# Complex synchronous behavior in interneuronal networks with delayed inhibitory and fast electrical synapses

###### Abstract

Networks of fast-spiking interneurons are crucial for the generation of neural oscillations in the brain. Here we study the synchronous behavior of interneuronal networks that are coupled by delayed inhibitory and fast electrical synapses. We find that both coupling modes play a crucial role by the synchronization of the network. In addition, delayed inhibitory synapses affect the emerging oscillatory patterns. By increasing the inhibitory synaptic delay, we observe a transition from regular to mixed oscillatory patterns at a critical value. We also examine how the unreliability of inhibitory synapses influences the emergence of synchronization and the oscillatory patterns. We find that low levels of reliability tend to destroy synchronization, and moreover, that interneuronal networks with long inhibitory synaptic delays require a minimal level of reliability for the mixed oscillatory pattern to be maintained.

###### pacs:

87.19.lg, 05.45.Xt, 89.75.Kd## I Introduction

Synchronization is an important phenomenon that occurs in many biological and physical systems r1 (). In the brain, the rhythmic oscillations of concerted electrical activity, which are representative for the synchronous firing of neurons, have been observed in different regions, including the neocortex, hippocampus and thalamus r2 (); r3 (). Since neural oscillations are associated with many high-level brain functions, the pertinent research has attracted considerable attention in the past decades r2 (); r3 (); r4 (). It has been proposed that these oscillations not only carry information by themselves, but that they may also regulate the flow of information and assist by its storage and retrieval in neural circuits r5 ().

In the brain, fast-spiking interneurons are mutually connected by both inhibitory chemical synapses as well as electrical synapses (gap junctions) r6 (). The evidence is mounting that networks composed of fast-spiking interneurons could provide synchronization mechanisms by means of which important rhythmic activities, such as the gamma (: 25-100 Hz) rhythm and the mixed theta (: 4-12Hz) and gamma rhythm r7 (); r7a1 (); r8 (), can be generated. Computational studies indicate that inhibitory and electrical synapses play an important role by the generation of these oscillations. For example, it has been shown that interneuronal networks with solely inhibitory synapses can produce gamma oscillations, but that adding gap junctions to the network can further increases their stability and coherence r7a1 (). It has also been reported that interneuronal networks coupled by both fast and slow inhibitory synapses can produce the mixed theta and gamma rhythmic activity r8 (). In addition, several theoretical studies have been performed to provide a deeper understanding of how the inhibitory and electrical synapses promote synchronization amongst coupled neurons r9 ().

Information transmission delays, which are due to the finite propagation speeds and due to time lapses occurring by both dendritic and synaptic processing r10 (), are also an inherent part of neuronal dynamics. In particular the transmission delays of chemical synapses are not to be neglected r10 (). Physiological experiments have revealed that they can be up to several tenths of milliseconds in length r11 (). On the other hand, the transmission delays introduced by electrical synapses are comparably short, usually not exceeding 0.05 milliseconds r11 (), so they are often not taken explicitly into account. In terms of dynamical complexity, the existence of time delays makes a nonlinear system with a finite number of degrees of freedom become an infinite-dimensional one, which may enrich the dynamics r12 (), enhance synchronization r13 (), and facilitate spatiotemporal pattern formation r14 ().

Although existing studies attest clearly to the fact that information transmission delays have a significant impact on the synchronization of interneuronal networks, to the best of our knowledge the focus has always been on considering only short inhibitory synaptic delays r15 (). However, since as noted above, the delays of chemical synapses may be substantial, it is also of interest to consider long inhibitory synaptic delays. Given that previous studies have shown that different time delay lengths may have rather different effects on the synchronization of coupled nonlinear oscillators r13 (); r14 (), we anticipate that the consideration of long inhibitory synaptic delays may lead to new and inspiring results.

To resolve this, we here study the synchronization in an interneuronal network that is coupled by both delayed inhibitory as well as fast electrical synapses, focusing specifically on the effects of inhibitory synaptic delays covering a wide window of values. Our simulations reveal that the delayed inhibition not only plays an important role in network synchronization, but that it can also lead to different oscillatory patterns. The comparatively fast gap-junctional coupling, on the other hand, contributes solely to the synchronization of the network but does not affect the emergence of oscillatory patterns. Most interestingly, we show that a sufficiently long inhibitory synaptic delay induces a rapid transition from the one-frequency to the two-frequency state, thus leading to the occurrence of a mixed oscillatory pattern. Moreover, we also show that the unreliability of inhibitory synapses has a significant impact on both the synchronization and the emergence of oscillatory patterns. Our findings thus add to the established relevance of time delays in neuronal networks and highlight the importance of synaptic mechanisms for the generation of synchronized neural oscillations.

## Ii Mathematical model

We consider a network composed of fast-spiking interneurons. Neurons in the network are randomly connected by inhibitory and electrical synapses with probability and , respectively. For simplicity, all synapses are bidirectional. We do not allow a neuron to be coupled to another neuron more than once by using the same type of synaptic coupling, or a neuron to be coupled with itself. We assume that all electrical synapses are fast, thus considering delays only by the inhibitory synapses. This assumption is reasonable, as we have argued in the Introduction. The dynamics of fast-spiking interneurons is described by the Wang-Buzsaki model r16 (). It has a form similar to the classical Hodgkin-Huxley model r10 (); r17 (), with details as follows r16 ():

(1) |

where the three gating variables obey the following equations

(2) |

Here is the neuron index, denotes the membrane potential of neuron , and the six rate functions are r16 (): , , , , , and . is the synaptic current of neuron due to the interactions with other neurons within the network (also referred to as internal synaptic current in this paper), and is an externally applied current representing the collective effect of inputs coming from the outside of the network. In this work, we model the externally applied current as , where is the mean current, is an independent Gaussian white noise with zero mean and unit variance, and is the intensity of stochastic fluctuations. The parameters of the Wang-Buzsaki model assume standard values r16 (): F/cm, ms/cm, mV, ms/cm, mV, ms/cm, mV, and . A spike is detected whenever the membrane potential exceeds the threshold of mV. Figure 1 shows the firing rate curve of the Wang-Buzsaki model when the later is driven solely by the externally applied current ().

For each neuron, the internal synaptic current consists of two terms, which are

(3) |

In this equation, the first and second outer sums run over all inhibitory and electrical synapses onto neuron , is the inhibitory synaptic strength from neuron to neuron , is the corresponding inhibitory synaptic variable, mV is the reversal potential for inhibitory synapses, and is the electrical synaptic strength from neuron to neuron . For inhibitory synapses, once a presynaptic neuron emits a spike, the corresponding is updated after a fixed spike transmission delay , according to . Otherwise decays exponentially with a fixed time constant . For simplicity, we set , and throughout this paper, implying that the coupling is identical for the same type of synapses.

To characterize the synchronization within the network, a dimensionless synchronization measure is introduced, following r18 (). We first compute the time fluctuations of the average membrane potential according to

(4) |

where the sign denotes the average over time and is the average membrane potential at time . Subsequently, the population-averaged variance of the activity of each individual neuron is determined according to

(5) |

Finally, the synchronization measure is computed as

(6) |

From this it follows that the larger the value of the better the synchronization in the network.

The described mathematical model is integrated numerically using the fourth-order Runge-Kutta algorithm with a fixed time step of ms. This is sufficiently small to ensure an accurate simulation of the Wang-Buzsaki model r16 (). For each set of parameters, the initial membrane potentials of neurons are uniformly distributed between -70 and 30 mV. The network size is , while and . We always generate both types of synapses, but use to denote the network without the fast gap-junctional coupling. The two parameters that determine the externally applied current are A/cm and A ms/cm. Under these conditions, the mean firing rate of the Wang-Buzsaki model is approximately 80 Hz in the absence of the internal synaptic current. We perform all simulations up to 3000 ms, and collect the data from 1000 to 3000 ms for further statistical analysis. The reported results, except for the spike raster diagrams, are averages over 30 independent runs.

## Iii Results

We first show elementary simulation results that reveal how the delayed inhibitory and fast gap-junctional coupling influence the synchronous behavior of the considered interneuronal network. In Figs. 2(a) and 2(b), several typical spike raster diagrams for different values of the inhibitory synaptic delay , without () and with () the fast electrical synapses, are plotted, respectively. Presented results show clearly that delayed inhibitory as well as fast electrical synapses play an important role by the synchronization of the network. Without the fast gap-junctional coupling, the neuronal firings at are rather disordered. Essentially this is because the considered network () is sparse, i.e., there are not much more links between the neurons (on average) than there are neurons constituting the network, and hence cannot be easily synchronized in the absence of additional mechanisms that promote the onset of synchronization. By introducing inhibitory synaptic delays, an improvement in the synchronization of the network can be observed. However, it can also be observed that this depends significantly on the length of the inhibitory synaptic delay. Only suitable delays can help the network to maintain a high level of synchronization (compare results obtained with , 18 and 30 ms, and ms in Fig. 2(a)). We demonstrate this quantitatively in Fig. 3(a), where also a near periodic oscillatory behavior in can be detected, which may be related to the matching of the inherent neuronal time scales with the duration of the delay, as proposed in r13 () (we will discuss this near-periodic oscillatory behavior further shortly). Furthermore, the results presented in Fig. 2(b) suggest that the fast electrical synapses provide a strong mechanism for fostering synchronization. With the fast gap-junctional coupling turned on, the high-quality synchronization may be observed even at . As the strength of the electrical synaptic coupling () is increased, the neuronal firings become more and more synchronized (see Fig. 3(b)). For sufficiently strong , the measure approaches 1, indicating that the synchronization is almost perfect. Indeed, several previous studies have concluded that gap-junctional coupling is more effective than chemical coupling in leading to highly synchronized states r19 (). One possible mechanism for this is that chemical synapses only act while the presynaptic neuron is spiking, whereas the electrical synapses are more efficient and can transmit the membrane potentials of presynaptic neurons to the corresponding postsynaptic neurons at all times.

Results presented in Fig. 2 reveal also that the oscillatory pattern is largely influenced by the inhibitory synaptic delay. For suitably short values of a regular oscillatory pattern can be observed. Interestingly however, if the delay is sufficiently long, we can observe the emergence of a mixed oscillatory pattern, which implies that there are two main oscillation frequencies present in the network. One is the low frequency of the whole mixed oscillatory pattern and the other is the high frequency of the fast oscillations within each periodic cycle (see Fig. 2). Theoretically, the mixed oscillatory pattern appears only for sufficiently long inhibitory synaptic delay. Under this condition, neurons in the synchronized or near-synchronized network have enough time to fire more than once during a whole periodic cycle, before the inhibitory synaptic currents caused by the first synchronous spiking group within the same periodic cycle start to suppress their firing. Obviously, the longer the inhibitory synaptic delay , the more groups of synchronous spikes might be contained in each periodic cycle (see and 30 ms in Figs. 2(a) and 2(b)). Moreover, once the inhibitory synaptic currents caused by the first synchronous spiking group start to have effect, these currents tend to decrease the membrane potentials of neurons and prolong their firing period. In this case, the following inhibitory synaptic bombardments due to one or more synchronous spiking groups from the same periodic cycle will further suppress neuronal firings, and therefore the neurons in the network can fire again only after these inhibitory effects wear down or become fully absent. This provides a viable mechanism for the emergence of the low-frequency component in the mixed oscillatory pattern. While performing additional simulations, we have discovered that without the long delayed inhibitory synapses, networks of interneurons cannot generate the mixed oscillatory pattern even if we consider the non-physiological case of long electrical synaptic delays (data not shown). The above results thus indicate that the inhibitory synaptic delay serves as an important control parameter for the selection of the oscillatory pattern in the network, and that long inhibitory synaptic delays provide a stable mechanism for the emergence of the mixed oscillatory pattern in interneuronal networks.

We now return to Fig. 3(a) and further discuss the near-periodic oscillatory behavior in . It can be determined that the frequency of this near-periodic behavior matches with the oscillation frequency of the high-frequency component quite well. Note that we will show later that the oscillation frequency of the high-frequency component is mainly influenced by the parameter . In the narrow regions of where the minima of appear (note that these regions also correspond to the transition points for the number of synchronously spiking groups contained in each periodic cycle), only a limited amount of neurons will participate in the last group of synchronous firings within each periodic cycle (see the red rectangle in the middle panel of Fig. 3(a)). This can be attributed to the matching between the synaptic delay and the high-frequency component, which ultimately causes the near-periodic behavior in the synchronization measure . We have also performed additional numerical simulations by using other models of neuronal dynamics, such as the model by Izhikevich with fast-spiking dynamics r20 () and the standard Hodgkin-Huxley model r10 (); r17 (), and we have observed qualitatively identical results, thus confirming the generality of this phenomenon (see also r13 ()). On the other hand, our results also reveal that the fast gap-junctional coupling tends to suppress the occurrence of such near-periodic oscillations, which may be attributed to the overall promotion of synchronization (see Figs. 3(b)). Indeed, if the electrical coupling is sufficiently strong this phenomenon disappears altogether because then the neurons in the network are perfectly synchronized.

In addition to the time delay , we also find that the synchronization depends significantly on the other two important inhibitory synaptic parameters, which are the strength and the time constant . Figures 4(a) and 4(b) depict the synchronization measure as a function of for different values of , without () and with () the fast electrical synapses, respectively. In the absence of fast gap-junctional coupling, there exists an optimal region of in each depicted dependence of , which implies that the network can support synchronization optimally only for intermediate values of (see also r17 () for related results). In this case, a strong inhibitory synaptic strength can drive the network towards a high-level of synchronization at the corresponding optimal value of . However, with the increasing of , it can also be observed that the top plateau region of the curve becomes narrower and shifts to the left (the direction of short ). An explanation why longer can no longer produce high values of is as follows. Due to the heterogeneity of connectivity, some neurons in the considered network will have more inhibitory synaptic inputs than others. For long , the slowly decaying synaptic inhibition accumulates in time and thus may lead to a tonic level of hyperpolarizing currents that cancel the external depolarizing currents by neurons with more inhibitory inputs r17 (). This will suppress or even fully disable the firing of such neurons, which in turn means that if the synaptic time constant is too long, the synchronization will deteriorate significantly. At a fixed , a large value of will introduce more inhibition to the network, and thus a relatively shorter will be needed to impair synchronization. As a result, although strong can enhance the synchronization in the corresponding optimal region, they may also reduce the size of this region. On the other hand, adding the fast gap-junctional coupling to the network, as expected, will enhance the synchronous firing of neurons. This enhancement is quite remarkable, even if the gap-junctional coupling is rather weak, as shown in Fig. 4(b). Further increasing the strength of gap-junctional coupling can lead to the perfect synchronization (data not shown). Several previous experiments have shown that some of the inhibitory synapses between interneurons can have rather slow synaptic kinetics r8 (). Therefore, to some extent, our results suggest that the fast electrical synapses might be essential for taming desynchronization if an interneuronal network contains a considerable number of slow inhibitory synapses.

Figure 5 shows how the oscillation frequency depends on the inhibitory synaptic delay for different levels of inhibition. As can be observed, the oscillation frequency of the considered interneuronal network is determined by both the inhibitory synaptic delay as well as the inhibition. In the short region, the oscillations are characterized by a single frequency. In this case, increasing and can reduce the oscillation frequency from the band to band (see Fig. 5). Once exceeds a critical time delay , we observe that the network oscillations transit from the one-frequency state to the two-frequency state, indicating the emergence of the mixed oscillatory pattern. This transition is rapid and stable with the aid of fast gap-junctional coupling. For the mixed oscillatory pattern, our results show that both and influence the low-frequency component, but only has a significant effect on the high-frequency component. When is short, the inhibitory synaptic currents from one periodic cycle decay fast, so that they may completely vanish before the firing of neurons enters into the next periodic cycle and almost do not influence the neuronal firing in the next periodic cycle. Thus, in this case, the critical time delay is approximately 12.5 ms, and based on the same reason, the number of synchronous spiking groups contained in each periodic cycle is also increased once about every 12.5 ms. The above analysis suggests that the oscillation frequency of the high-frequency component is around 80 Hz (in the band) for short (see ms in Fig 5), corresponding to the firing rate of a single Wang-Buzsaki neuron that is driven solely by the considered externally applied current. If is sufficiently long, the inhibitory synaptic currents from one periodic cycle can persist to a certain extent even after the neuronal firing enters into the next periodic cycle. These remaining inhibitory synaptic currents will suppress the neuronal firing in the next periodic cycle, and thus increase the firing interval between the first and the second synchronous spiking groups in the next periodic cycle. Therefore, the firing intervals of the high-frequency component are not perfectly identical, i.e., the first firing interval is slighter larger than the other firing intervals. This in turn yields a relatively smaller average frequency of the high-frequency component. As a result, for long the system needs a relatively longer to generate the mixed oscillatory pattern, and it also exhibits a slightly smaller frequency in the high-frequency component of its output (see ms in Fig 5). Moreover, our results also show that the frequency of the whole mixed oscillatory pattern is quite low, even in the case of weak inhibition (small and short values of ). For sufficiently long values of this oscillation frequency can be maintained in the band quite efficiently. The mixed theta and gamma rhythm is believed to play an important role in brain cognitive functions r21 (). The traditional viewpoint is that interneuronal networks with fast and slow inhibitory synaptic dynamics are the basic neural circuits to generate this special type of neural oscillations r8 (). Our results provide a new insight related to this, which is that long transmission delays of inhibitory synapses may also lead to the mixed theta and gamma rhythm in interneuronal networks. We note that this mechanism is still functional even if only some (not all) of the inhibitory synapses are considered to have long transmission delays, provided only that the delayed inhibitory synaptic currents are strong enough.

Finally, we examine how the unreliability of inhibitory synapses influences the synchronization and oscillatory patterns in the studied interneuronal network. This investigation is carried out because the synaptic transmission through real chemical synapses is indeed to a degree unreliable r22 (), and also because several previous studies have advocated that the unreliable synapses may play important functional roles in neural computation r23 (). In principle, the unreliability of chemical synapses can be explained by the phenomenon of probabilistic transmitter release, which has been confirmed by biological experiments r24 (). Typically, the synaptic unreliability is associated with synaptic depression, which can be simulated by a well-established phenomenological model proposed in r25 (). In this model, three parameters , , and , which denote the fractions of synaptic resources in the recovered, active, and inactive states, are employed and their dynamical equations are given by: , , and . Here is the Dirac delta function, gives the timing of presynaptic spikes, is the time constant of the inactive variable, is the time constant of the recovered variable, and describes the utilization of synaptic efficacy. Here we apply this model to modulate the updating of synaptic conductance as follows: whenever a presynaptic neuron fires a spike, the corresponding postsynaptic conductances are increased instantaneously after a fixed spike transmission delay , according to ; otherwise decays exponentially with a fixed time constant . In the following simulations, we set ms and , and change the variable to control the synaptic depression. A longer corresponds to a stronger synaptic depression, and therefore denotes a lower level of synaptic reliability.

In Figs. 6(a) and 6(b), we present several typical spike raster diagrams for different values of , without () and with () the fast electrical synapses, respectively. We choose the inhibitory synaptic delay to equal ms, ensuring that the network will be in the mixed oscillatory pattern when ms. Results presented in Fig. 6 demonstrate that the unreliability of inhibitory synapses has a great impact on both the network synchronization and the emergence of oscillatory patterns. In the absence of fast gap-junctional coupling, the synchronization reduces markedly with increasing . In this case, high synaptic unreliability (long ) leads to insufficient synaptic information interaction, which largely deteriorates network synchronization and causes the neural oscillations to disappear completely (see ms in Fig. 6(a)). With the fast gap-junctional coupling incorporated, we find that the network synchronization can be maintained in the majority of the region. Again, this is because the gap-junctional coupling itself can provide an effective mechanism for network synchronization. However, our results also show that the considered network needs a certain level of inhibitory synaptic reliability for the mixed oscillatory pattern to be preserved. For sufficiently long , it can be observed that the mixed oscillatory pattern transforms to the regular oscillatory pattern due to the lack of inhibition (see ms in Fig. 6(b)). The transitions in the oscillatory patterns can be observed more clearly from the data presented in Fig. 7. These findings suggest that the unreliability of inhibitory synapses might also provide a flexible mechanism for controlling the switch between different oscillatory patterns in interneuronal networks.

## Iv Discussion

In summary, we have employed a computational approach with the aim of investigating the complex synchronous behavior in interneuronal networks that are coupled by delayed inhibitory and fast electrical synapses. We have shown that these two types of synaptic coupling play an important role in warranting network synchronization. In particular, the considered network can achieve a high level of synchronization either by means of a suitable tuning of the inhibitory synaptic delay, by enhancing the strength of electrical synapses, or by means of both. On the other hand, our simulations have revealed that only delayed inhibition significantly influences the emergence of oscillatory patterns, while electrical synapses play at most a side role by this phenomenon. In particular, we have shown that short inhibitory delays evoke regular oscillatory patterns, while sufficiently long delays can lead to an abrupt emergence of mixed oscillatory pattern. By analyzing the oscillation frequencies, we found that the considered interneuronal network can generate both types of oscillations in physiologically relevant frequency bands, such as the gamma rhythm and the mixed theta and gamma rhythm. This fact might have biological implications as these rhythmic activities are frequently associated with fast-spiking interneurons, and are also believed to play prominent functional roles in cognitive tasks r7 (); r8 (); r21 (). Lastly, we have also demonstrated that the unreliability of inhibitory synapses plays an important role by the synchronization of the network as well as by the emergence of oscillatory patterns. More precisely, we have shown that high levels of unreliability destroy synchronization, and that a minimal level of reliability is needed for the emergence and stability of the mixed oscillatory pattern.

We hope that the presented results will improve our understanding of the synaptic mechanisms that are responsible for the generation of synchronous oscillations in the neural tissue. Indeed, our findings suggest that delayed inhibitory synapses are a viable candidate for controlling the emergence of oscillatory patterns. Depending on the actual biological circumstances, the same interneuronal ensembles may produce neural oscillations with different patterns in an adaptive way through the modulation of synaptic transmission. We also hope that this study will inspire further research on this topic, in particular by taking into account additional physiological properties of neuronal networks, such as the anatomical connectivity and distance-dependent synaptic information transmission delays.

###### Acknowledgements.

This research was supported by the National Natural Science Foundation of China (Grant No. 11172017) and the Slovenian Research Agency (Grant No. J1-4055). D. G. acknowledges the financial support from the University of Electronic Science and Technology of China.## References

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