# Effect of Adult Neurogenesis on Sparsely Synchronized Rhythms of The Granule
Cells 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

We are concerned about the main encoding granule cells (GCs) in the
hippocampal dentate gyrus (DG). Young immature GCs (imGCs) appear through
adult neurogenesis. In comparison to the mature GCs (mGCs) (born during
development), the imGCs show high activation due to lower firing threshold. On
the other hand, they receive low excitatory drive from the entorhinal cortex
via perforant paths and from the hilar mossy cells with lower connection
probability $p_{c}~{}(=20~{}x~{}\%)$ ($x:$ synaptic connectivity fraction;
$0\leq x\leq 1$) than the mGCs with the connection probability
$p_{c}~{}(=20~{}\%)$. Thus, the effect of low excitatory innervation (reducing
activation degree) for the imGCs counteracts the effect of their high
excitability. We consider a spiking neural network for the DG, incorporating
both the mGCs and the imGCs. With decreasing $x$ from 1 to 0, we investigate
the effect of young adult-born imGCs on the sparsely synchronized rhythms
(SSRs) of the GCs (mGCs, imGC, and whole GCs). For each $x$, population and
individual firing behaviors in the SSRs are characterized in terms of the
amplitude measure ${\cal M}_{a}^{(X)}$ ($X=m,~{}im,~{}w$ for the mGCs, the
imGCs, and the whole GCs, respectively) (representing the population
synchronization degree) and the random phase-locking degree ${\cal
L}_{d}^{(X)}$ (characterizing the regularity of individual single-cell
discharges), respectively. We also note that, for $0\leq x\leq 1,$ the mGCs
and the imGCs exhibit pattern separation (i.e., a process of transforming
similar input patterns into less similar output patterns) and pattern
integration (making association between patterns), respectively. Quantitative
relationship between SSRs and pattern separation and integration is also
discussed.

Hippocampal dentate gyrus, Adult neurogenesis, Immature granule cells (GCs),
Mature GCs, High excitability, Low excitatory innervation, Sparsely
synchronized rhythm, Pattern separation, Pattern integration

###### pacs:

87.19.lj, 87.19.lm, 87.19.lv

##  I Introduction

The hippocampus, consisting 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 . Here, we are concerned
about the DG which is the gateway to the hippocampus. Its excitatory granule
cells (GCs) 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 send the pattern-separated outputs to the
pyramidal cells in the CA3 via the mossy fibers (MFs) 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 ; PS . Then, the sparse, but strong
MFs play a role of “teaching inputs,” causing synaptic plasticity between the
pyramidal cells in the CA3. Thus, a new pattern may be stored in modified
synapses. In this way, pattern separation (transforming a set of input
patterns into sparser and orthogonalized patterns) in the DG may facilitate
pattern storage in the CA3.

The main encoding GCs in the DG are grouped into the lamellar clusters
Cluster1 ; Cluster2 ; Cluster3 ; Cluster4 . In each cluster, both one
inhibitory basket cell (BC) and one inhibitory HIPP (hilar perforant path-
associated) cell exist, together with excitatory GCs. During pattern
separation, the GCs exhibit sparse firing activity through 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 (arising from
strong feedback inhibition) has been considered to improve the pattern
separation efficacy Treves3 ; Oreilly ; Schmidt ; Rolls3 ; Knier ; Myers1 ;
Myers2 ; Myers3 ; Scharfman ; Chavlis ; Kassab ; PS ; WTA .

Most distinctly, adult neurogenesis occurs in the DG, which leads to
appearance of new young immature GCs (imGCs) during adulthood. Pioneering
studies of Altman in adult rat and cat brains for the adult neurogenesis were
made decades ago in the 1960s NG1 ; NG2 ; NG3 . Since then, adult neurogenesis
has been found 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 imGCs born in the subgranular zone
migrate into the granular layer of the DG. Thus, the whole population of GCs
consists of mature GCs (mGCs) born during the development and adult-born
imGCs. In comparison to the mGCs, the young adult-born imGCs are found to
exhibit marked properties such as high excitability, weak inhibition, and low
excitatory innervation NG7 ; NG8 ; NG9 ; NG10 ; NG11 .

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 both mature GCs (mGCs) and 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.

In this paper, we consider a spiking neural network for the adult neurogenesis
in the DG, including both mGCs and imGCs (fraction of the imGCs is 10 $\%$)
where the effect of adult-born imGCs on pattern separation was studied NG-PS .
In our DG network, both high excitability and low excitatory innervation for
the imGCs are considered; approximately no inhibition is provided to the
imGCs. The imGCs exhibit high activation because of lower firing threshold,
while they receive low excitatory drive from the entorhinal cortex (EC) via
perforant paths (PPs) and from the hilar mossy cells (MCs) with lower
connection probability $p_{c}~{}(=20~{}x~{}\%)$ ($x:$ synaptic connectivity
fraction; $0\leq x\leq 1$) than the mGCs with the connection probability
$p_{c}~{}(=20~{}\%)$. Thus, the effect of low excitatory innervation
(decreasing activation degree) for the imGCs counteracts the effect of their
high excitability.

We are concerned about population rhythms of the GCs in the DG. Sparsely
synchronized rhythm (SSR) was found to appear in the presence of only mGCs
(without imGCs) during pattern separation via winner-take-all competition PS ;
SSR . Here, with decreasing $x$ (synaptic connectivity fraction) from 1 to 0,
we investigate the effect of adult neurogenesis on SSRs of the GCs (mGCs,
pmGC, and whole GCs) in our DG spiking neural network. For each $x$,
population and individual firing behaviors of the mGCs, the imGCs, and the
whole GCs in their SSRs are characterized in terms of the amplitude measure
${\cal M}_{a}^{(X)}$ ($X=m,~{}im,~{}w$ for the mGCs, the imGCs, and the whole
GCs, respectively) (denoting the population synchronization degree) and the
random phase-locking degree ${\cal L}_{d}^{(X)}$ (characterizing the
regularity of individual single-cell firings), respectively AM ; SSR .

For $0\leq x\leq 1,$ the mGCs and the whole GCs were found to exhibit pattern
separation (i.e., a process of transforming similar input patterns into less
similar output patterns), while the imGCs were found to show pattern
integration (making association between patterns); efficacy of pattern
separation and pattern integration was characterized in terms of pattern
separation degree ${\cal S}_{d}^{(X)}$ ($X=m$ and $w$) and pattern integration
degree ${\cal I}_{d}$ ($X=im$), respectively PS ; NG-PS . Here, quantitative
relationship between ${\cal M}_{a}^{(X)}$ and ${\cal L}_{d}^{(X)}$ of the SSRs
and ${\cal S}_{d}^{(X)}$ and ${\cal I}_{d}$ of pattern separation and
integration is also discussed.

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 neurogenesis on SSRs of the
GCs (mGCs, imGCs, whole GCs) by varying $x$ (synaptic connectivity fraction)
from 0 to 1. Also, we study quantitative association between the SSRs and the
efficacy of pattern separation and integration. 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 hippocampal 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 SSR SSR , and the pattern separation PS . In our DG
spiking neural network for the adult neurogenesis NG-PS , the young adult-born
imGCs and the mGCs are incorporated, and more synaptic connections with a high
degree of anatomical and physiological realism are also included BN1 ; BN2 .

###  II.1 Framework of Our DG Spiking Neural Network for The Adult
Neurogenesis

We first describe framework of our DG spiking neural network for the adult
neurogenesis. The granular layer (GL) and the hilus constitute the DG. The GL
consists of the excitatory mGCs and imGCs and the inhibitory BCs. The hilus is
composed of the excitatory MCs and the inhibitory HIPP cells, whose axons
project to the upper molecular layer (ML). Figure 1(a) shows schematic
representation of major cells and synaptic connections in our DG network for
adult neurogenesis. In our DG network, fraction of imGCs is 10 $\%$ in the
whole population of GCs. High excitability and low excitatory innervation of
the imGCs are considered, but there are no inhibitory inputs into the imGCs
NG7 ; NG8 ; NG9 ; NG10 ; NG11 .

Based on the anatomical information given in Myers1 ; Myers2 ; Myers3 ;
Scharfman ; Chavlis , we choose the numbers of GCs, BCs, MCs, and HIPP cells
in the DG and the EC cells. As in our prior works WTA ; SSR ; PS , we consider
a scaled-down spiking neural network. 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 . Fraction of the imGCs in the whole population of the GCs
is $10~{}\%$; the number of the mGCs (imGCs) is 1800 (200). The whole GCs
(i.e., mGCs and imGCs) are grouped into the $N_{c}~{}(=20)$ lamellar clusters
Cluster1 ; Cluster2 ; Cluster3 ; Cluster4 . Then, in each GC cluster, there
are $n_{\rm{GC}}^{(c)}~{}(=100)$ GCs (i.e., 90 mGC and 10 imGCs) and one
inhibitory BC Myers2 ; Myers3 ; Scharfman . Next, we consider the hilus Hilus1
; Hilus2 ; Hilus3 ; Hilus4 ; Hilus5 ; Hilus6 ; Hilus7 . In our scaled-down DG
network, we choose the number of MCs and the number of HIPP cells as $N_{\rm
MC}=60$ and $N_{\rm HIPP}=20,$ respectively. Hence, in each cluster, there are
$n_{\rm{MC}}^{(c)}~{}(=3)$ MCs and one HIPP cell Myers2 ; Myers3 ; Scharfman .
Also, the number of EC cells (projecting the excitatory inputs to the mGCs,
the imGCs, and the BCs through the PPs via random connections) in our scaled-
down neural network is $N_{\rm EC}=400$, and their activation degree is chosen
as 10$\%$ ANA4 . Each active EC cell is modeled in terms of the Poisson spike
train with frequency of 40 Hz ANA5 .

Figure 1(b) shows the box diagram for our DG network with 3 types of lamellar
(blue), cross-lamellar (red), and random (black) synaptic connections The EC
provides the external excitatory inputs randomly to the mGCs, the imGCs, and
the inhibitory BCs (with dendrites extending to the outer ML) through PPs
Myers1 ; Myers2 ; Myers3 ; Scharfman ; Chavlis . Thus, both the mGCs and the
imGCs receive direct excitatory EC input through PPs (EC $\rightarrow$ mGC and
imGCs) via random connections. 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); $0\leq
x\leq 1$] because of low excitatory innervation. Furthermore, 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 all the
mGCs in the same cluster via lamellar connections. In the hilus, we also
consider lamellar organization for the MCs and HIPP cells Myers2 ; Myers3 ;
Scharfman ; Hilus6 . 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.

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 all the 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.

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). 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 .

The mGCs in a cluster show sparse firing activity through the winner-take-all
competition WTA1 ; WTA2 ; WTA3 ; WTA4 ; WTA5 ; WTA6 ; WTA7 ; WTA8 ; WTA9 ;
WTA10 ; WTA . Only strongly active mGCs can 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. In contrast, the imGCs
receive no inhibition. Particularly, because of their low firing threshold,
they become highly active, in contrast to the case of mGCs NG7 ; NG8 ; NG9 ;
NG10 . However, when taking into consideration their low excitatory
innervation from the EC cells and the MCs, their firing activity is reduced
NG11 .

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

As elements of our DG spiking neural network, we choose leaky integrate-and-
fire (LIF) spiking neuron models with additional afterhyperpolarization (AHP)
currents, determining refractory periods. This LIF spiking neuron model is one
of the simplest spiking neuron models LIF . Due to its simplicity, it can be
easily analyzed and simulated.

Our DG network consists of 5 populations of mGCs, imGCs, BCs, MCs, and HIPP
cells. The state of a neuron in each population is characterized by its
membrane potential. Then, time-evolution of the membrane potential is governed
by 4 types of currents into the neuron; the leakage current, the AHP current,
the external constant current, and the synaptic current.

We note that the equation for a single LIF neuron model (without the AHP
current and the synaptic current) describes a simple parallel resistor-
capacitor (RC) circuit. Here, the 1st type of leakage current is because of
the resistor and the integration of the external current is because of 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 Chavlis .

Detailed explanations on the leakage current and the AHP current, associated
with each type of single neuron (mGC, imGC, BC, MC, and HIPP cell), are given
in Appendix A. 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 I in WTA
. These parameter values are based on physiological properties of the GC, BC,
MC, and HIPP cell Chavlis ; Hilus3 .

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 $I^{*}=80$ pA. 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) in NG-PS .

Next, we consider the 4th type of synaptic current. Detailed explanations on
the synaptic current are given in Appendix B; here, we give a brief and clear
explanation on it. There are 3 kinds of synaptic currents from a presynaptic
source population to a postsynaptic neuron in the target population; 2 kinds
of excitatory AMPA and NMDA receptor-mediated synaptic currents and one type
of inhibitory GABA receptor-mediated synaptic current. In each $R$ (AMPA,
NMDA, and GABA) receptor-mediated synaptic current, the synaptic conductance
is given by the product of the synaptic strength per synapse, the average
number of afferent synapses (connected to a postsynaptic neuron), and fraction
of open ion channels.

The postsynaptic ion channels are opened via binding of neurotransmitters
(emitted from the source population) to receptors in the target population.
The time course of fraction of open ion channels is given by a sum of “double-
exponential” functions over presynaptic spikes. The double-exponential
function, corresponding to contribution of a presynaptic spike, is controlled
by the synaptic rising time constant, the synaptic decay time constant, and
the synaptic latency time constant; for details, refer to Eq. (20) in Appendix
B.

The parameter values for the synaptic strength per synapse, the synaptic
rising time constant, the synaptic decay time constant, the synaptic latency
time constant, and the synaptic reversal potential for the synaptic currents
into the GCs, the BCs, the MCs, and the HIPP cells are given in Tables I-III
in NG-PS . These parameter values are also based on the physiological
properties of the relevant cells Chavlis ; 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.

Figure 2: Characterization of SSR in the presence of only the mGCs without
imGCs. (a1) Raster plots of spikes of 120 active mGCs. (a2) instantaneous
population spike rate (IPSR) $R(t)$ of active mGCs. Band width for $R(t)$:
$h=20$ msec. (b) Plot of amplitude measure ${\cal M}_{a}(j)$ of the IPSR
$R(t)$ versus $j$ (spiking stripe). (c) ISI histogram of the 7th active mGC;
bin size = 2 msec. Vertical dotted lines in (c) represent the integer
multiples of the global period $T_{G}$ (= 54.9 msec) of $R(t)$. Plots of (d1)
normalized weight $w_{n}$ and (d2) random phase-locking degree ${\cal
L}_{d}^{(n)}$ for the $n$th peak of the ISI histogram for the 7th active mGC
versus $n$ (peak index).

##  III Effect of Adult Neurogenesis on Sparsely Synchronized Rhythms

In this section, we study the effect of adult neurogenesis on the SSRs of the
GCs (mGCs, imGCs, whole GCs) in our DG spiking neural network. With decreasing
$x$ (synaptic connectivity fraction) from 1 to 0, population and individual
firing behaviors of the mGCs, the imGCs, and the whole GCs in their SSRs are
investigated by employing the amplitude measure ${\cal M}_{a}^{(X)}$
($X=m,~{}im,~{}w$ for the mGCs, the imGCs, and the whole GCs, respectively)
(denoting the population synchronization degree) AM  and the random phase-
locking degree ${\cal L}_{d}^{(X)}$ (characterizing the regularity of
individual single-cell firings) SSR ; PS , respectively. For $0\leq x\leq 1$
the mGCs and the imGCs were found to exhibit pattern separation and pattern
integration, respectively NG-PS . We also discuss quantitative relationship
between the SSRs and the pattern separation and integration.

###  III.1 Characterization of Sparsely Synchronized Rhythm in The Presence of
Only The mGCs without The imGCs

We first consider the homogeneous population of mGCs (without the imGCs) SSR .
Population firing activity of the active mGCs may be well visualized in the
raster plot of spikes which is a collection of spike trains of individual
active mGCs. Figure 2(a1) shows the raster plot of spikes for 120 active mGCs
(activation degree $D_{a}$ of the mGCs is 6 $\%$); for convenience, only a
part from $t=300$ to 1,300 msec is shown in the raster plot of spikes. We note
that sparsely synchronized stripes (composed of sparse spikes and indicating
population sparse synchronization) appear successively.

As a population quantity showing collective behaviors, we employ an IPSR
(instantaneous population spike rate) which may be obtained from the raster
plot of spikes W_Review ; Sparse1 ; Sparse2 ; Sparse3 ; FSS ; SM . To get the
smooth IPSR, we employ the kernel density estimation (kernel smoother) Kernel
. Each spike in the raster plot is convoluted (or blurred) with a kernel
function $K_{h}(t)$ to get a smooth estimate of IPSR $R(t)$:

$R(t)=\frac{1}{N_{a}}\sum_{i=1}^{N_{a}}\sum_{s=1}^{n_{i}}K_{h}(t-t_{s,i}),$
(1)

where $N_{a}$ is the number of the active mGCs, $t_{s,i}$ is the $s$th spiking
time of the $i$th active mGC, $n_{i}$ is the total number of spikes for the
$i$th active mGC, and we use a Gaussian kernel function of band width $h$:

$K_{h}(t)=\frac{1}{\sqrt{2\pi}h}e^{-t^{2}/2h^{2}},~{}~{}~{}~{}-\infty<t<\infty,$
(2)

where the band width $h$ of $K_{h}(t)$ is 20 msec. The IPSR $R(t)$ is also
shown in Fig. 2(a2). We note that the IPSR $R(t)$ exhibits synchronous
oscillation with the population frequency $f_{p}~{}(=18.2$ Hz). The population
frequency $f_{p}$ is given by the reciprocal of the global period $T_{G}$
(i.e., $f_{p}=1/T_{G}$) which corresponds to the average “intermax” interval
(i.e., average interval between neighboring maxima) in the IPSR $R(t)$. Here,
we get $N_{\rm IMI}$ (= 545) intermax intervals during the stimulus period
$T_{s}$ ($=3\cdot 10^{4}$ msec), and get their average value (i.e., global
period) $T_{G}$ (= 54.9 msec). In this way, SSR with $f_{p}~{}(=18.2$ Hz)
appears in the (homogeneous) population of active mGCs.

The amplitude of the IPSR $R(t)$ may represent synchronization degree of the
SSR. Here, we characterize the synchronization degree of the SSR in terms of
the amplitude measure ${\cal M}_{a}$, given by the time-averaged amplitude of
$R(t)$ AM :

${\cal M}_{a}={\overline{{\cal M}_{a}(j)}};{\cal M}_{a}(j)=\frac{[R_{\rm
max}^{(j)}(t)-R_{\rm min}^{(j)}(t)]}{2},$ (3)

where the overline represents time average, ${\cal M}_{a}(j)$ is the amplitude
measure in the $j$th global cycle (corresponding to the $j$th spiking stripe),
and $R_{\rm max}^{(j)}(t)$ and $R_{\rm min}^{(j)}(t)$ are the maximum and the
minimum of $R(t)$ in the $j$th global cycle, respectively. As ${\cal M}_{a}$
increases (i.e., the time-averaged amplitude of $R(t)$ is increased),
synchronization degree of the SSR becomes higher. Figure 2(b) shows plot of
the amplitude ${\cal M}_{a}(j)$ versus the spiking stripe index $j$
(corresponding to the global cycle index). We follow the 546 stripes during
the stimulus period $T_{s}$ ($=3\cdot 10^{4}$ msec), and the amplitude measure
${\cal M}_{a}$ [corresponding to the time-averaged amplitude $\overline{{\cal
M}_{a}(j)}$] is thus found to be 3.83.

Next, we consider the individual firing behavior of the active mGCs. For each
active mGC, we get the inter-spike-interval (ISI) histogram by collecting the
ISIs during the stimulus period $T_{s}$ ($=3\cdot 10^{4}$ msec). Each active
mGC exhibits intermittent spikings, phase-locked to $R(t)$ at random multiples
of its global period $T_{G}$ (= 54.9 msec). This is in contrast to the case of
full synchronization where only one dominant peak appears at the global period
$T_{G}$; all cells fire regularly at each global cycle without skipping. As a
result of random spike skipping, there appear 19 distinct multiple peaks at
the integer multiples of $T_{G}$ in the ISI histogram. These peaks are called
as the random-spike-skipping peaks. Then, we get the population-averaged ISI
histogram by averaging the individual ISI histograms for all the active mGCs.
In this case, the population-averaged ISI ($\langle{\rm ISI}\rangle$) of all
the active mGCs in the population-averaged ISI histogram is 471.7 msec. Then,
the population-averaged mean firing rate (MFR) $\langle f_{i}\rangle$, given
by the reciprocal of $\langle{\rm ISI}\rangle$ (i.e., $\langle
f_{i}\rangle=1/\langle{\rm ISI}\rangle$), is 2.12 Hz, which is much less than
the population frequency $f_{p}~{}(=18.2$ Hz) of the SSR, in contrast to the
case of full synchronization where the population-averaged MFR is the same as
the population frequency.

As an example, we consider the case of the 7th ($i=7$) active mGC. Its ISI
histogram is shown in Fig. 2(c). In this case, the 8th-order peak is the
highest one, and hence spiking may occur most probably after 7-times spike
skipping. The $n$th-order random-spike-skipping peak in the ISI histogram is
located as follows:

$\displaystyle(n-\frac{1}{2})~{}T_{G}<{\rm
ISI}<(n+\frac{1}{2})~{}T_{G}~{}~{}~{}~{}~{}{\rm for}~{}n\geq 2,$ (4)
$\displaystyle 0<{\rm ISI}<{\frac{3}{2}}~{}T_{G}~{}~{}~{}~{}~{}{\rm
for}~{}n=1.$ (5)

For each $n$th-order peak, we obtain the normalized weight $w_{n}$, given by:

$w_{n}=\frac{N_{\rm ISI}^{(n)}}{N_{\rm ISI}^{(tot)}},$ (6)

where $N_{\rm ISI}^{(tot)}$ is the total number of ISIs obtained during the
stimulus period ($T_{s}$ $=3\cdot 10^{4}$ msec) and $N_{\rm ISI}^{(n)}$ is the
number of the ISIs in the $n$th-order peak. Figure 2(d1) shows plot of $w_{n}$
versus $n$ (peak index) for all the 19 peaks in the ISI histogram of the 7th
($i=7$) active mGC. For example, the highest 8th-order peak has $w_{8}=0.12$.

We now consider the sequence of the ISIs, $\\{{\rm
ISI}_{j}^{(n)},~{}j=1,\dots,N_{\rm ISI}^{(n)}\\}$, within the $n$th-order
peak, and get the random phase-locking degree ${\cal L}_{d}^{(n)}$ of the
$n$th-order peak (representing how well intermittent spikes make phase-locking
to the IPSR $R(t)$ at $t=nT_{G}$). As in the case of the pacing degree SM , we
give a phase $\psi$ to each ${\rm ISI}_{j}^{(n)}$ via linear interpolation:

$\psi(\Delta{\rm ISI}_{j}^{(n)})=\frac{\pi}{T_{G}}~{}\Delta{\rm
ISI}_{j}^{(n)}~{}~{}~{}~{}~{}{\rm for}~{}n\geq 2,$ (7)

where $\Delta{\rm ISI}_{j}^{(n)}={\rm ISI}_{j}^{(n)}-n~{}T_{G}$, leading to
$-{\frac{T_{G}}{2}}<\Delta{\rm ISI}_{j}^{(n)}<{\frac{T_{G}}{2}}$. However, for
$n=1$, $\psi$ changes depending on whether the ISI lies in the left or the
right part of the 1st-order peak:

$\psi(\Delta{\rm
ISI}_{j}^{(1)})=\left\\{\begin{array}[]{l}\frac{\pi}{2~{}T_{G}}~{}\Delta{\rm
ISI}_{j}^{(1)}~{}~{}~{}~{}~{}{\rm for}~{}-{T_{G}}<\Delta{\rm
ISI}_{j}^{(1)}<0,\\\ \frac{\pi}{T_{G}}~{}\Delta{\rm
ISI}_{j}^{(1)}~{}~{}~{}~{}~{}{\rm for}~{}0<\Delta{\rm
ISI}_{j}^{(1)}<{\frac{T_{G}}{2}},\end{array}\right.$ (8)

where $\Delta{\rm ISI}_{j}^{(1)}={\rm ISI}_{j}^{(1)}-T_{G}.$

Then, the contribution of the ${\rm ISI}_{j}^{(n)}$ to the locking degree
${\cal L}_{d}^{(n)}$ is given by $\cos(\psi_{j}^{(n)})$;
$\psi_{j}^{(n)}=\psi(\Delta{\rm ISI}_{j}^{(n)})$. An ${\rm ISI}_{j}^{(n)}$
makes the most constructive contribution to ${\cal L}_{d}^{(n)}$ for
$\psi_{j}^{(n)}=0$, while it makes no contribution to ${\cal L}_{d}^{(n)}$ for
$\psi_{j}^{(n)}={\frac{\pi}{2}}$ or $-{\frac{\pi}{2}}$. By averaging the
matching contributions of all the ISIs in the $n$th-order peak, we get:

${\cal L}_{d}^{(n)}={\frac{1}{N_{\rm ISI}^{(n)}}}\sum_{j}^{N_{\rm
ISI}^{(n)}}\cos(\psi_{j}^{(n)}).$ (9)

Figure 2(d2) shows plot of ${\cal L}_{d}^{(n)}$ versus $n$ (peak index) for
the 19 random-spike-skipping peaks in the ISI histogram of the 7th active mGC.
For example, the highest 8th-order ($n=8$) peak has the maximum value of
${\cal L}_{d}^{(n)}$ (= 0.961). Through weighted average of the random phase-
locking degrees ${\cal L}_{d}^{(n)}$ of all the peaks, we obtain the (overall)
random phase-locking degree ${\cal L}_{d}$

${\cal L}_{d}=\sum_{n=1}^{N_{p}}w_{n}\cdot{\cal L}_{d}^{(n)}={\frac{1}{N_{\rm
ISI}^{\rm(tot)}}}\sum_{n=1}^{N_{p}}\sum_{j=1}^{N_{\rm
ISI}^{\rm(tot)}}\cos(\psi_{j}^{(n)}),$ (10)

where $N_{p}$ is the number of peaks in the ISI histogram. We note that,
${\cal L}_{d}$ corresponds to the average of contributions of all the ISIs in
the ISI histogram. In the case of the 7th active mGC, the random phase-locking
degree ${\cal L}_{d}$, characterizing the sharpness of all the peaks, is 0.92.
Hence, the mGCs make intermittent spikes which are well phase-locked to $R(t)$
at random multiples of its global period $T_{G}$.

We repeat the above process in the ISI histogram of each $i$th
($i=1,\dots,120$) active mGC and get its random phase-locking degree ${\cal
L}_{d}(i)$. The range of $\\{{\cal L}_{d}(i)\\}$ is [0.65, 1.23]. Then, the
random phase-locking degree ${\cal L}_{d}$ of all the active mGCs ia given by
the average value (= 0.92) of the distribution $\\{{\cal L}_{d}(i)\\}$.

Figure 3: SSR and multi-peaked ISI histogram in each case of the mGCs, the
imGCs, and the whole GCs. (a1)-(a4) Raster plots of spikes and IPSRs
$R^{(X)}(t)$ for the active mGCs ($X=m$), the imGCs ($X=im$), and the whole
GCs ($X=w$) when $x$ (synaptic connectivity fraction) is 1.0, 0.7, 0.4, and
0.2, respectively. (b1)-(b4) Population-averaged ISI histograms for the active
mGC, imGC, and whole GCs when $x=$ 1.0, 0.7, 0.4, and 0.2, respectively; bin
size = 2 msec. Vertical dotted lines in (b1)-(b4) represent the integer
multiples of the global period $T_{G}^{(X)}$ of $R^{(X)}(t)$. In (a1)-(a4) and
(b1)-(b4), mGCs, imGCs, and whole GCs are denoted in green, red, and blue
color, respectively.

###  III.2 Effect of The Adult-Born imGCs on Sparsely Synchronized Rhythms

In this subsection, we consider a heterogeneous population, composed of mGCs
and imGCs; fraction of the imGCs in the whole population is 10 $\%$. As shown
in Fig. 2 in NG-PS , 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 imGC NG7 ; NG8 ; NG9 ;
NG10 . We also note that, the imGC has low excitatory innervation from the EC
cells and the hilar MCs, counteracting its high excitability NG11 . In the
case of the mGC, the connection probability $p_{c}$ from the EC cells and the
MCs to the mGC is 20 $\%$, while in the case of the imGC, $p_{c}$ 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 imGC becomes reduced. With decreasing $x$ from 1 to 0, we investigate
the effect of high excitability and low excitatory innervation for the imGC on
the population and individual firing behaviors of the mGCs, the imGCs, and the
whole GCs in their SSRs. We also note that, for $0\leq x\leq 1$ the mGCs and
the imGCs were found to exhibit pattern separation and pattern integration,
respectively NG-PS . Hence, we also study quantitative relationship between
SSRs and pattern separation and integration.

Here, as in the case of Fig. 2, we consider a long-term stimulus stage
(300-30,300 msec) (i.e., the stimulus period $T_{s}=30,000$ msec), because
long-term stimulus is necessary for analysis of dynamical behaviors.
Population firing activity of the active mGCs and imGCs may be well visualized
in the raster plot of spikes which is a collection of spike trains of
individual active GCs. Figures 3(a1)-3(a4) show the raster plots of spikes for
the active mGCs (green) and imGCs (red) for $x=$ 1.0, 0.7, 0.4, and 0.2,
respectively. For convenience, only a part from $t=300$ to 1,300 msec is shown
in each raster plot of spikes. We note that sparsely synchronized stripes
(composed of sparse spikes and indicating population sparse synchronization)
appear successively; overall, the pacing degree between spikes in the spiking
stripes is low. In the case of mGCs, with decreasing $x$ from 1 their spiking
stripes become clearer, while in the case of imGCs their stripes become more
smeared.

The instantaneous population spike rate [IPSR (showing population firing
behavior)] may be obtained from the raster plot of spikes [see Eq. (1)]. The
IPSRs $R^{(X)}(t)$ of the mGCs ($X=m:$ green), the imGCs ($X=im:$ red), and
the whole GCs ($X=w:$ blue) are shown in Figs. 3(a1)-3(a4) for $x=$ 1.0, 0.7,
0.4 and 0.2, respectively. We note that $R^{(X)}(t)$ exhibit synchronous
oscillations. But, the average amplitude of $R^{(X)}(t)$ in each case of mGCs,
imGCs, and whole GCs is smaller than that in the case of homogeneous
population of only mGCs in Fig. 2(a2), and variations in the amplitudes are
also large.

For $x=1$, imGCs fire spikings much more actively than mGCs because the imGCs
have high excitability. On the other hand, firing activity of mGCs becomes
much decreased due to strongly increased feedback inhibition from the BCs and
the HIPP cells. Hence, in the case of $x=1$ the amplitude of $R^{(im)}(t)$
(red) of the imGCs is larger than that of $R^{(m)}(t)$ (green) of the mGCs.
However, as $x$ is decreased from 1, firing activity of the imGCs becomes
rapidly reduced (i.e., the effect of imGCs decreases rapidly) because of low
excitatory innervation from the EC cells and the MCs. On the other hand,
firing activity of mGCs becomes enhanced due to decrease in the feedback
inhibition into the mGCs from the BCs and the HIPP cells. Thus, with
decreasing $x$ from 1, the amplitude of $R^{(m)}(t)$ of the mGCs makes an
increase because the pacing degree between spikes in each spiking stripe in
the rater plot of spikes becomes better (i.e., the spiking stripes in the
raster plot of spikes become clearer). In contrast, in the case of imGCs, the
amplitude of $R^{(im)}(t)$ decreases because the pacing degree of spikes in
the raster plot becomes worse (i.e., the spiking stripes in the raster plot of
spikes become smeared). Thus, for example, for $x=0.2$ the amplitude of
$R^{(m)}(t)$ becomes much larger than that of $R^{(im)}(t)$.

In the case of whole GCs, $R^{(w)}(t)$ (blue) gets an “average” value of
$R^{(m)}(t)$ and $R^{(im)}(t)$;
$R^{(w)}(t)=[N_{a}^{(m)}/N_{a}^{(w)}]R^{(m)}(t)+[N_{a}^{(im)}/N_{a}^{(w)}]R^{(im)}(t)$
($N_{a}^{(X)}$ is the number of active GCs in the $X$ population). Thus, for
any $x$, $R^{(w)}(t)$ follows the tendency of the larger one between
$R^{(m)}(t)$ and $R^{(im)}(t)$; for example, for $x=1$ $R^{(w)}(t)$ is close
to $R^{(im)}(t)$, while for other values of $x=$ 0.7, 0.4, and 0.2, it is near
to $R^{(m)}(t).$

In addition to the (above) population firing activity, we also study the
individual spiking activity of the active GCs. In each case of the mGCs
($X=m$), the imGCs ($X=im$), and the whole GCs ($X=w$), we get the ISI
histogram for each active GC by collecting the ISIs during the stimulus period
$T_{s}$ ($=3\cdot 10^{4}$ msec), and then obtain the population-averaged ISI
histogram by averaging the individual ISI histograms for all the active GCs.
Figures 3(b1)-3(b4) show the population-averaged ISI histograms for $x=$ 1,
0.7, 0.4, and 0.2, respectively.

We first consider the case of $x=1$ in Fig. 3(b1). For the mGCs (green), each
active mGC exhibits intermittent spikings, phase-locked to $R^{(m)}(t)$ at
random multiples of its global period $T_{G}^{(m)}$ (= 87.8 msec)
[corresponding to the average “intermax” interval between neighboring maxima
in $R^{(m)}(t)$]; vertical dotted lines represent integer multiples of the
global period $T_{G}^{(m)}$ of $R^{(m)}(t)$. As a result of random spike
skipping, there appear 12 multiple peaks in the ISI histogram. The middle
10th-order peak is the highest one, and hence spiking may occur most probably
after 9-times spike skipping. This is in contrast to the case of full
synchronization where only one dominant peak appears at the global period
$T_{G}$ of the IPSR $R(t)$; all cells fire regularly at each global cycle
without skipping. Next, we consider the case of imGCs (red). Its ISI histogram
has a single peak near the global period $T_{G}^{(im)}~{}(=34.3$ msec) of the
IPSR $R^{(im)}(t)$ and its distribution is broadly extended to $\sim
3~{}T_{G}^{(im)}$. The imGCs exhibit spikes mainly at $T_{G}^{(im)}$ (i.e.,
they fire mainly in each stripe), but they also show intermittent spikings at
2 $T_{G}^{(im)}$ or 3 $T_{G}^{(im)}$ (i.e., spike skippings also occur).

Overall, we consider the case of whole GCs (blue). In the case of $x=1$,
spikes of the imGCs are dominant, as shown in Fig. 3(a1). Thus, dominant major
peak, associated with the imGCs, appears near the global period $T_{G}^{(w)}$
(= 34.3 msec) of the IPSR $R^{(w)}(t)$, while fractions of multiple peaks,
related to the mGCs, are very small (not clearly seen).

As $x$ is decreased from 1 (i.e., considering low excitatory innervation for
the imGCs), the effect of the imGCs becomes weaker. In this case, the imGCs
show more irregular spiking behaviors. Hence, their single-peaked ISI
histograms become broader, as shown in Figs. 3(b2)-3(b4) for $x=$ 0.7, 0.4,
and 0.2, respectively. The order $n$ of peak also increases with decreasing
$x$ ($n=$ 2, 4, and 6 for $x=$ 0.7, 0.4, and 0.2, respectively). Hence, more
spike skippings occur. On the other hand, with decreasing $x$ from 1, the mGCs
exhibit more regular spiking behaviors. Hence, their ISI histograms become
clearer because multiple peaks become sharper and their heights become
increased [see Figs. 3(b2)-3(b4)].

In the case of whole GCs, both the peak (associated with the imGCs) and the
multiple peaks (related to the mGCs) coexist in the histogram. However, their
fractions vary depending on $x$. For $x=0.7,$ the effect of the imGCs becomes
reduced, and hence the height of the peak at $2~{}T_{G}^{(w)}$, associated
with the imGCs, becomes decreased. In this case, multiple peaks, related to
the mGCs, become clearly visible, because their fractions are increased.
However, with further decrease in $x$, spikes of mGCs become more and more
dominant. Thus, only the multiple peaks, related to the mGCs, are visible
because fraction of the peak, associated with the imGCs, becomes very small.

Figure 4: Population and individual firing behaviors in the SSRs of the mGCs
($X=m$), the imGCs ($X=im$), and the whole GCs ($X=w$). (a) Plots of the
population frequencies $f_{p}^{(X)}$ versus $x$ (synaptic connectivity
fraction). (b) Plots of the amplitude measures ${\cal M}_{a}^{(X)}$ versus
$x$. (c) Plots of the population-averaged mean firing rates $\langle
f_{i}^{(X)}\rangle$ versus $x$. (d) Plots of the random-phase-locking degrees
${\cal L}_{d}^{(X)}$ versus $x$. Horizontal dashed lines in (a)-(d) represent
$f_{p}$ (= 18.2 Hz), $M_{a}$ (= 3.83), $\langle f_{i}\rangle$ (= 2.12 Hz), and
${\cal L}_{d}$ (= 0.92) in the presence of only mGCs (without imGCs),
respectively.

From now on, in Fig. 4, we quantitatively characterize population and
individual firing behaviors in the SSRs of the mGCs ($X=m$), the imGCs
($X=im$), and the whole GCs ($X=w$). We first consider the population firing
behaviors which are well shown in $R^{(X)}(t)$. Figure 4(a) shows the plots of
the population frequency $f_{p}^{(X)}$ [i.e., the average oscillating
frequency of $R^{(X)}(t)$, corresponding to the reciprocal of the global
period $T_{G}^{(X)}$ of the SSRs for $X=m$ (green solid circles), $im$ (red
open circles), and $w$ (blue crosses)]. For $x=1$, $f_{p}^{(im)}$ (= 29.2 Hz)
for the imGCs is faster than $f_{p}^{(m)}$ (= 11.4 Hz) for the mGCs, as can be
well seen in Fig. 3(a1), mainly because for $x=1$ firing of the imGCs is much
more active than that of the mGCs (resulting from high excitability of the
imGCs).

However, as $x$ is decreased from 1 to 0, $f_{p}^{(im)}$ decreases to 0
rapidly due to rapid decrease in firing activity of the imGCs (resulting from
their low excitatory innervation). On the other hand, $f_{p}^{(m)}$ increases
to 18.6 Hz because of increase in firing activity of the mGCs (resulting from
decrease in the feedback inhibition into the mGCs). We note that,
$f_{p}^{(im)}$ and $f_{p}^{(m)}$ cross at $x^{*}\sim 0.4$; for $x>x^{*}$
$f_{p}^{(im)}>f_{p}^{(m)},$ while for $x<x^{*}$ $f_{p}^{(m)}>f_{p}^{(im)}.$ In
the case of whole GCs, the population frequency $f_{p}^{(w)}$ follows tendency
of the larger one between $f_{p}^{(m)}$ and $f_{p}^{(im)}$. Thus,
$f_{p}^{(w)}$ forms a well-shaped curve (i.e., $f_{p}^{(w)}$ decreases from
29.2 Hz to $\sim 18.1$ Hz for $1\geq x>x^{*}$, while it increases for
$x^{*}>x\geq 0$, and converges to a limit value (= 18.6 Hz) for $x=0$ which is
a little larger than the dashed horizontal line ($f_{p}=18.2$ Hz) in the
homogeneous population of only the mGCs (without imGCs), which may be
understood as follows. In the limiting case of $x=0,$ the imGCs become
completely inactive. Hence, the feedback inhibition (from the BCs and the HIPP
cells) to the mGCs becomes reduced in comparison to the homogeneous case
consisting of only mGCs, which results in increased firing activity of the
mGCs in the heterogeneous population of mGCs and imGCs.

The amplitude of the IPSR $R^{(X)}(t)$ may represent synchronization degree of
the SSR. Thus, we characterize the synchronization degree of the SSRs of the
mGCs ($X=m$), the imGCs ($X=im$), and the whole GCs ($X=w$) in terms of the
amplitude measure ${\cal M}_{a}^{(X)}$ of Eq. (3), given by the time-averaged
amplitude of $R^{(X)}(t).$ Figure 4(b) shows the plots of ${\cal M}_{a}^{(X)}$
versus $x$ for the mGCs (green solid circles), the imGCs (red open circles),
and the whole GCs (blue crosses). For $x=1$ (i.e., high excitability of the
imGCs), ${\cal M}_{a}^{(im)}$ (= 2.75) for the imGCs is larger than ${\cal
M}_{a}^{(m)}$ (= 0.99) for the mGCs, as can be seen well in Fig. 3(a1),
because the imGCs fire more actively and coherently than mGCs.

As $x$ is decreased from 1 to 0 (i.e., low excitatory innervation to the
imGCs) the effect of imGCs becomes decreased rapidly, which results in more
active and coherent firing activity of the mGCs (due to decreased feedback to
the mGCs from the BCs and the HIPP cells). Consequently, ${\cal M}_{a}^{(m)}$
increases to 3.79, while ${\cal M}_{a}^{(im)}$ decreases to 0. In the case of
whole GCs, ${\cal M}_{a}^{(w)}$ increases from 2.37 to 3.79 by following
tendency of the larger one between ${\cal M}_{a}^{(m)}$ and ${\cal
M}_{a}^{(im)}$, as can be well seen in Fig. 4(b). We note that the limit value
(= 3.79) of both ${\cal M}_{a}^{(m)}$ and ${\cal M}_{a}^{(w)}$ is a little
smaller than ${\cal M}_{a}$ (= 3.83) in the homogeneous population of only the
mGCs (without imGCs) [represented by the dashed horizontal line in Fig. 4(b)].
Hence, for all $x,$ ${\cal M}_{a}^{(X)}$ of the mGCs, the imGCs, and the whole
GCs is less than that (= 3.83) in the homogeneous case consisting of only
mGCs. Consequently, in the whole range of $x$, due to heterogeneity caused by
the imGCs, population firing behaviors (characterized in terms of ${\cal
M}_{a}^{(X)}$) of mGCs, imGCs, and whole GCs in their SSRs become
deteriorated, in comparison to that in the presence of only mGCs (without
imGcs).

Next, we consider the individual firing behaviors of the active mGCs, imGCs,
and whole GCs which are well shown in their ISI histograms. Figure 4(c) shows
the plots of the population-averaged MFRs $\langle f_{i}^{(X)}\rangle$ of the
individual mGCs (green solid circles), imGCs (red open circles), and whole GCs
(blue crosses); $\langle f_{i}^{(X)}\rangle$ corresponds to the reciprocal of
the population-averaged ISI (${\langle{\rm ISI}\rangle}^{(X)}$) (i.e.,
${\langle f_{i}^{(X)}\rangle}={1/{\langle{\rm ISI}\rangle}}^{(X)}$) in the
population-averaged ISI histogram of the $X$-population.

For $x=1$ $\langle f_{i}^{(im)}\rangle$ (= 28.7 Hz) for the imGCs is much
faster than $\langle f_{i}^{(m)}\rangle$ (= 1.12 Hz) for the mGCs, as can be
well seen in Fig. 3(b1). In this case, due to their high excitability, the
imGCs exhibit active firing activity, while the mGCs show very intermittent
spikings due to strong feedback inhibition (from the BCs and the HIPP cells).
Thus, in the case of mGCs the population-averaged MFR $\langle
f_{i}^{(m)}\rangle$ is much less than the population frequency
$f_{p}^{(m)}~{}(=11.4$ Hz) for the SSR, due to random spike skipping, which is
in contrast to the case of full synchronization where the population-averaged
MFR is the same as the population frequency. On the other hand, in the case of
imGCs, their population-averaged MFR $\langle f_{i}^{(im)}\rangle$ is close to
the population frequency $f_{p}^{(im)}~{}(=29.2$ Hz) for the SSR, and hence
the active imGCs show nearly fully synchronized rhythm (i.e., most of all
active imGCs fire in each spiking stripe) in the case of $x=1$.

However, as $x$ is decreased from 1 to 0, firing activity of imGCs is
decreased rapidly due to low excitatory innervation. Consequently, $\langle
f_{i}^{(im)}\rangle$ decreases so rapidly from 28.7 Hz to 0. Thus, for $x<1$,
active imGCs distinctly exhibit random spike skipping, leading to SSR with
$f_{p}^{(im)}>{\langle f_{i}^{(im)}\rangle}$. On the other hand, with
decreasing $x$ from 1, $\langle f_{i}^{(m)}\rangle$ of the mGCs increases
slowly from 1.12 to 2.30 Hz, because of decrease in feedback inhibition to the
mGCs. When passing a threshold ($\sim 0.3$), $\langle f_{i}^{(m)}\rangle$
crosses the horizontal dashed line (= 2.12 Hz), representing the population-
averaged MFR in the presence of only the mGCs (without the imGCs). It also
crosses the decreasing curve of $\langle f_{i}^{(im)}\rangle$ for $x\sim 0.2$,
and then converges to the limit value (= 2.30 Hz).

We also consider the case of whole GCs. For $x=1$, their population-averaged
MFR $\langle f_{i}^{(w)}\rangle$ (= 24.1 Hz) is high due to the effect of the
imGCs. However, with decreasing $x$ from 1 the effect of the imGCs is
decreased, and hence $\langle f_{i}^{(w)}\rangle$ is decreased until $x\sim
0.1$. Then, for $x<0.1$ $\langle f_{i}^{(w)}\rangle$ begins to increase slowly
and approach $\langle f_{i}^{(m)}\rangle$. Thus, in the limiting case of
$x=0,$ $\langle f_{i}^{(w)}\rangle=\langle f_{i}^{(w)}\rangle=$ 2.30 Hz which
is larger than the population-averaged MFR (= 2.12 Hz) in the case of
homogeneous population of only mGCs (without imGCs). For $x=0$ the imGCs
become completely inactive. Hence, the feedback inhibition to the mGCs becomes
decreased in comparison with the homogeneous case composed of only mGCs, which
leads to increase in firing activity of the mGCs in the heterogeneous
population of mGCs and imGCs.

Next, we characterize the degree of random spike skipping seen in the multi-
peaked ISI histogram in the case of $X=m,~{}im,$ or $w$ in terms of the random
phase-locking degree ${\cal L}_{d}^{(X)}$ of Eq. (10) (denoting how well
intermittent spikes make phase-locking to the IPSR $R^{(X)}(t)$ at random
multiples of its global period $T_{G}^{(X)}$). The sharper the random-spike-
skipping peaks in the ISI histogram are, the larger ${\cal L}_{d}^{(X)}$
becomes.

Figure 4(d) shows the plots of ${\cal L}_{d}^{(X)}$ versus $x$ for the mGCs
($X=m:$ green solid circles), the imGCs ($X=im:$ red open circles), and the
whole GCs ($X=w:$ blue crosses). In the case of mGCs, multi-peaked ISI
histograms appear due to random spike-skippings. As $x$ is decreased from 1,
their ISI histograms become clearer because multiple peaks become sharper and
their heights become increased. Thus, with decreasing $x$ from 1 to 0, ${\cal
L}_{d}^{(m)}$ is found to increase from 0.587 to 0.898. On the other hand, the
ISI histograms for the imGCs have single peaks. As $x$ is decreased from 1,
their single-peaked ISI histograms become broader, which leads to more random
spike skippings. But, at first, in the $n$th global cycle where the single
peak exists [e.g., $n=1$ (2) for $x=1.0$ (0.7)], the random phase locking
degree ${\cal L}_{d}^{(im)}(n)$ increases a little until $x$ is decreased to
$x^{*}(\sim 0.7)$. Thus, as $x$ is decreased from 1 to $x^{*},$ the overall
${\cal L}_{d}^{(im)}$ increases a little from 0.408 to 0.497. Then, for
$x<x^{*}$, ${\cal L}_{d}^{(im)}$ decreases rapidly to 0, in contrast to the
case of mGCs.

In the case of whole GCs, both the peak (associated with the imGCs) and the
multiple peaks (related to the mGCs) coexist in the histogram. Fractions of
the peaks vary depending on $x$. For $x=1$, dominant major peak, related to
the imGCs, appears, while fractions of multiple peaks, associated with the
mGCs, are very small. Thus, the value of ${\cal L}_{d}^{(w)}$ is close to that
of the imGCs. But, with decreasing $x$ from 1, the effect of the imGCs becomes
weakened, and fractions of the multiple peaks, associated with the mGCs,
become increased. Thus, as $x$ is decreased from 1, ${\cal L}_{d}^{(w)}$ is
found to increase from 0.457 and converge to ${\cal L}_{d}^{(m)}$ of the mGCs.
Thus, in the limiting case of $x=0,$ ${\cal L}_{d}^{(w)}={\cal
L}_{d}^{(m)}=0.898$ which is smaller than ${\cal L}_{d}$ (= 0.92) in the
homogeneous population of only the mGCs (without imGCs) [represented by the
dashed curve in Fig. 4(d)]. Hence, for all $x$ ${\cal L}_{d}^{(X)}$ of the
mGCs, the imGCs, and the whole GCs is smaller than that (= 0.92) in the
homogeneous case composed of only mGCs. As a result, in the whole range of
$x$, because of heterogeneity caused by the imGCs, individual firing behaviors
(characterized in terms of ${\cal L}_{d}^{(X)}$) of mGCs, imGCs, and whole GCs
in their SSRs become deteriorated, in comparison with the homogeneous case
consisting of only mGCs (without imGcs).

Figure 5: Quantitative relationship between SSRs and pattern separation and
integration in each case of mGCs, imGCs, and whole GCs. In the case of whole
GCs, plots of (a1) the pattern separation degree ${\cal S}_{d}^{(w)}$ versus
the amplitude measure ${\cal M}_{a}^{(w)}$ and (a2) ${\cal S}_{d}^{(w)}$
versus the random phase-locking degree ${\cal L}_{d}^{(w)}.$ In the case of
mGCs, plots of (b1) the pattern separation degree ${\cal S}_{d}^{(m)}$ versus
${\cal M}_{a}^{(m)}$ and (b2) ${\cal S}_{d}^{(m)}$ versus ${\cal
L}_{d}^{(m)}.$ In the case of imGCs, plots of (c1) the pattern integration
degree ${\cal I}_{d}$ versus ${\cal M}_{a}^{(im)}$ and (c2) ${\cal I}_{d}$
versus ${\cal L}_{d}^{(im)}.$ Fitted dashed lines are given in (a1)-(c1). In
(c2), fitted dashed and dotted lines are obtained from 8 data points for
$0.7\leq x\leq 0$ and 4 data points for $1\leq x\leq 0.7$, respectively. The
mGCs, imGCs, and whole GCs are denoted by green solid circles, red open
circles, and blue crosses, respectively.

Finally, we investigate quantitative association between SSRs and pattern
separation and integration. In NG-PS , by varying the synaptic connectivity
fraction $x$, we studied pattern separation (transforming similar input
patterns into less similar output patterns) of the mGCs ($X=m$) and the whole
GCs ($X=w$) in terms of the pattern separation degree ${\cal S}_{d}^{(X)}$
[see Fig. 4(e) in NG-PS ]. The pattern separation degree ${\cal S}_{d}^{(X)}$
is given by the ratio of $D_{p}^{(out)}$ (pattern distance for the output
pattern pair) to $D_{p}^{(in)}$ (pattern distance for the input pattern pair)
[see Eq. (17) in NG-PS ]. For ${\cal S}_{d}^{(X)}>1$ pattern separation
occurs, because the output pattern pair of the mGCs is more dissimilar than
the input pattern pair of the EC cells. On the other hand, for ${\cal
S}_{d}^{(X)}<1$ no pattern separation occurs; instead, pattern convergence
takes place. For the mGCs, with decreasing $x$ from 1, ${\cal S}_{d}^{(m)}$
was found to decrease from a high value (= 13.142) for $x=1$ to a limit value
(= 1.495) for $x=0$, as shown in Fig. 4(e) in NG-PS . Thus, in the whole range
of $0\leq x\leq 1$, the mGCs perform good pattern separation with ${\cal
S}_{d}^{(m)}>1$.

In contrast to the mGCs, the imGCs exhibit pattern integration (making
association between patterns), characterized in terms of the integration
degree ${\cal I}_{d}$. ${\cal I}_{d}$ is given by the average pattern
correlation degree for the output pattern pair of the imGCs to the average
pattern correlation degree for the input pattern pair of the EC cells [see Eq.
(18) in NG-PS ]. For $x=1$ the pattern integration degree ${\cal I}_{d}$ of
the imGCs is high (1.9559). With decreasing $x$ from 1 to 0, ${\cal I}_{d}$
was found to increase from 1.9559 to 2.2502, as shown in Fig. 4(f) in NG-PS .
Thus, in the whole range of $0\leq x\leq 1$, the imGCs are good pattern
integrators with ${\cal I}_{d}>1.$

Then, we consider the heterogeneous population of whole GCs (composed of mGCs
and imGCs). In this heterogeneous whole population, ${\cal S}_{d}^{(w)}$ for
$x=1$ is 0.659. Since ${\cal S}_{d}^{(w)}<1$ for $x=1$, no pattern separation
occurs, due to strong correlations between the imGCs. As $x$ is decreased from
1, the effect of the imGCs becomes reduced. Thus, when decreasing through a
threshold $x^{*}$ (= 0.92), pattern separation (with ${\cal S}_{d}^{(w)}>1$)
begins, and then the overall pattern separation degree ${\cal S}_{d}^{(w)}$
increases and converges to a limit value (= 1.577) for $x=0$ [see Fig. 4(e) in
NG-PS ].

Figures 5(a1)-5(a2) and 5(b1)-5(b2) show the plots of the pattern separation
degree ${\cal S}_{d}^{(X)}$ versus ${\cal M}_{a}^{(X)}$ and ${\cal
S}_{d}^{(X)}$ versus ${\cal L}_{d}^{(X)}$ for $X=w$ (whole GCs) and $m$
(mGCs), respectively. In the whole population of all the GCs, ${\cal
S}_{d}^{(w)}$ is found to be positively correlated with ${\cal M}_{a}^{(w)}$
and ${\cal L}_{d}^{(w)}$ of their SSR with the Pearson’s correlation
coefficients $r=0.9289$ and $0.9997$, respectively Pearson . Thus, in the
whole population of all the GCs, the larger ${\cal M}_{a}^{(w)}$ and ${\cal
L}_{d}^{(w)}$ in the SSR are, the better pattern separation efficacy becomes.
On the other hand, ${\cal S}_{d}^{(m)}$ for the mGCs are found to be
negatively correlated with ${\cal M}_{a}^{(X)}$ and ${\cal L}_{d}^{(X)}$ of
their SSR with the Pearson’s correlation coefficients $r=-0.9994$ and
$-0.9999$, respectively. Thus, in the population of the mGCs, the better
population and individual firing behaviors in their SSR are, the worse their
pattern separation becomes.

In the case of $X=im$ (imGCs), plots of the pattern integration degree ${\cal
I}_{d}$ versus ${\cal M}_{a}^{(im)}$ and ${\cal I}_{d}$ versus ${\cal
L}_{d}^{(im)}$ are shown in Figs. 5(c1)-5(c2). As in the case of mGCs, ${\cal
I}_{d}$ for the imGCs is also negatively correlated with ${\cal M}_{a}^{(im)}$
with the Pearson’s correlation coefficients $r=-0.8483$. Next we consider
quantitative relationship between ${\cal I}_{d}$ and ${\cal L}_{d}^{(im)}$. As
shown in Fig. 4(d), for $1.0\geq x\geq 0.7$, ${\cal L}_{d}^{(im)}$ makes a
small increase, and then for $x\leq 0.7$ it rapidly decreases to 0. Thus,
${\cal I}_{d}$ is also negatively correlated with ${\cal L}_{d}^{(im)}$ in the
range of $0.7\geq x\geq 0$ with the Pearson’s correlation coefficients
$r=-0.9159$, as in the case of ${\cal M}_{a}^{(im)}$, while in the initial
small range of $1.0\geq x\geq 0.7,$ ${\cal I}_{d}$ is positively correlated
with ${\cal L}_{d}^{(im)}$ with the Pearson’s correlation coefficients
$r=~{}0.9365$. Thus, in the population of the imGCs, for $0.7\geq x\geq 0$ the
better population and individual firing behaviors in their SSR are, the worse
their integration becomes.

##  IV Summary and Discussion

We studied the effect of adult neurogenesis on the SSRs of the GCs (mGCs,
imGCs, and whole GCs) in our DG spiking neural network. In comparison to the
mGCs, the imGCs show two competing distinct properties of high excitability
(causing high activation) and low excitatory innervation (reducing activation
degree). Thus, the effect of low excitatory innervation counteracts the effect
of high excitability. The connection probability $p_{c}$ from the EC cells and
the MCs to the imGCs is $20~{}x~{}\%$ [$x$ (synaptic connectivity fraction);
$0\leq x\leq 1$], in contrast to the case of mGCs with $p_{c}=20~{}\%$. With
decreasing $x$ from 1 to 0, population and individual firing behaviors of the
mGCs, the imGCs, and the whole GCs in their SSRs were characterized in terms
of the amplitude measure ${\cal M}_{a}^{(X)}$ ($X=m,~{}im,$ and $w$) and the
random phase-locking degree ${\cal L}_{d}^{(X)}$, respectively.

As shown in Fig. 4, as $x$ is decreased from 1, the amplitude measure ${\cal
M}_{a}^{(X)}$ and the random phase-locking degree ${\cal L}_{d}^{(X)}$ were
found to increase in the case of mGCs ($X=m$) and whole GCs ($X=w$). With
decreasing $x$ from 1, the effect of the imGCs became weaker, which resulted
in increase in ${\cal M}_{a}^{(X)}$ and ${\cal L}_{d}^{(X)}$ for the mGCs and
the whole GCs. In contrast, as $x$ decreases from 1, ${\cal M}_{a}^{(im)}$ of
the imGCs was found to monotonically decrease, and their ${\cal L}_{d}^{(im)}$
was found to first slowly a little increase and then rapidly decrease to zero
(i.e., for $x<0.7$, monotonic decrease to zero occurs). In this way, the
changing tendency for the imGCs was in contrast to those of the mGCs and the
whole GCs. We also note that in the heterogeneous population (consisting of
the mGCs and the imGCs), ${\cal M}_{a}^{(X)}$ and ${\cal L}_{d}^{(X)}$ ($X=m,$
$im,$ and $w$) were less than those in the homogeneous population of only mGCs
without imGCs. Due to heterogeneity caused by the imGCs, the population and
individual firing behaviors of the GCs in the SSRs became deteriorated, in
comparison with that in the presence of only mGCs (without imGCs).

Previously, in the whole range of $x$ (i.e., for $1\geq x\geq 0$), the mGCs
and the imGCs were found to exhibit pattern separation and pattern
integration, respectively NG-PS . As $x$ is decreased from 1, the pattern
separation degree ${\cal S}_{d}^{(m)}$ of the mGCs was found to decrease, as
shown in Fig. 4(e) in NG-PS , because their activation degree increased. In
contrast to the mGCs, the imGCs was found to show pattern integration, and its
degree ${\cal I}_{d}$ was found to increase as $x$ is decreased from 1, due to
increase in correlation between the imGCs [see Fig. 4(f) in NG-PS . Due to
presence of imGCs (good pattern integrators), the pattern separation efficacy
in the whole heterogeneous population (composed of mGCs and imGCs) was also
found to become deteriorated, as in the case of the SSR.

Quantitative association between SSRs and pattern separation and integration
was shown in Fig. 5. In the whole population of all the GCs, ${\cal
S}_{d}^{(w)}$ was found to be positively correlated with ${\cal M}_{a}^{(w)}$
and ${\cal L}_{d}^{(w)}$ of their SSR. Hence, in the whole population, the
larger ${\cal M}_{a}^{(w)}$ and ${\cal L}_{d}^{(w)}$ in the SSR are, the
better the pattern separation efficacy becomes, as in the homogeneous
population of only mGCs (without imGCs) PS . On the other hand, ${\cal
S}_{d}^{(m)}$ for the mGCs was found to be negatively correlated with ${\cal
M}_{a}^{(m)}$ and ${\cal L}_{d}^{(m)}$ of their SSR. Thus, in the population
of the mGCs, the better population and individual firing behaviors in their
SSR are, the worse their pattern separation efficacy becomes. Also, in the
case of imGCs, for $0.7\geq x\geq 0,$ ${\cal I}_{d}$ was found to be
negatively correlated to ${\cal M}_{a}^{(im)}$ and ${\cal L}_{d}^{(im)}$ of
their SSR. Hence, in the population of imGCs, for $0.7\geq x\geq 0,$ the
better population and individual firing behaviors in their SSR are, the worse
their pattern integration efficacy becomes.

Finally, we discuss limitations of our present work and future works. In the
present work, although correlations between the pattern separation and
integration degrees and the population synchronization and random phase-
locking degrees in the SSRs were found, this kind of correlations do not imply
causal relationship. Hence, in future work, it would be interesting to
intensively investigate their dynamical causation.

We also note that the pyramidal cells in the CA3 provide inhibitory
backprojections to the mGCs via polysynaptic connections, mediated by the BCs
and the HIPP cells (Myers2, ; Myers3, ; Scharfman, ). These inhibitory
backprojections may decrease the activation degree of the mGCs, resulting in
improvement of pattern separation in the population of the mGCs. Hence, as a
future work, it would be meaningful to study the effects of the
backprojections on pattern separation and SSR in the combined DG-CA3 network.

In the whole heterogeneous population of all the GCs (mGCs and imGCs), both
the pattern separation efficacy and the regularity of population and
individual firing activities in the SSR were found to get deteriorated, due to
presence of the imGCs (pattern integrators). But, 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 could be increased with pattern separation efficacy
Myers1 . On the other hand, in a heterogeneous population of mGCs (pattern
separators) and imGCs (pattern integrators), the memory storage capacity might
be optimally maximized via mixed encoding via pattern separation on similar
input patterns and pattern integration on very dissimilar input patterns NG9 ;
Hetero ; NG-PS . 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, although
regularity of population and individual firing activities in the SSR becomes
deteriorated. This speculation on increase in memory resolution via mixed
encoding (through cooperation of pattern separation and pattern integration)
must be examined in future works.

## 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).

##  Appendix A Leaky Integrate-and-Fire Models for Single Spiking Neurons

As elements of our DG spiking neural network, we choose LIF spiking neuron
models with additional AHP currents (determining the refractory period). The
following equations govern evolution of dynamical states of individual cells
in the $X$ population:

$\displaystyle C_{X}\frac{dv_{i}^{(X)}(t)}{dt}$ $\displaystyle=$
$\displaystyle-
I_{L,i}^{(X)}(t)-I_{AHP,i}^{(X)}(t)+I_{ext}^{(X)}-I_{syn,i}^{(X)}(t),$ (11)
$\displaystyle\;\;\;i=1,\cdots,N_{X},$

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 cell in the hilus. In
Eq. (11), $C_{X}$ (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 $I_{syn,i}^{(X)}(t)$. Here, we
consider a subthreshold case of $I_{ext}^{(X)}=0$ for all $X$ Chavlis .

In Eq. (11), the 1st type of leakage current $I_{L,i}^{(X)}(t)$ for the $i$th
neuron in the $X$ population is given by:

$I_{L,i}^{(X)}(t)=g_{L}^{(X)}(v_{i}^{(X)}(t)-V_{L}^{(X)}),$ (12)

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

$I_{AHP,i}^{(X)}(t)=g_{AHP}^{(X)}(t)~{}(v_{i}^{(X)}(t)-V_{AHP}^{(X)})~{}~{}~{}{\rm~{}for~{}}\;t\geq
t_{f,i}^{(X)}.$ (13)

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

$g_{AHP}^{(X)}(t)=\bar{g}_{AHP}^{(X)}~{}e^{-(t-t_{f,i}^{(X)})/\tau_{AHP}^{(X)}},$
(14)

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

##  Appendix B Three Types of Synaptic Currents

We consider the 4th type of synaptic current $I_{syn,i}^{(X)}(t)$ into the
$i$th neuron in the $X$ population in Eq. (11). The synaptic current
$I_{syn,i}^{(X)}(t)$ consists of the following 3 kinds of synaptic currents:

$I_{syn,i}^{(X)}(t)=I_{{\rm AMPA},i}^{(X,Y)}(t)+I_{{\rm
NMDA},i}^{(X,Y)}(t)+I_{{\rm GABA},i}^{(X,Z)}(t).$ (15)

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. On the other hand, $I_{{\rm
GABA},i}^{(X,Z)}(t)$ 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 in 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 neuron in the target $T$ population is
given by:

$I_{R,i}^{(T,S)}(t)=g_{R,i}^{(T,S)}(t)~{}(v_{i}^{(T)}(t)-V_{R}^{(S)}),$ (16)

where $g_{(R,i)}^{(T,S)}(t)$ and $V_{R}^{(S)}$ are 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:

$g_{R,i}^{(T,S)}(t)=K_{R}^{(T,S)}\sum_{j=1}^{N_{S}}w_{ij}^{(T,S)}~{}s_{j}^{(T,S)}(t),$
(17)

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.

On the other hand, 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 :

$g_{R,i}^{(T,S)}(t)={\widetilde{K}}_{R}^{(T,S)}~{}f(v^{(T)}(t))~{}\sum_{j=1}^{N_{S}}w_{ij}^{(T,S)}~{}s_{j}^{(T,S)}(t).$
(18)

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

$f(v^{(T)}(t))=\frac{1}{1+\eta\cdot[{\rm Mg}^{2+}]_{o}\cdot\exp(-\gamma\cdot
v^{(T)}(t))}.$ (19)

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. (17). 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. (17) in all the cases of $R=$ AMPA, NMDA,
and GABA.

The interpopulation synaptic connection from the source $S$ population (with
$N_{s}$ neurons) 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 neuron in the source $S$ population is pre-synaptic to the $i$th
neuron in the target $T$ population; otherwise $w_{ij}^{(T,S)}=0$.

The postsynaptic ion channels are opened through binding of neurotransmitters
(emitted from the source $S$ population) to receptors in the target $T$
population. 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
$E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)})$:

$s_{j}^{(T,S)}(t)=\sum_{f=1}^{F_{j}^{(s)}}E_{R}^{(T,S)}(t-t_{f}^{(j)}-\tau_{R,l}^{(T,S)}).$
(20)

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 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:

$E_{R}^{(T,S)}(t)=\frac{1}{\tau_{R,d}^{(T,S)}-\tau_{R,r}^{(T,S)}}\left(e^{-t/\tau_{R,d}^{(T,S)}}-e^{-t/\tau_{R,r}^{(T,S)}}\right)\cdot\Theta(t).$
(21)

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).

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