# Effect of Adult-Born Immature Granule Cells on Pattern Separation in The
Hippocampal Dentate Gyrus

Sang-Yoon Kim<EMAIL_ADDRESS>Woochang Lim<EMAIL_ADDRESS>Institute for
Computational Neuroscience and Department of Science Education, Daegu National
University of Education, Daegu 42411, Korea

###### Abstract

Young immature granule cells (imGCs) appear via adult neurogenesis in the
hippocampal dentate gyrus (DG). In comparison to mature GCs (mGCs) (born
during development), the imGCs exhibit two competing distinct properties such
as high excitability and low excitatory innervation. We develop a spiking
neural network for the DG, incorporating the imGCs, and investigate their
effect on pattern separation (i.e., a process of transforming similar input
patterns into less similar output patterns). We first consider the effect of
high excitability. The imGCs become very highly active due to their low firing
threshold. Then, because of high activation, strong pattern correlation
occurs, which results in pattern integration (i.e., making association between
events). On the other hand, the mGCs exhibit very sparse firing activity due
to strongly increased feedback inhibition (caused by the high activation of
the imGCs). As a result of high sparsity, the pattern separation efficacy
(PSE) of the mGCs becomes very high. Thus, the whole population of GCs becomes
a heterogeneous one, composed of a (major) subpopulation of mGCs (i.e.,
pattern separators) with very low activation degree $D_{a}^{(m)}$ and a
(minor) subpopulation of imGCs (i.e., pattern integrators) with very high
activation degree $D_{a}^{(im)}$. In the whole heterogeneous population, the
overall activation degree $D_{a}^{(w)}$ of all the GCs is a little reduced in
comparison to the activation degree $D_{a}^{(out)}$ in the presence of only
mGCs without imGCs. However, no pattern separation occurs, due to
heterogeneous sparsity, in contrast to the usual intuitive thought that
sparsity could improve PSE. Next, we consider the effect of low excitatory
innervation for the imGCs, counteracting the effect of their high
excitability. With decreasing the connection probability of excitatory inputs
to the imGCs, $D_{a}^{(im)}$ decreases so rapidly, and their effect becomes
weaker. Then, the feedback inhibition to the mGCs is also decreased, leading
to increase in $D_{a}^{(m)}$ of the mGCs. Accordingly, $D_{a}^{(w)}$ of the
whole GCs also increases. In this case of low excitatory connectivity, the
imGCs perform pattern integration. On the other hand, due to increase in
$D_{a}^{(m)}$, the PSE of the mGCs decreases from a high value to a limit
value. In the whole population of all the GCs, when the excitatory connection
probability decreases through a threshold, pattern separation starts, the
overall PSE increases and approaches that of the mGCs. However, due to
heterogeneity caused by the imGCs, the overall PSE becomes deteriorated, in
comparison with that in the presence of only mGCs.

Hippocampal dentate gyrus, Adult neurogenesis, Immature granule cells, High
excitability, Low excitatory innervation, Pattern separation efficacy

###### pacs:

87.19.lj, 87.19.lm, 87.19.lv

##  I Introduction

The hippocampus, composed of the dentate gyrus (DG) and the subregions CA3 and
CA1, plays important roles in memory formation, storage, and retrieval (e.g.,
episodic and spatial memory) Gluck ; Squire . In particular, the subregion CA3
has been considered as an autoassociative network, because of extensive
recurrent collateral synapses between the pyramidal cells in the CA3 Marr ;
Will ; Mc ; Rolls1 ; Rolls2a ; Rolls2b ; Treves1 ; Treves2 ; Treves3 ; Oreilly
. This autoassociative network operates in both the storage and the recall
modes. Storage capacity of the autoassociative network implies the number of
distinct patterns that can be stored and accurately recalled. Such storage
capacity could be increased if the input patterns into the CA3 are sparse
(containing few active elements in each pattern) and orthogonalized
(nonoverlapping: active elements in one pattern are unlikely to be active in
other patterns). This process of transforming a set of input patterns into
sparser and orthogonalized patterns is called pattern separation Marr ; Will ;
Mc ; Rolls1 ; Rolls2a ; Rolls2b ; Treves1 ; Treves2 ; Treves3 ; Oreilly ;
Schmidt ; Rolls3 ; Knier ; Myers1 ; Myers2 ; Myers3 ; Scharfman ; Yim ;
Chavlis ; Kassab ; PS1 ; PS2 ; PS3 ; PS4 ; PS5 ; PS6 ; PS7 .

Here, we are concerned about the DG which is the gateway to the hippocampus.
The excitatory granule cells (GCs) in the DG receive excitatory inputs from
the entorhinal cortex (EC) via the perforant paths (PPs). As a preprocessor
for the CA3, the principal GCs perform pattern separation on the input
patterns from the EC by sparsifying and orthogonalizing them, and provide the
pattern-separated outputs to the pyramidal cells in the CA3 through the mossy
fibers (MFs) Treves3 ; Oreilly ; Schmidt ; Rolls3 ; Knier ; Myers1 ; Myers2 ;
Myers3 ; Scharfman ; Yim ; Chavlis ; Kassab . Then, a new pattern may be
stored in modified collateral synapses between the pyramidal cells in the CA3.
In this way, pattern separation in the DG could facilitate pattern storage in
the CA3.

The whole GCs are grouped into the lamellar clusters Cluster1 ; Cluster2 ;
Cluster3 ; Cluster4 . In each cluster, there exist one inhibitory basket cell
(BC) and one inhibitory HIPP (hilar perforant path-associated) cell, together
with excitatory GCs. During pattern separation, the GCs show sparse firing
activity via the winner-take-all competition WTA1 ; WTA2 ; WTA3 ; WTA4 ; WTA5
; WTA6 ; WTA7 ; WTA8 ; WTA9 ; WTA10 ; WTA . Only strongly active GCs survive
under the feedback inhibitory inputs from the BC and the HIPP cell. We note
that, sparsity (resulting from strong feedback inhibition) has been considered
to improve the pattern separation efficacy Treves3 ; Oreilly ; Schmidt ;
Rolls3 ; Knier ; Myers1 ; Myers2 ; Myers3 ; Scharfman ; Chavlis ; Kassab .

One of the most distinctive characteristics of the DG is occurrence of adult
neurogenesis which results in the generation of new GCs during adulthood.
Altman’s pioneering studies in adult rat and cat brains for the adult
neurogenesis were done decades ago in the 1960s NG1 ; NG2 ; NG3 . Since then,
adult neurogenesis has been shown to be a robust phenomenon, occurring in most
mammals, mainly in the subgranular zone of the DG and the subventricular zone
of the lateral ventricles NG4 ; NG5 ; NG6 . The new GCs born in the
subgranular zone migrate into the granular layer of the DG. The whole
population of GCs is thus composed of mature GCs (mGCs) born during the
development and adult-born immature GCs (imGCs). In contrast to the mGCs, the
young adult-born imGCs are known to have marked properties such as high
excitability, weak inhibition, and low excitatory innervation NG7 ; NG8 ; NG9
; NG10 ; NG11 .

In this paper, we develop a spiking neural network for the DG, including both
mGCs and imGCs; the fraction of the imGCs is 10 $\%$. In our DG network, high
excitability of imGCs is considered, and approximately no inhibition is
provided to the imGCs. We first investigate the effect of adult-born imGCs
with high excitability on pattern separation NG7 ; NG8 ; NG9 ; NG10 . The
imGCs show high activation due to lower firing threshold [i.e., their
activation degree $D_{a}^{(im)}$ (= 45 $\%$) becomes very high]. As a result,
in the subpopulation of the imGCs, output patterns become highly overlapped
(i.e, their Pearson’s correlation coefficient is very high). Thus, instead of
pattern separation, pattern integration (i.e., making association between
events) occurs due to strong pattern correlation. On the other hand, the
activation degree $D_{a}^{(m)}$ (= 1.1 $\%$) of the mGCs becomes very low due
to strong feedback inhibition from the inhibitory basket cells (BCs) and HIPP
(hilar perforant path-associated) cells (caused by high activation of the
imGCs). As a result of high sparsity, the efficacy of pattern separation of
the mGCs becomes very high. In this way, the whole population of GCs is a
heterogeneous one, consisting of a (major) subpopulation of mGCs (pattern
separators) with very low $D_{a}^{(m)}$ and a (minor) subpopulation of imGCs
(pattern integrators) with very high $D_{a}^{(im)}$. In the whole
heterogeneous population, the overall activation degree $D_{a}^{(w)}$ of all
the GCs is 5.5 $\%$ [a little less than $D_{a}^{(out)}$ (= 6 $\%$) in the
presence of only mGCs without imGCs]. Although $D_{a}^{(w)}$ is a little
reduced (i.e., sparser firing activity), no pattern separation occurs, due to
heterogeneous sparsity, in contrast to the usual intuitive thought that
sparsity could improve pattern separation efficacy.

Next, we consider the effect of low excitatory innervation for the imGCs,
counteracting the effect of high excitability NG11 . In the case of mGCs, they
receive excitatory inputs from the entorhinal cortex (EC) via perforant paths
(PPs) and from the hilar mossy cells (MCs) with the connection probability
$p_{c}$ (= 20 $\%$). On the other hand, the imGCs receive low excitatory drive
from the EC via the PPs and from the MCs with lower connection probability
$p_{c}~{}(=20~{}x~{}\%)$ ($x:$ synaptic connectivity fraction; $0\leq x\leq
1$).

With decreasing $x$ from 1, $D_{a}^{(im)}$ of the imGCs decreases so rapidly,
and their effect becomes weaker. Then, the feedback inhibition to the mGCs is
also decreased, and hence $D_{a}^{(m)}$ of the mGCs becomes increased.
Accordingly, $D_{a}^{(w)}$ of the whole GCs also increases. In the whole range
of $0\leq x\leq 1,$ the imGCs are good pattern integrators with strong pattern
correlation. On the other hand, due to increase in $D^{(m)}$, the pattern
separation efficacy of the mGCs decreases from the high value for $x=1$ to a
limit value. In the whole population of all the GCs, due to decreased effect
of the imGCs, when $x$ decreases through a threshold, pattern separation
starts, and then the overall efficacy of pattern separation increases and
approaches that of the mGCs. In the limit case of $x=0$ where all imGCs are
silent, the limit efficacy of pattern separation in the whole population is
lower than that in the presence of only mGCs (without imGCs), mainly because
$D_{a}^{(w)}$ (= 7.3 $\%$) is larger than $D_{a}$ (= 6 $\%$) in the absence of
imGCs. In this way, due to heterogeneity caused by the imGCs (performing
pattern integration), the overall efficacy of pattern separation in the whole
heterogeneous population of the GCs becomes deteriorated.

This paper is organized as follows. In Sec. II, we describe a spiking neural
network for the adult neurogenesis in the hippocampal DG. Then, in the main
Sec. III, we investigate the effect of the adult-born imGCs on pattern
separation by varying $x$ (synaptic connectivity fraction). Finally, we give
summary and discussion in Sec. IV.

##  II Spiking Neural Network for The Adult Neurogenesis in The Dentate Gyrus

In this section, we describe our spiking neural network for the adult
neurogenesis in the DG. Based on the anatomical and the physiological
properties described in Myers1 ; Myers2 ; Chavlis , we developed the DG
spiking neural networks in the works for the winner-take-all competition WTA ,
the sparsely synchronized rhythm SSR , and the pattern separation PS . Here,
we first refine our prior spiking neural networks to include more synaptic
connections with a high degree of anatomical and physiological realism BN1 ;
BN2 , and then incorporate the young adult-born imGCs to complete structure of
our spiking neural network for the adult neurogenesis.

Obviously, our spiking neural network will not capture all the detailed
anatomical and physiological complexity of the DG. But, with a limited number
of essential elements and synaptic connections in our DG network, effect of
the imGCs on the pattern separation could be successfully studied. Hence, our
spiking neural network model would build a foundation upon which additional
complexity may be added and guide further research.

Figure 1: Spiking neural network for the hippocampal dentate gyrus (DG). (a)
Schematic representation of of major cells and synaptic connections in our DG
network incorporating adult-born immature GCs (imGCs). Fraction of the imGCs
is 10 $\%$ in the whole population of GCs. Note that there are no inhibitory
inputs into the imGCs, in contrast to the case of mGCs. Here, BC, MC, HIPP,
PP, GL, and ML represent the basket cell, the mossy cell, the hilar perforant
path-associated cell, perforant path, granular layer, and molecular layer,
respectively. (b) Box diagram for our DG network with 3 types of synaptic
connections. Blue, red, and black lines represent lamellar, cross-lamellar,
and random connections, respectively.

###  II.1 Architecture of The Spiking Neural Network of The DG

Figure 1 shows (a) schematic representation of major cells and synaptic
connections in our DG network incorporating adult-born imGCs and (b) the box
diagram for the DG network with 3 types of lamellar (blue), cross-lamellar
(red), and random (black) synaptic connections. In our DG network, the
fraction of imGCs is 10 $\%$ in the whole population of GCs, high excitability
of the imGCs is considered, there are no inhibitory inputs into the imGCs, and
their low excitatory innervation is also taken into consideration NG7 ; NG8 ;
NG9 ; NG10 ; NG11 .

In the DG, we consider the granular layer (GL), composed of the excitatory
mGCs and imGCs and the inhibitory BCs, and the underlying hilus, consisting of
the excitatory MCs and the inhibitory HIPP cells, whose axons project to the
upper molecular layer (ML). We note that there are two types of excitatory
cells, GCs and MCs, in contrast to the case of the CA3 and CA1 with only one
type of excitatory pyramidal cells.

From the outside of the DG, the EC provides the external excitatory inputs
randomly to the mGCs, the imGCs, and the inhibitory BCs (with dendrites
extending to the outer ML) via PPs Myers1 ; Myers2 ; Myers3 ; Scharfman ;
Chavlis . Thus, both the mGCs and the imGCs receive direct excitatory EC input
via PPs (EC $\rightarrow$ mGC and imGCs) through random connections in Fig.
1(b). The connection probability $p_{c}$ for EC $\rightarrow$ mGC and BC is 20
$\%$, while $p_{c}$ for EC $\rightarrow$ imGC is decreased to $20~{}x$ $\%$
[$x$ (synaptic connectivity fraction); $0\leq x\leq 1$] due to low excitatory
innervation. Moreover, only the mGCs receive indirect feedforward inhibitory
input, mediated by the BCs (EC $\rightarrow$ BC $\rightarrow$ mGC).

In the GL, the whole GCs (i.e., both the mGCs and the imGCs) are grouped into
lamellar clusters Cluster1 ; Cluster2 ; Cluster3 ; Cluster4 , and one
inhibitory BC exists in each cluster. Here, the BC (receiving excitation from
the whole GCs in the same cluster) provides the feedback inhibition to only
all the mGCs via lamellar connections in Fig. 1(b); a primary mGC-BC feedback
loop is formed. Thus, in each cluster the BC provides both the feedforward and
the feedback inhibition to all the mGCs in the same cluster.

In the hilus, we also consider lamellar organization for the MCs and HIPP
cells Myers2 ; Myers3 ; Scharfman ; Hilus6  (i.e., all the MCs and the HIPP
cells in the hilus also are grouped into lamellar clusters). As in the case of
BC, the HIPP cell receives excitation from the whole GCs in the same cluster,
and projects the feedback inhibition to all the mGCs in the same cluster
through lamellar connections; a secondary mGC-HIPP feedback loop is formed.
Thus, there appear two kinds of feedback loops of mGC-BC and mGC-HIPP.

In our DG network, the MCs play the role of “controller” for the activities of
the two feedback loops of mGC-BC and mGC-HIPP. Each MC in a cluster receives
excitation from the whole GCs in the same cluster (lamellar connection), while
it makes excitatory projection randomly to the mGCs and the imGCs in other
clusters via cross-lamellar connections Hilus6 . The connection probability
$p_{c}$ for MC $\rightarrow$ mGC is 20 $\%$, while $p_{c}$ for MC
$\rightarrow$ imGC is decreased to $20~{}x$ $\%$ ($0\leq x\leq 1$) because of
low excitatory innervation. Thus, the GC-MC driving loop for determining the
activities of the controller MCs is formed.

The MCs control the activities of the feedback loops of mGC-BC and mGC-HIPP.
Each MC in a cluster receives inhibition from the BC and the HIPP cell in the
same cluster (lamellar connection). Then, the MCs in the cluster project
excitation to the BCs in other clusters through cross-lamellar connections
(the connection probability $p_{c}$ for MC $\rightarrow$ BC is 20 $\%$) Hilus6
, while they provide excitation to the HIPP cell in the same cluster (lamellar
connection). Thus, two “control” loops of MC-BC and MC-HIPP, controlling the
activities of the two feedback loops of mGC-BC and mGC-HIPP, are formed.
Finally, the HIPP cell disinhibits the BC in the same cluster (lamellar
connection for HIPP $\rightarrow$ BC); there are no reverse synaptic
connections for HIPP $\rightarrow$ BC BN1 ; BN2 . Thus, the activity of the BC
in a cluster is controlled through excitation from the MCs in other clusters
(cross-lamellar connections) and inhibition from the HIPP cell in the same
cluster (lamellar connection).

The mGCs in a cluster exhibit sparse firing activity via the winner-take-all
competition WTA1 ; WTA2 ; WTA3 ; WTA4 ; WTA5 ; WTA6 ; WTA7 ; WTA8 ; WTA9 ;
WTA10 ; WTA . Only strongly active mGCs may survive under the feedback
inhibition from the BC and the HIPP cell in the same cluster. Here, the
activities of the BC and the HIPP cell are controlled by the controller MCs;
in the case of BC, the HIPP cell also disinhibits it. On the other hand, the
imGCs receive no inhibition. Particularly, due to their low firing threshold,
they become highly active, in contrast to the case of mGCs NG7 ; NG8 ; NG9 ;
NG10 . However, when considering their low excitatory innervation from the EC
cells and the MCs, their firing activity is reduced NG11 .

Based on the anatomical information given in Myers1 ; Myers2 ; Myers3 ;
Scharfman ; Chavlis , we choose the numbers of the GCs, BCs, MCs, and HIPP
cells in the DG and the EC cells. As in our prior works WTA ; SSR ; PS , we
develop a scaled-down spiking neural network where the total number of
excitatory GCs ($N_{\rm GC}$) is 2,000, corresponding to $\frac{1}{500}$ of
the $10^{6}$ GCs found in rats ANA1 . The fraction of imGCs in the whole
population of the GCs is $10~{}\%$, and hence the number of the imGCs (mGCs)
is 200 (1800). The whole GCs (i.e., mGCs and imGCs) are grouped into the
$N_{c}~{}(=20)$ lamellar clusters Cluster1 ; Cluster2 ; Cluster3 ; Cluster4 .
Then, in each cluster, there are $n_{\rm{GC}}^{(c)}~{}(=100)$ GCs (i.e., 90
mGC and 10 imGCs) and one inhibitory BC Myers2 ; Myers3 ; Scharfman . As a
result, the number of the BCs ($N_{\rm BC}$) in the whole DG network becomes
20, corresponding to 1/100 of $N_{\rm GC}$ GC-BC1 ; GC-BC2 ; GC-BC3 ; GC-BC4 ;
GC-BC5 ; BN2 .

The EC layer II projects the excitatory inputs to the mGCs, the imGCs, and the
BCs via the PPs through random connections Myers1 ; Myers2 ; Myers3 ;
Scharfman ; Chavlis . The estimated number of the EC layer II cells ($N_{\rm
EC}$) is about 200,000 in rats, which corresponds to 20 EC cells per 100 GCs
ANA3 . Hence, we choose $N_{\rm EC}=400$ in our DG network. Also, the
activation degree of the EC cells is chosen as 10$\%$ ANA4 . Thus, we randomly
choose 40 active ones among the 400 EC cells. Each active EC cell is modeled
in terms of the Poisson spike train with frequency of 40 Hz ANA5 .

Next, we consider the hilus, composed of the excitatory MCs and the inhibitory
HIPP cells Hilus1 ; Hilus2 ; Hilus3 ; Hilus4 ; Hilus5 ; Hilus6 ; Hilus7 . In
rats, the number of MCs ($N_{\rm MC}$) is known to change from 30,000 to
50,000, and the estimated number of HIPP cells ($N_{\rm HIPP}$) is about
12,000 ANA2 . In our scaled-down DG network, we choose $N_{\rm MC}=60$ and
$N_{\rm HIPP}=20$. All the MCs and the HIPP cells are also grouped into the 20
lamellar clusters, as in the case of the GCs and the BCs. Hence, in each
cluster, there are $n_{\rm{MC}}^{(c)}~{}(=3)$ MCs and one HIPP cell Myers2 ;
Myers3 ; Scharfman .

With the above information on the numbers of the relevant cells and the
synaptic connections between them, we develop a one-dimensional ring network
for the adult neurogenesis in the DG, as in our prior works WTA ; SSR ; PS ;
e.g., refer to Figs. 1(b1)-1(b3) in PS  for the schematic diagrams of the ring
networks. Due to the ring structure, our spiking neural network has advantage
for computational efficiency, and its visual representation may also be easily
made.

###  II.2 Single Neuron Models and Synaptic Currents in The DG Spiking Neural
Network

As elements of our DG spiking neural network for the adult neurogenesis, we
choose leaky integrate-and-fire (LIF) neuron models with additional
afterhyperpolarization (AHP) currents which determines refractory periods, as
in our prior DG networks WTA ; SSR ; PS . This LIF neuron model is one of the
simplest spiking neuron models LIF . Due to its simplicity, it may be easily
analyzed and simulated. It has thus been very popularly used as a spiking
neuron model.

The governing equations for evolutions of dynamical states of individual cells
in the $X$ population are as follows:

$\displaystyle C_{X}\frac{dv_{i}^{(X)}(t)}{dt}$ $\displaystyle=$
$\displaystyle-
I_{L,i}^{(X)}(t)-I_{AHP,i}^{(X)}(t)+I_{ext}^{(X)}-I_{syn,i}^{(X)}(t),$ (1)
$\displaystyle\;\;\;i=1,\cdots,N_{X},$

where $N_{X}$ is the total number of cells in the $X$ population, $X=$ mGC,
imGC, and BC in the granular layer and $X=$ MC and HIPP in the hilus. In Eq.
(1), $C_{X}$ (pF) represents the membrane capacitance of the cells in the $X$
population, and the dynamical state of the $i$th cell in the $X$ population at
a time $t$ (msec) is characterized by its membrane potential $v_{i}^{(X)}(t)$
(mV). We note that the time-evolution of $v_{i}^{(X)}(t)$ is governed by 4
types of currents (pA) into the $i$th cell in the $X$ population; the leakage
current $I_{L,i}^{(X)}(t)$, the AHP current $I_{AHP,i}^{(X)}(t)$, the external
constant current $I_{ext}^{(X)}$ (independent of $i$), and the synaptic
current $I_{syn,i}^{(X)}(t)$.

The equation for a single LIF neuron model (without the AHP current and the
synaptic current) describes a simple parallel resistor-capacitor (RC) circuit.
In this case, the 1st type of leakage current is due to the resistor and the
integration of the external current is due to the capacitor which is in
parallel to the resistor. When its membrane potential reaches a threshold, a
neuron fires a spike, and then the 2nd type of AHP current follows. As the
decay time of the AHP current is increased, the refractory period becomes
longer. Here, we consider a subthreshold case where the 3rd type of external
constant current is zero (i.e., $I_{ext}^{(X)}=0$) Chavlis .

The 1st type of leakage current $I_{L,i}^{(X)}(t)$ for the $i$th cell in the
$X$ population is given by:

$I_{L,i}^{(X)}(t)=g_{L}^{(X)}(v_{i}^{(X)}(t)-V_{L}^{(X)}),$ (2)

where $g_{L}^{(X)}$ and $V_{L}^{(X)}$ denote conductance (nS) and reversal
potential for the leakage current, respectively. The $i$th cell fires a spike
when its membrane potential $v_{i}^{(X)}$ reaches a threshold $v_{th}^{(X)}$
at a time $t_{f,i}^{(X)}$. Then, the 2nd type of AHP current
$I_{AHP,i}^{(X)}(t)$ follows after spiking (i.e., $t\geq t_{f,i}^{(X)}$), :

$I_{AHP,i}^{(X)}(t)=g_{AHP}^{(X)}(t)~{}(v_{i}^{(X)}(t)-V_{AHP}^{(X)})~{}~{}~{}{\rm~{}for~{}}\;t\geq
t_{f,i}^{(X)}.$ (3)

Here, $V_{AHP}^{(X)}$ represents the reversal potential for the AHP current,
and the conductance $g_{AHP}^{(X)}(t)$ is given by an exponential-decay
function:

$g_{AHP}^{(X)}(t)=\bar{g}_{AHP}^{(X)}~{}e^{-(t-t_{f,i}^{(X)})/\tau_{AHP}^{(X)}},$
(4)

where $\bar{g}_{AHP}^{(X)}$ and $\tau_{AHP}^{(X)}$ denote the maximum
conductance and the decay time constant for the AHP current, respectively.
With increasing $\tau_{AHP}^{(X)}$, the refractory period becomes longer.

The parameter values of the capacitance $C_{X}$, the leakage current
$I_{L}^{(X)}(t)$, and the AHP current $I_{AHP}^{(X)}(t)$ are the same as those
in our prior DG networks WTA ; SSR ; PS , and refer to Table 1 in WTA . These
parameter values are based on physiological properties of the GC, BC, MC, and
HIPP cell Chavlis ; Hilus3 .

Figure 2: Firing transitions of mature GCs (mGCs) and adult-born immature GCs
(imGCs). $f-I$ ($f:$ firing rate and $I:$ current) curve for the mature GC
(mGC) (red line) and the imGC (blue line).

We note that, the GC in Table 1 in WTA  corresponds to the mGC. The imGCs also
have the same parameter values as those of the mGC, except for the leakage
reversal potential $V_{L}$. The mGC with $V_{L}=-75$ mV exhibits a spiking
transition when passing a threshold $I^{*}=80$ mV. Here, we consider a case
that the imGC has an increased leakage reversal potential of $V_{L}=-72$ mV,
which could lead to intrinsic high excitability. Then, it shows a firing
transition when passing $I^{*}=69.7$ pA. In this way, the imGC may have a
lower firing threshold NG7 ; NG8 ; NG9 ; NG10 , which is well shown in Fig. 2
for the $f-I$ (i.e., firing rate-current) curves of the mGC (red curve) and
the imGC (blue curve).

Table 1: Parameters for the synaptic currents $I_{R}^{({\rm GC},S)}(t)$ into the GCs (granule cells). The whole population of the GCs is composed of a major subpopulation of mGCs (mature GCs) and a minor subpopulation of imGCs (immature GCs). Both the mGCs and the imGCs receive the excitatory inputs from the EC (entorhinal cortex) cells and the hilar MCs (mossy cells); synaptic parameters for the excitatory inputs are valid for both the mGCs and the imGCs. In addition, the mGCs receive the feedforward and feedback inhibitory inputs from the BCs (basket cells) and the feedback inhibitory input from the HIPP (hilar perforant-associated) cells, while there are no inhibitory inputs into the imGCs.  Target Cells ($T$) | GC
---|---
Source Cells ($S$) | EC | BC | HIPP | MC
Receptor ($R$) | AMPA | NMDA | GABA | GABA | AMPA | NMDA
$K_{R}^{(T,S)}$ | 0.89 | 0.15 | 15.0 | 3.0 | 0.07 | 0.01
$\tau_{R,r}^{(T,S)}$ | 0.1 | 0.33 | 0.9 | 0.5 | 0.1 | 0.33
$\tau_{R,d}^{(T,S)}$ | 2.5 | 50.0 | 6.8 | 6.0 | 2.5 | 50.0
$\tau_{R,l}^{(T,S)}$ | 3.0 | 3.0 | 0.85 | 1.6 | 3.0 | 3.0
$V_{R}^{(S)}$ | 0.0 | 0.0 | -86.0 | -86.0 | 0.0 | 0.0
Table 2: Parameters for the synaptic currents $I_{R}^{({\rm BC},S)}(t)$ into the BCs (basket cells). The BCs receive the excitatory inputs from the EC (entorhinal cortex) cells, the GCs (granulce cells; both mGCs and imGCs) and the MCs (mossy cells) and the inhibitory input from the HIPP (hilar perforant-associated) cells  Target Cells ($T$) | BC
---|---
Source Cells ($S$) | EC | GC | MC | HIPP
Receptor ($R$) | AMPA | NMDA | AMPA | NMDA | AMPA | NMDA | GABA
$K_{R}^{(T,S)}$ | 0.75 | 0.13 | 0.38 | 0.02 | 6.14 | 0.36 | 9.22
$\tau_{R,r}^{(T,S)}$ | 2.0 | 6.6 | 2.5 | 10.0 | 2.5 | 10.0 | 0.4
$\tau_{R,d}^{(T,S)}$ | 6.3 | 126.0 | 3.5 | 130.0 | 3.5 | 130.0 | 5.8
$\tau_{R,l}^{(T,S)}$ | 3.0 | 3.0 | 0.8 | 0.8 | 3.0 | 3.0 | 1.6
$V_{R}^{(S)}$ | 0.0 | 0.0 | 0 | 0 | 0.0 | 0.0 | -86.0
Table 3: Parameters for the synaptic currents $I_{R}^{(T,S)}(t)$ into the MCs (mossy cells) and the HIPP (hilar perforant-associated) cells. The MCs receive the excitatory inputs from the GCs (granule cells; both mGCs and imGCs) and the inhibitory inputs from the BCs (basket cells) and the HIPP (hilar perforant-associated) cells. The HIPP cells receive the excitatory inputs from the GCs (both mGCs and imGCs) and the MCs.  Target Cells ($T$) | MC | HIPP cell
---|---|---
Source Cells ($S$) | GC | BC | HIPP cell | GC | MC
Receptor ($R$) | AMPA | NMDA | GABA | GABA | AMPA | NMDA | AMPA | NMDA
$K_{R}^{(T,S)}$ | 9.58 | 1.71 | 3.08 | 2.05 | 0.08 | 0.004 | 4.09 | 0.25
$\tau_{R,r}^{(T,S)}$ | 0.5 | 4.0 | 0.3 | 0.5 | 0.3 | 1.2 | 0.9 | 3.6
$\tau_{R,d}^{(T,S)}$ | 6.2 | 100.0 | 3.3 | 6.0 | 0.6 | 22.2 | 3.6 | 133.7
$\tau_{R,l}^{(T,S)}$ | 1.5 | 1.5 | 1.5 | 1.0 | 1.5 | 1.5 | 3.0 | 3.0
$V_{R}^{(S)}$ | 0.0 | 0.0 | -86.0 | -86.0 | 0.0 | 0.0 | 0.0 | 0.0

Next, we consider the 4th type of synaptic current $I_{syn,i}^{(X)}(t)$ into
the $i$th cell in the $X$ population, composed of the following 3 types of
synaptic currents:

$I_{syn,i}^{(X)}(t)=I_{{\rm AMPA},i}^{(X,Y)}(t)+I_{{\rm
NMDA},i}^{(X,Y)}(t)+I_{{\rm GABA},i}^{(X,Z)}(t).$ (5)

Here, $I_{{\rm AMPA},i}^{(X,Y)}(t)$ and $I_{{\rm NMDA},i}^{(X,Y)}(t)$ are the
excitatory AMPA ($\alpha$-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid)
receptor-mediated and NMDA ($N$-methyl-$D$-aspartate) receptor-mediated
currents from the presynaptic source $Y$ population to the postsynaptic $i$th
neuron in the target $X$ population, respectively. In contrast, $I_{{\rm
GABA},i}^{(X,Z)}(t)$ is the inhibitory $\rm GABA_{A}$ ($\gamma$-aminobutyric
acid type A) receptor-mediated current from the presynaptic source $Z$
population to the postsynaptic $i$th neuron in the target $X$ population.

Like the case of the AHP current, the $R$ (= AMPA, NMDA, or GABA) receptor-
mediated synaptic current $I_{R,i}^{(T,S)}(t)$ from the presynaptic source $S$
population to the $i$th postsynaptic cell in the target $T$ population is
given by:

$I_{R,i}^{(T,S)}(t)=g_{R,i}^{(T,S)}(t)~{}(v_{i}^{(T)}(t)-V_{R}^{(S)}).$ (6)

Here, $g_{(R,i)}^{(T,S)}(t)$ and $V_{R}^{(S)}$ represent synaptic conductance
and synaptic reversal potential (determined by the type of the presynaptic
source $S$ population), respectively.

In the case of the $R$ (=AMPA and GABA)-mediated synaptic currents, we get the
synaptic conductance $g_{R,i}^{(T,S)}(t)$ from:

$g_{R,i}^{(T,S)}(t)=K_{R}^{(T,S)}\sum_{j=1}^{N_{S}}w_{ij}^{(T,S)}~{}s_{j}^{(T,S)}(t),$
(7)

where $K_{R}^{(T,S)}$ is the synaptic strength per synapse for the
$R$-mediated synaptic current from the $j$th presynaptic neuron in the source
$S$ population to the $i$th postsynaptic cell in the target $T$ population.
The inter-population synaptic connection from the source $S$ population (with
$N_{s}$ cells) to the target $T$ population is given by the connection weight
matrix $W^{(T,S)}$ ($=\\{w_{ij}^{(T,S)}\\}$) where $w_{ij}^{(T,S)}=1$ if the
$j$th cell in the source $S$ population is presynaptic to the $i$th cell in
the target $T$ population; otherwise $w_{ij}^{(T,S)}=0$. The fraction of open
ion channels at time $t$ is also represented by $s^{(T,S)}(t)$.

In contrast, in the NMDA-receptor case, some of the postsynaptic NMDA channels
are blocked by the positive magnesium ion ${\rm Mg}^{2+}$ NMDA . Hence, the
conductance in the case of NMDA receptor is given by Chavlis :

$g_{R,i}^{(T,S)}(t)={\widetilde{K}}_{R}^{(T,S)}~{}f(v^{(T)}(t))~{}\sum_{j=1}^{N_{S}}w_{ij}^{(T,S)}~{}s_{j}^{(T,S)}(t).$
(8)

Here, ${\widetilde{K}}_{R}^{(T,S)}$ is the synaptic strength per synapse, and
the fraction of NMDA channels that are not blocked by the ${\rm Mg}^{2+}$ ion
is given by a sigmoidal function $f(v^{(T)}(t))$:

$f(v^{(T)}(t))=\frac{1}{1+\eta\cdot[{\rm Mg}^{2+}]_{o}\cdot\exp(-\gamma\cdot
v^{(T)}(t))}.$ (9)

Here, $v^{(T)}(t)$ is the membrane potential of the target cell, $[{\rm
Mg}^{2+}]_{o}$ is the outer ${\rm Mg}^{2+}$ concentration, $\eta$ denotes the
sensitivity of ${\rm Mg}^{2+}$ unblock, $\gamma$ represents the steepness of
${\rm Mg}^{2+}$ unblock, and the values of parameters change depending on the
target cell Chavlis . For simplicity, some approximation to replace
$f(v^{(T)}(t))$ with $\langle f(v^{(T)}(t))\rangle$ [i.e., time-averaged value
of $f(v^{(T)}(t))$ in the range of $v^{(T)}(t)$ of the target cell] has been
done in SSR . Then, an effective synaptic strength $K_{\rm
NMDA}^{(T,S)}(={\widetilde{K}}_{\rm NMDA}^{(T,S)}\langle
f(v^{(T)}(t))\rangle$) was introduced by absorbing $\langle
f(v^{(T)}(t))\rangle$ into $K_{\rm NMDA}^{(T,S)}$. Thus, with the scaled-down
effective synaptic strength $K_{\rm NMDA}^{(T,S)}$ (containing the blockage
effect of the ${\rm Mg}^{2+}$ ion), the conductance $g$ for the NMDA receptor
may also be well approximated in the same form of conductance as the other
AMPA and GABA receptors in Eq. (7). Thus, we get all the effective synaptic
strengths $K_{\rm NMDA}^{(T,S)}$ from the synaptic strengths
$\widetilde{K}_{\rm NMDA}^{(T,S)}$ in Chavlis  by considering the average
blockage effect of the ${\rm Mg}^{2+}$ ion. Consequently, we can use the same
form of synaptic conductance of Eq. (7) in all the cases of $R=$ AMPA, NMDA,
and GABA.

The postsynaptic ion channels are opened through binding of neurotransmitters
(emitted from the source $S$ population) to receptors in the target $T$
population. The fraction of open ion channels at time $t$ is represented by
$s^{(T,S)}(t)$. The time course of $s_{j}^{(T,S)}(t)$ of the $j$th cell in the
source $S$ population is given by a sum of double exponential functions
$E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)})$:

$s_{j}^{(T,S)}(t)=\sum_{f=1}^{F_{j}^{(s)}}E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)}).$
(10)

Here, $t_{f}^{(j)}$ and $F_{j}^{(s)}$ are the $f$th spike time and the total
number of spikes of the $j$th cell in the source $S$ population, respectively,
and $\tau_{R,l}^{(T,S)}$ is the synaptic latency time constant for
$R$-mediated synaptic current. The double exponential-decay function
$E_{R}^{(T,S)}(t)$ (corresponding to contribution of a presynaptic spike
occurring at $t=0$ in the absence of synaptic latency) is given by:

$E_{R}^{(T,S)}(t)=\frac{1}{\tau_{R,d}^{(T,S)}-\tau_{R,r}^{(T,S)}}\left(e^{-t/\tau_{R,d}^{(T,S)}}-e^{-t/\tau_{R,r}^{(T,S)}}\right)\cdot\Theta(t).$
(11)

Here, $\Theta(t)$ is the Heaviside step function: $\Theta(t)=1$ for $t\geq 0$
and 0 for $t<0$, and $\tau_{R,r}^{(T,S)}$ and $\tau_{R,d}^{(T,S)}$ are
synaptic rising and decay time constants of the $R$-mediated synaptic current,
respectively.

In comparison with our prior DG networks WTA ; SSR ; PS , we include more
synaptic connections with a high degree of anatomical and physiological
realism BN1 ; BN2 , and incorporate the imGCs. Thus, a new feedforward
inhibition, mediated by the BCs, is provided to the mGCs, and there appear two
feedback loops of mGC-BC and mGC-HIPP, (projecting feedback inhibition to the
mGCs), the activities of which are controlled by the two control loops of MC-
BC and MC-HIPP (MCs: controllers).

Finally, we present the parameter values for the synaptic strength per synapse
$K_{R}^{(T,S)}$, the synaptic rising time constant $\tau_{R,r}^{(T,S)}$,
synaptic decay time constant $\tau_{R,d}^{(T,S)}$, synaptic latency time
constant $\tau_{R,l}^{(T,S)}$, and the synaptic reversal potential
$V_{R}^{(S)}$ for the synaptic currents into the GCs (i.e., both mGCs and
imGCs) and the BCs in the GL, in Tables 1 and 2, respectively, and for the
synaptic currents into the MCs and the HIPP cells in Table 3. These parameter
values are also based on the physiological properties of the relevant cells
Chavlis ; BN1 ; BN2 ; SynParm1 ; SynParm2 ; SynParm3 ; SynParm4 ; SynParm5 ;
SynParm6 ; SynParm7 ; SynParm8 .

All of our source codes for computational works were written in C programming
language. Numerical integration of the governing equation for the time-
evolution of states of individual spiking neurons is done by employing the
2nd-order Runge-Kutta method with the time step 0.1 msec.

##  III Effect of Immature Granule Cells Born via Adult Neurogenesis on
Pattern Separation

In this section, we study the effect of adult-born imGCs on pattern separation
in our spiking neural network, developed in Sec. II. Due to high excitability,
the imGCs become very active, while because of low excitatory innervation,
their activation degree is decreased. We investigate the effects of the two
competing properties of the imGCs on the activation degrees and the pattern
separation efficacy of the imGCs, the mGCs, and the whole GCs.

###  III.1 Characterization of Pattern Separation in The Presence of Only The
mGCs without The imGCs

In this subsection, we first consider the case of presence of only the mGCs
(without the imGCs) to present the methods characterizing the pattern
separation. As explained in the subsection II.1, the EC provides external
excitatory inputs to the mGCs via PPs [see Fig. 1(a)] (Myers1, ; Myers2, ;
Myers3, ; Scharfman, ; Chavlis, ; WTA, ; SSR, ). We characterize pattern
separation between the input patterns of the EC cells and the output patterns
of the mGCs via integration of the governing equations (1). In each
realization, we have a break stage ($0-300$ msec) (for which the network
reaches a stable state), and then a stimulus stage ($300-1,300$ msec) follows;
the stimulus period $T_{s}$ (for which network analysis is done) is 1,000
msec. During the stimulus stage, we get the output firings of the mGCs. For
characterization of pattern separation between the input and the output
patterns, 30 realizations are made.

The input patterns of the 400 EC cells and the output patterns of the 2,000
mGCs are given in terms of binary representations Myers1 ; Chavlis ; active
and silent cells are denoted by 1 and 0, respectively. Here, active cells
exhibit at least one spike during the stimulus stage. In each realization, we
first make a random choice of an input pattern $A^{(in)}$ for the EC cells,
and then construct another input patterns $B_{i}^{(in)}$ ($i=1,\dots,9$) from
the base input pattern $A^{(in)}$ with the overlap percentage $P_{OL}$
$=90~{}$%$,\dots,$ and $10$ $\%$, respectively, as follows Myers1 ; Chavlis .
Among the active EC cells in the pattern $A^{(in)}$, we randomly choose active
cells for the pattern $B^{(in)}$ with the probability $P_{OL}~{}\%$ (e.g., in
the case of $P_{OL}=60~{}\%$, we randomly choose 24 active EC cells among the
40 active EC cells in the base pattern $A^{(in)}$). The remaining active EC
cells in the pattern $B^{(in)}$ are randomly chosen in the subgroup of silent
EC cells in the pattern $A^{(in)}$.

We characterize pattern separation between the input and the output patterns
by changing the overlap percentage $P_{OL}$. For a pair of input ($l=in)$ or
output ($l=out$) patterns, $A^{(l)}$ and $B^{(l)}$, their pattern distance
$D_{p}^{(l)}$ is given by Chavlis ; PS :

$D_{p}^{(l)}={\frac{O^{(l)}}{D_{a}^{(l)}}}.$ (12)

Here, $D_{a}^{(l)}$ is the average activation degree of the two patterns
$A^{(l)}$ and $B^{(l)}$:

$D_{a}^{(l)}={\frac{(D_{a}^{(A^{(l)})}+D_{a}^{(B^{(l)})})}{2}},$ (13)

and $O^{(l)}$ is the orthogonalization degree between $A^{(l)}$ and $B^{(l)}$,
denoting their “dissimilarity” degree. Then, as the average activation degree
is lower and the orthogonalization degree is higher, the pattern distance
between the two patterns $A^{(l)}$ and $B^{(l)}$ increases.

Figure 3: Characterization of pattern separation between the input and the
output patterns in the presence of only the mGCs without imGCs. (a1) Raster
plots of spikes of ECs for the input patterns $A^{(in)}$ and $B^{(in)}$ in the
case of overlap percentage $P_{OL}=60\%$. (a2) Raster plots of spikes of GCs
for the output patterns $A^{(out)}$ and $B^{(out)}$. (b) Plots of average
activation degree $D_{a}^{(l)}$ versus $P_{OL}$ for the input ($l=in$; red
circle) and the output ($l=out$, blue cross) patterns. Plots of the diagonal
elements (0, 0) and (1, 1) and the anti-diagonal elements (1, 0) and (0, 1)
for the spiking activity (1: active; 0: silent) in the pair of (c1) input
($l=in$) and (c2) output ($l=out$) patterns $A^{(l)}$ and $B^{(l)}$ for
$P_{OL}=60\%$; sizes of solid circles, located at (0,0), (1,1), (1,0), and
(0,1), are given by the integer obtained by rounding off the number of
$5~{}\log_{10}(n_{p})$ ($n_{p}$: number of data at each location), and a
dashed linear least-squares fitted line is also given. Plots of (d) average
pattern correlation degree $C^{(l)}$, (e) average orthogonalization degree
$O^{(l)}$, (f) pattern distance $D_{p}^{(l)}$, and (g) pattern separation
degrees ${\cal S}_{d}$ versus $P_{OL}$ in the case of the input ($l=in$; red
circle) and the output ($l=out$, blue cross) patterns.

Let $\\{a^{(l)}_{i}\\}$ and $\\{b^{(l)}_{i}\\}$ ($i=1,\dots,N_{l}$) be the
binary representations [1 (0) for the active (silent) cell] of the two
patterns $A^{(l)}$ and $B^{(l)}$ ($l=in$ or $out$), respectively;
$N_{in}=N_{\rm EC}=400$ and $N_{out}=N_{\rm GC}=2,000$. Then, the Pearson’s
correlation coefficient $\rho^{(l)}$ between the two patterns $A^{(l)}$ and
$B^{(l)}$ is given by

$\rho^{(l)}=\frac{\sum_{i=1}^{N_{l}}\Delta a^{(l)}_{i}\cdot\Delta
b^{(l)}_{i}}{\sqrt{\sum_{i=1}^{N_{l}}\Delta{a^{(l)}_{i}}^{2}}\sqrt{\sum_{i=1}^{N_{l}}\Delta{b^{(l)}_{i}}^{2}}}.$
(14)

Here, $\Delta a^{(l)}_{i}=a^{(l)}_{i}-\langle{a^{(l)}_{i}}\rangle$, $\Delta
b^{(l)}_{i}=b^{(l)}_{i}-\langle{b^{(l)}_{i}}\rangle$, and
$\langle\cdots\rangle$ represents population average over all cells; the range
of $\rho^{(l)}$ is [-1, 1]. Then, the pattern correlation degree $C^{(l)}$,
representing the “similarity” degree between the two patterns, is given just
by their Pearson’s correlation coefficient $\rho^{(l)}$:

$C^{(l)}=\rho^{(l)}.$ (15)

Then, the orthogonalization degree $O^{(l)},$ denoting the dissimilarity
degree between the two patterns, is given by PS :

$O^{(l)}={\frac{(1-\rho^{(l)})}{2}},$ (16)

where the range of $O^{(l)}$ is [0, 1].

With $D_{a}^{(l)}$ and $O^{(l)}$, we can obtain the pattern distances of Eq.
(12), $D_{p}^{(in)}$ and $D_{p}^{(out)}$, for the input and the output pattern
pairs, respectively. Then, the pattern separation degree ${\cal S}_{d},$
representing the pattern separation efficacy, is given by the ratio of
$D_{p}^{(out)}$ to $D_{p}^{(in)}$:

${\cal S}_{d}={\frac{D_{p}^{(out)}}{D_{p}^{(in)}}}.$ (17)

If ${\cal S}_{d}>1$, the output pattern pair of the mGCs is more dissimilar
than the input pattern pair of the EC cells, which results in occurrence of
pattern separation. Otherwise (i.e., ${\cal S}_{d}<1$), no pattern separation
occurs; instead, pattern “convergence” (i.e., $D_{p}^{(out)}<D_{p}^{(in)}$)
takes place.

As a sample example, we consider the case of $P_{OL}=60~{}\%$. Figure 3(a1)
shows the raster plots of spikes of 400 EC cells (i.e. a collection of spike
trains of individual EC cells) for the input patterns $A^{(in)}$ and
$B^{(in)}$ for $P_{OL}=60~{}\%$. In this case, the activation degree
$D_{a}^{(in)}$ is chosen as 10 $\%$, independently of the input patterns.
Figure 3(a2) shows the raster plots of spikes of 2,000 mGCs for the output
patterns $A^{(out)}$ and $B^{(out)}$. As shown well in the raster plots of
spikes, the mGCs show sparser firings than the EC cells. In this case, the
average activation degree of Eq. (13), $D_{a}^{(out)}$, is 6 $\%$ (which is
obtained via 30 realizations). Figure 3(b) shows the plot of the average
activation degree $D_{a}^{(l)}$ versus the overlap percentage $P_{OL}$; red
circles represent the case of input patterns ($l=in$) and blue crosses denote
the case of output patterns ($l=out$). We note that $D_{a}^{(out)}=0.06$
(i.e., 6 $\%$), independently of $P_{OL}$. Then, the sparsity ratio, ${\cal
R}_{s}$ ($=D_{a}^{(in)}/D_{a}^{(out)}$), becomes 1.667; the firing activity in
the output patterns are 1.667 times as sparse as that in the the input
patterns.

Figures 3(c1) and (c2) show plots of the diagonal elements (0, 0) and (1, 1)
and the anti-diagonal elements (1, 0) and (0, 1) for the spiking activity (1:
active; 0: silent) in the pair of input ($l=in$) and output ($l=out$) patterns
$A^{(l)}$ and $B^{(l)}$ for $P_{OL}=60\%$, respectively. In each plot, the
sizes of solid circles, located at (0,0), (1,1), (1,0), and (0,1), are given
by the integer obtained by rounding off the number of $5~{}\log_{10}(n_{p})$
($n_{p}$: number of data at each location), and a dashed fitted line is also
given. In this case, the Pearson’s correlation coefficients of Eq. (14)
(obtained via 30 realizations) for the pairs of the input and the output
patterns are $\rho^{(in)}=0.5556$ and $\rho^{(out)}=0.3550$, which correspond
to the slopes of the dashed fitted lines. Then, from Eqs. (15) and (16), we
obtain the average pattern correlation degree $C^{(l)}$ and the average
orthogonalization degrees $O^{(l)}$ for the pairs of the input and the output
patterns; $C^{(in)}=0.5556,$ $C^{(out)}=0.3550$, $O^{(in)}=0.2222$ and
$O^{(out)}=0.3225$.

Figures 3(d) and 3(e) show plots of the average pattern correlation degree
$C^{(l)}$ and the average orthogonalization degree $O^{(l)}$ versus $P_{OL}$
in the case of the input (red circle) and the output (blue cross) patterns,
respectively. Obviously, $C^{(l)}$ and $O^{(l)}$ show oppositely-changing
tendencies. Hence, it is enough to discuss only the change in $O^{(l)}$. In
the case of the pairs of the input patterns, with decreasing $P_{OL}$ from 90
$\%$ to 10 $\%$, $O^{(in)}$ increases linearly from 0.0556 to 0.5. On the
other hand, in the case of the pairs of the output patterns, $O^{(out)}$
begins from a much larger value (0.2543), but slowly increases to 0.3507 for
$P_{OL}=10$ $\%$ (which is lower than $O^{(in)}$). Thus, the two lines of
$O^{(in)}$ and $O^{(out)}$ cross for $P_{OL}\simeq 40$ $\%$. Hence, for
$P_{OL}>40~{}\%$, $O^{(out)}$ is larger than $O^{(in)}$ (i.e., the pair of
output patterns is more dissimilar than the pair of input patterns). In
contrast, for $P_{OL}<40~{}\%$, $O^{(out)}$ is less than $O^{(in)}$ (i.e., the
pair of output patterns becomes less dissimilar than the pair of input
patterns).

With the average activation degrees $D_{a}^{(l)}$ and the average
orthogonalization degrees $O^{(l)}$, we can obtain the pattern distances
$D_{p}^{(l)}$ of Eq. (12) for the pairs of input and output patterns. Figure
3(f) shows plots of the pattern distance $D_{p}^{(l)}$ versus $P_{OL}$ in the
case of the input (red circle) and the output (blue cross) patterns. We note
that, for all values of $P_{OL}$, $D_{p}^{(out)}>D_{p}^{(in)}$ (i.e., the
pattern distance for the pair of output patterns is larger than that for the
pair of input patterns). However, with decreasing the overlap percentage
$P_{OL}$, the difference between $D_{p}^{(out)}$ and $D_{p}^{(in)}$ is found
to decrease.

Finally, we obtain the pattern separation degree ${\cal S}_{d}$ of Eq. (17)
via the ratio of $D_{p}^{(out)}$ to $D_{p}^{(in)}$. Figure 3(g) shows plots of
the pattern separation degree ${\cal S}_{d}$ (representing the pattern
separation efficacy) versus $P_{OL}$. As $P_{OL}$ is decreased from 90 $\%$ to
10 $\%$, ${\cal S}_{d}$ is found to decrease from 7.6273 to 1.1691. Hence, for
all values of $P_{OL}$, pattern separation occurs because ${\cal S}_{d}>1$.
However, the smaller $P_{OL}$ is, the lower ${\cal S}_{d}$ becomes.

###  III.2 Effect of The Adult-Born imGCs on Pattern Separation

In this subsection, we consider a population, composed of imGCs and mGCs; the
fraction of the imGCs in the whole population is 10 $\%$. As shown in Fig. 2,
as a result of increased leakage reversal potential $V_{L}$, the imGC has
lower firing threshold than the mGC (i.e., high excitability), which results
in high activation of the imGCs NG7 ; NG8 ; NG9 ; NG10 . We also note that,
the imGC has low excitatory innervation, counteracting the high excitability.
In the case of the mGCs, the connection probability $p_{c}$ from the EC cells
and the MCs to the mGCs is 20 $\%$, while in the case of the imGCs, $p_{c}$ is
decreased to $20~{}x~{}\%$ [$x$ (synaptic connectivity fraction); $0\leq x\leq
1$]. Due to low excitatory drive from the EC cells and the MCs, the activation
degree of the imGCs becomes reduced. With decreasing $x$ from 1 to 0, we
investigate the effect of high excitability and low excitatory innervation of
the imGCs on the pattern separation efficacy.

For a given $x$, we consider 9 pairs of input patterns
$(A^{(in)},B_{i}^{(in)})$ $(i=1,\dots,9$) with the overlap percentage
$P_{OL}=90~{}\%,\dots,$ and $10~{}\%$, respectively. All quantities for the
input patterns are independent of $x$. The activation degree $D_{a}^{(in)}$ is
0.1 (10 $\%$), independently of the pairs. Next, we get the average Pearson’s
correlation coefficient $\rho^{(in)}$ between the two input patterns in the
following way. We first obtain the realization-averaged Pearson’s correlation
coefficients $\\{\langle\rho^{(in)}(i)\rangle_{r}\\}$ ($i=1,\dots,9$
corresponds to $P_{OL}=90~{}\%,\dots,10~{}\%$, respectively) via 30
realizations; $\langle\cdots\rangle_{r}$ represents the average over 30
realizations. With decreasing $P_{OL}$ from 90 $\%$ to 10 $\%$,
$\langle\rho^{(in)}(i)\rangle_{r}$ decreases from 0.8889 to 0.0, respectively.
As a representative value, we get the average Pearson’s correlation
coefficient $\rho^{(in)}~{}(=0.4444),$ corresponding to the mean of
$\\{\langle\rho^{(in)}(i)\rangle_{r}\\}$ over all the 9 pairs. Then, from Eqs.
(15) and (16), we get the average pattern correlation degree $C^{(in)}$ (=
0.4444) and the average orthogonalization degree $O^{(in)}$ (= 0.2778). In
this way, $\rho^{(in)},$ $C^{(in)}$, and $O^{(in)}$ are obtained via double
averaging (i.e., averaging over 30 realizations and 9 pairs). Then, the
pattern distance $D_{p}^{(in)}$ of Eq. (12) between the two input patterns
(given by the ratio of the average orthogonalization degree to the average
activation degree) becomes 2.778.

Figure 4: Effect of adult-born immature GCs (imGCs) on the pattern separation.
(a) Plots of the average activation degree $D_{a}^{(X)}$ versus $x$ (synaptic
connectivity fraction). For clear presentation, we choose two different scales
for the vertical axis around $D_{a}^{(X)}=10$. (b) Plots of the average
pattern correlation degree $C^{(X)}$ versus $x$. (c) Plots of the average
orthogonalization degree $O^{(X)}$ versus $x$. (d) Plots of the pattern
distance $D_{p}^{(X)}$ versus $x$. For clear presentation, we choose two
different scales for the vertical axis around $D_{p}^{(X)}=5$. (e) Plots of
the pattern separation degree ${\cal S}_{d}^{(X)}$ versus $x$. For clear
presentation, we choose two different scales for the vertical axis around
${\cal S}_{d}^{(X)}=2$. In (a)-(e), imGCs ($X=im$), mGCs ($X=m$), and whole
GCs ($X=w$) are denoted by blue solid circles, red open circles, and green
crosses, respectively. Horizontal dashed lines in (a)-(e) represent
$D_{a}^{(out)}$ (= 6 $\%$), $C^{(out)}$ (= 0.3582), $O^{(out)}$ (= 0.3209),
$D_{p}^{(out)}$ (= 5.3483), and $S_{d}$ (=1.9252) in the presence of only mGCs
(without imGCs), respectively. (f) Plots of the pattern integration degree
${\cal I}_{d}$ of the imGCs versus $x$.

As in the above case of input patterns, through double averaging over 30
realizations and 9 pairs, we get the average activation degrees $D_{a}^{(X)},$
the average Pearson’s correlation coefficient $\rho^{(X)},$ the average
pattern correlation degree $C^{(X)}$, and the average orthogonalization
degrees ${\cal O}^{(X)}$ in each subpopulation of the imGCs ($X=im$) and the
mGCs ($X=m$) and in the whole population ($X=w$). Figures 4(a), 4(b), and 4(c)
show plots of $D_{a}^{(X)}$, $C^{(X)}$, and ${\cal O}^{(\rm X)}$ versus $x$
[$X=im$ (blue solid circles), $X=m$ (red open circles), and $X=w$ (green
crosses)], respectively. Then, we get the pattern distances $D_{p}^{(\rm X)}$
of Eq. (12) (given by the ratio of ${\cal O}^{(X)}$ to $D_{a}^{(X)}$), which
is shown in Fig. 4(d). Finally, we obtain the pattern separation degree ${\cal
S}_{d}^{(X)}$ of Eq. (17) via the ratio of $D_{p}^{(\rm X)}$ to
$D_{p}^{(in)}$. Figure 4(e) shows plots of ${\cal S}_{d}^{(X)}$ versus $x$. As
reference lines, horizontal dashed lines, representing $D_{a}^{(out)}$ (= 6
$\%$), $C^{(out)}$ (= 0.3582), $O^{(out)}$ (= 0.3209), $D_{p}^{(out)}$ (=
5.3483), and $S_{d}$ (=1.9252) in the presence of only the mGC (without the
imGCs) are given in Figs. 4(a)-4(e), respectively; these values are obtained
via averaging over 9 pairs in Fig. 3.

We first consider the case of $x=1$ (where the connection probability $p_{c}$
from the EC cells and the MCs to the imGCs and the mGCs are the same, 20
$\%$), and discuss the effect of adult-born imGCs with high excitability on
pattern separation NG7 ; NG8 ; NG9 ; NG10 . The imGCs exhibit high activation
due to lower firing threshold [i.e., their average activation degree
$D_{a}^{(\rm im)}$ (= 45 $\%$) becomes very high]. As a result, in the
subpopulation of the imGCs, output patterns become highly overlapped (i.e,
their average Pearson’s correlation coefficient is very high), which leads to
very high average pattern correlation degree $C^{(im)}~{}(=~{}0.8692)$ and
very low average orthogonalization degree ${\cal O}^{(\rm
im)}~{}(=~{}0.0654)$. Then, their pattern distance $D_{p}^{(\rm im)}$ (=
0.145), given by the ratio of ${\cal O}^{(\rm im)}$ to $D_{a}^{(\rm im)}$,
also becomes very low. Consequently, the pattern separation degree ${\cal
S}_{d}^{(\rm im)}$, given by the ratio of $D_{p}^{(\rm im)}$ to
$D_{p}^{(in)}$, is 0.052. Since ${\cal S}_{d}^{(\rm im)}<1$, no pattern
separation occurs, due to their high excitability. On the other hand, the
efficacy of pattern integration (i.e., making association between events) is
very high due to high pattern correlation degree $C^{(im)}$. We introduce the
pattern integration degree ${\cal I}_{d}$ of the imGCs, given by the ratio of
the average pattern correlation degree $C^{(im)}$ to the average pattern
correlation degree $C^{(in)}$ for the input patterns:

${\cal I}_{d}=\frac{C^{(im)}}{C^{(in)}},$ (18)

which is in contrast to the pattern separation degree ${\cal S}_{d}$ of Eq.
(17). For $x=1$ the pattern integration degree of the imGCs is high (i.e.,
${\cal I}_{d}=1.9559$). Figure 4(f) shows plots of ${\cal I}_{d}$ versus $x$
for the imGCs. With decreasing $x$ from 1 to 0, ${\cal I}_{d}$ is increased
from 1.9559 to 2.2502, because $C^{(im)}$ increases from 0.8692 to 1. In the
whole range of $0\leq x\leq 1$, the imGCs are good pattern integrators with
${\cal I}_{d}>1$.

In contrast, for $x=1$ the mGCs exhibit very sparse firing activity (i.e.,
their average activation degree $D_{a}^{(m)}$ (= 1.1 $\%$) of the mGCs becomes
very low) due to strong feedback inhibition from the BCs and the HIPP cells
(caused by the high activation of the imGCs). As a result of high sparsity,
the average Pearson’s correlation coefficient between the output-pattern pairs
becomes very low, which leads to high average orthogonalization degree ${\cal
O}^{(m)}$ (= 0.4016). Then, their pattern distance $P_{d}^{(m)}$ (=36.509)
becomes very high. Accordingly, the pattern separation degree ${\cal
S}_{d}^{(m)}$ is 13.142. Thus, the pattern separation efficacy of the mGCs
becomes very high (i.e., the mGCs become good pattern separators), due to high
sparsity.

In the above way, the whole population of all the GCs for $x=1$ is a
heterogeneous one, composed of a (major) subpopulation of sparsely active mGCs
(good pattern separators) with very low $D_{a}^{(m)}$ and a (minor)
subpopulation of highly active imGCs (good pattern integrators) with very high
$D_{a}^{(im)}$; most of active cells congregate in the subpopulation of the
imGCs. In the whole heterogeneous population, the overall activation degree
$D_{a}^{(w)}$ of all the GCs is 0.055 (5.5 $\%$) which is a little less than
$D_{a}^{(out)}$ (= 6 $\%$) in the presence of only mGCs (without imGCs).
Although $D_{a}^{(w)}$ is a little decreased (i.e., sparser firing activity),
the average Pearson’s correlation coefficient between the output-pattern pairs
becomes high, due to presence of strongly-correlated imGCs, which leads to low
orthogonalization degree ${\cal O}^{(w)}$ (=0.1004); ${\cal O}^{(w)}$ is also
much less than the average orthogonalization degree ${\cal O}^{(out)}$ (=
0.3209) in the presence of only mGCs. Then, we get the pattern distance
$D_{p}^{(w)}$ (= 1.825) which is also less than $D_{p}^{(in)}$ (= 2.778).
Consequently, the pattern separation degree ${\cal S}_{d}^{(w)}$ becomes
0.657. Since ${\cal S}_{d}^{(w)}<1$, no pattern separation occurs in the whole
heterogeneous population for $x=1$, due to heterogeneous sparsity, in contrast
to the usual intuitive thought that sparsity could improve pattern separation
efficacy; such intuitive thought might be applied only to homogeneous
sparsity. Instead of pattern separation, pattern “convergence” with
$D_{p}^{(w)}<D_{p}^{(in)}$ occurs for $x=1$ in the whole heterogeneous
population of all the GCs.

Next, with decreasing $x$ from 1, we consider the effect of low excitatory
innervation for the imGCs, counteracting the effect of high excitability NG11
. In the case of mGCs, they receive excitatory inputs from the EC via PPs and
from the hilar MCs with the connection probability $p_{c}$ (= 20 $\%$). On the
other hand, the imGCs receive low excitatory drive via the PPs and from the
MCs with lower connection probability $p_{c}~{}(=20~{}x~{}\%)$ ($x:$ synaptic
connectivity fraction; $0\leq x\leq 1$). As $x$ is decreased from 1,
$D_{a}^{(im)}$ of the imGCs decreases so rapidly, and their effect becomes
weaker. Then, the feedback inhibition to the mGCs is also decreased, and hence
$D_{a}^{(m)}$ of the mGCs becomes increased. Accordingly, $D_{a}^{(w)}$ of the
whole GCs also increases. In the whole range of $0\leq x\leq 1$, the average
pattern correlation degree $C^{(im)}$ of the imGCs are very high, and hence
they become good pattern integrators with the pattern integration degree
${\cal I}_{d}>1$ [see Fig. 4(f)]. On the other hand, due to increase in
$D^{(m)}$, the pattern separation efficacy of the mGCs decreases from the high
value (${\cal S}_{d}^{(m)}=$ 13.142) for $x=1$ to a limit value (${\cal
S}_{d}^{(m)}=$ 1.495) for $x=0$. In the whole population of all the GCs, due
to decreased effect of the imGCs, when $x$ decreases through a threshold
$x^{*}~{}(=0.92)$, pattern separation (with ${\cal S}_{d}^{(w)}>1$) starts,
and then the overall pattern separation degree ${\cal S}_{d}^{(w)}$ increases
and approaches a limit value (${\cal S}_{d}^{(w)}=$ 1.577) for $x=0$ which is
a little larger than the limit value of the mGCs. In the limit case of $x=0$
where all imGCs are silent, the limit pattern separation degree (${\cal
S}_{d}^{(w)}~{}=~{}1.577$) in the whole population is lower than that (${\cal
S}_{d}~{}=~{}1.9252$) in the presence of only mGCs (without imGCs), mainly
because $D_{a}^{(w)}$ (= 7.3 $\%$) is larger than $D_{a}^{(out)}$ (= 6 $\%$)
in the absence of imGCs. In this way, due to heterogeneity caused by the imGCs
(performing pattern integration), the overall efficacy of pattern separation
in the whole heterogeneous population of all the GCs becomes deteriorated.

##  IV Summary and Discussion

We investigated the effect of the adult-born imGCs on the pattern separation
in a spiking neural network, composed of both mGCs (born during development)
and imGCs. In contrast to the mGCs, the imGCs exhibit two competing distinct
properties of high excitability (causing high activation) and low excitatory
innervation (reducing activation degree). We first considered the effect of
high excitability. The activation degree $D_{a}^{(im)}$ (= 45 $\%$) of the
imGCs was found to be very high due to lower firing threshold. In this case,
the pattern correlation degree $C^{(im)}$ (= 0.8692) also became high, because
the outputs were highly overlapped. Consequently, the imGCs were found to
become good pattern integrators (i.e., making association between events) with
the pattern integration degree ${\cal I}_{d}~{}(=1.9559)$. In contrast, the
activation degree $D_{a}^{(m)}$ (= 1.1 $\%$) of the mGCs was found to be very
low due to strong feedback inhibition from the inhibitory BCs and HIPP cells
(caused by high activation of the imGCs). Due to high sparsity, the efficacy
of pattern separation of the mGCs became very high. Thus, the mGCs were found
to become good pattern separators with the pattern separation degree ${\cal
S}_{d}~{}(=13.142)$.

In the above way, the whole population of all the GCs became a heterogeneous
one, composed of a (major) subpopulation of mGCs (good pattern separators)
with very low $D_{a}^{(m)}$ and a (minor) subpopulation of imGCs (good pattern
integrators) with very high $D_{a}^{(im)}$; most of active cells congregated
in the subpopulation of imGCs. In the whole heterogeneous population, the
overall activation degree $D_{a}^{(w)}$ (= 5.5 $\%$) of all the GCs was found
to be a little less than $D_{a}^{(out)}$ (= 6 $\%$) in the presence of only
mGCs (without imGCs). However, in spite of sparser firing activity, no pattern
separation occurred, because of heterogeneous sparsity, in contrast to the
usual intuitive thought that sparsity could improve pattern separation
efficacy; such intuitive thought might be applied only to the case of
homogeneous sparsity. Instead, pattern convergence with ${\cal
S}_{d}~{}(=0.657)$ was found to occur because $D_{p}^{(w)}<D_{p}^{(in)}$.

Next, we studied the effect of low excitatory innervation of the imGCs,
counteracting the effect of their high excitability; the connection
probability $p_{c}$ from the EC cells and the MCs to the imGCs is
$20~{}x~{}\%$ [$x$ (synaptic connectivity fraction); $0\leq x\leq 1$]. As $x$
was decreased from 1 to 0, $D_{a}^{(im)}$ of the imGCs was found to decrease
so rapidly, and hence their effect became weaker. In contrast to the case of
the imGCs, $D_{a}^{(m)}$ of the mGCs became increased due to decrease in the
feedback inhibition from the BCs and the HIPP cells. Consequently,
$D_{a}^{(w)}$ of the whole GCs also increased. In the whole range of $0\leq
x\leq 1$, the imGCs were found to have high pattern correlation degree
($0.8692\leq C^{(im)}\leq 1.0$), and hence they became good pattern
integrators with the pattern integration degree ($1.9559\leq{\cal I}_{d}\leq
2.2502$).

On the other hand, due to increase in $D_{a}^{(m)}$, the pattern separation
degree ${\cal S}_{d}^{(m)}$ of the mGCs was found to decrease from the high
value (13.142) for $x=1$ to a limit value (1.495) at $x=0$. Thus, in the whole
range of $0\leq x\leq 1$, the mGCs performed pattern separation with ${\cal
S}_{d}^{(m)}>1$. In the whole population of all the GCs, when $x$ decreases
through a threshold $x^{*}~{}(=0.92)$, pattern separation (with ${\cal
S}_{d}^{(w)}>1$) was found to start, and then the overall pattern separation
degree ${\cal S}_{d}^{(w)}$ increased and approached a limit (1.577) which was
a little larger than the limit (1.495) of the mGCs. However, ${\cal
S}_{d}^{(w)}$ was found to be less than ${\cal S}_{d}$ (= 1.9252) in the
presence of only mGCs (without imGCs). Thus, due to heterogeneity caused by
the imGCs, the pattern separation efficacy in the heterogeneous population
became deteriorated, in comparison with that in the presence of only mGCs.

Figure 5: Pattern integration in the presence of only imGCs. (a) Plots of
pattern correlation degrees $C^{(l)}$ versus $P_{OL}$; $l=in$ (red) and
$l=out$ (blue). (b) Plots of integration degree ${\cal I}_{d}$ versus
$P_{OL}$; for $P_{OL}=10~{}\%$ ${\cal I}_{d}$ becomes infinity (not shown)
because $C^{(in)}=0$. In (a) and (b), the solid, dashed , and dotted lines
correspond to the cases of $x=1$, 0.6, and 0.1, respectively.

In Fig. 3, we characterized pattern separation by varying the overlap
percentage $P_{OL}$ in the homogeneous population of only the mGCs (without
the imGCs). Thus, the mGCs were found to perform pattern separation. It was
also found that, the smaller $P_{OL}$ is, the lower the pattern separation
degree ${\cal S}_{d}$ becomes (i.e., the pattern separation efficacy becomes
better for similar input patterns, while in the case of dissimilar input
patterns, the pattern separation efficacy becomes worse). For comparison, we
consider another homogeneous population of only the imGCs (without the mGCs)
to more clearly understand the role of the imGCs. Figure 5(a) shows the plots
of the pattern correlation degree ${\cal C}^{(l)}$ versus $P_{OL}$ for the
pair of input patterns [$l=in$ (red)] and output patterns [$l=out$ (blue)]; in
the case of $l=out$, the solid, dashed, and dotted lines correspond to the
cases of $x=1$, 0.6, and 0.1, respectively. Then, the pattern integration
degree ${\cal I}_{d}$ is given by the ratio of ${\cal C}^{(out)}$ to ${\cal
C}^{(in)}$. Figure 5(b) shows ${\cal I}_{d}$ versus $P_{OL}$. [We note that in
the case of $P_{OL}=10~{}\%$, ${\cal C}^{(in)}=0,$ and hence ${\cal I}_{d}$
becomes infinity (not shown).] We note that, as $P_{OL}$ is decreased, ${\cal
I}_{d}$ becomes increased, in contrast to the case of ${\cal S}_{d}$ in Fig.
3. Thus, the pattern integration efficacy becomes better for dissimilar input
patterns. Also, as $x$ is decreased from 1, the effect of imGCs becomes
weaker, leading to decrease in ${\cal I}_{d}$.

As discussed above, the pattern separation efficacy in the heterogeneous
population of all the GCs (composed of both mGCs and imGCs) was found to get
deteriorated, due to presence of the imGCs (good pattern integrators).
However, we note that the pattern separation may not always be a strict
requirement for accurate neural encoding. In the homogeneous population of
only the mGCs (without the imGCs), memory storage capacity (representing the
number of distinct patterns which may be stored and accurately recalled) could
be increased with pattern separation efficacy (facilitating the pattern
storage and retrieval) Myers1 . In contrast, in a heterogeneous population of
mGCs (pattern separators) and imGCs (pattern integrators), the memory storage
capacity might be optimally maximized via mixed encoding through pattern
separation on similar input patterns and pattern integration on very
dissimilar input patterns NG9 ; Hetero . Thus, through mixed encoding, memory
resolution (corresponding to the extent of information incorporated into
memories) could be increased, which would result in reduction in memory
interference. In this way, the imGCs (good integrators for very dissimilar
input patterns) could make contribution to increase in memory storage
capacity, although they have tendency to reduce the pattern separation
efficacy. Through cooperation of pattern separation for similar input patterns
and pattern integration for very dissimilar input patterns, the heterogeneous
population of the mGCs and the imGCs might achieve superior pattern encoding
than the homogeneous population of only the mGCs (performing purely sparse
coding). This speculation on increase in memory resolution via mixed encoding
(through cooperation of pattern separation and pattern integration) must be
examined in future works.

Finally, we discuss future works. During the pattern separation, sparsely
synchronized rhythms appear in the whole population of all the GCs and in each
subpopulation of the imGCs and the mGCs. Hence, it would be worthwhile to
investigate their population and individual firing behaviors and to discuss
their quantitative relationship with the pattern separation efficacy. As in
SSR ; PS , population and individual firing behaviors in the sparsely
synchronized rhythms in the subpopulations of the imGCs ($X=im$), the mGCs
($X=m$) and in the whole population ($X=w$) may be characterized in terms of
the amplitude measure ${\cal M}_{a}^{(X)}$ (representing the popupation
synchronization degree) AM  and the coefficient of variation $CV^{(X)}$
(characterizing the irregularity degree of individual single-cell discharges)
CV , respectively. Then, we could investigate the quantitative relationship
between ${\cal M}_{a}^{(X)}$ and $CV^{(X)}$ of the sparsely synchronized
rhythms and the pattern separation degree ${\cal S}_{d}^{(X)}$ (representing
the pattern separation efficacy). Next, we also note that the pyramidal cells
in the CA3 provide backprojections to the GCs via polysynaptic connections
(Myers2, ; Myers3, ; Scharfman, ). For example, the pyramidal cells send
disynaptic inhibition to the mGCs, mediated by the BCs and the HIPP cells in
the DG, and they provide trisynaptic inputs to the mGCs, mediated by the MCs
(pyramidal cells $\rightarrow$ MC $\rightarrow$ BC or HIPP $\rightarrow$ mGC).
These inhibitory backprojections may decrease the activation degree of the
mGCs, leading to improvement of pattern separation in the subpopulation of the
mGCs. Hence, in future work, it would be meaningful to take into consideration
the backprojection for the study of pattern separation in the combined DG-CA3
network. Moreover, in the DG-CA3 network, we could examine the memory storage
capacity by getting correct response percentage for a partial or noisy version
of cue input patterns in the homogeneous population of only the mGCs and in a
heterogeneous population of the mGCs and the imGCs Myers2 . Then, we could
determine which one of the purely sparse encoding (homogeneous case) and the
mixed encoding (heterogeneous case) would be superior.

## Acknowledgments

This research was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education (Grant No. 20162007688).

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