# Synchronised firing patterns in a random network of adaptive exponential integrate-and-fire

## Abstract

We have studied neuronal synchronisation in a random network of adaptive exponential integrate-and-fire neurons. We study how spiking or bursting synchronous behaviour appears as a function of the coupling strength and the probability of connections, by constructing parameter spaces that identify these synchronous behaviours from measurements of the inter-spike interval and the calculation of the order parameter. Moreover, we verify the robustness of synchronisaton by applying an external perturbation to each neuron. The simulations show that bursting synchronisation is more robust than spike synchronisation.

###### keywords:

synchronisation, integrate-and-fire, network^{1}

[cor]Corresponding author: antoniomarcosbatista@gmail.com

## 1 Introduction

The concept of synchronistion is based on the adjustment of rhythms of oscillating systems due to their interaction (1). Synchronisation phenomenon was recognised by Huygens in the 17th century, time when he performed experiments to understand this phenomenon (2). To date, several kinds of synchronisation among coupled systems were reported, such as complete (3), phase (4); (5), lag (6), and collective almost synchronisation (7).

Neuronal synchronous rhythms have been observed in a wide range of researches about cognitive functions (8); (9). Electroencephalography and magnetoencephalography studies have been suggested that neuronal synchronization in the gamma frequency plays a functional role for memories in humans (10); (11). Steinmetz et al. (12) investigated the synchronous behaviour of pairs of neurons in the secondary somatosensory cortex of monkey. They found that attention modulates oscillatory neuronal synchronisation in the somatosensory cortex. Moreover, in the literature it has been proposed that there is a relationship between conscious perception and synchronisation of neuronal activity (13).

We study spiking and bursting synchronisation between neuron in a neuronal network model. A spike refers to the action potential generated by a neuron that rapidly rises and falls (14), while bursting refers to a sequence of spikes that are followed by a quiescent time (15). It was demonstrated that spiking synchronisation is relevant to olfactory bulb (16) and is involved in motor cortical functions (17). The characteristics and mechanisms of bursting synchronisation were studied in cultured cortical neurons by means of planar electrode array (18). Jefferys Haas discovered synchronised bursting of CA1 hippocampal pyramidal cells (19).

There is a wide range of mathematical models used to describe neuronal activity, such as the cellular automaton (20), the Rulkov map (21), and differential equations (22); (23). One of the simplest mathematical models and that is widely used to depict neuronal behaviour is the integrate-and-fire (24), which is governed by a linear differential equation. A more realistic version of it is the adaptive exponential integrate-and-fire (aEIF) model which we consider in this work as the local neuronal activity of neurons in the network. The aEIF is a two-dimensional integrate-and-fire model introduced by Brette Gerstner (25). This model has an exponential spike mechanism with an adaptation current. Touboul Brette (26) studied the bifurcation diagram of the aEIF. They showed the existence of the Andronov-Hopf bifurcation and saddle-node bifurcations. The aEIF model can generate multiple firing patterns depending on the parameter and which fit experimental data from cortical neurons under current stimulation (27).

In this work, we focus on the synchronisation phenomenon in a randomly connected network. This kind of network, also called Erdös-Rényi network (28), has nodes where each pair is connected according to a probability. The random neuronal network was utilised to study oscillations in cortico-thalamic circuits (29) and dynamics of network with synaptic depression (30). We built a random neuronal network with unidirectional connections that represent chemical synapses.

We show that there are clearly separated ranges of parameters that lead to spiking or bursting synchronisation. In addition, we analyse the robustness to external perturbation of the synchronisation. We verify that bursting synchronisation is more robustness than spiking synchronisation. However, bursting synchronisation requires larger chemical synaptic strengths, and larger voltage potential relaxation reset to appear than those required for spiking synchronisation.

This paper is organised as follows: in Section II we present the adaptive exponential integrate-and-fire model. In Section III, we introduce the neuronal network with random features. In Section IV, we analyse the behaviour of spiking and bursting synchronisation. In the last Section, we draw our conclusions.

## 2 Adaptive exponential integrate-and-fire

As a local dynamics of the neuronal network, we consider the adaptive exponential integrate-and-fire (aEIF) model that consists of a system of two differential equations (25) given by

(1) |

where is the membrane potential when a current is injected, is the membrane capacitance, is the leak conductance, is the resting potential, is the slope factor, is the threshold potential, is an adaptation variable, is the time constant, and is the level of subthreshold adaptation. If reaches the threshold , a reset condition is applied: and . In our simulations, we consider pF, nS, mV, mV, mV, pA, ms, nS, and mV (27).

The firing pattern depends on the reset parameters and . Table 1 exhibits some values that generate five different firing patterns (Fig. 1). In Fig. 1 we represent each firing pattern with a different colour in the parameter space : adaptation in red, tonic spiking in blue, initial bursting in green, regular bursting in yellow, and irregular in black. In Figs. 1a, 1b, and 1c we observe adaptation, tonic spiking, and initial burst pattern, respectively, due to a step current stimulation. Adaptation pattern has increasing inter-spike interval during a sustained stimulus, tonic spiking pattern is the simplest regular discharge of the action potential, and the initial bursting pattern starts with a group of spikes presenting a frequency larger than the steady state frequency. The membrane potential evolution with regular bursting is showed in Fig. 1d, while Fig. 1e displays irregular pattern.

Firing patterns | Fig. | b (pA) | (mV) | Layout |
---|---|---|---|---|

adaptation | 1(a) | 60.0 | -68.0 | red |

tonic spiking | 1(b) | 5.0 | -65.0 | blue |

initial burst | 1(c) | 35.0 | -48.8 | green |

regular bursting | 1(d) | 40.0 | -45.0 | yellow |

irregular | 1(e) | 41.2 | -47.4 | black |

As we have interest in spiking and bursting synchronisation, we separate the parameter space into a region with spike and another with bursting patterns (Fig. 2). To identify these two regions of interest, we use the coefficient of variation (CV) of the neuronal inter-spike interval (ISI), that is given by

(2) |

where is the standard deviation of the ISI normalised by the mean (31). Spiking patterns produce . Parameter regions that represent the neurons firing with spiking pattern are denoted by gray colour in Fig. 2. Whereas, the black region represents the bursting patterns, which results in .

## 3 Spiking or bursting synchronisation

In this work, we constructed a network where the neurons are randomly connected (28). Our network is given by

(3) |

where is the membrane potential of the neuron , is the synaptic conductance, is the synaptic reversal potential, is the synaptic time constant, is the synaptic weight, is the adjacency matrix, is the external perturbation, and is randomly distributed in the interval .

The schematic representation of the neuronal network that we have considered is illustrated in Fig 3. Each neuron is randomly linked to other neurons with a probability by means of directed connections. When is equal to 1, the neuronal network becames an all-to-all network. A network with this topology was used by Borges et al. (32) to study the effects of the spike timing-dependent plasticity on the synchronisation in a Hodgkin-Huxley neuronal network.

A useful diagnostic tool to determine synchronous behaviour is the complex phase order parameter defined as (33)

(4) |

where and are the amplitude and angle of a centroid phase vector, respectively, and the phase is given by

(5) |

where corresponds to the time when a spike () of a neuron happens (). We have considered the beginning of the spike when mV. The value of the order parameter magnitude goes to 1 in a totally synchronised state. To study the neuronal synchronisation of the network, we have calculated the time-average order-parameter, that is given by

(6) |

where is the time window for calculating .

Figs. 4a, 4b, and 4c show the raster plots for nS, nS, and nS, respectively, considering mV, , and pA, where the dots correspond to the spiking activities generated by neurons. For nS (Fig. 4a) the network displays a desynchonised state, and as a result, the order parameter values are very small (black line in Fig. 4d). Increasing the synaptic conductance for nS, the neuronal network exhibits spike synchronisation (Fig. 4b) and the order parameter values are near unity (red line in Fig. 4d). When the network presents bursting synchronisation (Fig. 4c), the order parameter values vary between and (blue line in Fig. 4d). to the time when the neuron are firing.

In Fig. 5a we show as a function of for , pA (black line), pA (red line), and pA (blue line). The three results exhibit strong synchronous behaviour () for many values of when nS . However, for nS, it is possible to see synchronous behaviour only for pA in the range . In addition, we calculate the coefficient of variation (CV) to determine the range in where the neurons of the network have spiking or bursting behaviour (Fig. 5b). We consider that for CV (black dashed line) the neurons exhibit spiking behaviour, while for CV the neurons present bursting behaviour. We observe that in the range for pA there is spiking sychronisation, and bursting synchronisation for nS.

## 4 Parameter space of synchronisation

The synchronous behaviour depends on the synaptic conductance and the probability of connections. Fig. 6 exhibits the time-averaged order parameter in colour scale as a function of and . We verify a large parameter region where spiking and bursting synchronisation is strong, characterised by . The regions I and II correspond to spiking and bursting patterns, respectively, and these regions are separated by a white line with circles. We obtain the regions by means of the coefficient of variation (CV). There is a transition between region I and region II, where neurons initially synchronous in the spike, loose spiking synchronicity to give place to a neuronal network with a regime of bursting synchronisation.

We investigate the dependence of spiking and bursting synchronisation on the control parameters and . To do that, we use the time average order parameter and the coefficient of variation. Figure 7 shows that the spike patterns region (region I) decreases when increases. This way, the region I for pA and mV of parameters leading to no synchronous behaviour (Fig. 7a), becomes a region of parameters that promote synchronised bursting (Fig. 7b and 7c). However, a large region of desynchronised bursting appears for nS about mV and pA in the region II (Fig. 7b). For nS, we see, in Fig. 7c, three regions of desynchronous behaviour, one in the region I for pA, other in region II for pA, and another one is located around the border (white line with circles) between regions I and II for pA.

It has been found that external perturbations on neuronal networks not only can induce synchronous behaviour (34); (35), but also can suppress synchronisation (36). Aiming to study the robustness to perturbations of the synchronous behaviour, we consider an external perturbation (3). It is applied on each neuron with an average time interval of about ms and with a constant intensity during ms.

Figure 8 shows the plots for , where the regions I and II correspond to spiking and bursting patterns, respectively, separated by white line with circles, and the colour bar indicates the time-average order parameter values. In this Figure, we consider mV, pA, (a) pA, (b) pA, and (c) pA. For pA (Fig. 8a) the perturbation does not suppress spike synchronisation, whereas for pA the synchronisation is completely suppressed in region I (Fig. 8b). In Fig. 8c, we see that increasing further the constant intensity for pA, the external perturbation suppresses also bursting synchronisation in region II. Therefore,the synchronous behavior in region II is more robustness to perturbations than in the region I, due to the fact that the region II is in a range with high and values, namely strong coupling and high connectivity.

In order to understand the perturbation effect on the spike and bursting patterns, we consider the same values of and as Fig. 7a. Figure 9 exhibits the space parameter , where is equal to pA. The external perturbation suppresses synchronisation in the region I, whereas we observe synchronisation in region II. The synchronous behaviour in region II can be suppressed if the constant intensity is increased. Therefore, bursting synchronisation is more robustness to perturbations than spike synchronisation.

## 5 Conclusion

In this paper, we studied the spiking and bursting synchronous behaviour in a random neuronal network where the local dynamics of the neurons is given by the adaptive exponential integrate-and-fire (aEIF) model. The aEIF model can exhibit different firing patterns, such as adaptation, tonic spiking, initial burst, regular bursting, and irregular bursting.

In our network, the neurons are randomly connected according to a probability. The larger the probability of connection, and the strength of the synaptic connection, the more likely is to find bursting synchronisation.

It is possible to suppress synchronous behaviour by means of an external perturbation. However, synchronous behaviour with higher values of and , which typically promotes bursting synchronisation, are more robust to perturbations, then spike synchronous behaviour appearing for smaller values of these parameters. We concluded that bursting synchronisation provides a good environment to transmit information when neurons are strongly perturbed (large ).

## Acknowledgements

This study was possible by partial financial support from the following Brazilian government agencies: CNPq, CAPES, and FAPESP (2011/19296-1 and 2015/07311-7). We also wish thank Newton Fund and COFAP.

### Footnotes

- journal: Arxiv

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