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Edited by: Plamen Ch. Ivanov, Boston University, United States

Reviewed by: Bolun Chen, Brandeis University, United States; Grigory Osipov, Lobachevsky State University of Nizhny Novgorod, Russia

This article was submitted to Fractal and Network Physiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In the brain, the excitation-inhibition balance prevents abnormal synchronous behavior. However, known synaptic conductance intensity can be insufficient to account for the undesired synchronization. Due to this fact, we consider time delay in excitatory and inhibitory conductances and study its effect on the neuronal synchronization. In this work, we build a neuronal network composed of adaptive integrate-and-fire neurons coupled by means of delayed conductances. We observe that the time delay in the excitatory and inhibitory conductivities can alter both the state of the collective behavior (synchronous or desynchronous) and its type (spike or burst). For the weak coupling regime, we find that synchronization appears associated with neurons behaving with extremes highest and lowest mean firing frequency, in contrast to when desynchronization is present when neurons do not exhibit extreme values for the firing frequency. Synchronization can also be characterized by neurons presenting either the highest or the lowest levels in the mean synaptic current. For the strong coupling, synchronous burst activities can occur for delays in the inhibitory conductivity. For approximately equal-length delays in the excitatory and inhibitory conductances, desynchronous spikes activities are identified for both weak and strong coupling regimes. Therefore, our results show that not only the conductance intensity, but also short delays in the inhibitory conductance are relevant to avoid abnormal neuronal synchronization.

Network physiology reveals how organ systems dynamically interact (Bartsch et al.,

Time delay has been considered in several problems of biological interest (Glass et al.,

Dynamic brain behavior can be mimicked by means of neuronal network models (Protachevicz et al.,

Neurons can be modeled by differential equations. In 1907, Lapicque (Lapicque,

We build here a network composed of adaptive exponential integrate-and-fire (AEIF) neurons. The AEIF model was introduced by Brette and Gerstner (

In this work, we study AEIF neurons randomly connected by means of excitatory and inhibitory conductivities. The neurons can exhibit not only spike but also burst activities (Santos et al.,

The paper is organized as follows. In section 2, we introduce the neuronal network composed of AEIF neurons and delayed conductance. Section 3 shows our results about the effects of conduction delays in neuronal synchronization. We draw our conclusions in the last section.

We construct a neuronal network with 100 AEIF neurons, where the connections are randomly chosen with probability equal to 0.5. The connection probability is defined as

where _{T} is the total connection number of the network and

where _{i}, _{i}, and _{i} are the membrane potential, the adaptation current, and the conductance of the neuron _{L} = 12 nS (leak conductance), _{L} = −70 mV (resting potential), _{i} = 2·_{rheo} (constant input equal to two times the rheobase current Naud et al., _{T} = 2 mV (slope factor), _{T} = −50 mV (potential threshold), and τ_{w} = 300 ms (adaptation time constant). The level of subthreshold adaptation _{i} is randomly distributed in the interval [1.9, 2.1] nS. This set of parameters corresponds to the spike adaptation activity when neurons are uncoupled. In the model, the adaptation mechanism is able to generate burst activities when the neurons are connected by excitatory synapses (Fardet et al.,

where _{j} is the time delay in the conductance. We consider _{j} = _{inh} for inhibitory and _{j} = _{exc} for excitatory neurons. _{REV} = 0 mV for excitatory and _{REV} = −80 mV for inhibitory synapses). In the adjacency matrix (_{ij}), the element value is equal to 1 when the presynaptic neuron _{j} has an exponential decay with the synaptic time constant τ_{s} = 2.728 ms. When the membrane potential of the neuron _{i} > _{thres}) (Naud et al.,

where _{r} = −58 mV is the reset potential and _{s} assumes _{exc} and _{inh} for excitatory and inhibitory neurons, respectively. We define a relative inhibitory conductance as _{inh}/_{exc}.

Standard parameter set.

Number of AEIF on the network | 100 neurons | |

Capacitance membrane | 200 pF | |

_{L} |
Leak conductance | 12 nS |

_{L} |
Resting potential | −70 mV |

_{i} |
Constant input current | 2·_{rheo} |

Δ_{T} |
Slope factor | 2 mV |

_{T} |
Potential threshold | −50 mV |

τ_{w} |
Adaptation time constant | 300 ms |

_{i} |
Level of subthreshold adaptation | [1.9, 2.1] nS |

Level of triggered adaptation | 70 pA | |

_{r} |
Reset potential | −58 mV |

Excitatory synaptic reversal potential | 0 mV | |

Inhibitory synaptic reversal potential | −80 mV | |

_{ij} |
Adjacent matrix elements | 0 or 1 |

τ_{s} |
Synaptic time constant | 2.728 ms |

_{fin} |
Final time to analyses | 10 s |

_{ini} |
Initial time to analyses | 5 s |

_{s} |
Chemical conductance | _{exc} or _{inh} |

_{j} |
Time delay | _{exc} or _{inh} |

As a diagnostic tool to identify synchronization, we use the time average of the Kuramoto order parameter (Kuramoto,

where the final time in the simulation and initial time for analyses are _{fin} = 10 s and _{ini} = 5 s, respectively.

where _{j,m} is the time at which neuron _{j,m}, _{j,m+1}].

The AEIF neuron can exhibit spike or burst activities. To identify these activities, we compute the coefficient of variation of the inter-spike interval (ISI)

where σ_{ISI} and

We calculate the mean firing frequency

We also compute the instantaneous ^{syn}(

where _{inh} × _{exc} is computed by means of the average of 10 different initial conditions. The initial conditions of _{i} and _{i} are randomly distributed in the interval _{i} = [−70, −50] mV and _{i} = [0, 80] nA, respectively. The initial conductance _{i} is equal to 0 for all neurons. To solve the delayed differential equations, we consider an initial profile of the network (for _{j}, 0]) in which the neurons are not spiking.

Neuronal conductances play a key role in network responses to stimuli (di Volo et al., _{exc} for _{exc} = 0.2 nS, _{inh} = 5 ms. In _{exc} from 65 ms (blue) to 75 ms (red), the desynchronized spikes (_{exc} is increased to 85 ms (

_{exc}. Raster plots for _{exc} = 65 ms _{exc} = 75 ms _{exc} = 85 ms _{exc} = 0.2 nS, _{inh} = 5 ms, and according to the colored circles.

_{inh} × _{exc} for _{exc} = 0.2 nS (weak coupling), where the color bar corresponds to the average order parameter _{inh} constant, and varying _{exc}. Increasing the relative inhibitory conductance for

Colors represent _{exc} × _{inh} for _{exc} = 0.2 nS, where we consider

_{exc} ≤ 110 ms and 0 ≤ _{inh} ≤ 70 ms). In the domain with a synchronous pattern, we observe that _{exc} and _{inh} have a significant influence on the mean firing frequency and mean synaptic current, respectively. The dynamics of neurons for some values of _{exc}, indicated in the vertical line (blue circles) in _{exc} = 75 ms, are shown in ^{syn} (B,D,F), where we consider _{inh} = 70 ms (blue), _{inh} = 60 ms (red), and _{inh} = 10 ms (green).

Magnifications of the parameter spaces shown in the right column of _{exc} = 0.2 nS and _{exc} × _{inh}.

Raster plot (top) and ^{syn}(_{exc} = 0.2 nS, _{exc} = 75 ms for different values of _{inh} (blue circles in _{inh} = 70 ms (blue), _{inh} = 60 ms (red), and _{inh} = 10 ms (green).

In _{exc} ≈ _{inh}). ^{syn}(_{inh} = 30 ms (blue squares in _{exc} = 65 ms (blue), _{exc} = 75 ms (red), and _{exc} = 85 ms (green). The parameters correspond to the region where synchronization can occur. Furthermore, we observe that depending on the excitatory delay value, synchronization can be improved. We verify that the synchronization is improved for _{exc} = 75 ms, namely certain values of the delay can optimize the synchronization regime.

Raster plot (top) and ^{syn}(_{exc} = 0.2 nS, _{inh} = 30 ms for different values of _{exc} (blue squares in _{exc} = 65 ms (blue), _{exc} = 75 ms (red), and _{exc} = 85 ms (green).

Increasing _{exc} from 0.2 to 0.8 nS (strong coupling), in _{inh} × _{exc} (_{inh} ≈ _{exc}. _{inh} and _{exc} in which _{inh} ≈ _{exc}. ^{syn}(_{exc} = 0.8 nS, _{exc} and _{inh}, according to the parameters pointed by the symbols in

_{exc} × _{inh} for _{exc} = 0.8 nS and _{exc} × _{inh} correspond to _{exc} = _{inh} = 0 ms (cyan square), _{exc} = 0 ms and _{inh} = 50 ms (cyan circle), _{exc} = 70 ms and _{inh} = 50 ms (cyan hexagon), and _{exc} = 110 ms and _{inh} = 50 ms (cyan triangle).

Raster plots (top) and ^{syn}(_{exc} = 0.8 nS, _{exc} and _{inh}. Different delay values generate desynchronized spikes

In this paper, we investigate the influence of delayed conductance on the neuronal synchronization. The study of neuronal synchronization is of great importance in neuroscience, due to the fact that it has been related to cognition, as well as to brain pathology. The conductance between the neurons plays a crucial role in the synchronous behavior. Many studies investigated the effects of the conductance on the neuronal activities (Bezanilla,

We construct a network composed of adaptive exponential integrate-and-fire (AEIF) neurons. The AEIF neuron has been used to mimic spike and burst patterns. In our network, we consider that the neurons are randomly connected by means of inhibitory and excitatory synapses. We find that for some network parameters, it is possible to observe spikes or bursts synchronization. We use the mean order parameter (

In order to explore the effects of different delayed conductances on the neuronal synchronization, in the section 3, we consider delay in both inhibitory and excitatory conductances. When all neurons are spiking (weak coupling), the delays induce synchronization domains in the parameter space _{inh} × _{exc}. Inside the parameter domains with synchronized neurons, we observe separated parameter subdomains representing neurons with higher and lower values of the mean firing frequency _{inh} × _{exc}. However, we see synchronous and desynchronous activities with either spike and burst activities. We also observe a range of high values of _{exc} ≈ 0), responsible for turning desynchronous spikes into synchronous burst patterns. For _{exc} ≈ _{inh} and strong coupling, we also observed desynchronous spike activities. Desynchronous spike activities can be associated with lower mean firing frequency and synaptic currents for strong coupling.

For weak coupling, the size of the region with synchronized behavior in _{inh} × _{exc} decreases when the number of connections is decreased. In this situation, we observe that the size of the small regions can be increased by increasing _{exc}. In addition, for strong coupling and decreasing the number of connections, there is no burst activity and we verify the existence of synchronized and desynchronized spiking patterns, as shown for weak coupling and no sparse connectivity. Therefore, the connectivity and the synaptic conductance play an important role in the synchronization.

In conclusion, we verify that the delay in the conductances plays a crucial role in the behavior of the neurons in the neuronal network. For weak coupling, we uncover that not only the synchronous behavior, but also the mean firing frequency and the mean synaptic input depend on the delayed inhibitory and excitatory conductances. We identify which range of synaptic current allow the neuronal network to achieve and maintain synchronous activities. In the region with desynchronized activities, excitatory and inhibitory currents arrive in different times, consequently, high synchronization does not appear. For strong coupling, we see that also spike and burst patterns depend on the delayed conductances. The domain with synchronous pattern is characterized by having different delays in the inhibitory and excitatory conductances. Considering _{exc} ≈ _{inh}, we observe desynchronous spikes activities for both weak and strong coupling. In addition, our results demonstrate that not only intensity of synaptic conductance, but also a short delay in the inhibitory conductance are relevant to avoid abnormal neuronal synchronization.

Our results can be useful to clarify how synchronous and desynchronous activities are reached in a context of neuronal population with delayed conductance. In future works, we plan to analyse the influence of the connection probability between excitatory and inhibitory neurons in the neuronal synchronization, as well as the appearance of clusters synchronization.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

PP, FB, KI, EL, and MH designed the work, developed the theory, and performed the numerical simulations. AB wrote the manuscript with support from MB, IC, JS, and JK. The authors revised the manuscript several times and gave promising suggestions. All authors also contributed to manuscript revision, read, and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.