaper. Simulations of the up state showed an overall synaptic depression because the synaptic potentiation due to the information
term decreased with the mean firing rate for the reason describedabove. In addition, neither artificial postsynaptic depolarization
to the up state level in down states nor hyperpolarization to thedown state level in up states appreciably affected the synaptic
changes in the current setup (SI Appendix, Fig. S2), consistent withthe experimental results (20).
Next, if a postsynaptic spike was evoked before the presynaptic
stimulation (i.e., post-pre stimulation), the infomax rule caused
the synaptic depression both in the up and down states because
the induced postsynaptic spike before the presynaptic stimulation
reduced the value of R(t) and prevented the synaptic weights fromincreasing by the information term, whereas the cost term could
still decrease these synapses (Fig. 2A).
To investigate how the STDP window of the infomax rule differs
in the up and down states, the time difference between presynaptic and postsynaptic stimulations was systematically changed in
the single-neuron model. Consistent with the observations above,the entire STDP curve was biased toward synaptic depression in
the high mean firing rate condition (Fig. 2B). While the synaptic change caused by post-pre stimulation was relatively insensitive to mean firing rates, the high mean firing rates biased thesynaptic changes toward depression with the pre-post and preonly stimulations (Fig. 2C). These results were consistent withthe corresponding in-vivo experimental results (20). In addition,
while the experimental results are limited to a few representative time differences (within 10 ms and within −10 ms in thedown states and 10, 50, and −10 ms in the up states) (20), theinfomax model predicted the whole STDP curve. Although thistendency of synaptic depression induced by the high mean firing rates did not qualitatively depend on the choices of the ac-

tivation intensity function gE (u) except for the pure exponentialfunction (see SI Appendix, Fig. S1 and the “Methods” section),
the exact position of STDP curve and the activity threshold, separating potentiation and depression under the pre-only stimulations, depended on several parameters (SI Appendix, Figs. S3and S4). Especially, the pre-only stimulation with small synaptic
weights and a large number of the stimulated neurons tended toinduce potentiation even in up states, albeit to a lesser degree
than down states (SI Appendix, Fig. S4). In summary, the modulation of the infomax rule by the mean firing rate explained
the synaptic plasticity during the up and down states in slow
waves.

Firing rates of excitatory neurons are higher
during local slow waves than those during global
slow waves
To study how the above-mentioned findings would apply to memory reorganization during NREM sleep, we constructed the network models of cortical neurons that generate slow waves, which
we call the slow-wave model. We started by constructing a spatially homogeneous model similar to that in ref. (26) and then introduced spatial heterogeneity to produce local and global slow
waves. The slow-wave model consisted of recurrently connectedspiking neurons, including the excitatory and inhibitory neurons(see the “Methods” section). The spatially homogeneous model assumed no connections between two inhibitory neurons but all-toall connections between two excitatory neurons and between excitatory and inhibitory neurons. Each excitatory neuron had adaptation currents that accumulated with the spikes. Adaptation currents correspond to, for example, the potassium currents involvedin generating slow waves (27–30). The activation functions of excitatory and inhibitory neurons were both modeled by softplus
functions, but the threshold and slope were greater for the in-

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Fig. 1. The infomax rule in the single-neuron model. (A) A single-neuron model of synaptic plasticity. Stimulated neurons synchronously emit spikes
upon external presynaptic stimulation, and their synaptic weights change according to the infomax rule, whereas nonstimulated neurons
spontaneously emit Poisson spikes and their synaptic weights are fixed for simplicity. A postsynaptic neuron emits both spontaneous spikes andevoked spikes upon external postsynaptic stimulation. The red vertical bars represent evoked spikes. (B) Representative traces of a stimulated neuron’s
activity, the postsynaptic neuron’s activity, the ratio of the momentary and mean activation intensity gE (u)/ḡ, and changes of a synaptic weight from a
stimulated neuron in the down and up states. Synaptic changes by the infomax rule were computed by summing the effects of the information termdId
and cost term dw. The synaptic increase by the information term was smaller in the up state than that in the down state.
dw j
j

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PNAS Nexus, 2022, Vol. 2, No. 1

hibitory neurons than those for the excitatory neurons (Fig. 3B).
In these settings, the excitatory and inhibitory activities showedbistability of the up and down states, and the transitions werecaused by the slowly changing adaptation currents (SI Appendix,
Fig. S5A and B). This was consistent with the previous theoretical
study (26). The inhibitory population was mostly inactive in downstates because of the high threshold (Fig. 3B), but was active in theup state and stabilized the recurrent excitatory activity.
To consider the difference between global and local slow waves,
we extended the model by embedding four local networks, eachas described above, within the overall network (Fig. 3A). In casesof no between-network connections, each local network independently produced up and down cycles of slow waves (Fig. 3C). Next,
we introduced sparse long-range excitatory connections between
different local networks. We assumed that the long-range connections project to both the excitatory and inhibitory neurons,
as demonstrated in previous modeling studies (31). In this setting, each local network showed the transitions between the up
and down states, some of which were local, whereas others wereglobal in synchrony across the local networks (Fig. 3D). To objectively define global and local slow waves, we first classified the up
and down states of each local network based on the mean membrane potential averaged across excitatory neurons (Fig. 3C and
E). Transitions to the down and up states were detected when themean membrane potential decreased below −69.75 mV and exceeded −68.25 mV, respectively (the choice of these thresholds did

not affect the results; see SI Appendix, Fig. S6). We classified eachstate into a global or local state by counting the number of up anddown states across the local networks (Fig. 3D) (see the “Methods”
section). In the global up/down states, either all or all but one local
network simultaneously achieved the same state.
We then analyzed how the difference between global and local
slow waves affected the learning by the infomax rule. Since theoutcome of the infomax rule depends on the mean firing rates,
we examined the mean firing rates in the up and down states
during the global and local slow waves. The mean firing rates of
excitatory neurons followed global down < local down < global
up < local up in ascending order in the simulations (Fig. 3F).
The difference between the global and local down states was simply explained by the strength of the long-range excitation fromthe surrounding networks to the local excitatory population. Because the surrounding excitatory populations had elevated activity in their up state, the long-range excitation was stronger inthe local down states than that in the global down states. Notethat the local inhibitory population was mostly inactive in boththe local and global down states and did not contribute significantly to the difference. By contrast, the difference between the
global and local up states was mainly explained by the local inhibition to the excitatory population. While the local network wasin the up state, its inhibitory activity was more sensitive to thelong-range excitation than its excitatory activity because of thesteeper inhibitory activation function at high membrane poten-

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Fig. 2. The synaptic plasticity induced by the STDP stimulations. (A) The mean traces of a synaptic weight and the mean postsynaptic firing rate in the
pre-only stimulations or the post-pre stimulations with t = −10 ms, where the stimulated neurons emitted a synchronous spike upon the external
stimulation at t = 50 ms. In the post-pre stimulations, the postsynaptic neuron emitted an evoked spike at t = 40 ms and also responded to thepresynaptic stimulation at t = 50 ms with some delay due to refractoriness. After the pre-only stimulations, the synaptic weight changed little in thedown state, but was depressed in the up state. In the post-pre stimulation, the synaptic weight was depressed in both the down and up states. Thelines and shadows of the weight traces represent the means and SDs, respectively. (B) The synaptic changes by the STDP stimulations (blue points) andpre-only stimulations (orange dotted lines). As the value of t increased or decreased, the synaptic changes by the STDP stimulations converged to thechange by the pre-only stimulations. Synaptic plasticity was biased towards depression in the up state. (C) Synaptic changes were dependent on mean
activation intensity. The synaptic changes in pre-post stimulation with t = 10 ms and pre-only stimulation decreased with increasing mean
activation intensity, whereas the synaptic changes in post-pre stimulation with t = −10 ms were less sensitive to mean activation intensity.

Yoshida and Toyoizumi | 5tial (see Fig. 3B and the “Methods” section). Therefore, the strong
long-range excitation from the surrounding networks in the global
up state effectively reduced the local excitatory activity via local
inhibition. To verify this, we performed a phase plane analysis,
assuming a large number of neurons (see SI Appendix Methods).

The phase planes showed that the firing rates of the two stable
points (i.e., up and down states) changed depending on the stateof the surrounding networks (Fig. 3G and see SI Appendix Methods for parameter dependency). As expected, the membrane potential of the excitatory neurons was higher in local down states

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Fig. 3. The cortical network model exhibiting global and local slow waves. (A) The schematic description of the model. (Left) Model composed of four
local networks. Each local network included 200 excitatory and 50 inhibitory neurons. (Right) Within a network, connections existed between two
excitatory neurons, and between excitatory and inhibitory neurons, while two inhibitory neurons had no connections between them. Between
different networks, there were sparse connections from excitatory to excitatory or inhibitory neurons but none from the inhibitory neurons. (B) The
activation functions of excitatory and inhibitory neurons. Both functions were softplus functions, but the threshold and slope were greater for
inhibitory neurons than those for the excitatory neurons. (C) The dynamics of the slow-wave model in the case of no connections between different
local networks. The up and down states of the E1 population were classified using the mean membrane potential uE1 . The four networks
independently transited between the up and down states. (D) The dynamics of the slow-wave model in the case that there exist sparse connections
between different networks. The up and down states of the E1 population were classified into the global or local states depending on the states of theother populations. (E) The probability density of the membrane potential. The transition thresholds to the up and down states are indicated by red andgreen lines, respectively. (F) The mean firing rates of the E1 population in global down, local down, global up, and local up states. (G) The phase plane of
the E1 and I1 population in the case that other populations are in down states (upper) or in up states (lower) (see SI Appendix Methods for details).
Excitatory and inhibitory nullclines are shown as orange and blue lines, respectively. The four steady points corresponding to global down, local down,
global up, and local up states are shown; the up state of E1 population is global or local when the other populations are in the up or down states,
respectively, while the down state of E1 population is global or local when the other populations are in the down or up states, respectively. The uE1values of four steady points followed global down < local down < global up < local up in ascending order.

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with elevated long-range excitation than that in the global downstates. In addition, the membrane potential of excitatory neurons
was lower in the global up states than that in the local up states.
In this case, the long-range excitation shifted both the excitatory
and inhibitory nullclines. Since the shift of the inhibitory nullcline
was much larger than that of the excitatory nullcline, the membrane potential of excitatory neurons decreased with the longrange excitation (Fig. 3G). The observed higher local excitatory activity in the local up states than that of the global up states isa natural consequence of unstable recurrent excitatory dynamics being stabilized by the strong inhibition. A similar response toexternal input was previously demonstrated both experimentallyand theoretically as a property of inhibition-stabilized networks(ISNs) (31–33). Based on these results, we hypothesized that the
infomax rule, which is sensitive to the baseline firing rate, would
yield distinct learning outcomes in the global and local slow
waves.

Optimal synaptic plasticity in up and downstates of global and local slow waves
To study the outcome of the infomax rule in global and local slow
waves, we first explored the STDP window using the slow-wave
model in the previous section. We introduced multiple presynaptic excitatory neurons that spike synchronously when externallystimulated, as shown in Fig. 2. These presynaptic neurons projected onto a randomly selected postsynaptic neuron in the firstexcitatory population, E1, of the slow-wave model (Fig. 4A). We
assumed that these feedforward synaptic weights were updated
by the infomax rule, whereas the recurrent synaptic weights were
fixed. The changes in the feedforward synaptic weights averaged
over random realizations of the model are shown below.
We first studied the STDP by evoking a postsynaptic spike at a
fixed time, before or after the presynaptic stimulation to 20 external input neurons. Here, we assume strong synapses with 0.5 mV
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Fig. 4. STDP in different sleep states. (A) The schematic description of the simulation. Presynaptic neurons had a feedforward projection onto an
excitatory postsynaptic neuron in the E1 population. The feedforward synaptic weights were plastic, whereas the recurrent synaptic weights were
fixed. (B) The synaptic changes caused by the STDP stimulations (blue points) and pre-only stimulations (orange dotted lines) during each sleep state= 0.5 mV) and few stimulated neurons (Next = 20) (SF condition). As the value of t increased or decreased, thein the case of strong synapses (wextjsynaptic changes by the STDP stimulations converged to the change by the pre-only stimulations. The amount of synaptic changes follows global
down > local down > global up > local up, in descending order. (C) The synaptic changes caused by the STDP stimulations (blue points) and pre-only= 0.09 mV) and many stimulated neurons (Next = 40)
stimulations (orange dotted lines) during each sleep state in the case of weak synapses ( wextj
(WM condition). The synapses tend to be potentiated even in up states, while the amount of synaptic changes follows global down > local down >global up > local up, in descending order, as with (B). Note that the scale is different from (B). (D) The synaptic changes caused by the pre-only
stimulations. Error bars represent SEM. The synaptic changes were dominated by synaptic potentiation in the WM condition. Especially in the WM
condition, the pre-only stimulations during global and local up states induced synaptic potentiation and depression, respectively.

Yoshida and Toyoizumi | 7

postsynaptic neuron could generate other spikes triggered by the
network activity. In the slow-wave model, the mean activation intensity ḡ was computed by averaging the activation intensity gE (ui )of neuron i for neurons included within the same excitatory population (see the “Discussion” section for possible biological implementations). As expected from Figs. 2 and 3, the STDP results depended on the sleep states of the slow-wave model (Fig. 4B). These
results indicate two important points. First, the synaptic plasticity
in the up states is biased toward synaptic depression as comparedwith the down states, which is consistent with the experimental
findings (20). Second, synaptic plasticity in the local up and downstates is biased toward synaptic depression as compared with theglobal up and down states, respectively. This property results fromthe model prediction that the mean firing rates are higher in thelocal up and down states than those in the corresponding global
states. As expected from SI Appendix, Fig. S4, stimulating many
weak synapses (40 synapses with 0.09 mV EPSPs) tended to induce more potentiation (Fig. 4C, note that the scale is different

from Fig. 4B), although the difference between global and local up
and down states described above still existed (Fig. 4D). Especiallywhen many weak synapses were stimulated, as in Fig. 4C, the preonly stimulation during global up states caused synaptic potentiation, and that during local up states caused synaptic depression(Fig. 4D).
To further investigate the impact of sleep states on memory reorganization, we simulate how synaptic weights that contributeto task performance change during subsequent sleep. This time,
we simulated presynaptic neurons having small initial synaptic weights. The assumption is that relatively weak synapses aremainly involved in the in-vivo learning of a new task. During theawake condition, we assume that the presynaptic neurons emitted spikes synchronously upon the presentation of a task cue
and projected to a postsynaptic neuron (task neuron) in the E1
population of the slow-wave model (Fig. 5A). The simulation was
repeated over random realizations of the model parameters. Inspired by the brain–machine-interface task (13), we defined task

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Fig. 5. Changes in synaptic weights and task performance during post-learning NREM sleep. (A) The schematic description of the neuronal networks
related to the task. Presynaptic neurons were divided into two populations G and L. Both G and L populations emitted synchronous spikes (“task cue”)
during the task. The postsynaptic neuron (“task neuron”) is an excitatory neuron in the E1 population that is projected by the presynaptic neurons.
Task performance is defined as the firing rate increase of the task neuron during the task period. (B) As neuronal reactivation, the G and L populations
emitted synchronous spikes during the global and local up states of the E1 population during the post-learning NREM sleep, respectively, at the firing
rates decreasing from 7.5 Hz at the beginning of sleep to 5.0 Hz at the end of sleep. (C) The changes of the synaptic weights and the task neuron
reactivation strength during the post-learning sleep. The synaptic changes of the G and L populations are shown in red and purple, respectively(upper). The synapses of the population G were potentiated by reactivation during the global up states, whereas the synapses of the population L weredepressed by reactivation during the local up states. When synaptic plasticity during the local-up or global-up is blocked, the synaptic changes of thecorresponding population were inhibited. Hence, the sum of synaptic weights in two populations was global-up blocked < control < local-up blocked
in ascending order. The reactivation strength of the task neuron in global and local up states are shown in red and purple, respectively (lower). The
blue dotted line represents the assumed decrease in the task cue reactivation rate. The reactivation strengths of the task neuron in global and local up
states were preserved and diminished, respectively, reflecting the synaptic changes of the corresponding population. In the local-up-blocked andglobal-up-blocked plasticity conditions, the reactivation strength of the task neuron in local and global up states became the same as the time-courseof the task cue reactivation, respectively. The lines and shadows represent the means and SEMs in the 600 trials, respectively. (D) The comparison oftask performance before and after synaptic changes during sleep. This tendency is the same as that of the sum of synaptic weights in the G and L
populations shown in Fig. 5C. Error bars represent SEM.

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performance as an increase in the task neuron’s firing rate upon
the presentation of a task cue. Then, we considered the synaptic changes during post-learning NREM sleep, assuming that thefeedforward synaptic weights have already been potentiated toelevate the postsynaptic firing rate during the task. Experimentally, the triple coupling of slow waves, spindles, and reactivation is considered crucial for memory consolidation (2, 18, 34,35), in which spindles are considered to promote synaptic plasticity by facilitating dendritic activities (2, 35–38). Therefore, we
assumed that synaptic plasticity is induced by the reactivation
inputs from presynaptic neurons in the presence of spindles. Although we did not explicitly model spindles in this study, we assumed that spindles are nested in slow waves when memory reactivation occurs in up states of the task neuron (see the “Discussion” section for details). Presynaptic neurons were divided
into two populations, the global-up-reactivated neurons G and
local-up-reactivated neurons L, each consisting of 40 presynaptic
neurons and synchronously emitted spikes with Poisson statistics
during global up and local up states, respectively. We assumed
that the Poisson rate of the memory reactivation (17, 19) of thetask cue decreased from 7.5 Hz at the beginning of an NREM sleep
to 5.0 Hz at the end of an NREM sleep to reproduce experimental results of task neuron reactivation (Fig. 5C, see below for detail). We updated the feedforward synaptic weights during postlearning NREM sleep according to the infomax rule when presynaptic reactivation happened. In Fig. 5, we restricted memory reactivation to occur during the up states of the local network only
(Fig. 5B) because the intervention of the up states, not down states,
mainly affected the performance of the brain–machine-interface
task (13, 39). More generally, memory reactivation might also happen in down states depending on the experimental setup (40). Thispossibility was also investigated in SI Appendix, Fig. S7. Further
simulation details are described in SI Appendix Methods.
The feedforward synaptic weights of the populations G and L
were further potentiated and depressed in the simulation of postlearning NREM sleep, respectively (Fig. 5C), because global and local up states promote synaptic potentiation and depression, respectively, with weak and many stimulated synapses (Fig. 4D).
Task performance increased during sleep, reflecting the increased

sum of synaptic weights of the population G and L, consistent
with the experimentally suggested memory consolidation during
NREM sleep (Fig. 5D). The reactivation strength of the task neuron (i.e., the firing rate increase of the task neuron from baseline
firing rates when the task cue is reactivated) during global andlocal up states were kept constant and decreased, respectively(Fig. 5C). Under the gradual decrease of the task cue reactivation rate, synaptic potentiation of the population G and depression of the population L promoted the reactivation strength of thetask neuron during global and local up states to be kept constantand to be further decreased, respectively. This was consistent withthe experimental result that reactivation strengths in the spindles
nested in global and local up states were preserved and weakened,
respectively (13).
Next, we investigated the roles of global and local slow waves
in memory reorganization separately by inhibiting synaptic plasticity either during global or local up states. Note that in the simulation, synaptic plasticity within 50 mec after global or local up
states was also inhibited to eliminate synaptic plasticity during
the transition states that is sensitive to arbitrary model assumptions (see SI Appendix Methods for detail). The synaptic depression of the population L was inhibited when synaptic plasticity
was blocked during local up states. Further, synaptic potentiationof the population G was inhibited when synaptic plasticity was
blocked during global up states (Fig. 5C). As a result, the task performance showed a greater increase in the former case but a decrease in the latter case (Fig. 5D). The results are consistent withexperimental findings that global and local slow waves contributeto memory consolidation and forgetting, respectively (13). The reactivation strength of the task neuron exhibited changes monotonically related to synaptic weights. The decrease of the reactivation strength of the task neuron during local up states became slower when synaptic plasticity was blocked during local up
states. The reactivation strength of the task neuron during global
up states decreased when synaptic plasticity was blocked during
global up states. This was consistent with the experimental resultthat inhibition of global up states promoted a gradual decreaseof reactivation strength in the spindles nested in global up states(13).

Downloaded from https://academic.oup.com/pnasnexus/article/2/1/pgac286/6887287 by RIKEN CBS user on 12 January 2023

Fig. 6. The proposed role of NREM sleep is to bridge neural information coding, synaptic plasticity, and memory reorganization. (A) The relationshipbetween the mean firing rate and synaptic changes by the infomax rule. Synaptic potentiation by the information term was decreased at a high firing
rate owing to many background spikes, whereas synaptic depression by the cost term was unaffected. Therefore, the high firing rate induced synaptic
depression. Because the mean firing rates are global down < local down < global up < local up in ascending order, the amount of synaptic changes
follows the opposite order. (B) The possible distinct roles of global and local slow waves. Reactivated patterns during global slow waves induced
synaptic potentiation, whereas those during local slow waves induced synaptic depression. This could cause selective memory consolidation andforgetting of the reactivated patterns during the global and local slow waves, respectively.

Yoshida and Toyoizumi | 9
These results suggest that the balance of global and local slow
waves with distinct information transfer capacities regulates the
spectrum of memory consolidation and forgetting via the infomax
synaptic plasticity rule.

Discussion

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Using a top-down approach with the information theory, we provided a unified learning rule, the infomax rule, for the statedependent synaptic plasticity during NREM sleep. The infomax
rule is comprised of the synaptic changes by the information termand synaptic depression by the synaptic cost term (Fig. 6A). A highfiring rate condition biases the synaptic plasticity toward depression. The reason is that the signal-to-noise ratio for the synaptictransmission declines with the background firing rate of the postsynaptic neuron, and the cost term dominates the information
term under a high firing rate condition. The learning rule yields aninformation-theoretical interpretation of different STDP observedin the up and down states (20). Moreover, it also provides the distinct STDP during the global and local slow waves, suggesting a
possible mechanism for balancing memory consolidation and forgetting. These properties are consistent with the role of neuronal
reactivation in global and local slow waves (13).
The infomax rule not only reproduces the biased STDP toward
depression during up states (20), but also provides a reliable prediction of the entire STDP curve during up states; the original experiment measured the synaptic change using a few representative time differences (10, 50, and −10 ms) between the presynaptic and postsynaptic spikes. The infomax rule further predicts
that the STDP curve is sensitive to the initial synaptic weights
and number of synchronous inputs (SI Appendix, Figs. S3 andS4). These predictions are experimentally testable using protocols
similar to those of ref. (20), using various time differences, and altering the strength of presynaptic stimulations. Importantly, theinfomax rule also predicts that relatively weak synapses can bepotentiated, even in the up state, if the surrounding area is also
in the up state and if a large number of presynaptic neurons emit
synchronous spikes (SI Appendix, Fig. S4). Consistent with the predicted synaptic potentiation of weak synapses during sleep, recent
experimental findings suggest that synaptic potentiation and theformation of new spines play an essential role in memory consolidation (41–45). Therefore, the infomax rule provides a unifying
view by reproducing both the state-dependent STDP (Fig. 4) andthe wave-scale-dependent memory reorganization (Fig. 5). Notethat the STDP curve during up states was measured in urethaneanesthetized young mice (20). Since the animal age or anesthesia
could affect synaptic plasticity (46, 47), whether this STDP rule applies to physiological sleep in adult mice needs to be explored infuture studies.
The infomax rule requires estimating the expected firing rate ofeach postsynaptic neuron in real-time to set the activity threshold separating the synaptic potentiation and depression. There
are several biologically plausible implementations for computing this. The simplest estimate uses a temporally averaged firing rate. However, it tends to lag behind the true instantaneous
firing rate in the presence of slow waves. This possibility is also
at odds with the experimental observation, where manipulating
postsynaptic membrane potential to be depolarized during downstates or hyperpolarized during up states did not significantly affect the synaptic changes (20). This experimental result is reproduced if the expected instantaneous firing rate is accurately estimated using the average excitatory firing rate of the local network population because the artificial single-neuron manipula-tion does not significantly change the local population activity. It
is possible that the inhibitory neurons projected by nearby excitatory neurons compute the average firing rate, which is consistent
with the observation that inhibitory input could modulate the balance of synaptic potentiation and depression (48). Alternatively,
astrocytes may temporally and spatially integrate nearby synaptic inputs and regulate the activity threshold separating synaptic
potentiation and depression (49).
The infomax rule suggests the potential importance of downstates for memory consolidation. Although some studies have assumed down states as resting periods for cellular maintenance(50), it has been reported that the hyperpolarization might be essential for slow waves to induce synaptic potentiation (44). A recent study further suggested that the neuronal activities during
down states (“delta spikes”) coincided with the hippocampal ripple activities and might be important for memory consolidation(40). The infomax rule suggests that down states with fewer background spikes promote synaptic potentiation more than up states,
implying that the delta spikes could effectively induce synaptic
potentiation and memory consolidation. Many studies have also
focused on the involvement of neuronal activities in up states or
the transition period from down to up states during memory consolidation (13, 51–53). Such distinct roles of the neuronal reactivation during the up and down states warrant further study.
The infomax rule in our sleep model suggests the importanceof the spatial scale of slow waves (12, 54). Specifically, the present
model suggests that the synaptic changes induced by global slow
waves are dominated by potentiation as compared to local slow
waves because of different baseline firing rates. Although we did
not explicitly model spindles in this study, including a spindlegenerating mechanism to the model is an important future direction. In the current study, we simply assumed that slow-wavenested spindles would be generated upon memory reactivationin the task neuron’s up states and they are necessary for inducing synaptic plasticity. This assumption agrees with the argumentthat the triple coupling of slow waves, spindles, and reactivationis crucial for memory consolidation (2, 18, 34, 35), in which spindles promote synaptic plasticity by enhancing dendritic activities(2, 35–38). Our model behavior is consistent under this assumption with the experimental results by Kim et al. Optogenetic inhibition of the primary motor cortex during global and local up
states caused a decrease and increase in the ratio of the number of spindles nested in global slow waves to that in local slow
waves, respectively (13). This result is expected if cortical activity
in up states is needed to facilitate thalamic spindle generation (2,15, 55). An extreme condition with the total lack of nested spindles
during global or local up states corresponds to the lack of synaptic
plasticity then (as explored in Fig. 5) according to the assumption.
The suggested role of memory consolidation during global slow
waves and forgetting during local slow waves raises the possibility that there are mechanisms regulating the spatial scales of slow
waves for selecting a subset of reactivation events to be consolidated (Fig. 6B). One possibility is that some neuronal populations
projecting broadly to cortical neurons promote global synchronization. Previous studies have consistently suggested that thethalamus (56) and claustrum (57) play a role in synchronizing thedown states of multiple cortical neurons. If such neuronal populations are co-active with reactivation patterns, these patterns
could be selectively consolidated. Considering the high temporalcorrelation between the hippocampal sharp-wave ripples (SWRs)
and cortical slow waves (58), specific neuronal populations may
regulate both the generation of slow waves and neuronal reactivation. This possibility needs to be explored further.

10 |PNAS Nexus, 2022, Vol. 2, No. 1

Methods

tion (AHP) (65), and EPSP suppression by back-propagating action
potential (66).

Single-neuron model
We modeled a postsynaptic neuron that received the feedforward inputs from N presynaptic excitatory neurons. Each spike ofpresynaptic neuron j evoked EPSP of amplitude wj and exponential time course (t) = exp(−t/τmE )H(t) with time constant τmE = 25
ms and Heaviside step function H(t). The EPSP time course fromfpresynaptic neuron j was denoted by h j (t) = f (t − t j ), summingfthe influence of presynaptic spikes at time t j (f = 1, 2, …). Then,the postsynaptic membrane potential u(t) was denoted byu(t) = ur +Nwith resting membrane potential ur = −70 mV. Postsynaptic spikes
were generated probabilistically with density ρ(t) = gE (u(t))R(t), as
mentioned in the previous section.

Information maximizing learning rule
We used the infomax rule for synaptic plasticity (24, 25). The objective function L was described as
L = I − λ,with the mutual information term I, cost term  of the synaptic weights, and coefficient parameter λ = 0.32 [1/(mV)2 ]. I measures the mutual information between the pre and postsynaptic
spike trains. We omitted the homeostatic term included in previous studies because it did not contribute to our results. Each term
was denoted by
P(Y|X)
I = log P(Y )We introduced a stochastic spiking neuron model; each neuron
was either excitatory (E) or inhibitory (I). The spikes of each neuron in population P (P = E, I) were generated probabilistically withdensity ρ(t), as follows: = 12

2
j w j n j X ,with presynaptic and postsynaptic spike trains X and Y, respectively, and nj represents the number of presynaptic spikes atsynapse j during duration T. The presynaptic and postsynaptic
spike trains up to time t were formally denoted by X(t) = {x j (t  ) =


f




f δ(t − t j ) | j = 1, 2, . . . , N, 0 ≤ t < t} and Y(t) = {y(t ) =f δ(t −
ftpost ) | 0 ≤ t  < t}, respectively, where tpost represents the f-th (f =1, 2, …) postsynaptic spike timings. We specifically wrote X = X(T)and Y = Y(T) to represent the entire spike train from time t = 0 to t
= T. Note that the angular brackets  · Y, X and  · X represent the
averages over all possible Y, X, and X, respectively.
The optimal synaptic weight change followed the gradient ascent algorithm:

ρ(t) = gP (u(t))R(t),where gP (u) denoted a softplus activation intensity function, u(t)the membrane potential, and R(t) a refractory factor representing transient suppression of the instantaneous postsynaptic firing rate after a postsynaptic spike. The function gP (u) = r0P log(1 +exp((u − uP0 )/uP )) with r0E = 1.5 Hz, r0I = 6.0 Hz, uE0 = −69.4 mV,
uI0 = −62.5 mV, and uE = uI = 0.5 mV. The refractory factor ismodeled to be the same for excitatory and inhibitory neurons as
R(t) = (t − t̂)4 /(τR4 + (t − t̂)4 ), where t̂ denoted the last spike time ofthe postsynaptic neuron and time constant τ R = 30 ms. The suppression of this factor after spiking may reflect several mechanisms, including classical refractoriness (64), afterhyperpolariza-

,

Y,X

f

Spiking neuron modelw j h j (t),

j=1

dw j
∂L
,
=α
dt
∂w j
with a learning rate α = 0.01 (mV)2 . By calculating the gradient(24, 25), the infomax rule was described asdw j= α C j (t)Bpost (t) − λw j x j (t) ,
dt
with

 t+
− t−t
C j (t) = lim→+0 0 c j (t  )e τC dt  ,ρ(t)Bpost (t) = y(t) log ρ̄(t) − ρ(t) − ρ̄(t) ,Downloaded from https://academic.oup.com/pnasnexus/article/2/1/pgac286/6887287 by RIKEN CBS user on 12 January 2023

Our theory could be verified experimentally along with the following two points. First, excitatory firing rates during global up
states were lower than during local up states. Although the direct evidence supporting this prediction has not been reported to
our knowledge, it is indirectly supported by theoretical and experimental studies outside sleep research. Our network operates as an
ISN in its up states (31–33, 59). ISN models reproduce the experimentally observed surround suppression effect, i.e., the reduction
in both local excitatory and inhibitory firing rates upon the activation of surrounding populations (31, 33). This property is sufficient (but not necessary) for our model to exhibit lower excitatory
firing rate in the up states of more global slow waves. Notably, a
hallmark ISN property, the nonmonotonic inhibitory response, isexperimentally verified across cortical areas (60). Furthermore, recent technical advances in neuronal recordings from a large number of neurons (61) can annotate up and down states in each local
area more precisely. In the future, such recordings could directly
verify the model prediction that the mean firing rates of excitatory neurons decline with the spatial scale of the slow waves. Another testable model prediction is that global slow waves should
permit more efficient information transmission than local slow
waves. For example, one could optogenetically stimulate a small
group of neurons and quantify the accuracy of stimulus encoding
in neurons postsynaptic to the stimulated neurons during global
and local slow waves.
Another interesting perspective is the qualitative reorganization of memories during sleep (62). While our model focuseson synaptic plasticity and quantitative memory reorganization(i.e., consolidation vs. forgetting), a recent theory proposes thatthe learning cycle mimicking wakefulness, NREM sleep, and REM
sleep promote the formulation of new cortical representations,
not just strengthening or weakening experiences (63). Bridging
synaptic plasticity rules mainly obtained in the rodent experimentand qualitative memory reorganization proposed in the cognitive
study is an interesting future direction.
In summary, the proposed theory bridges neuronal information coding, synaptic plasticity, and memory reorganization. Our
normative framework provides a versatile learning rule for statedependent synaptic plasticity and memory reorganization during
NREM sleep.

Yoshida and Toyoizumi

described with auxiliary variable c j (t) =∂ log gE (u)∂uu=u(t)

y(t) −

ρ(t) h j (t), expected firing rate ρ̄(t) = ρ(t)X(t)|Y(t) , and time constant τ C = 100 ms. The expected firing rate was further computedasρ̄(t) = ḡ(t)R(t),Theoretical analysis of the STDP effect
For the theoretical analyses in this section, the activation function
is not restricted to the softplus function. We evaluated the synaptic changes due to the pre-post stimulation with synchronous
presynaptic spikes at t = 0 and a postsynaptic spike immediately afterward (at t = t > 0 with the limit of t → 0). To
estimate the effect of STDP analytically and qualitatively, we
made some simplifications. We approximated that the refractory factor R abruptly recovered from zero to one duration τ R after the postsynaptic spike, and that the spontaneous spiking of
the nonstimulated presynaptic neurons yielded a constant baseline membrane potential u0 . We also assumed that, after presynaptic stimulation at t = 0, the membrane potential u(t) = u0 +j ∈ stim wj hj (t) decayed back to the baseline u0 by the time thepostsynaptic neuron recovered from the refractoriness because
τmE was smaller than τ R . Here, stim denotes the set of stimulated neurons. The baseline intensity is denoted by ḡ = gE (u0 ),and the peak intensity increase after presynaptic stimulation
by g = gE (u0 + u) − ḡ, with u =j ∈ stim wj . In this case, we
found C j (t) = H(t)∂gE (u)/(ḡ + g) exp(−t/τC ) and Bpost (t) =∂uu=u +u0

gE (u(t))y(t) log ḡ , using the Heaviside step function H(t). Note that no

extra postsynaptic spike was possible when hj was significantly
positive because the refractory period was longer than that of
the stimulus-caused EPSP duration. Hence, the change in the j-th
synaptic weight due to the STDP stimulus was
C j (t)Bpost (t)dt
∂gE (u)g1=
log 1 +.
·ḡ∂u u=u0 +u ḡ + g
In the case that the activation function is a linear functiongE (u)


g0
log 1 + g
= g0 · (u − ur ), the synaptic change is denoted by ḡ+g,ḡwhere g does not depend on ḡ. This result indicated that the increase in synaptic weights due to the information term decreased
with the mean firing intensity ḡ.

Slow-wave model
We considered NE = 800 excitatory neurons, E, and NI = 200 inhibitory neurons, I, divided into four local networks. Each local
EI
network contained N4 excitatory neurons and N4 inhibitory neurons. Within each local network, no connections exist between
two inhibitory neurons, and all-to-all connections exist between
two excitatory neurons as well as between excitatory and inhibitory neurons. Between different local networks, the connection probability from E to E was pEE and the connection probability from E to I was pIE , whereas inhibitory neurons did not send
long-range connections to the other local networks. The connection probabilities pEE and pIE were set to 0.05 and 0.3, respectively,

except for those in Fig. 3C and SI Appendix, Fig. S5. There is noself-coupling in these neurons. The dynamics of the membranepotential uPi of neuron i in population P (P = E, I) were described
by
E

I

j=1

j=1

N
N

uE (t) − ur 
duEi (t)
E
I
+
wEE
wEI
=− i Ei j S j (t) +i j S j (t)dtτm+Iiext (t) − Iia (t),E

N
duIi (t)uI (t) − ur E+wIE=− i Ii j S j (t),
dt
τm
j=1

with the recurrent synaptic weight wEE, wEI, and wIEfrom exciij
ij
ij
tatory neuron j to excitatory neuron i, from inhibitory neuron j
to excitatory neuron i, and from excitatory neuron j to inhibitory
neuron i, respectively, membrane time constant τmE = 25 ms for excitatory neurons and τmI = 5 ms for inhibitory neurons, the restingfmembrane potential ur = −70 mV, spike train SEj (t) = f δ(t − t j )
for SIj (t) = f δ(t − t j ) of excitatory or inhibitory neuron j describedfwith its f-th spike time t j , external current Iiext (t), and adaptation current Iia (t). The spikes of neuron i in population P (P = E,I) were generated probabilistically with an instantaneous firing
rate gP (uPi (t))Ri (t), as mentioned in the previous section, with its
refractory factor Ri . The recurrent synaptic weight was fixed at
wEE= 0.16 mV, wEI= −0.14 mV, and wIE= 0.66 mV if synaptic conij
ij
ij
nections existed from neuron j to neuron i, whereas it was set to
0 mV if there were no synaptic connections. The external current
was a feedforward input only to the excitatory neuron 1, described
 ext
Sext
(t), with the Kronecker delta δ i1 , synapas Iiext (t) = δi1 Nj=1 wextjjext
tic weights wext
fromN
presynaptic neurons, and presynaptic
j

f
f
ext
spike train S j (t) = t f δ(t − t j ), where t j represented the spikejtiming. Presynaptic neurons emitted the synchronous spikes upon
STDP stimulation (Fig. 4) or task stimulation (Fig. 5 and SI Appendix, Fig. S7), although they did not emit spontaneous spikes.
The STDP and task simulations (Figs. 4 and 5 and SI Appendix,
Fig. S7) changed the feedforward synaptic weights wj according to
the infomax rule. The dynamics of the adaptation current Iia (t) of
neuron i were described as follows
Ia (t)
dIia (t)+ βSEi (t),
=− i
dt
τa
with a time constant of τ a = 1,500 ms and a constant value of β =0.0077 mV/ms.

Stage classification
First, we classified the up and down states of each population using two transition thresholds. When the mean membrane potential of excitatory neurons in each local network exceeded the uptransition threshold θ up = −68.25 mV, this moment was judged
as a state transition to the up state. When the mean membrane potential fell below the down-transition threshold θ down =−69.75 mV, this moment was judged as a state transition to thedown state. We then classified each state into the global or local
states. When a local network was in the down state, it was classified into the global down state if the number of other local networks in the down states was two or three, while it was classified
as the local down state otherwise. Likewise, when a focused population was in the up state, it was classified into the global up state
if the number of other local networks in the up states was two or
three, while it was classified as the local up state otherwise.

Downloaded from https://academic.oup.com/pnasnexus/article/2/1/pgac286/6887287 by RIKEN CBS user on 12 January 2023

using the expected intensity ḡ(t) = gE (u(t))X(t)|Y(t) ; however, thisvalue can be difficult to calculate if ḡ is time dependent. We estimated ḡ using slightly different methods for the single-neuron
and slow-wave models, as described in the corresponding sections (SI Appendix Methods).

| 11

12 |

PNAS Nexus, 2022, Vol. 2, No. 1

Simulation environment

9.

All numerical calculations were performed using the customwritten Python codes. The model was simulated in discrete timewith time steps of 1 ms. Synaptic weights did not change during
the first 10,000 ms (200,000 ms in Fig. 5 and SI Appendix, Fig. S7)
in all simulations to avoid the effects of initial values. The initialvalue of the last spike time was set to −10,000 ms for all neurons.

Acknowledgments

10.

11.
12.

13.

Supplementary Material
Supplementary material is available at PNAS Nexus online.

14.

Funding

15.

This study was supported by RIKEN Center for Brain Science,
Brain/MINDS from AMED under Grant no. JP15dm0207001 (T.T.),
KAKENHI Grant-in-Aid JP18H05432 (T.T.) and JP21J10564 (K.Y.)
from JSPS, RIKEN Junior Research Associate Program (K.Y.), andMasason Foundation (K.Y.).

16.

17.

Authors’ Contributions
K.Y. and T.T. conceived the project. K.Y. conducted numerical simulations. K.Y. and T.T. wrote the manuscript.

18.

Preprint
A preprint version of the manuscript has been posted on https://www.biorxiv.org/content/10.1101/2022.03.22.485283v1.

19.
20.

Data Availability
The
codes
used
for
simulationsarehttps://github.com/kkyoshida/SleepInfomax.

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