## Plastic neural network with transmission delays
promotes equivalence between function and structure

P. R. Protachevicz1,2∗, F. S. Borges3,4, A. M. Batista5,6, M. S. Baptista2, I.
L. Caldas1, E. E. N. Macau7, E. L. Lameu8

1Physics Institute, University of São Paulo, São Paulo, SP, Brazil.
2Institute for Complex Systems and Mathematical Biology, SUPA, University of
Aberdeen, Aberdeen, United Kingdom.
3Department of Physiology and Pharmacology, State University of New York
Downstate Health Sciences University, Brooklyn, NY, USA
4Center of Mathematics, Computation and Cognition, Federal University of ABC,
São Bernardo do Campo, SP, Brazil.
5Post-Graduation in Science, State University of Ponta Grossa, Ponta Grossa,
PR, Brazil.
6Mathematics and Statistics Department, State University of Ponta Grossa,
Ponta Grossa, PR, Brazil.
7Federal University of São Paulo, São José dos Campos, SP, Brazil.
8 Cell Biology and Anatomy Department, Hotchkiss Brain Institute, University
of Calgary, Calgary, Alberta, Canada.
<EMAIL_ADDRESS>

The brain is formed by cortical regions that are associated with different
cognitive functions. Neurons within the same region are more likely to connect
than neurons in distinct regions, making the brain network to have
characteristics of a network of subnetworks. The values of synaptic delays
between neurons of different subnetworks are greater than those of the same
subnetworks. This difference in communication time between neurons has
consequences on the firing patterns observed in the brain, which is directly
related to changes in neural connectivity, known as synaptic plasticity. In
this work, we build a plastic network of Hodgkin-Huxley neurons in which the
connectivity modifications follow a spike-time dependent rule. We define an
internal-delay among neurons communicating within the same subnetwork, an
external-delay for neurons belonging to distinct subnetworks, and study how
these communicating delays affect the entire network dynamics. We observe that
the neuronal network exhibits a specific connectivity configuration for each
synchronised pattern. Our results show how synaptic delays and plasticity work
together to promote the formation of structural coupling among the neuronal
subnetworks. We conclude that plastic neuronal networks are able to promote
equivalence between function and structure meaning that topology emerges from
behaviour and behaviour emerges from topology, creating a complex dynamical
process where topology adapts to conform with the plastic rules and firing
patterns reflect the changes on the synaptic weights.
Key-words: Synaptic plasticity; Hodgkin-Huxley model; Neural dynamics;
Synchronization

##  1 Introduction

Signal transmission delays are an intrinsic property of neuronal communication
and are extremely relevant in brain activities [1, 2]. Asl et al. highlighted
the importance of realistic time delays, emphasising that this property has
just been marginally taking into account the used models over the decades [3].
The synaptic conduction delay is proportional to the axonal distance from the
soma [4] and the experimental conduction velocity is 300 $\mu$m/ms in rat [5].
Therefore, neurons that are closer together have lower latencies, while
communication between neurons in different hemispheres of the brain may have
delays of a few hundred milliseconds [6]. Lameu et al. [7] showed that small
values of the synaptic delay are related to synchronisation while non-trivial
topology is presented in networks with high delay. However, in some cases, the
synaptic delay can be associated with synchronisation suppression [8, 9].
Different transmission or synaptic delays can generate synchronous and
asynchronous neuronal activities [10, 11].

Maps of neuronal circuits obtained from functional magnetic resonance imaging
(fMRI) show the existence of modules in brain networks [12]. Sporns and Betzel
[13] discussed how this modular formation may be related to brain evolution
favouring the emergence of functional specialisation and complex dynamics. Lin
et al. [14] observed the fMRI and electrophysiological timing information
delays during a visuomotor reaction-time task across five brain regions.
Latency matrices of 9321 brain subregions from fMRI signals showed structures
associated with different sensory states and brain functions [15]. Sun et al.
demonstrated that certain intra and inter-networks delays can facilitate fast
regular firings [16]. Know and Choe [17] pointed to a possible role of
facilitation dynamics to compensate such delays in motor neurons by a
systematic approach.

Synaptic plasticity is the fundamental ability of the neurons to change their
connection intensities due to spike activities [18]. It has been reported that
memory and learning functions are supported by synaptic plasticity in the
brain [19]. Experimental data suggest that synchronous spike dynamics patterns
can be intensified due to the plasticity, being associated with long-term
memory processing. However, it is not completely clear how plasticity and
synchronisation are related [20]. Synaptic plasticity occurs by updating the
connection strength depending on the difference of spiking times between pre
and postsynaptic neurons. This function bounds topology with spiking time
intervals and therefore with synchronisation. Different from networks that
have fixed coupling, where synchronous behaviour emerges from topology, in the
plastic network, topology emerges from behaviour and behaviour emerges from
topology. This mutual influence creates a complex dynamical process where
topology adapts to conform with the plastic rules and firing patterns reflect
the changes on the synaptic weights that is not completely understood.

Kim and Lim [21] studied spike synchronisation in the presence of excitatory
spike time-dependent plasticity (STDP). They observed a Matthew effect in
synaptic plasticity due to a positive feedback mechanism. In a similar
framework, they also study the effect of inhibitory STDP for the same network
topology showing that both depression and potentiation of inhibitory
connections can occur [22]. Studying the cerebellar ring network with synaptic
plasticity, they still found that phase, anti-phase, and complex out-of-phase
activities are involved in the long-term depression [23]. Soltoggio and
Stanley [24] reported the relationship between local Hebbian plasticity and
learning using a computational approach focused on the noise and weight
saturation.

Aoki [25] demonstrated the self-organisation phenomena in a recurrent network
of oscillators in the presence of synaptic plasticity identifying phase, anti-
phase, coherent and chaotic activities. Phase, anti-phase and phase-lock
activities are the main types of synchronisation observed between brain
regions [26, 27, 28, 29]. Klimesch et al. [30] reported the importance of
phase synchronisation in various cognitive processes. Phase synchronisation
has been observed in distant cortical areas with long conduction delays [31].
Some works also showed that time delay, synaptic types and connection
densities play an important role in anti-phase synchronisation [32, 33].

In this work, we study how the emergence of synchronised symmetric patterns
between the subnetworks depends on the absence and presence of long-term
plasticity and on internal and external transmission delays. The anatomy of
neurons is directly related to the values of internal and external delays
between subnetworks. The transmission velocity of the signal depends on the
morphology of the dendrites and axons [34, 35] and the cell processing time
[36]. Moreover, the time required for neurons to communicate may be
significantly extended due to the physical distance between the sending and
receiving cells [37]. Neurons in the same subnetwork have propagation delays
associated with dendritic trees, with values ranging from submilliseconds to a
few milliseconds [38, 39]. In connections between different subnetworks, the
axonal propagation delay contributes greatly to signal latency [40].
Considering that internal delays among neurons inside a subnetwork are smaller
than the external ones between subnetworks, we find that distinct patterns of
synchronisation can be achieved by changing the delays. In the presence of
plasticity, one of main motivations is to investigate how the connections
between the subnetworks evolve from initially random to a more structured
configuration. In this context, we are able to identify the stronger
connections between the subnetworks related to specific synchronised patterns.

The increase of internal delays reduces the synchronous patterns inside the
subnetworks. External delays, on the contrary, can promote collective
synchronisation. Subnetworks in one group symmetry whose neurons are connected
with small external delays are less synchronous than those neurons connected
via large delays. In addition to that, the synchronisation in one group is
higher than for two, three, and four group symmetry. In particular, we note
that different types of neuronal synchronisation are related to the network
structure between the subnetworks. Our results also suggest that synchronous
behaviour reveals the structure being created due to the plasticity. More
specifically, functional communities and their connections inferred by
measurements of neural phase synchronisation reflect the subnetworks and their
linkage structure provided by the structural topology of the synapses.

The paper is organised as follows: In Section 2, we introduce our plastic
neuronal network of coupled HH neurons and the diagnostic tool to identify
synchronisation. In Section. 3, we present the results of our study about the
effects of synaptic delays between neuronal subnetworks. In the last section,
we draw our conclusions.

##  2 Plastic neuronal network

###  2.1 Hodgkin-Huxley model

We consider the type-II neuron model proposed by [41]. The individual dynamics
of each Hodgkin-Huxley (HH) neuron in the network is given by

$\displaystyle C\dot{V_{i}}$ $\displaystyle=$ $\displaystyle I_{i}-g_{\rm
K}n_{i}^{4}(V_{i}-E_{\rm K})$ (1) $\displaystyle-g_{\rm
Na}m_{i}^{3}h_{i}(V_{i}-E_{\rm Na})-g_{L}(V_{i}-E_{\rm L})$
$\displaystyle+(V_{\rm
r}^{\rm+}-V_{i})\sum_{j=1}^{N}g_{ij}f_{j}(t-\tau_{ij}),$
$\displaystyle\dot{n}_{i}$ $\displaystyle=$
$\displaystyle\alpha_{n_{i}}(v_{i})(1-n_{i})-\beta_{n_{i}}(v_{i})n_{i},$ (2)
$\displaystyle\dot{m}_{i}$ $\displaystyle=$
$\displaystyle\alpha_{m_{i}}(v_{i})(1-m_{i})-\beta_{m_{i}}(v_{i})m_{i},$ (3)
$\displaystyle\dot{h}_{i}$ $\displaystyle=$
$\displaystyle\alpha_{h_{i}}(v_{i})(1-h_{i})-\beta_{h_{i}}(v_{i})h_{i},$ (4)
$\displaystyle\dot{f}_{i}$ $\displaystyle=$
$\displaystyle\frac{-f_{i}}{\tau_{\rm s}}.$ (5)

Equation (1) represents the membrane dynamics of the neuron $i$. $C$
($\mu$F/cm2) is the membrane capacitance and $I_{i}$ ($\mu$A/cm2) is a
constant current density chosen in the interval $[10,11]$. Depending on the
$I_{i}$ value, the HH model can exhibit silence, bistability, and repetitive
spike firings [42, 43, 44, 45]. Once one understands well how bifurcations and
dynamical behaviour happens as a parameter is changed in a nonlinear dynamical
system, it is natural to take that parameter as the one to induce
heterogeneity in a network. Following this idea, we have considered the
heterogeneity in the constant current in the regime of repetitive spike
firings as done in the references [46, 47, 48, 49]. The parameters $g_{\rm
K}$, $g_{\rm Na}$, and $g_{\rm L}$ are the conductance of the potassium,
sodium, and leak ion channels, respectively. $E_{\rm K},E_{\rm Na}$, and
$E_{\rm L}$ are the reversal potentials for these ion channels. $V_{\rm
r}^{\rm+}$ corresponds to the excitatory reversal potential. $g_{ij}$ is the
excitatory coupling strength from the presynaptic neuron $j$ to the
postsynaptic neuron $i$ with maximum and minimum value within the interval
$[0.0,0.01]$. We consider that the neuron has no self-connections, implying
$g_{ii}=0$. $f_{i}(t)$ is the normalised synaptic current from the neuron $j$
to $i$. The state variable $f_{i}$ decays exponentially and it is updated to
the unity ($f_{i}\to 1$) at the spike time $t_{i}$ of the neuron $i$ [50, 51].
Equation (5) corresponds to an exponential decay on the evolution of
$f_{i}(t)$. The parameter $\tau_{\rm s}$ is the synaptic time decay and
$\tau_{ij}$ the delay on the synaptic transmission [52, 53, 54]. The intensity
of synaptic current with time delay on the signal transmission depends on the
state of the pre and postsynaptic neuron. $\tau_{ij}$ assumes values
$\tau_{\rm int}$ and $\tau_{\rm ext}$ for internal and external connections
between the subnetworks, respectively. In Eqs. (2) and (3), the functions
$m(v_{i})$ and $n(v_{i})$ represent the sodium and potassium activation,
respectively. In Eq. (4), $h(v_{i})$ is the function for sodium inactivation.
The functions $\alpha_{n}$, $\beta_{n}$, $\alpha_{m}$, $\beta_{m}$,
$\alpha_{h}$, and $\beta_{n}$ are given by

$\displaystyle\alpha_{n}(v)$ $\displaystyle=$
$\displaystyle\frac{0.01v+0.55}{1-\exp\left(-0.1v-5.5\right)},$ (6)
$\displaystyle\beta_{n}(v)$ $\displaystyle=$ $\displaystyle
0.125\exp\left(\frac{-v-65}{80}\right),$ (7) $\displaystyle\alpha_{m}(v)$
$\displaystyle=$ $\displaystyle\frac{0.1v+4}{1-\exp\left(-0.1v-4\right)},$ (8)
$\displaystyle\beta_{m}(v)$ $\displaystyle=$ $\displaystyle
4\exp\left(\frac{-v-65}{18}\right),$ (9) $\displaystyle\alpha_{h}(v)$
$\displaystyle=$ $\displaystyle 0.07\exp\left(\frac{-v-65}{20}\right),$ (10)
$\displaystyle\beta_{h}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp\left(-0.1v-3.5\right)},$ (11)

where $v=V/[{\rm mV}]$. In our simulations, we consider $C=1$ $\mu$F/cm2,
$E_{\rm Na}=50$ mV, $E_{\rm K}=-77$ mV, $E_{\rm L}=-54.4$ mV, $g_{\rm Na}=120$
mS/cm2, $g_{\rm K}=36$ mS/cm2, $g_{\rm L}=0.3$ mS/cm2, and $\tau_{\rm
s}=2.728$ ms. The reversal potential for excitatory connections is
$V^{\rm+}_{\rm r}=20$ mV [55]. For the numerical integration, we use the
Runge-Kutta fourth-order method with a time step equal to $\delta t=0.01$ ms.

###  2.2 Spike-time dependent plasticity

Spike-time dependent plasticity (STDP) is a process that produces changes in
the synaptic strength. It is calculated taking into consideration the times
between the spikes of the postsynaptic neuron $t_{i}$ and the presynaptic
neuron $t_{j}$. For each synaptic connection, the presynaptic neuron is that
one which sends the signal while the postsynaptic neuron receives such signal.
The intensity of the excitatory synaptic weight is defined by the coupling
strength. The change in the excitatory synaptic weights $\Delta g_{ij}$ due to
the time difference $\Delta t_{ij}=t_{i}-t_{j}$ is given by [56, 57, 58, 59]

$\Delta g_{ij}=\left\\{\begin{array}[]{ll}\displaystyle A_{1}{\rm e}^{(-\Delta
t_{ij}/\tau_{1})}\;\;\;,\;\;\;{\rm if}\;\;\;\Delta t_{ij}\geq 0\\\
\displaystyle-A_{2}{\rm e}^{({\Delta t_{ij}/\tau_{2}})}\;\;,\;\;\;{\rm
if}\;\;\;\Delta t_{ij}<0\\\ \end{array}\right.$ (12)

where $A_{1}=1$, $A_{2}=0.5$, $\tau_{1}=1.8$ ms, and $\tau_{2}=6$ ms. The
synaptic weights are updated according to Eq. (12), where $g_{ij}\rightarrow
g_{ij}+G\cdot\Delta g_{ij}$. The change rate of the synaptic weight is
considered as $G=10^{-5}$ mS/cm2. The initial value of all excitatory synaptic
weights is given by $g_{ij}=0.001$ mS/cm2. The minimal and maximal excitatory
synaptic weight are considered in the interval [$g_{\rm min}$, $g_{\rm max}$]
= [0,0.1] nS.

Figure 1 displays the plasticity curves described by Eq. (12) (red line) as a
function of $\Delta t_{ij}$. This figure shows that the difference on the
spike times in which plastic rule generates synaptic depression or synaptic
potentiation. High values of the time difference generate synaptic changes
close to zero, as shown in the figure. To exemplify such update protocol,
suppose that there is a synaptic connection from the neuron $i$ to neuron $j$,
as shown in the inset of Figure 1. Neurons $i$ and $j$ are represented by the
circles while the synaptic connection from neuron $i$ to neuron $j$ is
represented by the arrow. If neuron $j$ spikes before the neuron $i$, $\Delta
t_{ij}$ will be negative, and the change in the synaptic connection from
neuron $i$ to neuron $j$ ($\Delta g_{ij}$) will be negative (for small
$|\Delta t_{ij}|$) or zero (for large $|\Delta t_{ij}|$). In other way,
considering the same synaptic connection from neuron $i$ to neuron $j$, but
with neuron $i$ spiking before that neuron $j$, $\Delta t_{ij}$ will be
positive, and the change in the synaptic connection ($\Delta g_{ij}$) will be
positive (for small $|\Delta t_{ij}|$) or zero (for large $|\Delta t_{ij}|$).

Figure 1: Plasticity curves as a function of $\Delta t_{ij}$ for excitatory
synapses. Negative and positive plus zero values of $\Delta t_{ij}$ correspond
to synaptic depression and potentiation, respectively. The inset shows a
schematic representation of the calculation of time difference $\Delta t_{ij}$
for a connection from neuron $i$ to neuron $j$, for the case of synaptic
depression ($\Delta t_{ij}<0$) and synaptic potentiation ($\Delta t_{ij}\geq
0$).

###  2.3 A network of subnetworks

We consider $N=400$ non-identical HH neurons separated into $S=4$ subnetworks
with $N_{\rm sub}=100$ neurons each one ($N=S\cdot N_{\rm sub}$). The
heterogeneity in the system is given by the neuron currents $I_{i}$. To
facilitate the visualisation and interpretation of our results, we sorted
neurons in each subnetwork in ascending order according to their spiking
frequency (or $I$). Therefore, the neuron $i=1$ has the slowest spiking
frequency and the neuron $i=100$ has the highest one. The neurons are
connected by means of excitatory synapses. For the initial coupling
configuration, each subnetwork has an internal all-to-all topology without
self-connections (autapses) [60]. The connections between subnetworks or
external ones are randomly distributed with a certain probability. Thus, the
internal and external probability of connections are given by $p_{\rm int}=1$
and $p_{\rm ext}=0.05$, respectively [61, 62]. New connections are not allowed
between subnetworks, however, changes in the weights of initial external
connections are permitted. The network does not evolve to a configuration of
only one community due to the plasticity due to the fixed internal connection
probability between the subnetwork. With regard to the subnetworks, we
consider an internal and external transmission delay given by $\tau_{\rm int}$
and $\tau_{\rm ext}$, respectively.

###  2.4 Measuring synchronisation and symmetries

In order to study neuronal synchronisation and symmetries, we compute the
order parameter. Firstly, we use the traditional Kuramoto order parameter as a
diagnostic tool for the whole network, that is given by [63]

$R_{\rm T}(t)=\left|\frac{1}{N}\sum_{j=1}^{N}e^{{\rm i}\phi_{j}(t)}\right|,$
(13)

where “${\rm i}$” is the imaginary unit $\sqrt{-1}$ and $\phi_{j}(t)$ is the
neural phase associated with the spikes of each neuron $j$, given by

$\phi_{j}(t)=2\pi\frac{t-t_{j,k}}{t_{j,k+1}-t_{j,k}},$ (14)

$t_{j,k}$ is the time when a $k$-th spike ($k=0,1,2,\dots$) happens in the
neuron $j$ ($t_{j,k}<t<t_{j,k+1}$).

The time-average order parameter for the network is given by

${\bar{R}}=\frac{1}{t_{\rm fin}-{t_{\rm ini}}}\sum_{t_{\rm ini}}^{t_{\rm
fin}}R_{\rm T}(t),$ (15)

in which $t_{\rm fin}-t_{\rm ini}$ is the time window set to measure the
phases, where $t_{\rm ini}$ and $t_{\rm fin}$ correspond to the initial and
final time of the analyses, respectively. In our simulations, we consider
$t_{\rm ini}=80$ s and $t_{\rm fin}=100$ s. The magnitude of the time-average
order parameter tends to the unity when the network has a globally
synchronised behaviour. For uncorrelated spiking phases, the order parameter
is close to $0$.

The traditional Kuramoto order parameter for each subnetwork, $s=1,2,3,$ and
$4$, is described as

$R_{(s)}(t)=\left|\frac{1}{N_{\rm sub}}\sum_{j=(s-1)\cdot N_{\rm
sub}+1}^{s\cdot N_{\rm sub}}e^{{\rm i}\phi_{j}(t)}\right|.$ (16)

Figure 2: Raster plots of all neurons $i$ for fixed internal ($\tau_{\rm
int}$) and external ($\tau_{\rm ext}$) time delays when synaptic plasticity is
active. Different colours denote each subnetwork. Figures (a-d), (e-h), and
(i-l) display the raster plots for the internal time delays equal to
$\tau_{\rm int}=0$ ms, $\tau_{\rm int}=3$ ms, and $\tau_{\rm int}=6$ ms,
respectively. Figures (a,e,i), (b,f,j), (c,g,k), (d,h,l) display the raster
plots for external time delays equal to $\tau_{\rm ext}=0$ ms, $\tau_{\rm
ext}=4$ ms, $\tau_{\rm ext}=6$ ms, and $\tau_{\rm ext}=10$ ms, respectively.
For small internal delays, high synchronous patterns are observed in each
subnetwork, while bigger internal ones generate less synchronised patterns.

In order to quantify and distinguish the different symmetric synchronisation
patterns, we calculate the so-called $m$-th moment of the order parameter
$R^{m}$ ($m$ is an index), that is a variation of Eq. (13) with $m=1,2,...,S$
[64, 65, 66]

$R^{m}=\left|\frac{1}{N}\sum_{j=1}^{N}{\rm e}^{{\rm i}m\phi_{j}(t)}\right|.$
(17)

The summation in Eq. (17) considers the phases of all neurons in the network.
This measure allows us to quantify the number of synchronised neuronal groups
and consequently the symmetry of their phase distributions. The calculation of
the $m-th$ moment is similar to the traditional order parameter with the
difference that considers $m=$1, 2, 3, or 4, multiplying each neuron phase.
For the particular case where $m$=1, Eq. (17) is the same as Eq. (13). For the
network composed of $S=4$ subnetworks, we calculate all the moments
$m\in[1,4]$. The moment $R^{m}$ with the highest intensity (closer to 1)
provides information about how the subnetworks are synchronised among them.
They can vary from one big synchronised group to two or four groups, showing a
fix phase difference among them. In this framework, the highest moment of
order parameter gives us the information about synchronisation and type of
symmetry configuration. For instance, if $R^{1}$ ($m=1$) has the highest value
(close to 1), the subnetworks have neurons forming effectively a single large
network ( small phase difference between subnetworks). If $R^{2}$ ($m$=2) is
the highest value, the neurons in subnetworks are synchronised in $2$ groups
in an anti-phase pattern (phase difference of $\pi$). The same idea applies
for $m=3$ and $m=4$, where there are 3 and 4 groups, and the neurons in the
groups have approximately $2\pi/3$ and $\pi/2$ phase differences,
respectively. Table 1 exhibits the standard range of parameters that we
consider in our simulations.

Table 1: Descriptions of the standard parameters and range values considered in our simulations. Descriptions | Parameter | Value
---|---|---
Number of subnetworks | $S$ | 4
Neurons per subnetwork | $N_{\rm sub}$ | $100$
Internal connect. probab. | $p_{\rm int}$ | $1.0$
External connect. probab. | $p_{\rm ext}$ | $0.05$
Internal time delay | $\tau_{\rm int}$ |  $[0,6]$ ms
External time delay | $\tau_{\rm ext}$ |  $[0,12]$ ms
Exc. synaptic conductance | $g_{\rm exc}$ |  $[0,0.01]$ mS/cm2
Membrane capacity | $C$ | 1.0 $\mu$F/cm2
Potassium conductance | $g_{\rm K}$ | 36 mS/cm2
Sodium conductance | $g_{\rm Na}$ | 120 mS/cm2
Leak conductance | $g_{\rm l}$ | 0.3 mS/cm2
Potassium rev. potential | $V_{\rm K}$ | -77 mV
Sodium reversal potential | $V_{\rm Na}$ | 50 mV
Leak reversal potential | $V_{\rm l}$ | -54.4 mV
Excitatory reversal potential | $V_{\rm r}^{\rm+}$ | 20 mV
Constant current | $I_{i}$ | [10,11] $\mu$A/cm2
Change rate of synap. weight | $G$ |  $10^{-5}$ mS/cm2.
Time step integration | $\delta t$ |  $10^{-2}$ ms
Initial time for analyses | $t_{\rm ini}$ | 80 s
Final time for analyses | $t_{\rm fin}$ | 100 s
Internal time delay | $d_{\rm int}$ | [0,1] ms
External time delay | $d_{\rm ext}$ | [0,12] ms
Time step integration | $\delta t$ |  $10^{-2}$ ms
Figure 3: The panel (a) displays the initial matrix configuration for all
simulations, where the red colour represents the initial intensity of non null
synaptic connections. Black indicates no synapses and yellow approximates to
the maximal synaptic conductance $g_{ij}^{\rm max}$=0.01 mS/cm2. The panel (b)
shows the initial phases of all neurons in the network. The top panel in (c)
exhibits the resultant matrix for fixed internal and external time delays. In
bottom in panel (c), the traditional Kuramoto order parameter for the entire
network is represented by a black arrow ($R_{\rm T}(t)$), as well as the same
order parameter considering each subnetwork is denoted by the red,
violet,cyan, and green arrows ($R_{(S)}$).

##  3 Results and Discussions

In this work, we consider internal ($\tau_{\rm int}$) and external ($\tau_{\rm
ext}$) delays in the subnetworks with and without the presence of STDP.
Without plasticity, for small $\tau_{\rm int}$ and varying $\tau_{\rm ext}$,
we observe different patterns of synchronisation between the subnetworks.
However, our main goal is to investigate how these patterns of synchronisation
affect the weights of network connections when plasticity is active. To do
this, we consider different delay values which modify the dynamics of the
network. We restricted the network to only excitatory neurons since the
presence of inhibition in the current simulation generates very similar
results.

Figure 2 shows the raster plots for fixed internal and external time delays
($\tau_{\rm int}$ and $\tau_{\rm ext}$) when the synaptic plasticity is on. In
the left side, we consider $\tau_{\rm int}=0$ ms, (a) $\tau_{\rm ext}=0$ ms,
(b) $\tau_{\rm ext}=4$ ms, (c) $\tau_{\rm ext}=6$ ms, and (d) $\tau_{\rm
ext}=10$ ms. For small internal time delays, we verify synchronised symmetric
patterns. In the center column, we consider $\tau_{\rm int}=3$ ms, (e)
$\tau_{\rm ext}=0$ ms, (f) $\tau_{\rm ext}=4$ ms, (g) $\tau_{\rm ext}=6$ ms,
and (h) $\tau_{\rm ext}=10$ ms. For these parameters, we observe no firing
coherence, but the fastest neurons (higher $I_{i}$) in each subnetwork start
firing and subsequently the slower neuron. In the right side, we use
$\tau_{\rm int}=6$ ms, (i) $\tau_{\rm ext}=0$ ms, (j) $\tau_{\rm ext}=4$ ms,
(k) $\tau_{\rm ext}=6$ ms, and (l) $\tau_{\rm ext}=10$ ms. Although some
synchronisation can be noticed, it is lower than in the case for $\tau_{\rm
int}=0$ ms. For $\tau_{\rm int}=0$ ms, we identify an equal pattern for the
case with and without plasticity, as shown in Figure 2(a-d).

We focus on the most synchronised symmetric patterns. In Figure 2(a), neuron
spikes in a single group (almost complete phase synchronisation) without delay
between the subnetworks. Highest order parameter is the one with order $m$=1,
indicating all neurons spiking nearly synchronously. Figure 2(b) displays
neurons spiking in two groups for an external time delay equal to $d_{\rm
ext}=4$ ms. Highest order parameter is the one with order $m$=2. In Figure
2(c), the neurons spike in four groups for an external time delay equal to
$d_{\rm ext}=6$ ms. Highest order parameter is the one with order m=4. For a
delay close to the average period between spikes ($\approx 14$ ms) the network
returns to a single group, as shown in Figure 2(d). In this case, the neurons
of all subnetworks exhibit a strong phase synchronisation. These patterns can
also be obtained for different parameters when there is no plasticity.

Figure 3(a) displays the initial matrix connections $g_{ij}$ of the
subnetworks. In Figure 3(b), we plot the initial neuronal phases where each
colour represents a subnetwork. The coloured arrows are the initial order
parameters showing that initially the neurons are not synchronised. Figure
3(c) exhibits the final coupling matrix after the plasticity actuates by $100$
s, considering different values of the external delays and fixed internal
delay $\tau_{\rm int}=0$ ms. For $\tau_{\rm ext}=0$ ms, we see a strong
coherent dynamics among the subnetworks, all organised in a single group. The
resulting network presents some hierarchical organisation where the directed
connections between some subnetworks were reinforced as highlighted by the
dashed blue squares in Figure 3(c) (first column). For $\tau_{\rm ext}=4$ ms,
the network effectively forms two pairs of subnetworks with neurons belonging
to different subnetworks having a constant phase difference around $\pi$
radians (anti-phase synchronisation). In this case, connections between in-
phase subnetworks are potentiated and depressed for the anti-phase ones. For
$\tau_{\rm ext}=6$ ms, the network effectively presents four groups where
neurons within each pair of groups have a phase difference around $\pi/2$
radians and the network is set to a configuration of subnetworks being
connected under a ring topology. Considering $\tau_{\rm ext}=10$ ms, the
network goes effectively to one phase group formation, and the network
topology shows no preferential connection among the subnetworks. Therefore,
when the delay values come close to the time period of the spikes, the
behaviour is similar to that observed when the connection delay between the
subnetworks is close to zero and the synchronisation is improved. As the
difference for one group synchronisation with small time delays, all
connections between the subnetworks are potentiated.

Figure 4: Resultant mean synaptic weight (left) and schematic representation
of resultant connections between the subnetworks (right) for $\tau_{\rm
int}=0$ ms. The edges in the graphs plotted on the right column panel
represent the presence of strong connections from one subnetwork to another.
We consider (a) $\tau_{\rm ext}=0$ ms, (b) $\tau_{\rm ext}=4$ ms, (c)
$\tau_{\rm ext}=6$ ms and (d) $\tau_{\rm ext}=10$ ms.

In the synchronised regimes, the phase difference between groups is roughly
constant, as seen in Figure 3(c). The traditional Kuramoto order parameter is
not capable to detect and classify all those synchronised configurations in
addition to a single group. For this reason, we consider the $m$-th moments of
the order parameter. The moments of the order parameter is a suitable
diagnostic of symmetry between the synchronised subnetworks.

Figure 5: The panels (a) and (b) display the symmetric patterns found in the
parameter space of $\tau_{\rm int}$ and $\tau_{\rm ext}$ without and with
plasticity. The panels (c) and (d) show the highest value of the order
parameters moments in the space parameter of $\tau_{\rm int}$ and $\tau_{\rm
ext}$ for the case without and with synaptic plasticity.

The left side of Figure 4 shows the mean synaptic coupling between the
subnetworks computed with the matrices from Figure 3(c). The schematic
representations of the stronger mean connection between the subnetworks are
displayed on the right side. The strongest weight connections between the
subnetworks can be associated to the average matrix, as highlighted by the
blue dashed squares. The black line with a triangle at the end corresponds to
the connection direction from a presynaptic subnetwork to a postsynaptic one.
In Figure 4(a), the strong phase synchronisation potentiates connections
between the subnetworks in an asymmetric way. In this case, the neurons in the
subnetwork $1$ spike first, followed by subnetworks 2, 4, and 3. STDP is
responsible for reshaping the network leading to the configurations depicted
in Figure 4(a), where the connection reinforcement follows the order of
spikes. All connections from subnetwork 1 to 2, 3, and 4 are reinforced. The
same happens with the connections from subnetwork 2 to 3 and 4, and from
subnetwork 4 to 3. Figure 4(b) exhibits two groups in anti-phase. The
connections are reinforced between in-phase subnetworks and weakened between
the two groups. In Figure 4(c), the subnetworks show a phase-lock
synchronisation with average phase difference of $\pi$ radians. These patterns
lead to a ring network configuration. The connections are potentiated in a
cyclic way. The subnetworks are in-phase synchronisation in Figure 4(d) in a
more coherent state than in Figure 4(a). This strong synchronisation promotes
an all-to-all connection organisation among subnetworks with an average
reinforcement of the connections.

To better understand how plasticity promotes synchronised patterns, in Figure
5, we calculate the $m$-th moments for $\tau_{\rm int}=0$ ms, (a) $\tau_{\rm
ext}=0$ ms, (b) $\tau_{\rm ext}=4$ ms and (c) $\tau_{\rm ext}=6$ ms. To
calculate the $R^{m}$, we consider the phase temporal evolution of all neurons
of the network in Equation 17, independent of their respective subnetwork, for
each moment $m$ = 1, 2, 3, and 4. We observe a higher 1-st moment in Figure
5(a), which corresponds to a one-group phase synchronisation between the
subnetworks. In this case, we verify that other moments are relatively lower
than the first one. Figure 5(b) displays a higher 2-nd moment due to an anti-
phase synchronisation in two major groups. In Figure 5(c), we identify a
higher 4-th moment, which is associated with a phase-lock synchronisation
between the subnetworks, as shown in Figure 3(c). The higher moment of the
order parameter indicates the symmetry of the synchronised patterns. It is
worth to mention, that as can be seen in Figures 3(a-c), the moments present a
constant value overtime. Figures 5 (d) and (e) exhibit the symmetric
synchronised patterns in the parameter space $\tau_{\rm ext}\times\tau_{\rm
int}$ without and with synaptic plasticity, respectively. In the case without
plasticity and small external time delay, we find one group symmetry with
phase synchronisation. Increasing the external delays, we identify
predominantly 2 and 4 groups symmetry, respectively. For larger external
delays (around 10 ms), the network returns to a one synchronised group. These
cases are found for small internal time delays (less than 1 ms). For higher
internal time delays (more than 1 ms), we observe less synchronised patterns.
On the other hand, synaptic plasticity shifts to the left in $\tau_{\rm ext}$,
in which the regions where one, two, and four group symmetries are found.
Moreover, the area in parameter space covering the existence of order 2 and 4
phase patterns is enlarged by plasticity. Then, not only plasticity allows for
more complex patterns to emerge for smaller $\tau_{\rm ext}$, but also the
livelihood of its appearance for a large range of delays. Thus, plasticity
promotes complexity as measured by the emergence of symmetric synchronous
patterns. Figures 5(f) and 5(g) show the higher values of the order parameters
moments for the case without and with synaptic plasticity, respectively. We
verify that the highest values of $R^{m}$ correspond to one group symmetry.

In summary, spiking synchronous patterns are strongly related to the network
connection between subnetworks in a plastic neuronal network with time delay.
For small external delays, one group configuration exhibits subnetworks
spiking in a specific order, and this order promotes synaptic potentiation
from the subnetworks spiking first to the subsequent ones. Increasing external
delay, two group patterns generate potentiation in in-phase subnetworks, while
a synaptic depression among the ones from distinct groups. For major delays,
four groups (a phase-lock synchronisation between subnetworks) show a
synchronised state, in which all subnetworks spiking almost at the same time
and without any preferential order. This dynamical behaviour promotes a final
global network configuration with an average increase in all synaptic
strengths. In all cases, the potentiation between the neuronal areas can also
depend on the internal time delay. The less recurrent pattern found in our
simulations is the 3 groups organisation, which can be associated with a
transient behaviour.

##  4 Conclusions

In this work, we consider a network of subnetworks to study the effects of
internal and external transmission delays on the generation of symmetric
dynamics patterns, as well as the potentiation and depression on the synaptic
weights due to the presence of plasticity. To do that, we consider Hodgkin-
Huxley neurons coupled by means of excitatory chemical synapses and a time
dependent plastic rule. To achieve synchronised patterns, the internal delayed
transmission of neuron communication in the subnetworks assumes small values
while the external one assumes higher values. When the internal transmission
delay of neurons in the subnetworks assumes higher values, nonsynchronised
patterns are observed.

Without plasticity and depending on the delay transmission between the
subnetworks, we verify that synchronisation among subnetworks can be observed
in different patterns. These regimes can be detected by means of the $m$-th
moment of the order parameter, which provides the information about how the
dynamics of neurons in the subnetworks are correlated in the phase space of
spiking times. Due to the plasticity effect, we verify that the final
connectivity reflects the symmetric synchronous patterns on the emergence of
the strongest connections between the subnetworks. Thus, subnetworks that are
strongly connected, are also strongly synchronous, and the same symmetric
patterns observed in terms of synchronisation are also found in the final
connecting topology. We show that the synaptic transmission delays play an
important role in the generation of symmetric synchronised patterns. In
addition, we also show that the phase, anti-phase, and symmetric phase-lock
synchronised firing patterns influence the synaptic changes of the weight
connections among the subnetworks. We observe the relationship of each
symmetric synchronised firing pattern with the induced potentiation between
the subnetworks. As a consequence, our results suggest that firing patterns
can induce different topologies in addition to that topology induced firing
patterns. We show that it is possible to identify spiking correlations and
delayed excitatory connectivities between different neuronal groups.

A recent work [44] has proposed the construction of modular, scalable and
adaptable neuronal networks with neurons possessing different dynamic
properties and distinct firing patterns. To achieve that the authors
considered neurons with different external input. In this work, we also
consider heterogeneity in the external parameter. Our goal however was focused
not on the design but on the relation between topology and synchronisation
patterns. Whenever networks with STDP are designed with neurons that can have
external currents controllable and accessible, our work shows that topology
and synchronisation are strongly related.

Topology and behaviour is extensively studied in the literature. The novelty
in our results was to study the emergency of different synchronous regimes,
phase, anti-phase and shift phase. The emergency of these distinct regimes has
been found in the brain. Experimental evidence shows that phase
synchronisation is able to support memory processes and changes in the
connection strengths [20, 67]. In particular, results of Jutras and Buffalo
suggest that phase synchronisation can lead to potentiation of the synaptic
connections [68]. Our results can be linked to these experimental works,
providing a possible explanation about how these synchronous firing patterns
leads to a topology. Anti-phase oscillations are observed during rest state in
humans [69] and in anaesthetised monkeys [70]. In addition, shift-phase
synchronisation was observed in primary visual cortex [71] and could play a
function to stimulus selection [72]. Our work shows that all these synchronous
phenomena induce topology and emerge from it.

All these paving the way for us to conclude that plasticity - described by a
pairwise function that regulates synapse strength by the time intervals
between two spiking neurons - in fact promotes the creation of evolved network
structures whose subnetworks of intra connected neurons and their inter
connections is strongly reflected in the global synchronisation patterns
measured by the phase dynamics of the neurons. Thus, the plastic neural
network has a strong match between phase activity and graph structure.

## Conflict of Interest Statement

The authors declare that there is no conflict of interest.

## Acknowledgements

The authors acknowledge the financial support from São Paulo Research
Foundation (FAPESP, Brazil) (Grants Nos. 2016/23398-8, 2017/13502-5,
2018/03211-6, 2020/04624-2, 2022/05153-9, 2022/13761-9), National Council for
Scientific and Technological Development (CNPq), Fundação Araucária and
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES).

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